Optsim

OptSim Models Reference

Volume II Block Mode

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OptSim Models Reference: Block Mode Contents •••• iii

Contents

Chapter 1: Signal Generators 15 PRBS Pattern Generator ..........................................................................................................16

PreBits and PostBits ..................................................................................................16 Properties...................................................................................................................16

Electrical Signal Generator ......................................................................................................18 Properties...................................................................................................................19

Optical Signal Generator .........................................................................................................21 Optical Parameters.....................................................................................................22 Polarization................................................................................................................22 Multi-Line Output .....................................................................................................22 Properties...................................................................................................................22

Custom Signal Generator.........................................................................................................24 Properties...................................................................................................................25

Analog Sine Generator.............................................................................................................27 Properties...................................................................................................................29

Sawtooth Generator .................................................................................................................31 Properties...................................................................................................................31

Frequency Sweep Generator ....................................................................................................32 Properties...................................................................................................................32

Expression Signal Generator ...................................................................................................34 Properties...................................................................................................................34

Chapter 2: Electrical Modules 37 1x1 Expression Signal Operator ..............................................................................................38

Properties...................................................................................................................38 2x1 Expression Signal Operator ..............................................................................................39

Properties...................................................................................................................39 Summer....................................................................................................................................40

Properties...................................................................................................................40 Mixer .......................................................................................................................................41

Properties...................................................................................................................41 Electrical Filter ........................................................................................................................42

Definitions of filter types...........................................................................................42 Noise filtering............................................................................................................44 Test display................................................................................................................44 Properties...................................................................................................................45

Feed-Forward Equalization Filter ............................................................................................47 Properties...................................................................................................................47

Decision-Feedback Equalizer for Electronic Dispersion Compensation with Minimum Mean Square Error Optimization (MMSE DFE-EDC)......................................................................48

References .................................................................................................................49 Properties...................................................................................................................49

Standard S-Parameter Block....................................................................................................50 Properties...................................................................................................................51

iv Contents OptSim Models Reference: Block Mode

Modulator-Driving S Block .....................................................................................................52 Properties...................................................................................................................52

Electrical Gain .........................................................................................................................54 Properties...................................................................................................................54

Electrical Amplifier .................................................................................................................55 Properties...................................................................................................................56

Electrical Noise Adder.............................................................................................................58 Properties...................................................................................................................59

Crosstalk Block........................................................................................................................61 Properties...................................................................................................................61

Electrical Integrate And Dump ................................................................................................62 Properties...................................................................................................................62

Chapter 3: Special Functions and Logical Models 65 Shift Signal ..............................................................................................................................66

Properties...................................................................................................................68 Boolean Operator.....................................................................................................................69

Properties...................................................................................................................69 D Flip-Flop ..............................................................................................................................70

Properties...................................................................................................................70 T Flip-Flop...............................................................................................................................71

Properties...................................................................................................................71 DQPSK Precoder .....................................................................................................................72

Properties...................................................................................................................72 Electrical Signal Resampler.....................................................................................................73

Properties...................................................................................................................73 Optical Signal Resampler ........................................................................................................74

Properties...................................................................................................................74

Chapter 4: Optical Sources and Modulators 75 Direct Modulated Laser ...........................................................................................................76

Driving Source ..........................................................................................................76 Parasitics....................................................................................................................77 Laser Cavity ..............................................................................................................77 Polarization................................................................................................................80 Multi-Line Output .....................................................................................................81 Numerical Settings ....................................................................................................81 Test Parameters .........................................................................................................81 Compatibility with LinkSIM version 2.1 VCSEL Model...........................................83 References .................................................................................................................83 Properties...................................................................................................................83

Mode-Locked Laser.................................................................................................................87 Polarization................................................................................................................88 Multi-Line Output .....................................................................................................88 References .................................................................................................................89 Properties...................................................................................................................89

CW Laser .................................................................................................................................91 Linewidth ..................................................................................................................91 Polarization................................................................................................................91 Multi-Line Output .....................................................................................................91 References .................................................................................................................93 Properties...................................................................................................................93

Fabry Perot CW Laser .............................................................................................................95 Linewidth ..................................................................................................................95

OptSim Models Reference: Block Mode Contents •••• v

Polarization................................................................................................................95 References .................................................................................................................96 Properties...................................................................................................................96

VCSEL.....................................................................................................................................99 Driving Source ..........................................................................................................99 Parasitics..................................................................................................................100 VCSEL Cavity.........................................................................................................100 Polarization..............................................................................................................104 Multi-Line Output ...................................................................................................104 Numerical Settings ..................................................................................................104 Test Parameters .......................................................................................................104 References ...............................................................................................................105 Properties.................................................................................................................105

Light Emitting Diode (LED)..................................................................................................109 LED Optical Response ............................................................................................109 Linewidth ................................................................................................................110 LED Electrical Model..............................................................................................110 Driving Source ........................................................................................................110 Polarization..............................................................................................................111 Multi-Line Output ...................................................................................................111 Test Functions .........................................................................................................111 References ...............................................................................................................112 Properties.................................................................................................................112

Modulator ..............................................................................................................................115 Mach-Zehnder .........................................................................................................115 Ideal.........................................................................................................................116 Amplitude................................................................................................................116 Phase........................................................................................................................116 Properties.................................................................................................................117

Electroabsorption Modulator .................................................................................................119 Properties.................................................................................................................121

Chapter 5: Optical Fibers 123 Nonlinear Fiber......................................................................................................................124

Background .............................................................................................................124 Complete model ......................................................................................................127 Noise........................................................................................................................138 Skew........................................................................................................................139 Interior Property Maps ............................................................................................139 Simulation Features and Settings ............................................................................139 Test functions ..........................................................................................................140 References ...............................................................................................................141 Properties.................................................................................................................142 Appendix: File formats............................................................................................146

Bi-directional Nonlinear Fiber (Raman Amplifier) ...............................................................149 Introduction .............................................................................................................149 Background .............................................................................................................150 Model Operation......................................................................................................151 User-Specified Profiles............................................................................................153 Power Solution Equations and Calculation .............................................................153 Coherent Solution Calculation.................................................................................157 Internal Plots............................................................................................................158 References ...............................................................................................................160 Properties.................................................................................................................160

Fiber Delay ............................................................................................................................167

vi Contents OptSim Models Reference: Block Mode

Properties.................................................................................................................167

Chapter 6: Optical Amplifiers 169 Black Box Optical Amplifier .................................................................................................170

Defined Gain Model ................................................................................................170 Custom Gain Model ................................................................................................170 Defined ASE Noise Models ....................................................................................170 Custom ASE Noise Model ......................................................................................171 References ...............................................................................................................171 Properties.................................................................................................................171

Physical EDFA ......................................................................................................................173 Background .............................................................................................................173 Model Implementation ............................................................................................175 References ...............................................................................................................184 Properties.................................................................................................................184 Appendix: File formats............................................................................................190

Physical EYCDFA.................................................................................................................194 Background .............................................................................................................194 Model Implementation ............................................................................................197 References ...............................................................................................................207 Properties.................................................................................................................207 Appendix: File formats............................................................................................214

Semiconductor Optical Amplifier (SOA) ..............................................................................218 Model Description...................................................................................................218 Noise Treatment ......................................................................................................220 Simulation Techniques ............................................................................................221 Additional Notes......................................................................................................221 References ...............................................................................................................222 Properties.................................................................................................................222

Controlled SOA .....................................................................................................................224 Properties.................................................................................................................224

Semiconductor Optical Amplifier (SOA) – Comprehensive Model ......................................226 Model Description...................................................................................................227 Numerical Settings ..................................................................................................232 Test Functions .........................................................................................................232 References ...............................................................................................................233 Properties.................................................................................................................234

Optical Noise Adder ..............................................................................................................238 Direct addition of noise ...........................................................................................238 Normalization..........................................................................................................239 Stochastic representation of noise ...........................................................................239 Properties.................................................................................................................240

Linewidth Adder....................................................................................................................243 References ...............................................................................................................243 Properties.................................................................................................................243

Liekki LAD Interface ............................................................................................................245 Properties.................................................................................................................245

Chapter 7: Optical Components 247 Optical Attenuator .................................................................................................................248

Properties.................................................................................................................248 Optical Power Normalizer .....................................................................................................249

Properties.................................................................................................................249 Optical Phase Shift ................................................................................................................250

OptSim Models Reference: Block Mode Contents •••• vii

Properties.................................................................................................................250 Optical Phase Conjugator ......................................................................................................251

Properties.................................................................................................................251 Polarization Transformer .......................................................................................................252

Properties.................................................................................................................253 Ideal Frequency Converter.....................................................................................................255

Properties.................................................................................................................255 Sagnac Effect model for Interferometric Fiber Optic Gyroscope (I-FOG)............................257

References ...............................................................................................................257 Properties.................................................................................................................257

Chapter 8: WDM Components 259 Optical Splitter (1xN) ............................................................................................................260

Properties.................................................................................................................260 Optical Coupler (2x2) ............................................................................................................261

Properties.................................................................................................................261 Controlled Optical Coupler (2x2) ..........................................................................................263

Properties.................................................................................................................263 Optical Filter..........................................................................................................................264

Carrier shifting ........................................................................................................265 Test Parameters .......................................................................................................266 Properties.................................................................................................................266

Fiber Bragg Grating Filter .....................................................................................................268 Properties.................................................................................................................271

Optical Multiplexer (Nx1 MUX) ..........................................................................................272 Properties.................................................................................................................272

Optical DeMultiplexer (1xN DEMUX) ................................................................................274 Properties.................................................................................................................274

Optical Add Multiplexer ........................................................................................................276 Properties.................................................................................................................276

Optical Drop Multiplexer....................................................................................................... 278 Properties.................................................................................................................278

Optical Add/Drop Multiplexer...............................................................................................280 Properties.................................................................................................................281

General Multiport Optical Device (NxM and WDM)............................................................283 Properties.................................................................................................................284

Jones Matrix Transfer Function .............................................................................................286 Theoretical Background ..........................................................................................286 Using the Model ......................................................................................................287 Test Features............................................................................................................287 File Formats.............................................................................................................287 Properties.................................................................................................................290

Chapter 9: Optical Receivers 291 Compound Optical Receiver..................................................................................................292

Model Parameters....................................................................................................293 Noise Representation and Effects............................................................................293 Calibrating Receiver Sensitivity..............................................................................296 Test Function...........................................................................................................297 References ...............................................................................................................297 Properties.................................................................................................................297

Photodetector .........................................................................................................................302 Detection Process ....................................................................................................302 Test function............................................................................................................305

viii Contents OptSim Models Reference: Block Mode

References ...............................................................................................................305 Properties.................................................................................................................305

Chapter 10: Analyzers 309 Interior Property Map ............................................................................................................311

Model Usage............................................................................................................311 Available Maps........................................................................................................312 Scanning ..................................................................................................................313 Display Parameters..................................................................................................313 Properties.................................................................................................................313

Property Map .........................................................................................................................315 Model Usage............................................................................................................315 Model Parameters....................................................................................................315 Scanning Behavior...................................................................................................316 Limitations...............................................................................................................316 Properties.................................................................................................................316

Electrical Monitor ..................................................................................................................318 Model Use ...............................................................................................................318 Support for Scanning...............................................................................................318 Measurement Definitions ........................................................................................318 Properties.................................................................................................................319

Optical Monitor .....................................................................................................................320 Model Use ...............................................................................................................320 Support for Scanning...............................................................................................320 Property Definitions ................................................................................................320 References ...............................................................................................................323 Properties.................................................................................................................323

Gain/NF Analyzer..................................................................................................................325 Model Use ...............................................................................................................325 Support for Scanning...............................................................................................325 Gain/NF Definitions ................................................................................................325 References ...............................................................................................................326 Properties.................................................................................................................326

Polarization Monitor ..............................................................................................................327 Model Use ...............................................................................................................327 Support for Scanning...............................................................................................327 Property Definitions ................................................................................................328 Poincaré Sphere Plot................................................................................................329 Properties.................................................................................................................330

Optical Eye Analyzer.............................................................................................................332 Properties.................................................................................................................333

Bit Error Rate Tester..............................................................................................................335 BER Estimation Techniques....................................................................................335 Confidence Limits for the BER and Q ....................................................................341 Complete BER Estimation Algorithm.....................................................................342 Additional BER Features.........................................................................................345 Forward Error Correction ........................................................................................345 Performance Budget ................................................................................................347 NON-BER facilities.................................................................................................348 Eye Mask.................................................................................................................349 Scanning Facilities ..................................................................................................350 Validation: 10 Gbps Example..................................................................................351 References ...............................................................................................................359 Properties.................................................................................................................360

Karhunen-Loeve BER Estimator ...........................................................................................364

OptSim Models Reference: Block Mode Contents •••• ix

Model Use ...............................................................................................................364 Support for Scanning...............................................................................................364 Model of the Direct-Detection Optical Receiver .....................................................365 Model of Optical Binary Differential Receiver .......................................................366 Model of DQPSK Receiver .....................................................................................367 Brief Description of the Semi-Analytical Technique ..............................................367 Karhunen-Loeve Technique (KLT).........................................................................368 Filter Data File Formats...........................................................................................370 References ...............................................................................................................370 Properties.................................................................................................................371

Monte Carlo DPSK BER Estimator.......................................................................................376 MC DPSK BER Estimation Technique...................................................................376 Model Inputs............................................................................................................377 Scanning Facilities ..................................................................................................379 References ...............................................................................................................379 Properties.................................................................................................................379

XY-Plotter .............................................................................................................................381 Properties.................................................................................................................383

Signal Analyzer .....................................................................................................................384 Properties.................................................................................................................384

Constellation Diagram Analyzer............................................................................................386 Properties.................................................................................................................386

Eye Diagram Analyzer...........................................................................................................387 Properties.................................................................................................................387

Signal Spectrum Analyzer .....................................................................................................389 Properties.................................................................................................................389

Optical Frequency/Wavelength Chirp Analyzer....................................................................392 Properties.................................................................................................................392

Optical Autocorrelator Analyzer............................................................................................395 Properties.................................................................................................................395

Multiplot ................................................................................................................................397 Usage .......................................................................................................................397 Properties.................................................................................................................399

Transfer Function Analysis Tool ...........................................................................................403 Properties.................................................................................................................405

Chapter 11: Data Storage and Meta Blocks 407 Save and Load Signal To/From File ......................................................................................408

File Formats.............................................................................................................408 Properties.................................................................................................................412

Repeat Loop and Typed Repeat Loop ...................................................................................413 Properties.................................................................................................................413

Delay Block ...........................................................................................................................415 Properties.................................................................................................................415

Fork........................................................................................................................................416 Properties.................................................................................................................416

Typed Fork ............................................................................................................................417 Properties.................................................................................................................417

Hierarchical Input Signal Port Block.....................................................................................418 Properties.................................................................................................................418

Hierarchical Output Signal Port Block ..................................................................................419 Properties.................................................................................................................419

Write Once Read Many (WORM) Block ..............................................................................420 Properties.................................................................................................................420

Null Signal Block ..................................................................................................................421

x Contents OptSim Models Reference: Block Mode

Properties.................................................................................................................421

Chapter 12: Transient Modules 423 Transient Pulse Generator...................................................................................................... 424

Properties.................................................................................................................424 Compact Transient EDFA .....................................................................................................425

Background .............................................................................................................425 Model Implementation ............................................................................................426 References ...............................................................................................................432 Properties.................................................................................................................433 Appendix: File formats............................................................................................438

Transient EDFA.....................................................................................................................442 Background .............................................................................................................442 Model Implementation ............................................................................................445 References ...............................................................................................................452 Properties.................................................................................................................452 Appendix: File formats............................................................................................457

Static Optical Switch (2x2)....................................................................................................460 Properties.................................................................................................................460

Transient Optical Switch (2x2)..............................................................................................462 Properties.................................................................................................................462

Dynamic Optical Switch (2x2) ..............................................................................................464 Properties.................................................................................................................464

Transient Plotter.....................................................................................................................466 Properties.................................................................................................................466

Chapter 13: Multimode Modules 469 Spatial Adder .........................................................................................................................471

General Options.......................................................................................................471 Mode-Attachment Styles .........................................................................................471 Mode Types.............................................................................................................472 Multimode File Format............................................................................................473 Test Parameters .......................................................................................................474 Properties.................................................................................................................474

Spatial Direct Modulated Laser .............................................................................................478 Driving Source ........................................................................................................478 Parasitics..................................................................................................................479 Laser Cavity ............................................................................................................479 Polarization..............................................................................................................483 Spatial Effects..........................................................................................................483 Multi-Line Output ...................................................................................................485 Numerical Settings ..................................................................................................486 Test Parameters .......................................................................................................486 Compatibility with OptSim version 2.1 VCSEL Model ..........................................487 References ...............................................................................................................488 Properties.................................................................................................................488

Spatial Mode-Locked Laser...................................................................................................495 Polarization..............................................................................................................496 Spatial Effects..........................................................................................................496 Multi-Line Output ...................................................................................................499 Test Parameters .......................................................................................................499 References ...............................................................................................................500 Properties.................................................................................................................500

Spatial CW Laser ...................................................................................................................505

OptSim Models Reference: Block Mode Contents •••• xi

Linewidth ................................................................................................................505 Polarization..............................................................................................................505 Spatial Effects..........................................................................................................506 Multi-Line Output ...................................................................................................508 Test Parameters .......................................................................................................509 References ...............................................................................................................509 Properties.................................................................................................................510

Spatial VCSEL.......................................................................................................................514 Driving Source ........................................................................................................514 Parasitics..................................................................................................................515 VCSEL Cavity.........................................................................................................515 Polarization..............................................................................................................519 Spatial Effects..........................................................................................................519 Multi-Line Output ...................................................................................................522 Numerical Settings ..................................................................................................522 Test Parameters .......................................................................................................522 References ...............................................................................................................523 Properties.................................................................................................................524

Spatial Light Emitting Diode (LED)......................................................................................530 LED Optical Response ............................................................................................530 Linewidth ................................................................................................................531 LED Electrical Model..............................................................................................531 Driving Source ........................................................................................................531 Polarization..............................................................................................................532 Spatial Effects..........................................................................................................532 Multi-Line Output ...................................................................................................535 Test Functions .........................................................................................................535 References ...............................................................................................................536 Properties.................................................................................................................536

Thin Lens ...............................................................................................................................541 Numerical Effects....................................................................................................541 References ...............................................................................................................541 Properties.................................................................................................................541

Vortex Lens ...........................................................................................................................543 Numerical Effects....................................................................................................543 References ...............................................................................................................543 Properties.................................................................................................................544

Spatial Coupler ......................................................................................................................545 References ...............................................................................................................548 Properties.................................................................................................................548

Spatial BeamPROP Interface.................................................................................................549 Properties.................................................................................................................549

Multimode Fiber ....................................................................................................................550 Library Configuration..............................................................................................550 Parabolic Configuration...........................................................................................557 Numerical Configuration.........................................................................................559 Step-Index Configuration ........................................................................................561 Differential Mode Attenuation ................................................................................561 Corrections to the Modal Delay Due to Dispersion.................................................562 Linear Configuration ...............................................................................................563 General ....................................................................................................................564 Test ..........................................................................................................................564 References ...............................................................................................................568 Properties.................................................................................................................569 Appendix: Library Generation.................................................................................573

Spatial Aperture .....................................................................................................................577

xii Contents OptSim Models Reference: Block Mode

Properties.................................................................................................................578 Spatial Compound Optical Receiver......................................................................................579

Model Parameters....................................................................................................580 Noise Representation and Effects............................................................................580 Test display..............................................................................................................584 References ...............................................................................................................584 Properties.................................................................................................................584

Spatial Photodetector .............................................................................................................588 Spatial Effects..........................................................................................................588 Detection Process ....................................................................................................588 Frequency Response................................................................................................589 Noise Response .......................................................................................................590 Test display..............................................................................................................591 References ...............................................................................................................591 Properties.................................................................................................................592

Spatial Analyzer.....................................................................................................................594 Properties.................................................................................................................597

Encircled Flux Analysis Tool ................................................................................................598 References ...............................................................................................................601 Properties.................................................................................................................601

Differential Mode Delay Analysis Tool ................................................................................603 References ...............................................................................................................605 Properties.................................................................................................................606

Signal Band Converter...........................................................................................................607 Properties.................................................................................................................607

Spatial Gridded Field Converter ............................................................................................609 Properties.................................................................................................................609

Chapter 14: Predefined Compound Components 611 Dual-Arm Mach-Zehnder Modulator ....................................................................................613

Properties.................................................................................................................613 Mach-Zehnder Delay Interferometer (MZDI) .......................................................................615

Properties.................................................................................................................615 Tunable Mach-Zehnder Interferometer (Tunable_MZI)........................................................616

Properties.................................................................................................................616 Non-Return-to-Zero (NRZ) Transmitter (NRZ_TX) .............................................................618

Properties.................................................................................................................618 Return-to-Zero (RZ) Transmitter (RZ_TX)...........................................................................620

Properties.................................................................................................................620 Chirped Return-to-Zero (CRZ) Transmitter (CRZ_TX)........................................................621

Properties.................................................................................................................621 Carrier-Suppressed Return-to-Zero (CSRZ) Transmitter (CSRZ_TX)..................................623

Properties.................................................................................................................623 Soliton Transmitter (Soliton_TX)..........................................................................................625

Properties.................................................................................................................625 Duobinary Transmitter (Doubi_TX)......................................................................................627

Properties.................................................................................................................627 Differential Phase-Shift-Keying Transmitter (DPSK_TX)....................................................628

Properties.................................................................................................................628 Return-to-Zero DPSK Transmitter (RZ-DPSK_TX) .............................................................629

Properties.................................................................................................................629 Differential Phase-Shift-Keying Transmitter – Advanced (DPSK_TX_adv) ........................631

Properties.................................................................................................................632 Return-to-Zero Differential Phase-Shift-Keying Transmitter – Advanced (RZ_DPSK_TX_adv) ............................................................................................................633

OptSim Models Reference: Block Mode Contents •••• xiii

Properties.................................................................................................................634 Differential Quadrature Phase-Shift-Keying Transmitter (DQPSK_TX) ..............................636

Properties.................................................................................................................637 Return-to-Zero Differential Quadrature Phase-Shift-Keying Transmitter (RZ_DQPSK_TX)639

Properties.................................................................................................................640 WDM Transmitters ................................................................................................................642

Properties.................................................................................................................643 Fiber Link 1 ...........................................................................................................................645



Properties.................................................................................................................649 Free-Space Optical Channel (Weak Turbulence) ..................................................................651

Properties.................................................................................................................651 Differential Phase-Shift-Keying Receiver (DPSK_RX)........................................................653

Properties.................................................................................................................653 Differential Phase-Shift-Keying Receiver – Gaussian, Advanced (DPSK_RX_gauss_adv).655

Properties.................................................................................................................655 Differential Phase-Shift-Keying Receiver – Raised Cosine, Advanced (DPSK_RX_rcos_adv)657

Properties.................................................................................................................657 Differential Quadrature Phase-Shift-Keying Receiver – Gaussian (DQPSK_RX_gauss) .....659

Properties.................................................................................................................660 Differential Quadrature Phase-Shift-Keying Receiver – Raised Cosine (DQPSK_RX_rcos)661

Properties.................................................................................................................662 WDM Receivers ....................................................................................................................663

Properties.................................................................................................................664

Index 665

OptSim Models Reference: Block Mode Chapter 1: Signal Generators •••• 15

Chapter 1: Signal Generators

Models in this chapter generate binary, optical or electrical signals:

• PRBS Pattern Generator generate binary signals

• Analog Sine Generator generate electrical sine waves

• Electrical Signal Generator convert binary to electrical signals

• Optical Signal Generator convert binary to optical signals

• Custom Signal Generator specify an electrical or optical signal with a data file.

• Sawtooth Generator generate sawtooth electrical signals

• Frequency Sweep Generator generate chirped electrical signals

• Expression Signal Generator specify an electrical signal with a mathematical function

16 Chapter 1: Signal Generators OptSim Models Reference: Block Mode

PRBS Pattern Generator

This model generates a binary sequence of several different types. A single model instance may be used to provide multiple pattern outputs, optionally offset from each other, to drive different channels of a WDM or parallel optical bus simulation. Or, each channel may have its own model instance configured to provide a different pattern than the other model instances. The different pattern types are described according to their name in the user parameter list:

• PRBS

Produces a maximal length pseudo-random binary sequence.

• Alternating

Produces a series of bits alternating between 0 and 1.

• Single

Produces a single 1 bit in the center of a series of 0 bits.

• One

Produces a series of 1 bits.

• Zero

Produces a series of 0 bits.

• Custom

Produces a bit sequence as specified by the user in a data file. This file is to be placed in the directory in which the simulation will be performed, which is the directory in which the topology data file is located. This data file must be formatted as follows: Each line must contain 8 binary numbers separated by spaces. Each bit is represented as either ‘0’ or ‘1’. There must be as many bits in the data file as are specified to be generated by the model.

PreBits and PostBits The bit sequence can be modified such that the first few bits (prebits) and the last few bits (postbits) are set to 0. This is useful in simulations because it increases the accuracy of the FFT when the begin and end of the sequence match. The default values should be sufficient for most applications.

Properties

Inputs None.

Outputs #1-N: Binary signal

Parameter Values


Name Type Default Range Units patternType enumerated PRBS PRBS, alternating,

single, one, zero, custom

filename string

custompattern

bitRate double 10e9 [ 1, 1e15 ] bps patternLength integer 7 [ 0, 18 ] 2^x_bits

startTime double 0 [ -1e32, 1e32 ] sec

offset integer 0 [ 0, 262144 ] bits

shift integer 0 [ 0, 262144 ] bits preBits integer 2 [ 0, 1024 ] bits

postBits integer 3 [ 0, 1024 ] bits

Parameter Descriptions patternType The type of bit pattern to be generated. filename The name of the file containing the custom binary sequence bitRate The bit rate of the binary sequence generated patternLength The number of bits in the generated bit sequence is 2x where x is the parameter value offset The number of bits to offset the first signal output vs. the standard binary sequence. The

sequence is rotated by the number of bits specified. shift The number of bits to shift each successive output’s binary sequence relative to the previous

output. This applies to the second and subsequent output ports.

startTime The start time of the bit sequence preBits

The number of zero bits at the start of the sequence postBits The number of zero bits at the end of the sequence


Electrical Signal Generator

This model converts an input binary signal into an output electrical signal. The output signal may be specified as either voltage or current. The user parameters are used to configure the electrical signal output. Four different electrical drive types are modeled, as described below.

• On_off

A square wave electrical signal is generated. No jitter is included in this square wave. For each bit period, the output signal data is set according to the value of the input bit and the voltage levels specified by the user: Vmax for a 1 bit, and Vmin for a 0 bit. After generation of this signal, it may optionally be passed through a ringing generation filter according to the user parameter setting for filterType.

• On_off_exp

A square wave electrical signal is generated using the specified timing jitter. After the signal is generated, it is optionally passed through a ringing generation filter according to the user parameter setting for filterType with specified rise time.

• On_off_ramp

An electrical signal is generated using the specified rise time, fall time, and timing jitter. The signal rise and falls are ramped between the high and low levels. After the signal is generated, it is optionally passed through a ringing generation filter according to the user parameter setting for filterType.

• RaisedCosine

An electrical signal is generated using a raised cosine shape to represent the binary signal. This signal includes the timing jitter, but does not include the user-specified rise and fall times because the signal shape is specified as a raised cosine. This signal type is not passed through the ringing generation filter. The duty cycle of the pulse may be varied from nearly 0% to 50% using the alpha parameter. The raisedCosine electrical signal is genenerated using the following formula:

+⋅=

BTtAtA

απcos1

21)( max

where α is the pulsewidth parameter (αααα = 1 for TP= 0.5 TB ), TP is the pulse width and TB is the bit period.

Three different modulation formats are also available. These are the NRZor non-return to zero format, the RZ or return to zero format, and Manchester format.

A real electrical signal is usually not a pure square on-off or ramped on-off waveform. Most of the time there is ‘ringing’ on both the leading and the trailing edges. This ‘ringing’ effect is modeled by letting the ideal on-off signal pass through ringing generating filters for on_off type signals. The transfer function of the ringing generating filter is,


⋅+

+−⋅

⋅+=RCFilter''

211

'RingFilter' 2/21

1

)(22

0

20

RCfj

fjfff

RCfjfH

π

πγπ

where f0 is the resonant frequency, γ is the damping factor, while RC is the RC time constant corresponding to the rise time, RC = tr/ln(9).

Properties

Inputs #1: Binary signal

Outputs #1: Electrical signal

Parameter Values Name Type Default Range Units driveType enumerated on_off_ramp on_off, on_off_ramp,

on_off_exp, raisedCosine

signalType enumerated VOLTAGE VOLTAGE, CURRENT

modulationType enumerated NRZ RZ, NRZ, Manchester

pointsPerBit integer 5 [ 1, 27 ] 2^x_points Vmax double 1 [ -1000, 1000 ] volts or amps

Vmin double 0 [ -1000, 1000 ] volts or amps

tr double 10e-12 [ 0, 100 ] sec

tf double 10e-12 [ 0, 100 ] sec timeJitter double 0 [ 0, 1 ] sec

filterType enumerated None None, RCFilter, RingFilter

f0 double 1e18 [ 0, 1e18 ] Hz

gamma double 0.0 [ 0, 1e18 ] rad/sec alpha double 0.5 [ 1e-32, 0.5 ] none

Parameter Descriptions driveType The type of model used for generating the signal modulationType The type of modulation (RZ, NRZ, or Manchester) pointsPerBit Number of sampling points per bit in electrical signal signalType Whether the output value is specified as voltage (volts) or current (amps) Vmax Maximum value of the output electrical signal (binary one value)


Vmin Minimum value of the output electrical signal (binary zero value) tr Rise time of the output electrical signal tf Fall time of the output electrical signal timeJitter Timing jitter to add to output electrical signal filterType Filter type for ring generator f0 Resonant frequency of the ring filter gamma Damping frequency of the ring filter alpha Duty cycle for raisedCosine signal pulse


Optical Signal Generator

This model converts an input binary signal into an output optical signal, specified in Watts. The user parameters are used to configure the optical signal output. Four different drive types are modeled, as described below.

• On_off

A square wave optical signal is generated. No jitter is included in this square wave. For each bit period, the output signal data is set according to the value of the input bit and the power levels specified by the user: Pmax for a 1 bit, and Pmin for a 0 bit. After generation of this signal, it may optionally be passed through a ringing generation filter according to the user parameter setting for filterType.

• On_off_exp

A square wave optical signal is generated using the specified timing jitter. After the signal is generated, it is optionally passed through a ringing generation filter according to the user parameter setting for filterType.

• On_off_ramp

An optical signal is generated using the specified rise time, fall time, and timing jitter. The signal rise and falls are ramped between the high and low levels. After the signal is generated, it is optionally passed through a ringing generation filter according to the user parameter setting for filterType.

• RaisedCosine

An optical signal is generated using a raised cosine shape to represent the binary signal. This signal includes the timing jitter, but does not include the user-specified rise and fall times because the signal shape is specified as a raised cosine. This signal type is not passed through the ringing generation filter. The duty cycle of the pulse may be varied from nearly 0% to 50% using the alpha parameter. The raisedCosine optical signal is genenerated using the following formula:

+⋅=

BTtAtA

απcos1

21)( max

where α is the pulsewidth parameter (αααα = 1 for TP= 0.5 TB ), TP is the pulse width and TB is the bit period.

Three different modulation formats are also available. These are the NRZor non-return to zero format, the RZ or return to zero format, and Manchester format.

A real signal is usually not a pure square on-off or ramped on-off waveform. Most of the time there is ‘ringing’ on both the leading and the trailing edges. This ‘ringing’ effect is modeled by letting the ideal on-off signal pass through ringing generating filters for on_off type signals. The transfer function of the ringing generating filter is,

⋅+

+−⋅

⋅+=RCFilter''

211

'RingFilter' 2/21

1

)(22

0

20

RCfj

fjfff

RCfjfH

π

πγπ


where f0 is the resonant frequency, γ is the damping factor, while RC is the RC time constant corresponding to the rise time, RC = tr/ln(9).

Optical Parameters The wavelength of the output signal is specified via the parameter wavelength. The initial output phase may be set via the parameter initial_phase, or can be determined stochastically by setting randomInitialPhase to yes, and selecting an appropriate seed value (phaseSeed). The signal RIN is specified via the parameter RIN.

Polarization By default, the signal generator emits an output field polarized along the X axis, with no corresponding Y-polarized component. A zero-valued Y-polarized component may be included in this case by setting the parameter force_Ey to yes. The field polarization itself may be changed via the parameters azimuth and ellipticity. These parameters correspond to the azimuth and ellipticity angles of the polarization ellipse, respectively.

Multi-Line Output It is frequently necessary to produce several signals with similar properties but different wavelengths. In DWDM simulations especially, a series of regularly spaced optical sources is common. The Optical Signal Generator provides a number of convenient facilities so that many or all required lines can be generated from a single icon. To produce a series of lines spaced equally in wavelength or frequency, set the parameter mode=LambdaGrid or mode=FreqGrid, respectively. The number of sources is controlled by the number of binary inputs, and in both modes the parameter wavelength specifies the first line in a series of ascending wavelengths or ascending frequencies, respectively. The source separation in wavelength or frequency is specified with deltaFreq.

Properties

Inputs #1-#512: Binary signal

Outputs #1-#512: Optical signal

Parameter Values Name Type Default Range Units wavelength double 850e-9 [ 0, 1e32 ] m mode enumerated FreqGrid FreqGrid,

LambdaGrid

deltaFreq double 100e9 [ 0, 1e18 ] meters or Hz

randomInitialPhase enumerated no no, yes

initial_phase double 0 [ -180, 180 ] degrees

phaseSeed integer 0 [ -1e8, 1 ] none RIN double -150.0 [ -1e32, 1e32 ] dB/Hz

azimuth double 0 [ -90, 90 ] degrees


ellipticity double 0 [ -45, 45 ] degrees

force_Ey enumerated no yes, no driveType enumerated on_off_ramp on_off,

on_off_ramp, on_off_exp, raisedCosine

modulationType enumerated NRZ RZ, NRZ, Manchester

pointsPerBit integer 5 [ 1, 27 ] 2^x_points

Pmax double 1 [ 0, 1e32 ] W Pmin double 0 [ 0, 1e32 ] W

tr double 10e-12 [ 0, 100 ] sec

tf double 10e-12 [ 0, 100 ] sec

timeJitter double 0 [ 0, 1 ] sec alpha double 0.5 [ 1e-32, 0.5 ] none

filterType enumerated None None, RCFilter, RingFilter

f0 double 1e18 [ 0, 1e18 ] Hz

gamma double 0.0 [ 0, 1e18 ] rad/sec

Parameter Descriptions wavelength Signal wavelength mode Type of wavelength grid deltaFreq Frequency or wavelength grid spacing for multi-line output randomInitialPhase Select randomization of initial phase initial_phase Initial optical phase phaseSeed Seed for phase randomization RIN Signal RIN value azimuth Polarization azimuthal angle ellipticity Polarization ellipticity force_Ey Force Y-polarization driveType The type of model used for generating the signal modulationType The type of modulation (RZ, NRZ, or Manchester) pointsPerBit Number of sampling points per bit in electrical signal Pmax Maximum value of the output electrical signal (binary one value) Pmin Minimum value of the output electrical signal (binary zero value) tr Rise time of the output electrical signal tf Fall time of the output electrical signal timeJitter Timing jitter to add to output electrical signal filterType Filter type for ring generator f0 Resonant frequency of the ring filter gamma Damping frequency of the ring filter alpha

Duty cycle for raisedCosine signal pulse


Custom Signal Generator

This model creates either an electrical or optical signal and corresponding binary signal according to the user input signal waveform. The binary signal is provided for use as an input for the BER Tester block in the link. If the binary signal output is not desired, it may be left unconnected. For the BER Tester, it is usually best to specify the exact bit sequence the data file represents explicitly by supplying a data file containing the bit sequence to the PRBS block and setting it to use the data file as a custom bit sequence. When using the PRBS to supply the bit sequence this way, the output binary signal from the custom signal generator block is unnecessary.

The user must specify the waveform in a data file. This file is to be placed in the directory in which the simulation will be performed, which is the directory in which the topology data file is located. The data file must conform to the following format:

The first line must specify the number of data points in the custom waveform.

Each subsequent line’s entries depend upon the signalType parameter of the model. These are described below.

• VOLTAGE

Contains two numerical entries in scientific notation which specify the real value of one data point. The first entry on this line is the time of the data point in units of seconds, and the second entry is the value of the data point in units of V.

• CURRENT

Contains two numerical entries in scientific notation which specify the real value of one data point. The first entry on this line is the time of the data point in units of seconds, and the second entry is the value of the data point in units of A.

• OPTICALWATT

Contains two numerical entries in scientific notation which specify the real value of one data point. The first entry on this line is the time of the data point in units of seconds, and the second entry is the value of the data point in units of W.

• OPTICALDBM

Contains two numerical entries in scientific notation which specify the real value of one data point. The first entry on this line is the time of the data point in units of seconds, and the second entry is the value of the data point in units of dBm.

• OPTICAL_REAL_IMAG

Contains three numerical entries in scientific notation which specify the complex value of one data point. The first entry on this line is the time of the data point in units of seconds, and the second and third entries are the values of the real and imaginary portions of the data point, respectively, in units of sqrt(W).

• OPTICAL_AMP_PHASE

Contains three numerical entries in scientific notation which specify the complex value of one data point. The first entry on this line is the time of the data point in units of seconds, the second entry is the value of the amplitude magnitude of the data point in units of sqrt(W), and the third entry is the value of the phase of the data point in units of degrees.

• OPTICAL_AMP_PHASE_RAD


Contains three numerical entries in scientific notation which specify the complex value of one data point. The first entry on this line is the time of the data point in units of seconds, the second entry is the value of the amplitude magnitude of the data point in units of sqrt(W), and the third entry is the value of the phase of the data point in units of radians.

• OPTICAL_POWER_PHASE

Contains three numerical entries in scientific notation which specify the complex value of one data point. The first entry on this line is the time of the data point in units of seconds, the second entry is the value of the power magnitude of the data point in units of W, and the third entry is the value of the phase of the data point in units of degrees.

• OPTICAL_POWER_CHIRP

Contains three numerical entries in scientific notation which specify the complex value of one data point. The first entry on this line is the time of the data point in units of seconds, the second entry is the value of the power magnitude of the data point in units of W, and the third entry is the value of the frequency chirp of the data point in units of Hz.

OPTICAL_CHIRP_POWER_FMT2

This uses a different format for the data altogether compared to the other formats. This data format is as follows: The first line of the file starts with the keyword "POINTS:" followed by a space and the number of data points in the file. The second line of the file starts with the keyword "WAVELENGTH:" followed by a space and the wavelength of the optical signal in units of meters. The third line of the file starts with the keyword "DATE:" followed by a space and the time and date the data was taken in an arbitrary format. The actual data begins on the fourth line, and consists of the following columns separated by commas: The first column is the time in units of seconds. The second column is the frequency chirp in units of Hz. The third column is the power in units of watts. This is the format produced by the Agilent 86146B Optical Spectrum Analyzer (OSA) time-resolved chirp (TRC) test solution.

The user must specify the appropriate bit rate (bitRate), number of samples per bit (pointsPerBit), and total number of bits in the signal (patternLength). The model then takes the signal data from the data file, and interpolates it to fill the model’s output signal with appropriately sampled data. Note that if the input data does not contain enough data points to fill the output signal, the last data entry in the data file will be used to fill in the missing data points through to the end of the signal. For example, if you specify that the output signal is to have 27 bits, with 25 samples per bit, and at a bit rate of 109 bps, then the data file should contain at least 128 bits at a data rate of 109 bps which cover a time period of 12.8 ns. There are no requirements on the sampling rate in the data file, but it is generally best to have a reasonably large number of samples.

Properties

Inputs None

Outputs #1: Electrical or Optical Signal

#2: Binary Signal

Parameter Values Name Type Default Range Units signalType enumerated VOLTAGE VOLTAGE,


CURRENT, OPTICALDBM, OPTICALWATT, OPTICAL_REAL_IMAG, OPTICAL_AMP_PHASE, OPTICAL_AMP_PHASE_RAD, OPTICAL_POWER_PHASE, OPTICAL_POWER_CHIRP, OPTICAL_CHIRP_POWER_FMT2

filename string

bitRate double 10e9 [ 1, 1e15 ] bps

pointsPerBit integer 5 [ 1, 27 ] 2^x_pointsPerBit

patternLength integer 7 [ 1, 27 ] 2^x_bits wavelength double 1550e-9 [ 1e-15, 1e-5 ] m

Parameter Descriptions filename File name of the custom waveform data file signalType Signal type: VOLTAGE (V), CURRENT (A), OPTICALDBM (dBm), OPTICALWATT

(W), OPTICAL_REAL_IMAG (sqrt(W)), OPTICAL_AMP_PHASE (sqrt(W), degrees), OPTICAL_AMP_PHASE_RAD (sqrt(W), radians), OPTICAL_POWER_PHASE (W, degrees), OPTICAL_POWER_CHIRP (W, Hz), OPTICAL_CHIRP_POWER_FMT2 (Hz, W)

bitRate Bit rate of the signal, bit/sec pointsPerBit Number of data points used to represent each bit in output signal patternLength Total number of bits in output signal wavelength Wavelength of output optical signal (only for OPTICAL types)


Analog Sine Generator

This model creates an analog electrical signal representing a sine wave. It may be used to create a single sine wave, or a frequency comb of sine waves. In general, the output signal of this block can be described by the following equation:

∑ ++=N

onnppo VtfVts1

))2sin(()( φπ

where so is the output signal value (whether it be in units of volts or amps depends on the setting of the signalType parameter). Vpp is the peak to peak value of each sine wave in the frequency comb. N is the total number of frequencies summed together in the frequency comb. fn is the frequency of each of the individual sine waves present in the frequency comb. t is the time. φn is the phase offset for the nth element in the frequency comb. Vo is the offset value, which is added to the total signal after all sine waves in the frequency comb have been summed together.

Single sine wave generator When used as a single sine wave generator, the numChannels parameter is set to 1. The parameters in the Comb tab of the component parameters editing window are then unused.

There are several ways in which this sine wave may be specified: as a Periodic sine wave, in which the signal start and end meet to produce a continuous periodic sine wave; or as an Aperiodic sine wave, in which the signal start and end do not necessarily meet to produce a continuous periodic sine wave.

The primary difference between these approaches is how the time step (or alternatively, the sampling rate) and the number of data samples representing the signal are determined. The Periodic sine wave must be specified by setting the number of samples per period and number of periods. These settings along with the frequency determine the time step and the number of data samples. The Aperiodic sine wave must be spcified by setting the time step, and number of data samples to represent the output signal. The number of periods which appear in the signal are then determined by the specified frequency, and the number of periods may be a fractional amount.

The Periodic approach is most convenient except in cases where the signal will interact with other signals in the simulation which may have different frequencies and number of periods, resulting in different time steps and number of data samples. In these cases, the Aperiodic approach is most convenient because it allows the time step and number of data samples to be specified explicitly to match the other signals. When the Periodic approach is used, the user may easily view eye diagrams as the sine wave is mapped to a digital signal with a bit rate of twice the sine wave frequency and a number of bits equal to twice the number of periods specified. To override this behavior and set the bit rate equal to the frequency and the number of bits equal to the number of periods, set the parameter halfBitRate to Yes.

Frequency comb generator When used as a frequency comb generator, the numChannels is set to a number greater than 1.

The sine wave may be specified as a Periodic or Aperiodic sine wave, however, it should be noted that it will not be truly periodic when specified as a Periodic sine wave. This is because the time step and number of samples will be determined based on the first frequency of the comb, but the additional frequencies may not be periodic with these settings. To create a truly period signal, you will need to experiment or calculate the necessary time step and number of samples manually. It is best to use a periodic signal to minimize numerical artifacts in the simulation.


The frequency comb starts with the frequency value specified in parameter frequency, and continues up to frequency+(numChannels-1)*frequencyStep. Up to three channels in the frequency comb may be omitted, as specified by integer values of n (see equation at top of this model description) in parameters deleteChannel1, deleteChannel2, and deleteChannel3. This is a useful way to remove channels from the frequency plan to evaluate distortions such as composite second order (CSO), composite triple beat (CTB), and various orders of intermodulation (IM) distortions in the system at those frequencies.

The phases of each of the sine waves in the frequency comb may be identical (set to the value in parameter phase) or statistically determined according to the settings of the randomPhase and phaseSeed parameters. The settings of the phaseSeed parameter are according to the standard OptSim seed convention, defined below for reference: –1e8 <= seed <= 1 with the following behavior:

• seed < 0

The generator is seeded with the actual value of seed on every run of the simulation. This is useful for obtaining repeatable results.

• seed = 0

The generator is seeded with an integer hashed from the string value of the component’s name.

• seed = 1

The generator is seeded with a random number obtained from the system clock. This is essentially unrepeatable.

Notes on sampling rates Setting an appropriate sampling rate is very important, especially when evaluating a system utilizing a frequency comb with narrow channel spacing. You want to choose a sampling rate and total number of samples that will provide good resolution of the frequency spectrum as well as the time domain waveform. You want to make sure you have a sufficient number of samples between the frequencies in the frequency comb to evaluate distortions. An equation to keep in mind when choosing your sampling rate and total number of samples are the following:

imeWindowsimulatedTTsnoPo sf

1int1 =

⋅=∆

This equation says that the frequency resolution (∆f) is inversely proportional to the simulated time window. If you want a high frequency resolution, you need a large simulated time window.

You also need to be careful to have sufficient simulation bandwidth for your time domain signal resolution. The simulated bandwidth should be 3 times the total signal bandwidth. For example, if your highest sine wave frequency is 1 GHz, you should have 3 GHz simulated bandwidth. The following equation defines the simulated bandwidth:

sPerPeriodpo

ssim frequency

TBW int21 •==

where BWsim is the simulated bandwidth, Ts is the time step, frequency is the sine wave frequency, and pointsPerPeriod are the number of sampled points per period of the sine wave.


Properties

Inputs None

Outputs #1: Electrical Signal

Parameter Values Name Type Default Range Units numChannels integer 1 100000 none

signalType enumerated VOLTAGE VOLTAGE, CURRENT

mode enumerated Periodic Periodic, Aperiodic

Vpp double 1.0 [ -1e10, 1e10 ] Volts or Amps

Voffset double 0.5 [ -1e10, 1e10 ] Volts or Amps frequency double 10e9 [ 1e-15, 1e15 ] Hz

phase double -90 [ -1e10, 1e10 ] degrees

noPeriods integer 0 [ 0, 27 ] 2^x_noPeriods

ptsPerPeriod integer 5 [ 1, 27 ] 2^x_ptsPerPeriod noSamples integer 1 [ 1, 27 ] 2^x_samples

timeStep double 0.0 [ 0, 1 ] seconds

timeStart double 0.0 [ 0, 1e32 ] seconds

halfBitRate enumerated No Yes, No randomPhase enumerated FALSE TRUE, FALSE none

phaseSeed integer [ -1e8, 1 ] none

deleteChannel1 integer [ 0, 100000 ] none

deleteChannel2 integer [ 0, 100000 ] none deleteChannel3 integer [ 0, 100000 ] none

frequencyStep double [ 1e-15, 1e15 ] Hz

Parameter Descriptions numChannels Number of channels in frequency comb signalType Whether the output value is specified as voltage (Volts) or current (Amps) mode Determines how sine wave is specified Vpp Peak to peak value, in Volts or Amps Voffset Offset value, in Volts or Amps frequency Frequency of the sine wave, in Hz. Phase Phase of the sine wave, in degrees noPeriods Total number of periods in output signal ptsPerPeriod Number of data points used to represent each period in output signal noSamples Determines number of data samples in Aperiodic mode


timeStep Determines time step of sampled signal in Aperiodic mode timeStart Determines the start time of the output signal halfBitRate Determines how the signal bit rate and pattern length are calculated randomPhase Whether each channel in comb has a random phase or not phaseSeed Seed for random phase deleteChannel1 First channel to delete in frequency comb deleteChannel2 Second channel to delete in frequency comb deleteChannel3 Third channel to delete in frequency comb frequencyStep Frequency difference between channels in frequency comb (Hz)


Sawtooth Generator

This model creates an analog sawtooth signal. The signal’s sampling rate and duration are specified by setting the number of samples per period and number of periods. These settings along with the frequency determine the time step and the number of data samples. The output values may be either in voltage (Volts) or current (Amps). The Voffset parameter defines the midpoint output value. The Vpp parameter defines the output value difference between the maximum and minimum values in the sawtooth signal. The phase parameter is used to set the relative times when the signal wraps from its high to its low point. The frequency determines the slope of the sawtooth signal waveform.

Properties

Inputs None



CURRENT

Vpp double 1 [ -1e32, 1e32 ] V or A

Voffset double 0 [ -1e32, 1e32 ] V or A


ptsPerPeriod integer 5 [ 1, 27 ] 2^x_ptsPerPeriod frequency double 1e9 [ 0, 1e32 ] Hz

phase double 0 [ -180, 180 ] degrees

timeStart double 0 [ 0, 1e32 ] seconds

Parameter Descriptions signalType Whether the output value is specified as voltage (Volts) or current (Amps) frequency Frequency of sawtooth signal, in Hz. Phase Phase of the sawtooth signal, in degrees Vpp Peak to peak voltage, in Volts Voffset Offset voltage, in Volts ptsPerPeriod Number of data points used to represent each period in output signal noPeriods Total number of periods in output signal timeStart Determines the start time of the output signal


Frequency Sweep Generator

This model creates an analog frequency sweep signal. This is a periodic signal which chirps its frequency from a start frequency to a stop frequency value. In addition, the chirp itself can be specified to be periodic at a particular frequency. The signal’s sampling rate and duration are specified by setting the number of samples per period and number of periods. These settings along with the frequency (of the chirps) determine the time step and the number of data samples. The output may be specified to be in voltage (Volts) or current (Amps). The Voffset parameter defines the output midpoint value. The Vpp parameter defines the output value difference between the maximum and minimum values in the frequency sweep signal. The phase parameter is used to set the relative times when the signal wraps from its start to its end frequency point. The frequency determines the frequency of repetition of the chirp.

Properties

Inputs None



CURRENT

Vpp double 1 [ -1e32, 1e32 ] V or A

Voffset double 0 [ -1e32, 1e32 ] V or A

noPeriods integer 0 [ 0, 27 ] 2^x_noPeriods ptsPerPeriod integer 5 [ 1, 27 ] 2^x_ptsPerPeriod

frequency double 1e9 [ 0, 1e32 ] Hz


freqStart double 1e9 [ 0, 1e32 ] Hz freqStop double 1e10 [ 0, 1e32 ] Hz


Parameter Descriptions signalType Whether the output value is specified as voltage (Volts) or current (Amps) frequency Frequency of repetitions of the swept signal, in Hz. Phase Phase of the swept signal, in degrees Vpp Peak to peak voltage, in Volts Voffset Offset voltage, in Volts


freqStart Start frequency of the swept signal freqStop Stop frequency of the swept signal ptsPerPeriod Number of data points used to represent each period in output signal noPeriod Total number of periods in output signal timeStart Determines the start time of the output signal


Expression Signal Generator

This model creates an analog electrical signal representing a periodic function of time as specified. The output signal may be specified to be in voltage (Volts) or current (Amps). The signal is assumed to contain no noise. If desired, noise may be added to the signal by using the expression signal operator block following this block.

The user inputs for this model are the expression, the number of periods in the output signal, the number of data points used to represent each period, the frequency of repetition of the periods, the phase of the periodic signal, and the start time of the output signal. The sampling rate and time step are computed from the periodic frequency and the number of data points per period. The total number of output data points are the number of data points per period times the number of periods. For compatibility with digital signals and analysis options, the bit period associated with the electrical signal is half the signal period, the pattern length is twice the number of periods, and the bit rate is twice the signal’s periodic frequency. This allows a sine wave to represent alternating one and zero bit values.

The model creates a user variable named t. If this user variable name already exists, it is replaced with the one the model creates just for the duration of the model’s execution. Therefore, a user variable named t created outside this model cannot be used by this model.

The syntax for the expression is described in the Parameter Expressions section of Chapter 4 in the OptSim User Guide.

The value of t in the execution of the formula ranges from zero to the period time (1/frequency) minus the time step, then subtracts the period time to start the next period. After the signal is generated, the specified start time of the signal is applied, effectively shifting the signal in time by that amount.

Properties

Inputs None


Parameter Values

Name Type Default Range Units signalType enumerated VOLTAGE VOLTAGE,

CURRENT

expression string sin(360e9*t)


ptsPerPeriod integer 5 [ 1, 27 ] 2^x_ptsPerPeriod frequency double 1e9 [ 0, 1e32 ] Hz




Parameter Descriptions signalType Whether the output value is specified as voltage (Volts) or current (Amps) expression Expression which may involve user variables representing the periodic signal frequency Frequency of periods ptsPerPeriod Number of data points used to represent each period in output signal noPeriods Total number of periods in output signal phase Phase of output periodic signal timeStart Start time of signal

OptSim Models Reference: Block Mode Chapter 2: Electrical Modules •••• 37

Chapter 2: Electrical Modules

This chapter describes the electrical models:

• 1x1 Expression Signal Operator perform operations on an electrical signal

• 2x1 Expression Signal Operator perform operations on two electrical signals

• Summer add two electrical signals

• Mixer mix two electrical signals

• Electrical Filter filter an electrical signal

• Feed-Forward Equalization (FFE) Filter model a feed-forward (also known as linear) equalization filter

• Decision Feedback Equalization (DFE) Filter model a decision feedback equalization filter

• Standard S-Parameter Block implement an S-parameter

• Modulator-Driving S Block implement two coupled S-parameters.

• Electrical Amplifier amplify electrical signals

• Electrical Noise Adder

add electrical noise stochastically

• Crosstalk Block model cross-talk interference between electricals

• Electrical Integrate And Dump implement integrate-and-dump operations

38 Chapter 2: Electrical Modules OptSim Models Reference: Block Mode

1x1 Expression Signal Operator

This model operates on the input analog electrical signal according to the specified expressions. There is one expression for the noiseless electrical signal, and another expression which may either be applied to the standard deviations or the signal to noise ratio of each signal data point.

The user inputs for this model are the expression for the signal, the expression for the noise, and the mode of the noise expression functionality. The syntax for the expressions are described in the Parameter Expressions section of Chapter 1 of the user guide.

The model creates user variables named “s” representing the input electrical signal at each time point, “n” representing either the standard deviation or signal to noise ratio of the electrical signal at each time point, and “t” representing the time value at each data point. If these user variable names already exists, they are replaced with the ones the model creates just for the duration of the model’s execution. Therefore, user variables with these names created outside this model cannot be used by this model.

Properties

Inputs #1: Electrical Signal


Parameter Values Name Type Default Range Units expression_sig string s expression_noise string n

noise_mode enumerated StdDev StdDev, SNR

Parameter Descriptions expression_sig Expression which may involve user variables representing the operation on the input

electrical signal expression_noise Expression which may involve user variables representing the operation on the noise

of the input electrical signal noise_mode Whether the noise expression applies to the SNR or the standard deviation of the

input signal


2x1 Expression Signal Operator

This model operates on two input analog electrical signals according to the specified expressions to produce a single output signal. There is one expression for the noiseless electrical signals, and another expression which may either be applied to the standard deviations or the signal to noise ratio of each signal data point.

The user inputs for this model are the expression for the signal, the expression for the noise, and the mode of the noise expression functionality. The syntax for the expressions are described in the Parameter Expressions section of Chapter 3 of this document.

The model creates user variables named s1 and s2 representing the input electrical signals #1 and #2 respectively at each time point, n1 and n2 representing either the standard deviation or signal to noise ratio of the input electrical signals #1 and #2 at each time point, and t representing the time value each each data point. If these user variable names already exists, they are replaced with the ones the model creates just for the duration of the model’s execution. Therefore, user variables with these names created outside this model cannot be used by this model.

Properties


#2: Electrical Signal


Parameter Values Name Type Default Range Units expression_sig string s1+s2

expression_noise string n1+n2

noise_mode enumerated StdDev StdDev, SNR

Parameter Descriptions expression_sig Expression which may involve user variables representing the operation on the input

electrical signals

expression_noise Expression which may involve user variables representing the operation on the noise of the input electrical signals

noise_mode Whether the noise formula applies to the SNR or the standard deviation of the input signals


Summer

This model operates on two input analog electrical signals to produce an output signal consisting of the sum of the two input signals.

The user inputs for this model are the loss of each of the input signals to be included prior to summing the signals together.

Properties




Parameter Values Name Type Default Range Units loss1 double 0 [ 0, 1000 ] dB

loss2 double 0 [ 0, 1000 ] dB

Parameter Descriptions loss1 Signal loss to be applied to first input signal prior to summing operation loss2 Signal loss to be applied to second input signal prior to summing operation


Mixer

This model mixes the two input analog electrical signals to produce an output signal.

The user inputs for this model are the loss of each of the input signals to be included prior to mixing the signals together

Properties




Parameter Values Name Type Default Range Units loss1 double 0 [ 0, 1000 ] dB

loss2 double 0 [ 0, 1000 ] dB

Parameter Descriptions loss1 Signal loss to be applied to first input signal prior to mixing operation loss2 Signal loss to be applied to second input signal prior to mixing operation


Electrical Filter

This model implements a variety of standard electrical filters – Butterworth, Chebyshev, Bessel and Ideal – each in low pass (LP), high pass (HP) and band pass (BP) configurations.

Definitions of filter types The operation of the filter is performed in the Fourier domain in terms of the amplitude response function

( )H f :

( ) ( ) ( )out inE f H f E f= .

Since the electrical signal is a real quantity, the filter functions are hermitian and need only be defined for positive frequencies.

The class of filter response is controlled by the parameter type. The 3dB bandwidth B of the filter is set by the parameter bandwidth, and the filter order N with order. The gain coefficient G is related to the parameter lossGain by / 2010G = lossGain . The nominal center of a band pass filter is given by cf (geometricCenter).

We provide complete but concise definitions of the filter responses here. The reader is referred to any good text on filter design to explore the complete theory of these filters. The response functions ( )H f for each filter type are defined as follows.

Ideal response • Low Pass:

( )ideal ,0 ,LP

G f BH f

f B<=

= >

• High Pass:

( )ideal 0 ,,HP

f BH f

G f B<=

= >

• Band Pass:

( )ideal,

0 ,c

BPc

G f f BH f

f f B

− <== − >

Butterworth response The low pass power response is defined as


( )( )

2

21

1 fB

H f =+

.

Defining the filter poles as

( )exp 2 12ks j j N k

Nπ = + −

,

for 1k N= K , we have:

• Low Pass:

( ) ( )But

1

1/

N

LPk k

H fjf B s=

=−∏ .

• High Pass: The transformation ( ) ( )/ /jf B B jf→ applied to the low pass response gives

( ) ( )But

1

1/

N

HPk k

H fB jf s=

=−∏ .

• Band Pass: The transformation /cf f f f→ − applied to the low pass response gives

( ) ( )But

1

1/

N

HPk c k

H fj f f fB s=

=− −∏ .

Chebyshev response The low pass power response is defined as

( ) ( )Cheb

2 2

1 ,1 /

N

LPH fV f Bε

=+

where the Chebyshev polynomials are defined as ( ) ( )1cos cosNV x N x−= . The passband rippleε is

controlled with the parameter passbandRipple. The filters are implemented with the same relations as the Butterworth filters but with poles defined as

( ) 1 1cos 2 1 sinh2k

js j kN N

πε

− = − +

Bessel response The Bessel response is defined in general as:

( ) ∑=

=

N

kk

kBessel

fsfH

1

1


( )( )!!2

!2kNk

kNs kNk−

−=−

Where sk is derived from the Bessel polynomials [1]. The previously described transformations [2], [3] are used to obtain the low-pass, high-pass, and band-pass implementations:

• Low Pass:

( ) ∑=

−

⋅=

N

k

k

normkBessLP B

fjfsfH1

1

• High Pass:

( ) ∑=

−

⋅=

N

k

k

normkBessHP jf

BfsfH1

1

• Band Pass:

( ) ( )∑=

−

−⋅=

N

k

kc

normkBessBP B

fffjfsfH

1

1

The frequency normalization parameter fnorm is generally represented by:

( )12)2ln( −⋅= Nfnorm but is more accurately represented internally for values of 10≤N [1].

Noise filtering As discussed elsewhere, noise in electrical signals may be represented in two ways – as a stochastic component to the sampled voltage or current values, or as a companion vector of time-dependent standard deviations of a Gaussian noise source. In the former case, the noise is of course directly filtered along with the signal. However, for noise stored as a sequence of standard deviations, there is no phase information available and it is impossible to convert the noise to the frequency domain. Consequently, the standard deviation representation of noise is unchanged by this model, and the stochastic representation is generally a better choice. If the filter is to be used as a part of an optical receiver and a quasi-analytic treatment of noise is needed, the Compound Receiver model should be used instead.

Test display The test button for the model displays the spectral response generated by the current filter settings. The format for display of the complex filter function H(f) is controlled by the parameter test_display. Writing

( ) | ( ) | exp[ ( )]H f H f j fφ= , the display choices for left and right axes are ( ) 2[| | , ( )]H f fφ

(norm_phase), ( ) 2[| | , arg ( )]H f H f (norm_phase_wrap), ( )10[20log | |, ( )]H f fφ (dB_phase),

( )10[20log | |, arg ( )]H f H f (dB_phase_wrap) and ( ) ( )[Re ( ) , Im ( ) ]H f H f (real_imag).


References [1] L. Weinberg, Network Analysis and Synthesis. New York, NY: McGraw-Hill Book Company, 1962.

[2] A. D. Poularikas and S. Seely, Signals and Systems. Boston, MA: PWS-Kent Publishing Company, 1991.

[3] M. E. VanValkenberg, Analog Filter Design. New York, NY: Oxford University Press, 1982

Properties

Inputs #1: Electrical signal (voltage or current)

Outputs #1: Electrical signal (voltage or current)

Parameter Values Name ype Default Range Units type enumerated LPbessel LPbessel,

LPbutterworth, LPchebyshev, LPideal, HPbessel, HPbutterworth, HPchebyshev, HPideal, BPbessel, BPbutterworth, BPchebyshev, BPideal

preserve_alignment enumerated YES NO, YES bandwidth double 10e9 [ 0, 1e18 ] Hz

order integer 4 [ 0, 128 ] none

lossGain double 0 [ -1e32, 1e32 ] dB

passbandRipple double 1e-15 [ 1e-15, 1e32 ] dB geometricCenter double 0.0 [ 0, 1e32 ] Hz

test_display enumerated norm_phase norm_phase, norm_phase_wrap, dB_phase, dB_phase_wrap, real_imag

Parameter Descriptions type Filter type preserve_alignment Preserve alignment of incoming bits


bandwidth Filter 3dB bandwidth geometricCenter Geometric center frequency for bandpass filters order Order of the filter lossGain Filter gain or loss passbandRipple Passband ripple for Chebyshev filter type Filter type bandwidth Filter 3dB bandwidth geometricCenter Geometric center frequency for bandpass filters test_display Format for display of complex filter function


Feed-Forward Equalization Filter

This module simulates a Feed-Forward Equalization (FFE) filter (also known as forward-feedback or linear equalization). This equliazer structure is based on transversal finite-impulse response (FIR) filter. The signal transformation is described by the following equation:

)( )( TitxCtyN

Nii ∆−⋅= ∑

−=

(1)

where x(t) is incoming electrical signal, y(t) - outgoing signal, (2N+1) – number of taps, Ci - weight coefficients or taps, and ∆T – taps delay.

The model takes three parameters: number of taps 2N + 1 (number_of_taps), taps values (taps) specified as an array w1, w2, …, wk, and taps delay (taps_delay) specified as fraction of bit period, i.e.

∆T = TB / 2^( taps_delay )

For example, the taps_delay =1 correspond to delay of half of the bit period - ∆T =1/2 TB.

Properties

Inputs #1: Electrical signal

Outputs #1: Electrical l signal

Parameter Values Name Type Default Range Unit number_of_taps Integer 5 [ 1, 27 ] None

taps Double Array 0, 0, 1, 0, 0 [ -1, 1 ] None taps_delay Double 1 [ 0, 27] 2^x_points

Parameter Descriptions number_of_taps Number of taps taps Taps weights taps_delay Taps delays as fraction of bit period


Decision-Feedback Equalizer for Electronic Dispersion Compensation with Minimum Mean Square Error Optimization (MMSE DFE-EDC)

This block models a Decision-Feedback Equalizer for Electronic Dispersion Compensation (DFE-EDC) including the coefficient optimization with the Minimum Mean Square Error criterion.

The DFE-EDC is a nonlinear equalizer with two sections; a first Feed-Forward section filters the channel output y and the noise n with a Finite Impulse Response (FIR) filter with coefficients FeedForward_taps; a second Feed-Back section filters the decision threshold output with an FIR filter with coefficients FeedBack_taps. The equalizer structure is depicted in Fig. 1.

Figure 1 depicts the structure of a DFE.

Fig. 1 DFE-EDC structure

Using the past decisions to correct the error at the decision threshold helps the equalizer in compensating the Inter-Symbolic Interference. For this reason the DFE is widely used in the most modern receiver for single- and multi-mode electronic dispersion compensation.

This block includes the coefficients optimization with a Minimum Mean Square Error criterion [1]. The user selects the length of the Feed-Forward and Feed-Back filters using the corresponding parameters FeedForward_taps_number and FeedBack_taps_number. In case the number of Feed-back taps is set to zero, the block degenerates in a Feed-Forward Equalizer (FFE) and the optimization finds the optimal FFE.

The model creates an output file containing the values of the Feed-Forward and Feed-Back taps, as well as the Delay and the Offset. In case of MMSE-optimization this file can be used to know the optimal coefficient values so to re-use them in some following simulation with some perturbation to analyze the performance of a non-ideal equalizer.

When the Monte-Carlo technique is used to estimate the BER, the noise samples are mixed together with the signal. If the MMSE optimization is selected, the DFE-EDC model will optimize its coefficients on the addition of noise and signal, resulting in an excessive compensation of the noise. In fact, the noise actual samples are unknown in reality and for this reason they shouldn’t be used in the optimization process. For this reason in case of Monte-Carlo simulation the model should executed a first time with no noise, to only optimize its coefficients; then these coefficients should be used, the MMSE optimization should be turned off, and a second simulation including noise should be executed.


References [1] Paul A. Voois, InKyu Lee, John M. Cioffi, “The Effect of Decision Delay in Finite-Length Decision Feedback Equalization”, IEEE Transactions on Information Theory, vol 42, no. 2, March 1996

[2] J. G. Proakis, Digital Communications, New York: McGraw Hill, 4th edition (August 15, 2000)

Properties

Inputs #1: Electrical Signal at the output of the receiver

#2: Binary Signal transmitted

Outputs #1: Electrical Signal equalized

Parameter Values Name Type Default Range Units FeedForward_taps_number Integer 14 [ 1, 100 ]

FeedBack_taps_number Integer 5 [ 0, 100 ]

MMSE_optimization enumerated yes yes, no

FeedForward_taps Double Array 1.0, 0.0, .. [ -1e32, 1e32 ] FeedBack_taps Double Array 1.0, 0.0, .. [ -1e32, 1e32 ]

Offset Double 0 [ -1e32, 1e32 ]

Delay Integer 0 [ 0 Bit_number) Bit_periods

Parameter Descriptions FeedForward_taps_number Number of taps of the Feed-Forward Finite Impulse Filter FeedBack_taps_number Number of taps of the Feed-Back Finite Impulse Filter MMSE_optimization If Yes, the Feed-Forward and Feed-Back taps, the offset and the delay will be

optimized with the Minimum Mean Square criterion FeedForward_taps Array containing the values of the Feed-Forward taps

FeedBack_taps Array containing the values of the Feed-Back taps Offset Offset added to the signal before the decision threshold, scaled on the ratio

between the electrical and binary signal power Delay Delay in number of bit periods before filtering the signal with the Feed-Forward

FIR


Standard S-Parameter Block

This block models a standard S-parameter block shown in the following diagram.

a1

b1

a2

b2

SS

Where

=

2

1

aa

a represents the incident waves and

=

2

1

bb

b represents the reflection waves.

,Sab =

=

2221

1211

SSSS

S .

This model is actually implemented as

a1

a2

b1

b2

SS

i.e., a represents input signals and b represents the output signals. This model calculates the reflection signals from the incident signals.

The S parameters are input in a text data file. An example of the data file format is as follows:

! Frequency S11 S21 S12 S22 # HZ S DB R 50 50000000 –12.429 164.41 –40.377 –123.19 –50.881 172.86 –1.4557 76.855 149750000 –14.815 140.64 –37.81 –26.561 –65.337 58.476 –1.3314 –130.63 249500000 –19.546 132.07 –36.337 56.143 –71.422 –175.4 –1.3279 21.993 349250000 –23.769 167.4 –36.412 146.36 –75.215 –171.29 –1.2779 174.28

In this example file, the first line are comments. The comments must follow the symbol “!”. The line starting with the symbol “#” is standard and should not be changed. The remaining lines are data and have 9 columns : col. 1 is frequency (Hz); col. 2 & 3 are the amplitude (dB) and phase (degrees) of S11, respectively; col. 4 & 5 are the amplitude (dB) and phase (degrees) of S21, respectively; col. 6 & 7 are the amplitude (dB) and phase (degrees) of S12, respectively; col. 8 & 9 are the amplitude (dB) and phase (degrees) of S22, respectively.

There can be any number of lines of S parameter data, and no additional characters are required after the last line, to terminate the file. The Frequency values need not be uniformly spaced.

When the S parameters at the frequencies lower than the minimum frequency or higher than the maximum frequency given in the data file are needed, the algorithm will truncate the S values at those frequencies either to zero or to the values at the lowest frequency for the lower frequency end and the highest frequnecy


for the higher frequency end. The way of truncation will be controlled by setting the input parameters “hf_truncate” and “lf_truncate”.

This block can be used as a custom electrical filter by only using input 1 (a1) and output 2 (b2) and keeping input 2 (a2) and output 1 (b1) disconnected. The transfer functon of the electrical filter will be represented by the S21 parameters. Other S parameters are not needed for the calculation but the S data file must be complete to avoid the error of reading files.

Properties


#2: Electrical signal



Parameter Values Name Type Default Range Units filename string lf_truncate enumerated FirstValue FirstValue, Zero

hf_truncate enumerated Zero Zero, LastValue

Parameter Descriptions filename File name of the S parameters data file hf_truncate The way of truncating the S parameters in the higher frequency end at the frequencies

higher than the maxmum frequency given in the S data file (Zero, LastValue) lf_truncate The way of truncating the S parameters in the lower frequency end at the frequencies

lower than the minimum frequency given in the S data file (FirstValue, Zero)


Modulator-Driving S Block

This model containing two S-parameter blocks calculates the electrical characteristics of a modulator and its driving circuits. Normally this model is followed by a modulator model in simulating systems. It can be described by the following diagram.

a1 a2’

b2’

S τ S’b1

a2

b2 a1’

b1’

IN

OUT

In this model, two S blocks and a time delay τ are included. The first S block represents the effect from the source, i.e., the driving circuit of the modulator, and the second S block S’ represents the effect from the load, i.e., the electrical

characteristics of the modulator. Where

=

2

1

aa


=

2

1

bb

b represents the

reflection waves, ,Sab =

=

2221

1211

SSSS

S . In the implementation, a2’ was set as zero and b1 was not calculated.

The interactions between the two S blocks have been taken into consideration.

The S parameters are input in an text data file. An example of the data file format is as follows. ! Frequency S11 S21 S12 S22 # HZ S DB R 50 50000000 –12.429 164.41 –40.377 –123.19 –50.881 172.86 –1.4557 76.855 149750000 –14.815 140.64 –37.81 –26.561 –65.337 58.476 –1.3314 –130.63 249500000 –19.546 132.07 –36.337 56.143 –71.422 –175.4 –1.3279 21.993 349250000 –23.769 167.4 –36.412 146.36 –75.215 –171.29 –1.2779 174.28

In this example file, the first line are comments. The comments must follow the symbol “!”. The line starting with the symbol “#” is standard and should not be changed. The remaining lines are data and have 9 columns : col. 1 is frequency (Hz); col. 2 & 3 are the amplitude (dB) and phase (degrees) of S11, respectively; col. 4 & 5 are the amplitude (dB) and phase (degrees) of S21, respectively; col. 6 & 7 are the amplitude (dB) and phase (degrees) of S12, respectively; col. 8 & 9 are the amplitude (dB) and phase (degrees) of S22, respectively.

There can be any number of lines of S parameter data, and no additional characters are required after the last line to terminate the file. The Frequency values need not be uniformly spaced.

When the S parameters at the frequencies lower than the minimum frequency or higher than the maximum frequency given in the data file are needed, the algorithm will truncate the S values at those frequencies either to zero or to the values at the lowest frequency for the lower frequency end and the highest frequnecy for the higher frequency end. The way of truncation will be controlled by setting the input parameters “hf_truncate” and “lf_truncate”.

Properties




Parameter Values Name Type Default Range Units SFile_src string

SFile_load string

lf_truncate enumerated FirstValue FirstValue, Zero hf_truncate enumerated Zero Zero, LastValue

TimeDelay double 0.0 [ 0.0, 1.0e32 ] Sec

Parameter Descriptions SFile_load File name of the S parameters for the load. If no file name is given, the signals will go through

without being affected. SFile_src File name of the S parameters for the source. If no file name is given, the signals will go through

without being affected. TimeDelay Time delay introduced in this blcok. hf_truncate The way of truncating the S parameters in the higher frequency end at the frequencies higher than the

maxmum frequency given in the S data file (Zero, LastValue) lf_truncate The way of truncating the S parameters in the lower frequency end at the frequencies lower than the

minimum frequency given in the S data file (FirstValue, Zero)


Electrical Gain

This model simulates an ideal electrical amplifier or attenuator (negative gain is allowed). The gain (Gain) is specified in dB, where GaindB = 20log10(Gainlinear).

Properties



Parameter Values Name Type Default Range Units Gain double 0 -1e32≤ x ≤ 1e32 dB

Parameter Descriptions Gain Ideal electrical gain in dB


Electrical Amplifier

This component provides a phenomenological model for an electrical amplifier, converting an incoming current signal to an outgoing voltage signal. It is equally applicable to transimpedance amplifiers (commonly used as the front end amplifier in optical receivers), and high-impedance amplifiers. This component is usually used in conjunction with the photodetector and electrical filter component to model a complete receiver. The model accounts for thermal noise in a stochastic fashion. If a Quasi-Analytic treatment of noise is required, the monolithic Receiver model should be used.

Premplifier Model and Parameter Values

The model converts a current signal to a voltage signal through a transfer function ( )H f , such that in the Fourier domain the output voltage satisfies

( ) ( ) ( ) ( )thˆ

j j j jv f H f i f N f = +

where ( )thN f is an optional thermal noise source discussed below. The transfer function is specified either as a simple pole-zero expression or with a user-defined file as selected by the parameter model_type:

• Pole-zero representation (model_type=defined) The transfer function is defined as

( ) pT

z p

ffH f Zj f f j f f

= ⋅ + +

,

in terms of the transimpedance ZT , low frequency zero fz, and high frequency pole fp. These quantities are set by tZ, zero and pole respectively. Users requiring additional poles and zeroes should contact RSoft Design Group.

• User-defined representation (model_type=custom) The user provides a file filename specifiying the real and imaginary parts of the filter function ( )H f with the following format: <num_pts> freq_1 real(H(freq_1)) imag(H(freq_1)) freq_2 real(H(freq_2)) imag(H(freq_2)) freq_3 real(H(freq_3)) imag(H(freq_3)) … The frequencies must be monotonically decreasing or increasing. The parameters lo_trunc and hi_trunc control the extrapolation behavior of the custom filter at low and high frquencies respectively, if the supplied data does not cover the whole numerical bandwidth. With these parameters set to zero, any outlying points are set to zero, while if the value extend is chosen, the extreme value of the supplied data is applied to all outlying frequencies.

Note that if a photodetector is connected prior to the electrical amplifier, it is often convenient to lump their responses together into the present model's transfer function. In this case, the parameter detect in the photodetector model would be set to false. See the documentation of that model for details. Often, only rudimentary high-frequency information such as the receiver’s 3-dB bandwidth is available. In these situations, it is useful to assign this frequency to the parameter pole, set zero = 0. This will result in a frequency response with a 3-dB point that closely approximates the desired one.


Thermal Noise Thermal noise is added to the signal if include_thermal=YES. This noise is represented stochastically by adding a complex random number to the Fourier representation of the incoming current before the transfer function is applied. The added noise source is given by

( ) ( )thˆˆ

j j jN f S f fξ= ∆ ,

where ˆjξ is a complex Gaussian random variable of zero mean and unit variance, f∆ is the frequency spacing of the

sampled spectrum, and the the noise spectral density function ( )thS f measured in A2/Hz is defined as

( ) 2 4 6th 0 2 4 6S f a a f a f a f= + + +

with the coefficients ja set by the parameters n_a0, n_a2, n_a4 and n_a6. This is a more general form of the commonly accepted expression:

2

2th

4 (2 )( ) 4 T

f m

kT CS f kT fR g

π= + Γ

which describes the thermal contribution of the feedback resistor in the transimpedance amplifier and the thermal channel noise in the preamplifier input transistor. In this expression, k is Boltzmann’s constant, T is the temperature,

fR is the amplifier feedback resistance, mg is the transconductance of the preamplifier input transistor, Γ is the excess

channel noise factor, and TC is the total input capacitance. The generalized polynomial representation is chosen to allow the user to tailor the noise spectral density as he sees fit. It is also useful when actual noise spectra are available since it allows the noise to be represented by simply fitting the polynomial coefficients to measured data. To model white noise, simply set the coefficients a 2, a4, and a6 to zero.

The random number seed for the thermal noise source is controlled with the parameter random_seed, which supports the standard OptSim convention for random number seeds.

Test Function The test function can display either the transfer function of the amplifier or the thermal noise spectral density as determined by the parameter test_output. For the former case, the parameter test_display selects the representation for the complex transfer function.

Properties

Inputs #1: Electrical signal (current)

Outputs #1: Electrical signal (voltage)

Parameter Values Name Type Default Range Units modeltype enumerated defined defined, custom

filename string


tZ double 1.0 [ 0, 1e18 ] Ohms

zero double 0.0 [ 0, 1e18 ] Hz pole double 1e18 [ 0, 1e18 ] Hz

lo_trunc enumerated extend extend, zero

hi_trunc enumerated extend extend, zero

n_a0 double 5.4617e-23 [ 0, 1e32 ] A^2/Hz n_a2 double 2.924e-43 [ 0, 1e32 ] A^2/Hz^3

n_a4 double 1.1118e-63 [ 0, 1e32 ] A^2/Hz^5

n_a6 double 0 [ 0, 1e32 ] A^2/Hz^7

include_thermal enumerated YES NO, YES random_seed integer 0 [ -1e8, 1 ] none


test_output enumerated noise_spectrum noise_spectrum, amp_response

Parameter Descriptions model_type Select parameterized or user-defined transfer function filename Filename for user-defined transfer function tZ Transimpedance coefficient zero Single zero low frequency rolloff pole Single zero high frequency rolloff lo_trunc Low frequency truncation behavior for user-defined transfer function hi_trunc High frequency truncation behavior for user-defined transfer function n_a0 Thermal noise coefficient n_a2 Thermal noise coefficient n_a4 Thermal noise coefficient n_a6 Thermal noise coefficient include_thermal Enable/disable thermal noise random_seed Random number seed for thermal noise (obeys OptSim random seed convention) test_display Select display format for complex transfer function test_output Select test function display of noise spectrum or transfer function


Electrical Noise Adder

This model provides a mechanism for directly adding the electrical noise to the signal with various profiles. With the parameter distribution, the user may select noise distributions with Gaussian, Uniform, or Rayleigh profiles. The noise is added stochastically to the electrical signal (see Chapter 5 of OptSim User Guide for more details). So a “quiet” electrical signal could be converted to a noisy one as follows:

ˆqtensek kkV V N= +

where kV denote real-valued time-sampled electrical signal (voltage or current); and the added stochastic noise term ˆkN

follows one of the following distributions as selected by user:

• Gaussian

2

2( )

21( )2

V V

P V e σπ

−−=

k

k

0000

,

with the mean 0V specified by mean, and standard deviation s specified by std_dev.

• Uniform

( ) 0

0

1 , / 2

0, / 2

kk

k

V V rrP V

V V r

− ≤= − >

with the mean 0V specified by mean and range r is specified by range.

• Rayleigh

2

2-2

2e

( )

kVs

kVP V

sκ=

where s is a scale parameter specified by rayleigh_S.

Note that once noise has been added stochastically to an electrical signal, it is no longer suitable for Quasi-Analytic noise analysis. The receiver and BER must be set to Monte-Carlo noise analysis.

Seed for stochastic noise The user may control the seeding of the random number generator used to implement the stochastic representation. In the standard OptSim convention, this satisfies –1e8 <= seed <= 1 with the following behavior:

• seed < 0

The generator is seeded with the actual value of seed on every run of the simulation. This is useful for obtaining repeatable results.

• seed = 0

The generator is seeded with an integer hashed from the string value of the component’s name.

• seed = 1



Properties



Parameter Values Name Type Default Range Units distribution enumerated Gaussian Gaussian, Uniform,

Rayleigh

mean double 0 [ 0, 1000 ] V or A

std_dev double 1 [ 0, 1000 ] V or A range double 1 [ 0, 1e32 ] V or A

rayleigh_S double 1 [ 0, 1e32 ] V or A

seed integer 1 [ -1e8, 1 ] none

Parameter Descriptions distribution Select appropriate noise distribution mean Mean value of selected noise distribution std_dev Standard deviation of selected noise distribution range Range of uniform noise distribution rayleigh_S Rayleigh scale parameter seed Seed for random number generation


Crosstalk Block

This model adds crosstalk to the input electrical signals according to the specified crosstalk amounts.

Properties

Inputs #1-N: Electrical signal

Outputs #1-N: Electrical signal

Parameter Values Name Type Default Range Units Mode enumerated Complex Magnitude, Complex

ParameterType enumerated Simple Simple

InChannelTransmission

double 0 [ -1e32, 1e32 ] dB

NearestNeighborXtlk double -20 [ -1e32, 1e32 ] dB

NextNearestNeighborXtlk

double -30 [ -1e32, 1e32 ] dB

Parameter Descriptions Mode Complex applies the crosstalk to the full complex electrical signals; Magnitude applies the

crosstalk to the magnitude of the input electrical signals ParameterType Simple uses above three parameters; exists for future expansion InChannelTransmission Amount of input signal to transmit to corresponding output signal NearestNeighborXtlk Amount of crosstalk from nearest neighbor signal (NN) NextNearestNeighborXtlk Amount of crosstalk from next nearest neighbor signal (NNN)


Electrical Integrate And Dump

This model represents an integrate and dump operation which may be performed on an electrical signal. The model integrates the input electrical signal over a specified period, multiplies it by a specified constant, and dumps that value at the end of the integration period. It then starts the integration over again for the next period.

Prior to starting the first integration period, the model waits the amount of time specified in InitialDelay. Then the first integration period begins, which consists of three regions. The first region is the initial ignored time of the period. From the start of the integration period until the time specified in IntegrateBegin, the input signal value is not integrated. The second region is the integration region. From the time specified in IntegrateBegin until the time specified in IntegrateEnd, the input signal is integrated according to the following equation:

( )∑=

=2

1

τ

τtdttxay

where y is the value dumped at the output at the end of the integration period (and held through to the end of the next integration period), a is the constant, x(t) is the value of the input signal at time t, dt is the time step between input signal samples, τ1 is the start of the integration region specified by IntegrateBegin, and τ2 is the last sample time in the integration region prior to the time specified by IntegrateEnd.

A pictorial representation of the operation of this model follows:

...

t=0 t=td t=td+t1 t=td+t2 t=td+tp

InitialDelayIgnore

IntegrateIgnore

Ignore

where each color coded tick mark represents a sample in the input signal, t=td is the InitialDelay value, t1 is the IntegrateBegin value, t2 is the IntegrateEnd value, and tp is the IntegratePeriod value.

The input signal’s noise may be handled in one of two ways. It may be ignored by setting the NoiseMode to Ignore. This is useful if it is known that the integration will average the noise to zero over the integration period, or if the input signal has no quasianalytical noise component. Alternatively, the model may be set to convert the quasianalytical noise representation of the input signal to a Monte Carlo noise representation prior to performing the integration. In either case, the output signal has no quasianalytical noise component.

The block will output the integrated value at the end of the signal even if it has not yet reached the end of the specified integration period. The output result is provided at the last data point of the period.

Properties




Parameter Values Name Type Default Range Units NoiseMode enumerated Ignore Ignore, MonteCarlo

a double 1e9 [ 0, 1e32 ] none

InitialDelay double 0 [ 0, 1e32 ] s IntegrateBegin double 0 [ 0, 1e32 ] s

IntegrateEnd double 1e-9 [ 0, 1e32 ] s

IntegratePeriod double 1e-9 [ 0, 1e32 ] s

Parameter Descriptions NoiseMode Whether to ignore the noise array in input signal, or convert to MonteCarlo prior to

integration a Integration constant InitialDelay Time before integration periods begin IntegrateBegin Time when integration phase of integration period begins IntegrateEnd Time when second ignore phase of integration period begins IntegratePeriod Time when next integration period begins

OptSim Models Reference: Block Mode Chapter 3: Special Functions and Logical Models •••• 65

Chapter 3: Special Functions and Logical Models

The models in this chapter fall into no other general categories:

• Shift Signal shift a signal of any type in time

• Boolean Operator perform element boolean operations on binary signals

• D Flip-Flop model a delay (D) flip-flop

• T Flip-Flop model a toggle (T) flip-flop

• Electrical Signal Resampler resample input electrical signal to modify time-step and number of samples

• Optical Signal Resampler resample input optical signal to modify time-step and number of samples

66 •••• Chapter 3: Special Functions and Logical Models OptSim Models Reference: Block Mode

Shift Signal

This model may be used to do one of two primary functions: delay the output signal relative to the input signal, or shift the signal data around in the data structure’s array without modifying the signal timing. Note that when signal data is shifted by this block, it wraps around the edges of the data array. These two functions are described below.

Time Delay Function The ‘timeDelay’ mode (set by the parameter shiftType) delays the output signal relative to the input signal by a specified amount by either adjusting the start time of the signal while leaving the signal data array alone (when keepStartTime is set to ‘No’), or by shifting the signal data around within the data array to implement the delay (when keepStartTime is set to ‘Yes’). Time delays are specified in units of time only. Use of the time delay feature is useful whenever a time delay needs to be modeled for any type of signal, whether it be binary, electrical, or optical. When setting keepStartTime to ‘Yes’ for bit-streams, the requested time shift will be rounded to the nearest multiple of the bit period. For example, a 10Gbps bit sequence can only be shifted by 0, ± 100ps, ± 200ps, etc. Thus, a 10ps shift effectively gets rounded to zero. This is necessary since binary signals are only stored as ones and zeros, and so fractional bit shifts can't be taken into account while also keeping the start time unchanged.

Shift Function The ‘timeShift’, ‘bitShift’, and ‘percentShift’ modes (set by the parameter shiftType) shift the signal data array without changing the actual signal timing. The user may specify the amount of shift in time units, percentage of the signal array, or number of bits. Appropriate adjustments are made to the start time of the signal to maintain the original signal timing. A positive shift will advance the start time of the signal while rotating the data to the left in the data array.

The use of this block to shift signal data around in the data array is not generally required for simulations, but is useful in cases where the location of a bit in the data array is significant. For example, if it is desired to simulate a single Gaussian pulse and view the flat phase of the frequency spectrum, this block could be used to center the pulse around time = 0. Note that the relation between shifts in time and the frequency spectrum are described by the following basic Fourier transform pair:

( ) 020

ftjett πδ −=−

Example The following example demonstrates the impact that the Time Delay function of the Shift Signal block has on a signal waveform. The following topology contains a PRBS generator block that generates the bit sequence ‘0100’ with a bit period of 0.1 ns. Following this, the Optical Signal Generator block is used to convert this binary sequence to an optical signal with 32 samples (data points) per bit. The optical signal then goes through the Shift Signal block. The signal is plotted at the output of each of these blocks, plus there is a combined plot of the signal going into and coming out of the Shift Signal plot.


In the first exercise, we set the Shift Signal block to apply a time shift of 1e-10 seconds to the input signal. The signal plot coming into the Shift Signal block, coming out of the Shift Signal block, and combined are shown below, respectively. Note that the signal is shifted to the left by one bit period (1e-10 seconds) in the array, but the absolute time of the single ‘1’ bit in the output signal is not changed by the Shift Signal block. The signal going into and coming out of the Shift Signal block overlap one another perfectly in the combined plot. The start time of the input signal’s data array is 0 seconds, while the start time of the output signal’s data array is 1e-10 seconds. The start time of the signal data array has been shifted forward by 1e-10 seconds without affecting the timing of the actual signal. Note that the plot does not display the actual signal start time, but this information is stored with the signal data structure and can be viewed using the signal summary feature or with the Save Signal To File block.

Now we change the above simulation by setting it to apply a time delay of 1e-10 seconds, and set keepStartTime to ‘Yes’. Since we are applying a delay of 1e-10 seconds, but are not changing the start time of the signal, we must shift the ‘1’ bit to the right in the output signal data array to implement the delay. The signal plot coming into the Shift Signal block, coming out of the Shift Signal block, and combined are shown below, respectively. In the combined plot below, the delayed signal’s ‘1’ bit now follows the original signal’s ‘1’ bit by one bit period, but the start time of the signal data array has not been changed. The actual signal data arrays and start time values can be inspected in detail by looking at the *.sgd files generated by the Save Signal To File blocks before and after the Shift Signal block.


Note that positive values for the time shift result in the signal being shifted to the left in the data array, while negative values for the time shift result in the signal being shifted to the right in the data array. For the time delay option with keepStartTime set to ‘Yes’, the opposite is true: positive values for the time delay result in the signal being shifted to the right in the data array, while negative values for the time delay result in the signal being shifted to the left in the data array. Also note that for this example, 1e-10 second shift = 1 bit period shift = 25% signal array shift.

Properties

Inputs #1: Electrical, Binary, or Optical signal

Outputs #1: Electrical, Binary, or Optical signal, matching input signal type

Parameter values Name Type Default Range Units shiftType enumerated percentShift percentShift, bitShift,

timeShift, timeDelay

keepStartTime enumerated No No, Yes

percentShift double 0 [ -100, 100 ] percent timeShift double 0 [ -1e32, 1e32 ] s

bitShift integer 0 [ -1000000, 1000000 ] bits

Parameter Descriptions shiftType Whether the block implements a time delay or a signal shift which specified as a

percentage of the signal array, in units of time, or as a number of bits. keepStartTime Only when shiftType is timeDelay. If No, the start time of the signal is adjusted by

adding the timeShift value to it. If Yes, the start time of the signal is not modified, and a shift of the data in the signal array is performed to implement the specified delay.

percentShift The percentage of the signal array to be shifted. timeShift The amount of time to shift the signal in the signal array. bitShift The number of bits to shift the signal by.


Boolean Operator

This model represents a boolean operator. It accepts either one or two binary input signals, and outputs a binary output signal. The operations include NOT, AND, NAND, OR, NOR, and XOR. This block may be used to perform logical operations on binary signals for various applications such as modeling a duobinary transmission scheme.

There are several conditions for the binary input signals that must be satisified when using this model due to possible confusion about the expected outcome when these conditions are not satisified. All input binary signals must be of equal number of bits, equal bit rates, and equal starting times. If these conditions present problems for your application, you may follow the following suggestion or contact RSoft Design Group for assistance.

If it is desired to operate on two binary signals which do not have identical start times but do have identical number of bits and bit rates, the user may use the Shift Signal block prior to the Boolean Operator block to shift the signals such that the start times are identical. Then following the Boolean Operator, use the Shift Signal block again to shift the resulting binary signal to the desired start time. By following this approach, the user will have no confusion regarding the expected output of the boolean operator.

Properties


#2: Binary signal (not used for NOT operation)

Outputs #1: Binary signal

Parameter Values Name Type Default Range Units operation enumerated NOT NOT, AND, OR,

NAND, NOR, XOR

Parameter Descriptions operation Boolean operation to perform: NOT, AND, OR, NAND, NOR, or XOR.


D Flip-Flop

This block models a D Flip-Flop. A D Flip-Flop sets its output to “1” when the data input is high, and to “0” when the data input is low. If the gate is clocked, the data input is only checked during the user-specified clock transitions. Both regular and inverted binary outputs are available.

When the parameter Clocked is set to no, the output signal is set to “1” whenever the data signal is “1”, and to “0” whenever the data signal is “0”.

When the parameter Clocked is set to yes, the output signal is set to “1” for a high data input and “0” for a low data input only during a clock transition. The user specifies which clock transitions to use via the parameter Pulse_Transition. Positive_Edge indicates that the output can change state during the rising edge of the clock; Negative_Edge uses the falling edge. Note that if the data signal changes state simultaneously with the clock, the new data signal value is used to determine how to set the flip-flop’s outputs.

The user must specify the initial output state of the flip-flop via the parameter Initial_State.

Properties

Inputs #1: Binary Signal

#2: Binary Signal

Outputs #1: Binary Signal

#2: Binary Signal

Parameter Values Name Type Default Range Units Clocked enumerated yes yes, no None

Pulse_Transition enumerated Positive_Edge Positive_Edge, Negative_Edge

None

Initial_State enumerated 1 1, 0 None

Parameter Descriptions Clocked Flag to control whether the flip-flop is clocked or not

Pulse_Transition Flag to determine which clock transition to use when the flip-flop is clocked

Initial_State Sets initial state of the flip-flop


T Flip-Flop

This block models a T Flip-Flop. A T Flip-Flop toggles its output (from “1” to “0”, or “0” to “1”) whenever the toggle input is high and, if the gate is clocked, during the user-specified clock transitions. Both regular and inverted binary outputs are available.

When the parameter Clocked is set to no, the output signal changes whenever the toggle signal is “1”. With Force_Transition set to no, the toggle signal must return to a “0” state before another change of state at the output can occur. With Force_Transition set to yes, a change of state will occur whenever the toggle signal is “1”.

When the parameter Clocked is set to yes, the output signal changes state whenever the toggle signal is “1” during a clock transition. The user specifies which clock transitions to use via the parameter Pulse_Transition. Positive_Edge indicates that the output can change state during the rising edge of the clock; Negative_Edge uses the falling edge. Note that if the toggle signal changes state simultaneously with the clock, the new toggle signal value is used to determine whether or not to toggle the flip-flop’s outputs.

The user must specify the initial output state of the flip-flop via the parameter Initial_State.

Properties


#2: Binary Signal


#2: Binary Signal

Parameter Values Name Type Default Range Units Clocked enumerated yes yes, no None

Pulse_Transition enumerated Positive_Edge Positive_Edge, Negative_Edge

None

Force_Transition enumerated no yes, no None

Initial_State enumerated 1 1, 0 None

Parameter Descriptions Clocked Flag to control whether the flip-flop is clocked or not Pulse_Transition Flag to determine which clock transition to use when the flip-flop is clocked

Force_Transition Flag to determine if the unclocked flip-flop should always toggle on a 1 bit

Initial_State Sets initial state of the flip-flop


DQPSK Precoder

This model converts a pair of bit streams into a pair of encoded P and Q DQPSK bit streams suitable for controlling a DQPSK modulator. Given input bit streams a and b, and encoded output bit streams p and q, the kth output bits satisfy the relationships:

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

1 1

1 1

k k k k k k k k k

k k k k k k k k k

p a b a p a b b q

q a b b q a b a p− −

− −

= ⊕ ⋅ ⊕ + ⊕ ⋅ ⊕

= ⊕ ⋅ ⊕ + ⊕ ⋅ ⊕

The initial states of the output bits should be specified via the parameters P_Initial_State and Q_Initial_State. In general, these initial values do not affect the received bit sequences at a DQPSK receiver, but they are useful if you want to generate a specific set of bit patterns from the Precoder. It should also be noted that due to the nature of the signal representations in block mode, and the differential character of DQPSK modulation, the demodulated bit streams at the DQPSK receiver will likely show errors in the first or last bits due to phase discontinuities at the signal boundaries. For this reason, it is strongly recommended that sufficient pre- and postbits be used in the original bit-stream sources.

Properties


#2: Binary Signal


#2: Binary Signal

Parameter Values Name Type Default Range Units P_Initial_State enumerated 0 0, 1

Q_Initial_State enumerated 0 0, 1

Parameter Descriptions P_Initial_State Sets initial state of the encoded P output

Q_Initial_State Sets initial state of the encoded Q output


Electrical Signal Resampler

The Electrical Signal Resampler allows the user to modify an electrical signal’s time step (timeStep) and number of samples (2^noSamples). Note that this conversion ignores any logical information contained in the signal. If the user specifies a set of values for timeStep and noSamples that causes the signal to be lengthened, then the additional signal data is obtained by assuming that the original signal is periodic. The user may also truncate the signal if the time range specified by timeStep and noSamples is less than that of the original signal.

Properties



Parameter Values Name Type Default Range Units timeStep double 1e-12 1e-32≤ x ≤ 1e32 s

noSamples integer 0 0 ≤ x ≤ 27 2^noSamples

Parameter Descriptions timeStep New time step of the signal

noSamples New number of samples in the signal, specified as 2^noSamples


Optical Signal Resampler

The Optical Signal Resampler allows the user to modify an optical signal’s time step (timeStep) and number of samples (2^noSamples). Note that this conversion ignores any logical information contained in the signal. If the user specifies a set of values for timeStep and noSamples that causes the signal to be lengthened, then the additional signal data is obtained by assuming that the original signal is periodic. The user may also truncate the signal if the time range specified by timeStep and noSamples is less than that of the original signal.

Properties

Inputs #1: Optical Signal

Outputs #1: Optical Signal

Parameter Values Name Type Default Range Units timeStep double 1e-12 1e-32≤ x ≤ 1e32 s

noSamples integer 0 0 ≤ x ≤ 27 2^noSamples

Parameter Descriptions timeStep New time step of the signal

noSamples New number of samples in the signal, specified as 2^noSamples

OptSim Models Reference: Block Mode Chapter 4: Optical Sources and Modulators •••• 75

Chapter 4: Optical Sources and Modulators

This chapter describes lasers and a variety of modulators:

• Direct Modulated Laser generate an optical signal using direct modulation

• Mode-Locked Laser generate a train of mode-locked pulses

• CW Laser generate one or more CW signals

• Fabry Perot CW Laser

• generate a Fabry Perot CW signal with multiple longitudinal modes

• VCSEL generate an optical signal from a VCSEL

• Light Emitting Diode (LED) create an optical signal from an LED

• Modulator modulate an optical signal using external modulation

• Electro-Absorption Modulator modulate an optical signal using electro-absorption modulation

76 •••• Chapter 4: Optical Sources and Modulators OptSim Models Reference: Block Mode

Direct Modulated Laser

This block models a semiconductor laser directly modulated with an electrical signal. It computes the electrical current injected into the laser’s optical cavity and solves the laser rate equations for the optical output. The behavior of the model can be partitioned into three blocks, as shown in Fig. 1.

The driving source consists of the electrical signal input into the model. The parasitics consist of a bond inductance and shunting capacitance. Finally, the laser cavity is modeled via a simplified current-voltage (IV) relationship and the laser rate equations.

Figure 1: Main components of the Direct Modulated Laser model

Driving Source The Direct Modulated Laser is driven by a combination of the electrical signal at its input and, when applicable, a dc bias current specified by Io (Io). Io can also be specified via a dc bias output power Po (Po). The user determines which will be used via the parameter Bias_Value. The other parameter is then calculated automatically.

The input electrical signal can be interpreted in a variety of ways, based on the settings of the model parameter Drive_Scheme. This parameter can take on values of direct_drive, bias_tee, or bias_tee_old.

• Direct_drive The input signal is assumed to come from either an ideal current source or zero-impedance voltage source. In other words, the laser is assumed to be undergoing direct-drive modulation. If the input signal is a current, then it is combined with the bias current Io to form the total input current. Fig. 2(a) illustrates this scenario. If the input signal is a voltage, then the bias current Io is ignored. Note that the input voltage should be larger than the laser’s turn-on voltage Von; otherwise, the resulting current is zero. This arrangement is illustrated in Fig. 2(b).

Figure 2: Direct-drive modulation schemes: (a) current and (b) voltage


• bias_tee The input signal is assumed to be generated by a source with output impedance Rs (Rs), connected to the laser via an ideal bias tee. The bias current Io is similarly connected. The input signal can be either a current or a voltage. Figure 3 depicts a bias tee setup for both the voltage- and current-source cases.

Figure 3: Bias-tee modulation schemes: (a) voltage source and (b) current source

• bias_tee_old This choice implements the driving scheme from OptSim versions prior to 3.0. Note that these previous implementations intended to model the bias tee as described above, but did not remove any dc component which may have been present in the electrical input signal that would naturally occur via the tee’s capacitive leg. While future releases of OptSim may no longer support this option, it is included here for compatibility.

Parasitics The parasitics consist of a bond inductance Lb (Lb) and shunting capacitance Cp (Cp), as shown in Fig. 1. These can be turned on or off via the parameter Parasitics.

Laser Cavity Both electrical and optical effects are modeled within the laser cavity.

Electrical The electrical model of the laser cavity is that of a simplified diode IV relationship, consisting of a series resistance Rd (Rd) and turn-on voltage Von (Von). During solution of the cavity current I, the model ensures that negative currents are effectively limited to zero.

Rate Equations The core of the Direct Modulated Laser block are the semiconductor laser rate equations, which determine the optical output in response to the cavity current I. Relative intensity noise is modeled via a constant value RIN (RIN). The rate equations, based largely on those discussed in [1], are:

( ) ( ) ( , )isp nr p p

IdN R N R N G N N Ndt qV

η= − − − ⋅

(1)


( ) ( , )p psp sp p p

p

dN NR N G N N N

dtβ

τ= − + Γ + Γ ⋅

(2)

1( , )2

pp

p p

Nd G N Ndt

εφ ατ τ

′= ⋅ Γ − +

(3)

g mirout fact p

v AhcP c N V Fλ

= ⋅ = ⋅ ⋅ ⋅Γ

(4)

where I is the injection current, N is the carrier density, Np is the photon density, φ is the optical phase, ηI (effint) is the current injection efficiency, q is the electron charge, V is the cavity volume, τp is the photon lifetime, Γ (K) is the confinement factor, βsp (b) is the spontaneous emission coupling coefficient, α (a) is the linewidth enhancement factor, ε ′ is a modified gain saturation factor (see Gain section below), h is Planck’s constant, c is the speed of light in a vacuum, λ (wavelength) is the lasing wavelength, vg is the group velocity (c/n, where n (indx) is the material index), Amir is the mirror, or facet, loss, F is the fraction of light that escapes the output facet, Rsp(N) is the radiative recombination, Rnr(N) is the nonradiative recombination, and G(N,Np) is the laser gain. The recombination terms are modeled using:

2( )spR N BN=

(5)

3( )nrR N AN CN= +

(6)

where A (A) is the unimolecular recombination coefficient, B (B) is the radiative recombination coefficient, and C (C) is the Auger recombination coefficient.

Device Geometry The device geometry can be modeled in a number of ways via the parameter geometry, which can take on values of rectangular, cylindrical, or volumetric.

• Rectangular The laser is assumed to be an edge emitting laser with cavity width Lstp (Lstp), cavity thickness Lact (Lact), and cavity length Lcav (Lcav). The volume V is then computed as:

stp act cavV L L L= ⋅ ⋅

(7)

• cylindrical The laser is assumed to be a vertical-cavity device with cavity diameter W (W), cavity thickness Lact, and total cavity length Lcav. The volume V is then computed as:

2

4actWV L π

= ⋅

(8)

• volumetric The volume is directly specified via the parameter V (V).


Mirror Effects When the parameter mirror_effects is set to defined, the mirror parameters Amir and F are directly specified by the parameters Amir and F, respectively. When mirror_effects is set to calculated, the mirror parameters are calculated using Lcav and the mirror reflectivities R1 (R1) and R2 (R2), where the optical output is assumed to exit from mirror #1 [2]:

1 2ln( )2mir

cav

R RA

L= −

(9)

1

1 2 1 2

11 (1 ) /

RF

R R R R−

=− + − ⋅

(10)

Intrinsic Loss If the parameter intrinsic_loss is set to defined, then the intrinsic loss of the cavity is directly specified as Aint (Aint). When intrinsic_loss is set to calculated, then Aint is defined in relation to Amir via the scaling factor loss_ratio (loss_ratio):

int _ mirA loss ratio A= ⋅

(11)

Photon Lifetime When parameter photon_lifetime is set to defined, the photon lifetime is directly specified via τp (tp). When photon_lifetime is set to calculated, the photon lifetime is calculated using [2]:

int

1( )p

gr mirv A Aτ =

⋅ +

(12)

Gain The laser gain G(N,Np) consists of the material gain g(N) and the optical gain saturation Φ(Np), with both terms able to take on a number of forms via the setting of the gain and saturation parameters:

( , ) ( ) ( )p gr pG N N v g N N= ⋅ ⋅Φ

(13)

The gain parameter can take on values of logarithmic:N, logarithmic:R(N), or linear:

• Logarithmic:N The material gain is modeled as a logarithmic function of N [1]:

( ) ln so

tr s

N Ng N G

N N +

= +

(14)

where Go (Go) is the gain coefficient, Ntr (Ntr) is the transparency density, and Ns (Ns) is an adjustable correction parameter [1].

• Logarithmic:R(N)


The material gain is modeled as a logarithmic function of the carrier recombination [3],[4]. This approach is a more general version of the simple logarithmic gain.

( ) ( )

( ) ln( ) ( )

sp nr so

sp tr nr tr s

R N R N Ng N G

R N R N N

+ += + +

(15)

• linear

The material gain is assumed to be linearized about the transparency density Ntr, and is described by [2]:

( ) ( )o trg N G V N N= ⋅ ⋅ −

(16)

Gain Saturation The saturation parameter can also take on three values: Channin, Agrawal, or linear.

• Channin

The gain saturation is modeled following the approach in [1] and [5]:

1( )1p

pN

NεΦ =

+

(17)

where ε (e) is the gain saturation factor. In this case, the modified gain saturation factor is simply ε ε=′ .

• Agrawal

The gain saturation is modeled following the approach in [4] and [6], and is applicable when intraband effects are important:

1

( )1p

pN

NεΦ =

+

(18)

In this case, the modified gain saturation factor is / 2ε ε=′ .

• Linear

The gain saturation is assumed to be linear, though the expression is strictly only valid when Np < 1/ε. The modified gain saturation factor is ε ε=′ .

( ) 1p pN NεΦ = −

(19)

Polarization By default, the laser emits an output field polarized along the X axis, with no corresponding Y-polarized component. A zero-valued Y-polarized component may be included in this case by setting the parameter force_Ey to yes. The field polarization itself may be changed via the parameters azimuth and ellipticity. These parameters correspond to the azimuth and ellipticity angles of the polarization ellipse, respectively. Some typical values for these parameters are [7]:


• ellipticity = 0: Linear polarization, with azimuth describing the tilt of the field vector relative to the X axis.

• ellipticity = +45 degrees, azimuth = 0: Right-handed circular polarization.

• ellipticity = -45 degrees, azimuth = 0: Left-handed circular polarization.

Multi-Line Output It is frequently necessary to produce several signals with similar properties but different wavelengths. In DWDM simulations especially, a series of regularly spaced optical sources is common. The Direct Modulated Laser model provides a number of convenient facilities so that many or all required lines can be generated from a single icon. To produce a series of lines spaced equally in wavelength or frequency, set the parameter mode=LambdaGrid or mode=FreqGrid, respectively. The number of sources is controlled by the number of electrical inputs to the laser, and in both modes the parameter wavelength specifies the first line in a series of ascending wavelengths or ascending frequencies, respectively. The source separation in wavelength or frequency is specified with deltaFreq. To set the number of electrical inputs to the laser, select the direct modulated laser icon and open the menu item Properties. In the “Ports” tab, “number_input_ports” field, enter the number of inputs.

Multi-Line Multi-Node Output The comb of sources may be emitted either through a single node (the default) or with one channel per output node. To achieve the latter behavior, select the direct modulated laser icon and open the menu item Properties. In the “Ports” tab, set the value of the number_output_ports field to that of the number_input_ports field.

Numerical Settings During simulation, the laser rate equations are numerically solved. To control the accuracy of these calculations, the user has access to three parameters. Eps adjusts the overall tolerance level, or accuracy, of the solution. Initial_tstep is the initial time step used by the model’s ODE solver. Min_tstep is the smallest time step that this solver is allowed to take.

Test Parameters In order to ascertain whether the parameter settings for the Direct Modulated Laser block provide the component performance desired, the user may test them from the component parameter editing window. Depending on the setting of the parameter test_function, this test produces a light-current (LI) curve (test_function set to LI), small-signal frequency response curves (ac), or a performance report (report). These are typically used to fit rate-equation laser model parameters to the performance of actual semiconductor lasers. The test can be controlled via model parameters carrying a prefix Test_. These parameters allow the user to set sweep limits, bias conditions, etc.

LI Curve The LI curve is controlled via the parameters Test_LItype, Test_Imin, Test_Imax, and Test_Lipoints. When Test_Litype is set to auto-scaled, an LI curve is generated over currents ranging from 0 to Imax, where Imax is the larger of 3Ith or 2Io, Ith being the calculated laser threshold current. With Test_Litype set to user-specified, the LI curve is generated over currents ranging from Test_Imin to Test_Imax. The total number of points in the LI curve is determined by Test_Lipoints. A sample LI curve is shown in Fig. 4.


Figure 4: Sample test LI curve, generated using automatic scaling

Small-Signal Frequency Responses Two sets of frequency response curves are generated. The first set depicts the device’s transfer function Tf, i.e. the small-signal output power vs. small-signal input current. This calculation includes parasitics if any are present. The second set of curves depicts the device’s normalized S21 response using the relationship S21 = 2Tf/(Zin+50), where Zin is the device input impedance and a 50-Ω test setup is assumed. Both sets of curves are generated at five different bias currents. With the parameter Test_actype set to Ith-based, these currents are 1.1Ith, 1.5Ith, 2.0Ith, 3.0Ith, and Io. When Test_actype is set to user-specified, these currents are defined by the parameters Test_I1ac, Test_I2ac, Test_I3ac, Test_I4ac, and Test_I5ac. Parameters Test_acxscale and Test_acyscale control the type of scales on the x- and y-axes, respectively. Log results in logarithmic scaling, and linear in linear scaling. Test_freqlow and Test_freqhigh specify the range of frequencies over which to generate the response curves, and Test_acpoints defines the number of points in each curve. Sample frequency-response curves are shown in Fig. 5.

Figure 5. Sample transfer-function and S21 frequency-response curves at threshold-based bias values

Performance Report In addition to the above plots, a report is generated highlighting some of the important performance metrics of the laser. These include the threshold current, differential quantum efficiency, power, or slope,


efficiency, bias values Io and Po, as well as bandwidths and relaxation-peak locations for the transfer-function, S21, and intrinsic (i.e., no parasitics) frequency responses at the five bias currents of interest.

Compatibility with LinkSIM version 2.1 VCSEL Model With this release of OptSim, the LinkSIM 2.1 VCSEL model can be invoked through the use of the Direct Modulated Laser block. In most cases, the old VCSEL parameters can be mapped into the present ones with little or no change. However, the old parameters τn, τs, a, and ε can be converted to new parameters by setting ( ) ( ) ( ) ( )( ) /( )n old s old n old s oldA τ τ τ τ= + , ( ) /o oldG a V= , and ( ) /oldVε ε= Γ .

References [1] L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (John Wiley & Sons, New York, 1995).

[2] G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd. ed. (Van Nostrand Reinhold, New York, 1993).

[3] T. A. DeTemple and C. M. Herzinger, “On the semiconductor laser logarithmic gain-current density relation,” IEEE Journal of Quantum Electronics, 29, 1246 (1993).

[4] P. V. Mena, S.-M. Kang, and T. A. DeTemple, “Rate-equation-based laser models with a single solution regime,” Journal of Lightwave Technology, 15, 717 (1997).

[5] D. J. Channin, “Effect of gain saturation on injection laser switching,” Journal of Applied Physics, 50, 3858 (1979).

[6] G. P. Agrawal, “Effect of gain and index nonlinearities on single-mode dynamics in semiconductor lasers,” IEEE Journal of Quantum Electronics, 26, 1901 (1990).

[7] M. Born and E. Wolf, Principles of Optics, 7th. Ed. (Cambridge University Press, Cambridge, 1999).

Properties

Inputs #1-#512: Electrical signal


Parameter Values Name Type Default Range Units wavelength double 1550e-9 [ 0, 1 ] m

mode enumerated FreqGrid FreqGrid, LambdaGrid



ellipticity double 0 [ -45, 45 ] degrees force_Ey enumerated no yes, no

effint double 0.9 [ 0, 1 ] none

K double 0.1 [ 0, 1 ] none


indx double 4.1 [ 0, 1e32 ] none

geometry enumerated rectangular rectangular, cylindrical, volumetric

Lact double 500e-8 [ 0, 1e32 ] cm

Lstp double 2.5e-4 [ 0, 1e32 ] cm Lcav double 345e-4 [ 0, 1e32 ] cm

W double 1.5e-3 [ 0, 1e32 ] cm

V double 4.3125e-11 [ 0, 1e32 ] cm^3

mirror_effects enumerated defined defined, calculated Amir double 33 [ 0, 1e32 ] 1/cm

F double 1 [ 0, 1 ] none

R1 double 0.9966 [ 0, 1e32 ] none

R2 double 0.9989 [ 0, 1e32 ] none intrinsic_loss enumerated defined defined, calculated

Aint double 40 [ 0, 1e32 ] 1/cm

loss_ratio double 2.0 [ 0, 1e32 ] none

b double 1e-4 [ 0, 1 ] none photon_lifetime enumerated calculated calculated, defined

tp double 1e-12 [ 0, 1e32 ] s

A double 0 [ 0, 1e32 ] 1/s

B double 1e-10 [ 0, 1e32 ] cm^3/s C double 30e-30 [ 0, 1e32 ] cm^6/s

gain enumerated logarithmic_N logarithmic_N, logarithmic_RN, linear

Go double 1537 [ 0, 1e32 ] 1/cm

Ntr double 1.5e18 [ 0, 1e32 ] 1/cm^3

Ns double 0 [ 0, 1e32 ] none

saturation enumerated Channin Channin, Agrawal, linear

e double 10e-17 [ 0, 1e32 ] cm^3 a double 2 [ 0, 1e32 ] none

RIN double -150 [ -1e32, 1e32 ] dB/Hz

Rd double 5 [ 0, 1e32 ] ohm Von double 2.0 [ 0, 1e32 ] V

Drive_Scheme enumerated bias_tee_old direct_drive, bias_tee, bias_tee_old

Rs double 50 [ 0, 1e32 ] ohm

Bias_Value enumerated Io Io, Po

Io double 30e-3 [ 0, 1e32 ] A

Po double 0 [ 0, 1e32 ] W Parasitics enumerated on on, off

Lb double 0.3e-9 [ 0, 1e32 ] H


Cp double 2e-12 [ 0, 1e32 ] F

eps double 1e-6 [ 0, 1e32 ] none initial_tstep double 1e-13 [ 0, 1e32 ] s

min_tstep double 0 [ 0, 1e32 ] s

test_function enumerated LI LI, ac, report

Test_LItype enumerated auto-scaled auto-scaled, user-specified

Test_Imin double 0 [ 0, 1e32 ] A Test_Imax double 30e-3 [ 0, 1e32 ] A

Test_LIpoints integer 201 [ 0, 100000 ] none

Test_actype enumerated Ith-based Ith-based, user-specified

Test_I1ac double 1e-3 [ 0, 1e32 ] A

Test_I2ac double 2.5e-3 [ 0, 1e32 ] A Test_I3ac double 5e-3 [ 0, 1e32 ] A



Test_acxscale enumerated log log, linear Test_acyscale enumerated linear log, linear

Test_freqlow double 1e7 [ 0, 1e32 ] Hz

Test_freqhigh double 1e12 [ 0, 1e32 ] Hz

Test_acpoints integer 201 [ 0, 100000 ] none

Parameter Descriptions wavelength Wavelength of the laser, λ mode Type of wavelength grid deltaFreq Frequency or wavelength grid spacing for multi-line output effint Current injection efficiency, ηI K Optical confinement factor, Γ indx Optical mode index, n geometry Device geometry: rectangular, cylindrical, volumetric Lact Laser active region thickness, Lact Lstp Laser active region width, Lstp Lcav Laser cavity length, Lcav W Laser diameter (for cylindrical geometries), W V Laser cavity volume, V mirror_effects Mirror effects: defined, calculated Amir Laser mirror loss, Amir F Fraction of power that escapes from the output mirror, F R1 Reflectivity of mirror #1, R1 R2 Reflectivity of mirror #2, R2 intrinsic_loss Intrinsic loss: defined, calculated Aint Laser internal loss, Aint loss_ratio Ratio of intrinsic loss to mirror loss, loss_ratio b Spontaneous emission coupling coefficient, βsp


photon_lifetime Photon lifetime definition: calculated, defined tp Photon lifetime, τp A Unimolecular recombination coefficient, A B Radiative recombination coefficient, B C Auger recombination coefficient, C gain Gain definition: logarithmic:N, logarithmic:R(N), linear Go Gain coefficient, Go Ntr Carrier transparency density, Ntr Ns Logarithmic gain correction factor, Ns saturation Gain saturation definition: Channin, Agrawal, linear e Gain saturation factor, ε a Linewidth enhancement factor, α RIN Laser relative intensity noise, RIN Rd Laser cavity resistance, Rd Von Laser turn-on voltage, Von Drive_Scheme Drive signal definition: direct_drive, bias_tee, bias_tee_old Rs Source impedance, Rs Bias_Value Bias definition: Io, Po Io Laser bias current, Io Po Laser dc power level, Po Parasitics Parasitics flag: on, off Lb Laser bond inductance, Lb Cp Laser parasitic capacitance, Cp eps Accuracy of ODE solver initial_tstep Initial time step taken by ODE solver min_tstep Minimum time step taken by ODE solver test_function Test function selection: LI curve, ac curves, or performance report Test_LItype LI curve definition: auto-scaled, user-specified Test_Imin Minimum current for LI curve Test_Imax Maximum current for LI curve Test_Lipoints Number of points in LI curve Test_actype Frequency-response curve definition: Ith-based, user-specified Test_I1ac Bias current for first ac curve Test_I2ac Bias current for second ac curve Test_I3ac Bias current for third ac curve Test_I4ac Bias current for fourth ac curve Test_I5ac Bias current for fifth ac curve Test_acxscale Scale for x axis: log, linear Test_acyscale Scale for y axis: log, linear Test_freqlow Minimum frequency for ac curves Test_freqhigh Maximum frequency for ac curves Test_acpoints Number of points per ac curve


Mode-Locked Laser

This models a mode-locked laser.

The following pulse types are presently supported for the output optical signal (u t( ) is the field amplitude):

• gaussian

( )[ ]205.0exp)( Tttu −=

• sech

( )0sec)( Tthtu =

• on_off

|u(t)|2 = linear ramp to 1 for tr; flat at 1 for tFWHM-tr/2-tf/2; linear ramp to 0 for tf.

• raisedCosAmp

( )02cos5.05.0)( Tttu π+=

• raisedCosPow

( ) ( )02 2cos5.05.0 Tttu π+=

• supGaussian

( )[ ]mTttu 205.0exp)( −=

supGaussian (Super Gaussian) is an approximation to square pulses but with rounded edges. The m parameter is determined from the rise time of the pulse. Although tr and tf may be different for on_off pulse type, they must be the same for supGaussian type. The 10%-90% rise time rule is used for the supGaussian while for the on_off it is the total rise or fall time.

The FWHM of these pulse types are:

• gaussian

02ln2 TTFWHM ⋅=

• sech

)21ln(2 0 +⋅= TTFWHM

• raisedCosAmp

)12cos(10 −⋅

π= aTTFWHM

• raisedCosPow

2/0TTFWHM =

• supGaussian


mFWHM TT 21

0 )2(ln2 ⋅=

Arbitrary amounts of linear chirp may be added to each of these pulse forms. The chirp is included via a chirp factor C. The expression for the field amplitude of a chirped pulse uchirp(t) is given by (this is a general result irrespective of the pulse shape, see G. P. Agrawal, Nonlinear Fiber Optics, Academic Press),

−= 2

202

exp)()( TTCjtutuchirp

The expression for the additional spectral width due to pulse chirp is given by,

λ−

π

−λ

=λ∆−

0

1

00 412

cTC

In the case of mode-locked sources, phase shift between adjacent bits is allowed so that the phase shifted soliton transmission technique can be studied.

Note that this model’s output can be disabled by setting its peak-power value to 0.





Multi-Line Output It is frequently necessary to produce several signals with similar properties but different wavelengths. In DWDM simulations especially, a series of regularly spaced optical sources is common. The Mode-Locked Laser model provides a number of convenient facilities so that many or all required lines can be generated from a single icon. To produce a series of lines spaced equally in wavelength or frequency, set the parameter mode=LambdaGrid or mode=FreqGrid, respectively. The number of sources is controlled by noSources, and in both modes the parameter wavelength specifies the first line in a series of ascending wavelengths or ascending frequencies, respectively. The source separation in wavelength or frequency is specified with deltaFreq.

For applications in which the sources are not regularly spaced or the peak power or phase must be controlled independently, the option mode=File is available. In this case, the user provides an ascii file with the name filename detailing the frequency, power and phase of the sources. The format of the file is as follows:

MultiModeLockedLaserFormat1 <freq_format> <power_format>

<freq_1> <power_1>

<freq_2> <power_2>

etc..


Here <freq_format> indicates the units for the frequency data in the first column and must be one of [nm], [um], [m], [Hz], [MHz], [GHz], [THz], [cm^(-1)], [m^(- 1)] or [rad/s]. Similarly, <power_format> indicates the units for the power data in the second column and must be one of [W], [mW], [uW] or [dBm]. The phase information must be entered in degrees.

Multi-Line Multi-Node Output The comb of sources may be emitted either through a single node (the default) or with one channel per output node. To achieve the latter behavior, first set multiNodeOutput=YES. Then select the mode-locked laser icon and open the menu item Properties. In the Ports tab, number_output_ports field, enter the number of lines that the model will generate (i.e. either noSources or the number of lines in the user data file if mode=File).

References [1] M. Born and E. Wolf, Principles of Optics, 7th. Ed. (Cambridge University Press, Cambridge, 1999).

Properties

Inputs None


Parameter Values Name Type Default Range Units pattern enumerated Multiple Single, Multiple

type enumerated gaussian sech, gaussian, supGaussian, on_off, raisedCosPow, raisedCosAmp

peakPower double 1e-3 [ 0, 1e32 ] Watts

wavelength double 1550e-9 [ 0, 1e18 ] meters

mode enumerated Single Single, LambdaGrid, FreqGrid, File

multiNodeOutput enumerated NO NO, YES noSources integer 10 [ 1, 1000 ] none


filename string azimuth double 0 [ -90, 90 ] degrees


force_Ey enumerated no yes, no

width double 10e-12 [ 0, 1e32 ] sec patternLength integer 7 [ 0, 27 ] 2^x_bits

pointsPerBit integer 5 [ 1, 27 ] 2^x_bits

repRate double 10e9 [ 1, 1e32 ] Bit/s



ChirpFactor double 0 [ -1e32, 1e32 ] none phaseShift double 0 [ -6.284, 6.284 ] rad

riseTime double 10e-12 [ 0, 1e32 ] sec

fallTime double 10e-12 [ 0, 1e32 ] sec

Parameter Descriptions mode Type of wavelength grid multiNodeOutput Select multi-line output on a single or multiple output nodes noSources Number of lines in multi-line output deltaFreq Frequency or wavelength grid spacing for multi-line output filename Filename for user-specified source grid type Mode-locked source type: gaussian, supGaussian, sech, on_off,

raisedCosAmp, raisedCosPow repRate Repetition rate of the source patternLength Bit pattern length = 2x, x is input pointsPerBit Number of sampling points per bit period in output optical signal peakPower Peak power wavelength Wavelength of the laser RIN Relative intensity noise of the laser width Pulse FWHM Parameter ChirpFactor Chirp factor C phaseShift Phase Shift Between Adjacent Pulses pattern Single or multiple pulses in bit stream riseTime Pulse Rise Time for on-off and supGaussian types fallTime Pulse Fall Time for on-off type


CW Laser

This model produces the optical signal output of one or more CW lasers. It is most commonly used in conjunction with the external modulator model to encode a binary signal upon the CW source.

In this model, the CW source is characterized completely by its power, wavelength, linewidth, relative intensity noise (RIN) and phase. These are controlled directly through the parameters peakPower, wavelength, linewidth, RIN and phase. The laser can also be assigned a random phase by setting randomPhase=YES. Note that by setting peakPower to 0, you can disable this model’s output.

There are two options for the temporal representation of the laser output selected by the parameter signalType. For topologies in which a CW laser model provides direct input to a modulator model or the pumps of a Raman fiber amplifier, the PowerValue signal representation is most convenient. In this representation, the optical signal holds a single complex value describing the field amplitude and phase. For topologies in which a CW laser model’s output is used as input to other component models or is to be multiplexed with different optical signals from other types of sources, the TimeSequence signal representation should be used. This is the standard time-sampled representation of optical signals. For this representation, the timeStep and noSamples parameters must be set appropriately to match the sampling rate and number of data samples of any other signals with which it will interact in the simulation. The nominalBitRate parameter should also be set to an appropriate data rate.

Linewidth Inclusion of a source linewidth in the laser output is controlled via the parameter linewidth_model, which by default is set to none (for a linewidth of zero). If linewidth_model=phase_noise, and the laser output uses the TimeSequence signal representation, then linewidth is added to the output via phase noise. These random phase variations (seeded via the same phaseSeed parameter that controls random initial phase values) result in a Lorentzian output power spectrum [1]. If linewidth_model=value, then the constant linewidth value is attached to each output.

The specific value for the source linewidth is set by the parameter linewidth. If the parameter linewidth_units=frequency, then linewidth is specified in Hz. If linewidth_units=wavelength, then linewidth is specified in meters.





Multi-Line Output It is frequently necessary to produce several CW signals with similar properties but different wavelengths. In DWDM simulations especially, a series of regularly spaced optical sources is common. The CW Laser


model provides a number of convenient facilities so that many or all required lines can be generated from a single icon. To produce a series of lines spaced equally in wavelength or frequency, set the parameter mode=LambdaGrid or mode=FreqGrid, respectively. The number of sources is controlled by noSources, and in both modes the parameter wavelength specifies the first line in a series of ascending wavelengths or ascending frequencies, respectively. The source separation in wavelength or frequency is specified with deltaFreq.


MultiCWLaserFormat1 <freq_format> <power_format>

<freq_1> <power_1> <phase_1>


etc..


Multi-Line Multi-Node Output The comb of sources may be emitted either through a single node (the default) or with one channel per output node. To achieve the latter behavior, first set multiNodeOutput=YES. Then select the cw laser icon and open the menu item Properties. In the Ports tab, number_output_ports field, enter the number of lines that the model will generate (i.e. either noSources or the number of lines in the user data file if mode=File). Figure 1 indicates the use of multi-node output mode to produce a series of WDM sources.

Figure 1: CW laser in multi-node output mode for WDM sources.


References [1] G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd. ed. (Van Nostrand Reinhold, New York, 1993).


Properties

Inputs None


Parameter Values Name Type Default Range Units peakPower double 1e-3 [ 0, 1e18 ] Watts



multiNodeOutput enumerated NO NO, YES noSources integer 10 [ 1, 1000 ] none


filename string azimuth double 0 [ -90, 90 ] degrees



linewidth_model enumerated none none, phase_noise, value

linewidth_units enumerated frequency frequency, wavelength

linewidth double 100e6 [ 0, 1e32 ] Hz or m RIN double -150 [ -1e32, 1e32 ] dB/Hz

signalType enumerated PowerValue PowerValue, TimeSequence

timeStep double 0.0 [ 0, 1 ] seconds

nominalBitRate double 10e9 [ 0, 1e32 ] Hz

noSamples integer 0 [ 0, 27 ] 2^x_samples randomPhase enumerated NO NO, YES


phaseSeed integer 0 [ -1e8, 1 ] none

Parameter Descriptions mode Type of wavelength grid multiNodeOutput Select multi-line output on a single or multiple output nodes


peakPower Peak power (also average power for CW laser) wavelength Laser wavelength noSources Number of lines in multi-line output deltaFreq Frequency or wavelength grid spacing for multi-line output filename Filename for user-specified source grid linewidth_model Select linewidth representation linewidth_units Select linewidth units linewidth Linewidth value RIN Relative intensity noise of the laser signalType Select representation of a power or time sequence timeStep For TimeSequence representation, the signal sampling time nominalBitRate For TimeSequence representation, the signal bit rate noSamples For TimeSequence representation, the number of signal samples randomPhase Select randomization of laser phase phase Select amplitude of phase randomization phaseSeed Random number seed for phase randomization. (Standard OptSim seed convention).


Fabry Perot CW Laser

This block models a Fabry Perot CW laser with multiple longitudinal modes. The user specifies a center wavelength (wavelength),the total output power (totalPower), the total spontaneous output power (spontaneousPower), the total number of side modes (numSidemodes), and the spacing in Hz between adjacent modes (modeSpacing). Based on these settings, the power in each mode is calculated as in [1]:

21 1

o

o

spo

P

P mP M

+ − ⋅

where Po is the power in the central mode, Pspo is the spontaneous power in each mode (assumed to be the same for all modes, and thus equal to the total spontaneous power divided by the total number of modes), m is the mode number in the range [-M,+M], and the total number of modes is 2M+1. Po is calculated such that the total power over all modes is equal to the total output power specified by the user.

The user may also control the representation of the output signal. There are two options for the temporal representation of the laser output selected by the parameter signalType. In the PowerValue signal representation, the optical signal holds a single complex value describing the field amplitude and phase. The TimeSequence signal representation is the standard time-sampled representation of optical signals. For this representation, the timeStep and noSamples parameters must be set appropriately to match the sampling rate and number of data samples of any other signals with which it will interact in the simulation. The nominalBitRate parameter should also be set to an appropriate data rate. If the TimeSequence representation is chosen, then the user may further determine the representation of the laser’s multiple output modes. Setting bandType to single forces the output to be a single-band optical signal. When bandType is set to multi, each mode has its own output signal. Note that for PowerValue signals, the output will always be multi.

The user can also set values for the laser output’s relative-intensity noise (RIN), as well as its optical phase (phase). A random value for the phase can be selected by setting randomPhase to yes and choosing an appropriate seed value (randomSeed).

Linewidth Inclusion of a source linewidth for each mode in the laser output is controlled via the parameter linewidth_model, which by default is set to none (for a linewidth of zero). If linewidth_model is set to phase_noise, and the laser output uses the TimeSequence signal representation, then linewidth is added to the output via phase noise. These random phase variations (seeded via the same randomSeed parameter that controls random initial phase values) result in a Lorentzian output power spectrum [1]. If linewidth_model is set to value, then the constant linewidth value is attached to each output.

The specific value for the linewidth is set by the parameter linewidth. If the parameter linewidth_units=frequency, then linewidth is specified in Hz. If linewidth_units=wavelength, then linewidth is specified in meters.

Polarization By default, the laser emits an output field polarized along the X axis, with no corresponding Y-polarized component. A zero-valued Y-polarized component may be included in this case by setting the parameter force_Ey to yes. The field polarization itself may be changed via the parameters azimuth and ellipticity.


These parameters correspond to the azimuth and ellipticity angles of the polarization ellipse, respectively. Some typical values for these parameters are [2]:






Properties

Inputs None

Outputs #1: Optical signal

Parameter Values Name Type Default Range Units totalPower double 1e-3 [ 0, 1e32 ] W

wavelength double 1550e-9 [ 0, 1e32 ] m

spontaneousPower double 0 [ 0, 1e32 ] W

numSidemodes integer 0 [ 0, 100000 ] none modeSpacing double 100e9 [ 0, 1e32 ] Hz



linewidth double 100e6 [ 0, 1e32 ] Hz or m



bandType enumerated single single, multi timeStep double 0.0 [ 0, 1 ] seconds

nominalBitRate double 10e9 [ 0, 1e32 ] Hz

noSamples integer 0 [ 0, 27 ] 2^x_samples

randomPhase enumerated no no, yes phase double 0 [ -180, 180 ] degrees

randomSeed integer 0 [ -1e8, 1 ] none

azimuth double 0 [ -90, 90 ] degrees ellipticity double 0 [ -45, 45 ] degrees



Parameter Descriptions totalPower Total output power wavelength Center mode wavelength spontaneousPower Total output power due to spontaneous emission numSidemodes Number of side modes modeSpacing Spacing between modes in Hz linewidth_model Select linewidth representation linewidth_units Select linewidth units linewidth Linewidth value RIN Relative intensity noise of the laser signalType Select representation of a power or time sequence bandType Select single- or multi-band spectral representation timeStep For TimeSequence representation, the signal sampling time nominalBitRate For TimeSequence representation, the signal bit rate noSamples For TimeSequence representation, the number of signal samples randomPhase Select randomization of laser phase phase Optical phase value randomSeed Random number seed azimuth Polarization azimuthal angle ellipticity Polarization ellipticity force_Ey Select forcing of Y-polarization even if zero


VCSEL

This block models a vertical-cavity surface-emitting laser (VCSEL) directly modulated with an electrical signal. It computes the electrical current injected into the laser’s optical cavity and solves the laser rate equations for the optical output. Important VCSEL behaviors such as spatial hole burning, lateral carrier diffusion, thermally dependent gain, and thermal carrier leakage are all accounted for. The behavior of the model can be partitioned into three blocks, as shown in Fig. 1.

Figure 1: Main components of the VCSEL model

The driving source consists of the electrical signal input into the model. The parasitics consist of a bond inductance and shunting capacitance. Finally, the VCSEL cavity is modeled via a simplified current-voltage (IV) relationship and spatially independent VCSEL rate equations.

Driving Source The VCSEL is driven by a combination of the electrical signal at its input and, when applicable, a dc bias current specified by Io (Io). The input electrical signal can be interpreted in a variety of ways, based on the settings of the model parameter Drive_Scheme. This parameter can take on values of direct_drive, direct_ user_iv, or bias_tee.

• direct_drive

The input signal is assumed to come from either an ideal current source or zero-impedance voltage source. In other words, the laser is assumed to be undergoing direct-drive modulation. If the input signal is a current, then it is combined with the bias current Io to form the total input current. Figure 2(a) illustrates this scenario. If the input signal is a voltage, then the bias current Io is ignored. Note that the input voltage should be larger than the laser’s turn-on voltage Von; otherwise, the resulting current is zero. This arrangement is illustrated in Fig. 2(b).


• direct_user_iv

Equivalent to direct_drive, but with support for a user-specified equation for the cavity voltage (see below).


• bias_tee

The input signal is assumed to be generated by a source with output impedance Rs (Rs), connected to the laser via an ideal bias tee. The bias current Io is similarly connected. The input signal can be either a current or a voltage. Figure 3 depicts a bias tee setup for both the voltage- and current-source cases.


Parasitics The parasitics consist of a bond inductance Lb (Lb) and shunting capacitance Cp (Cp). These can be turned on or off via the parameter Parasitics.

VCSEL Cavity Both electrical and optical effects are modeled within the VCSEL cavity.

Electrical In both the direct_drive and bias_tee modes of operation, the electrical model of the VCSEL cavity is that of a simplified diode IV relationship, consisting of a series resistance Rd (Rd) and turn-on voltage Von (Von). During solution of the cavity current I, the model ensures that negative currents are effectively limited to zero.

For the direct_user_iv mode of operation, the user may specify a nonlinear equation for the cavity voltage via the parameter voltage_equation. This equation should be a function of cavity current I and device temperature T. For example, to implement a simple diode, the user might set voltage_equation equal to “I*100+17.2e-5*T*log(I/1e-18+1.0)”, where we have assumed a 100-Ω series resistance, a saturation current of 10-18 A, and a diode ideality factor of 2.

Rate Equations At the core of the VCSEL block are spatially independent semiconductor laser rate equations, which determine the optical output in response to the cavity current I [1]. Relative intensity noise is modeled via a constant value RIN (RIN), and the optical emission frequency is set by λ (wavelength). The model rate equations are based on the following single-mode spatially dependent equations [1]:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2

2, , , ,, ,effi l

n n

LN r t I r t N r t I N TG r t S t r N r t

t q qη

ψτ τ

∂= − − + ∇ −

∂

r r rr r r


(1)

( ) ( ) 1 1

( , ) ( , ) ( ) ( )p n V V

S t S t N r t dv G r t S t r dvt V V

∂ β ψ∂ τ τ

= − + ⋅ ⋅ + ⋅∫ ∫v v v

(2)

In (1), I is the spatially dependent injection current, N is the carrier density scaled by the effective active-layer volume V, S and ψ are the total photon number and normalized transverse mode profile, T is the device temperature, G is the gain, Il is the thermal leakage current, ηI (effint) is the current-injection efficiency, τn (tn) is the carrier lifetime, Leff is the effective carrier diffusion length, and q is the electron charge. In (2), β (b) is the spontaneous-emission coupling coefficient, and τp (tp) is the photon lifetime. Following the approach taken in [1], we can eliminate the explicit spatial dependence from (1) and (2) by assuming a cylindrical geometry, neglecting azimuthal variations, and adopting a two-term Bessel-series expansion for the carrier profile:

0 1 0 1( / )N N J r Rσ−

(3)

where σ1 is the first nonzero root of J1(x), and R is the active-layer effective radius. If we further assume a uniform current distribution and linear gain, we can eliminate the explicit spatial dependence from (1) and (2), thereby obtaining the following spatially independent VCSEL rate equations:

0 0 00 0 01 1 0( ) [ ( ( )) ] ( , )1

i t l

n

dN I N G T N N T N I N TS

dt q S qη γ γ

τ ε⋅ − −

= − − ⋅ −+

(4)

100 0 101 11 1 ( ) [ ( ( )) ](1 )

1t

diffn

G T N N T NdN Nh S

dt Sφ φ

τ ε⋅ − −

= − ⋅ + + ⋅+

(5)

0 00 0 01 1( ) [ ( ( )) ]1

t

p n

N G T N N T NdS S Sdt S

β γ γτ τ ε

⋅ − −= − + + ⋅

+

(6)

In (4)-(6), the spatial dependence of the gain is now accounted for via the overlap coefficients γ00 (gam00), γ01 (gam01), φ100 (phi100), and φ101 (phi101), while diffusive effects are taken care of via hdiff (hdiff), which is equal to 2

1( / )effL Rσ . The thermal dependence of the gain is taken into account via the thermally dependent gain constant G(T) and transparency number Nt(T), while gain saturation is modeled via ε (e), the gain saturation factor. Furthermore, the leakage Il is now a function of N0 (the average carrier number), as opposed to N. Finally, the photon number S is converted to an output power via the output-power coupling coefficient kf (kf) using the expression Pout = kfS.

Phase Rate Equation A rate equation for the optical phase is also included in the model, and is based on work presented in [7] and [8]:

00 0 01 1( ) [ ( ) ]2 1

thG T N N Nddt S

γ γφ αε

⋅ − −= ⋅

+

(7)

φ is the optical phase, α (alpha) is the linewidth enhancement factor, and Nth is the room-temperature threshold carrier number (the model assumes that the specified laser wavelength is defined at room temperature).

Mode-Carrier Overlap If the transverse mode profile ψI is normalized such that


20

2( ) 1r r dr

Rψ

∞⋅ ⋅ =∫

(8)

then the mode overlap coefficients can be calculated as [1]:

0 020

2( / ) ( )

R

i iJ r R r r drR

γ σ ψ= ⋅ ⋅ ⋅∫

(9)

10 0 0 12 20 1 0

2( / ) ( / ) ( )

( )

R

i iJ r R J r R r r drR J

φ σ σ ψσ

= ⋅ ⋅ ⋅ ⋅∫

(10)

where I = 0 or 1, and σ0 = 0. By setting the parameter overlap_calculation to off, the user is free to calculate specific values for the overlap coefficients depending on their particular choice of mode profile.

In many situations, one can model the mode profile of a single mode device as a Gaussian with characteristic radius Rm. In this case, it can be shown that the overlap coefficients reduce to functions of ρ (overlap), where ρ = Rm/R [2]. By setting overlap_calculation to on, the overlap coefficients are calculated as functions of ρ, overriding the specified values.

Thermally Dependent Gain In order to account for a VCSEL gain’s unique thermal dependence, the gain is modeled as a linear function of carrier number N, with thermally dependent gain constant and transparency number. The gain constant G(T) and transparency number Nt(T) are described via the following empirical relationships [1]:

2

0 1 22

0 1 2( ) g g g

og g g

a a T a TG T G

b b T b T

+ += ⋅

+ +

(11)

20 1 2( ) ( )t tr n n nN T N c c T c T= ⋅ + +

(12)

where Go (Go) is a gain constant, Ntr (Ntr) is a transparency number, and ag0-ag2 (ag0-ag2), bg0-bg2 (bg0-bg2), and cn0-cn2 (cn0-cn2) are fitting parameters. Generally, the gain constant will be peaked about some optimal temperature value, as a result of the temperature-dependent mismatch between lasing wavelength and gain peak. An example of the gain constant based on the model’s default values is illustrated in Fig. 4. The transparency number generally increases with temperature.


Figure 4: Sample thermally dependent gain constant based on default model values

Thermal Carrier Leakage Thermally dependent carrier leakage is modeled using the following empirical relationship [1], based on the analysis of [3]:

0 1 0 2 0 3 00

/( , ) expl lo

a a N a N T a NI N T I

T− + + − = ⋅

(13)

where Ilo (Ilo) is the leakage current factor, and a0-a3 (a0-a3) are fitting parameters. This expression accounts for the interdependence of the carrier number and temperature in determining the total leakage current. An example of (13) for various temperatures is illustrated in Fig. 5.

Figure 5: Sample thermal leakage current based on default model values

Temperature Rate Equation To complete the thermal aspect of the model, a rate equation for the device temperature as a function of dissipated heat is included. Heat generation is assumed to arise from any power not dissipated as part of the optical output. The resulting equation is [1],[4]:

( )o tot out th thdTT T I V P Rdt

τ= + − ⋅ − ⋅

(14)


where To (To) is the ambient temperature, Itot is the total current flowing through the VCSEL (including current through the intrinsic parasitic shunting capacitance Cp), Rth (Rth) is the device thermal impedance, and τth (tth) is the thermal time constant.





Multi-Line Output It is frequently necessary to produce several signals with similar properties but different wavelengths. In DWDM simulations especially, a series of regularly spaced optical sources is common. The VCSEL model provides a number of convenient facilities so that many or all required lines can be generated from a single icon. To produce a series of lines spaced equally in wavelength or frequency, set the parameter mode=LambdaGrid or mode=FreqGrid, respectively. The number of sources is controlled by the number of electrical inputs to the laser, and in both modes the parameter wavelength specifies the first line in a series of ascending wavelengths or ascending frequencies, respectively. The source separation in wavelength or frequency is specified with deltaFreq. To set the number of electrical inputs to the laser, select the VCSEL icon and open the menu item Properties. In the Ports tab, number_input_ports field, enter the number of inputs.

Multi-Line Multi-Node Output The comb of sources may be emitted either through a single node (the default) or with one channel per output node. To achieve the latter behavior, select the VCSEL icon and open the menu item Properties. Set the value of the number_output_ports field in the Ports tab to that of the number_input_ports field.


Test Parameters In order to ascertain whether the parameter settings for the VCSEL block provide the component performance desired, the user may test them from the component parameter editing window. This test produces a family of light-current (LI) curves at different ambient temperature To. The test can be controlled via model parameters carrying a prefix Test_. These parameters allow the user to set sweep limits, etc.

LI Curve The LI curve is controlled via the parameters Test_Imin, Test_Imax, Test_Lipoints, Test_Tomin, Test_Tomax, and Test_Topoints. Based on these parameters, a family of LI curves is generated over a range of ambient temperatures from Test_Tomin to Test_Tomax. The number of curves is determined by Test_Topoints. Each LI curve is generated over currents ranging from Test_Imin to Test_Imax, with the total number of points per curve specified by


Test_LIpoints. A sample family of LI curves is shown in Fig. 6. Note that if a user-specified cavity voltage equation is being used, a plot of cavity voltage versus current will also be displayed.

Figure 6: Sample test family of LI curves at ambient temperatures of 25-75 C

References [1] P. V. Mena, J. J. Morikuni, S.-M. Kang, A. V. Harton, and K. W. Wyatt, “A comprehensive circuit-level model of vertical-cavity surface-emitting lasers,” Journal of Lightwave Technology, 17, 2612 (1999).

[2] P. V. Mena, J. J. Morikuni, and K. W. Wyatt, “Compact representations of mode overlap for circuit-level VCSEL models,” IEEE/LEOS Annual Meeting Conference Proceedings, 234 (2000).

[3] J. W. Scott, R. S. Geels, S. W. Corzine, and L. A. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE Journal of Quantum Electronics, 29, 1295 (1993).

[4] N. Bewtra, D. A. Suda, G. L. Tan, F. Chatenoud, and J. M. Xu, “Modeling of quantum-well lasers with electro-opto-thermal interaction,” IEEE Journal of Selected Topics in Quantum Electronics, 1, 331 (1995).


[6] G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd. Ed. (Van Nostrand Reinhold, New York, 1993).

[7] M. X. Jungo, D. Erni, and W. Bächtold, “VISTAS: A comprehensive system-oriented spatiotemporal VCSEL model,” IEEE Journal of Selected Topics in Quantum Electronics, 9, 939 (2003).

[8] L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (John Wiley and Sons, Inc., New York, 1995).

Properties




Parameter Values Name Type Default Range Units wavelength double 850e-9 [ 0, 1e32 ] m


deltaFreq double 100e9 [ 0, 1e18 ] meters or Hz azimuth double 0 [ -90, 90 ] degrees



effint double 1.0 [ 0, 1e32 ] none kf double 1.5e-8 [ 0, 1e32 ] W

b double 1e-3 [ 0, 1e32 ] none

tp double 2.5e-12 [ 0, 1e32 ] s tn double 2.5e-9 [ 0, 1e32 ] s

Go double 3e4 [ 0, 1e32 ] 1/s

ag0 double -0.4 [ -1e32, 1e32 ] none

ag1 double 0.00147 [ -1e32, 1e32 ] 1/K ag2 double 7.65e-7 [ -1e32, 1e32 ] 1/K^2

bg0 double 1.3608 [ -1e32, 1e32 ] none

bg1 double -0.00974 [ -1e32, 1e32 ] 1/K

bg2 double 1.8e-5 [ -1e32, 1e32 ] 1/K^2 Ntr double 1e7 [ 0, 1e32 ] none

cn0 double -1 [ -1e32, 1e32 ] none

cn1 double 0.008 [ -1e32, 1e32 ] 1/K

cn2 double 6e-6 [ -1e32, 1e32 ] 1/K^2 e double 5e-7 [ 0, 1e32 ] none

alpha double 0 [ 0, 1e32 ] none

Ilo double 9.61 [ 0, 1e32 ] A

a0 double 4588.24 [ -1e32, 1e32 ] K a1 double 2.12e-5 [ -1e32, 1e32 ] K

a2 double 8e-8 [ -1e32, 1e32 ] none

a3 double 9.01e9 [ -1e32, 1e32 ] K

overlap_calculation enumerated off on, off overlap double 1 [ 0.01, 10.00 ] none

gam00 double 1 [ -1e32, 1e32 ] none

gam01 double 0.37978 [ -1e32, 1e32 ] none

phi100 double 2.3412 [ -1e32, 1e32 ] none phi101 double 1.8193 [ -1e32, 1e32 ] none

hdiff double 15 [ 0, 1e32 ] none


To double 25 [ -1e32, 1e32 ] C Rth double 900 [ 0, 1e32 ] K/W

tth double 1e-6 [ 0, 1e32 ] s


Rd double 105 [ 0, 1e32 ] ohm

Von double 1.75 [ 0, 1e32 ] V Drive_Scheme enumerated direct_drive direct_drive, bias_tee


Io double 1.5e-3 [ 0, 1e32 ] A

Parasitics enumerated on on, off Lb double 0.3e-9 [ 0, 1e32 ] H

Cp double 2e-12 [ 0, 1e32 ] F

eps double 1e-6 [ 0, 1e32 ] none

initial_tstep double 1e-13 [ 0, 1e32 ] s min_tstep double 0 [ 0, 1e32 ] s

test_function enumerated LI LI

Test_Imin double 0 [ 0, 1e32 ] A

Test_Imax double 30e-3 [ 0, 1e32 ] A Test_LIpoints integer 201 [ 0, 100000 ] none

Test_Tomin double 25 [ 0, 1e32 ] C

Test_Tomax double 75 [ 0, 1e32 ] C

Test_Topoints integer 3 [ 0, 20 ] none

Parameter Descriptions wavelength Wavelength of the laser, λ mode Type of wavelength grid deltaFreq Frequency or wavelength grid spacing for multi-line output effint Current injection efficiency, ηI kf Output power coupling coefficient, kf b Spontaneous emission coupling coefficient, β tp Photon lifetime, τp tn Carrier lifetime, τn Go Gain coefficient, Go ag0-ag2, bg0-bg2 Gain coefficient empirical parameters, ag0-ag2, bg0-bg2 Ntr Carrier transparency number, Ntr cn0-cn2 Transparency number empirical parameters, cn0-cn2 e Gain saturation factor, ε alpha Linewidth enhancement factor, α Ilo Leakage current factor, A a0-a3 Leakage current empirical parameters, a0-a3 overlap_calculation Overlap calculation flag: on, off overlap Overlap parameter, ρ gam00, gam01 Overlap coefficients for N0 and S rate equations, γ00 and γ01 phi100, phi101 Overlap coefficients for N1 rate equation, φ100 and φ101 hdiff Diffusion parameter, hdiff

RIN VCSEL relative intensity noise, RIN To Ambient temperature, To

Rth VCSEL thermal impedance, Rth


tth Thermal time constant, τth Rd VCSEL cavity resistance, Rd Von VCSEL turn-on voltage, Von Drive_Scheme Drive signal definition: direct_drive, bias_tee Rs Source impedance, Rs Io Laser bias current, Io Parasitics Parasitics flag: on, off Lb VCSEL bond inductance, Lb Cp VCSEL parasitic capacitance, Cp eps Accuracy of ODE solver initial_tstep Initial time step taken by ODE solver min_tstep Minimum time step taken by ODE solver test_function Test-function selection: LI curve Test_Imin Minimum current for LI curves Test_Imax Maximum current for LI curves Test_LIpoints Number of points per LI curve Test_Tomin Minimum ambient temperature for LI curve generation Test_Tomax Maximum ambient temperature for LI curve generation Test_Topoints Number of LI curves (one per ambient temperature setting)


Light Emitting Diode (LED)

This block models a Light Emitting Diode (LED) directly modulated with an electrical signal. It computes the electrical current injected into the LED’s optical cavity and generates the optical response, including linewidth and relative intensity noise. The behavior of the model can be partitioned into three blocks, as shown in Fig. 1.

The driving source consists of the electrical signal input into the model. The parasitics consist of a bond inductance and shunting capacitance. Finally, the LED optical cavity is modeled via a simplified current-voltage (IV) relationship and linear carrier rate equation.

Figure 1: Main components of the Light Emitting Diode model.

LED Optical Response The LED cavity’s optical response is modeled via a linear rate equation for the cavity carrier density. Relative intensity noise is modeled via a constant value RIN (RIN). The carrier rate equation, following the treatment in [1], is:

n

dN I Ndt qV τ

= −

(1)

where I is the injection current, N is the carrier density, q is the electron charge, V is the cavity volume, and nτ is the carrier lifetime. The optical output power at wavelength λ (wavelength) is proportional to N/ nτ . When the parasitic response time is much shorter than nτ , the carrier lifetime is proportional to the intrinsic LED rise/fall time rfτ

(rise_fall_time), where ln(9)rf nτ τ= ⋅ . Because (1) is linear and the optical output is proportional to N, we can readily convert (1) into a frequency-domain transfer function for the output power versus input current, whose dc value is simply the LED responsivity ℜ (responsivity), measured in W/A. Thus, the intrinsic optical response is (in the frequency domain):

( ) ( )1out

nP I

jω ω

ωτℜ= ⋅

+

(2)

where Pout is the optical output power and I is the cavity current. Note that when the parasitic response time contributes significantly to the rise/fall time, the value of the parameter rise_fall_time should be adjusted accordingly by removing any parasitic contributions which are accounted for via the model’s parasitic element parameters.

In order to add a constant optical phase to this output, the user may specify either a fixed value (initial_phase), or request a randomly generated value (by setting randomInitialPhase to yes). In the latter case, the user must provide an integer seed (randomSeed) in the range –1e8 to 1. If randomSeed < 0, then randomSeed itself acts as the seed value


for random number generation. If randomSeed = 0, then the LED component name acts as the seed. In both cases, the same random number is generated in successive simulations. Alternatively, if randomSeed = 1, then the system clock setting serves as the seed value, in which case different random numbers will be generated in successive simulations.

Linewidth Inclusion of a source linewidth in the LED output is controlled via the parameter linewidth_model, which by default is set to none (for a linewidth of zero). If linewidth_model=phase_noise, then linewidth is added to the output via phase noise. These random phase variations (seeded via the same randomSeed parameter that controls the random initial phase value) result in a Lorentzian output power spectrum [2]. If linewidth_model=value, then the constant linewidth value is attached to each output.


LED Electrical Model The electrical model of the LED’s optical cavity is that of a simplified diode IV relationship, consisting of a series resistance Rd (Rd) and turn-on voltage Von (Von). During solution of the cavity current I, the model ensures that negative currents are effectively limited to zero.

Device parasitics may consist of a bond inductance Lb (Lb) and shunting capacitance Cp (Cp), as shown in Fig. 1. These can be turned on or off via the parameter parasitics.

Driving Source The Light Emitting Diode is driven by a combination of the electrical signal at its input and, when applicable, a dc bias current specified by Ibias (Ibias).

The input electrical signal can be interpreted in a variety of ways, based on the settings of the model parameter drive_scheme. This parameter can take on values of direct_drive or bias_tee.

• direct_drive

The input signal is assumed to come from either an ideal current source or zero-impedance voltage source. In other words, the LED is assumed to be undergoing direct-drive modulation. If the input signal is a current, then it is combined with the bias current Ibias to form the total input current. Figure 2(a) illustrates this scenario. If the input signal is a voltage, then Ibias is ignored. Note that the input voltage should be larger than the LED’s turn-on voltage Von; otherwise, the resulting current is zero. This arrangement is illustrated in Fig. 2(b).

• bias_tee

The input signal is assumed to be generated by a source with output impedance Rs (Rs), connected to the LED via an ideal bias tee. The bias current Ibias is similarly connected. The input signal can be either a current or a voltage. Figure 3 depicts a bias tee setup for both the voltage- and current-source cases.

Figure 2: Direct-drive modulation schemes: (a) current and (b) voltage.


Figure 3: Bias-tee modulation schemes: (a) voltage source and (b) current source.





Multi-Line Output It is frequently necessary to produce several signals with similar properties but different wavelengths. In DWDM simulations especially, a series of regularly spaced optical sources is common. The LED model provides a number of convenient facilities so that many or all required lines can be generated from a single icon. To produce a series of lines spaced equally in wavelength or frequency, set the parameter mode=LambdaGrid or mode=FreqGrid, respectively. The number of sources is controlled by the number of electrical inputs to the laser, and in both modes the parameter wavelength specifies the first line in a series of ascending wavelengths or ascending frequencies, respectively. The source separation in wavelength or frequency is specified with deltaFreq. To set the number of electrical inputs to the laser, select the LED icon and open the menu item Properties. In the Ports tab, number_input_ports field, enter the number of inputs.

Multi-Line Multi-Node Output The comb of sources may be emitted either through a single node (the default) or with one channel per output node. To achieve the latter behavior, select the LED icon and open the menu item Properties. Set the value of the number_output_ports field in the Ports tab to that of the number_input_ports field.

Test Functions In order to ascertain whether the parameter settings for the Light Emitting Diode block provide the component performance desired, the user may test them from the component parameter editing window. This test produces either a light-current curve or small-signal frequency response curves. By setting the test_function parameter to the desired output and clicking the Test button in the component-parameter editing window, the user may display the LED


characteristics summarized below. Furthermore, the default plot ranges for each characteristic may be overridden by setting test_default_settings to no, and specifying values for test_function_x_low, test_function_x_high, and test_function_points (the number of data points to plot). For frequency response curves, the user may also select whether or not to display the x-axis in log scale (test_log_x).

• LI: Plots the LED’s light-current (LI) characteristic. The plot range should be specified in amperes. The plot units can be set via the parameters power_units and current_units.

• transfer_function: Displays the LED’s small-signal optical transfer function (optical power versus current), including parasitic effects. The plot range should be specified in Hertz. The plot units can be set via the parameters response_units and frequency_units.

• S21: Displays the LED’s small-signal S21 response. The plot range should be specified in Hertz. Plot units can be set via the parameters response_units and frequency_units.

References [1] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, New York, 1991).



Properties





deltaFreq double 100e9 [ 0, 1e18 ] meters or Hz azimuth double 0 [ -90, 90 ] degrees



responsivity double 1.0 [ 0, 1e32 ] W/A rise_fall_time double 1e-9 [ 0, 1e32 ] s

linewidth_model enumerated phase_noise none, phase_noise, value



randomInitialPhase enumerated no no, yes initial_phase double 0 [ -180, 180 ] degrees


RIN double -150.0 [ -1e32, 1e32 ] dB/Hz


Rd double 5.0 [ 0, 1e32 ] ohm

Von double 2.0 [ 0, 1e32 ] V drive_scheme enumerated direct_drive direct_drive, bias_tee

Rs double 50.0 [ 0, 1e32 ] ohm

Ibias double 25e-3 [ 0, 1e32 ] A

parasitics enumerated no no, yes Cp double 2e-12 [ 0, 1e32 ] F

Lb double 0.3e-9 [ 0, 1e32 ] H

test_function enumerated LI LI, transfer_function, S21

test_default_settings enumerated yes yes, no

test_function_x_low double 0 [ 0, 1e32 ] none test_function_x_high double 100e-3 [ 0, 1e32 ] none

test_function_points integer 201 [ 2, 100000 ] none

test_log_x enumerated yes yes, no

power_units enumerated mW uW, mW, W, dBm current_units enumerated mA nA, uA, mA, A

frequency_units enumerated Hz Hz, kHz, MHz, GHz, THz

response_units enumerated dB linear, dB

Parameter Descriptions wavelength Wavelength of the LED, λ

mode Type of wavelength grid deltaFreq Frequency or wavelength grid spacing for multi-line output responsivity Optical response coefficient ℜ

rise_fall_time Intrinsic rise/fall time ln(9)rf nτ τ= ⋅

linewidth_model Linewidth model selection linewidth_units Linewidth units selection linewidth Linewidth value randomInitialPhase Random phase setting initial_phase Constant phase value, when randomInitialPhase = no randomSeed Random-number-generation seed value RIN Relative intensity noise, RIN Rd Cavity resistance Rd Von Turn-on voltage Von drive_scheme Drive signal setting Rs Source impedance Rs Ibias Bias current Ibias parasitics Parasitics setting Cp Parasitic capacitance Cp Lb Bond inductance Lb

power_units Units for optical power current_units Units for electrical current


frequency_units Units for modulation frequency response_units Units for small-signal response data test_function Test function output selection. test_default_settings Selection for default settings in test function output test_function_x_low Lowest x-value for test function output test_function_x_high Highest x-value for test function output test_function_points Number of points in test function output test_log_x Switch for using log scale for test function x-values


Modulator

This models an electro-optic modulator. Several types of modulators may be modeled with this block, including the Mach-Zehnder type. When using the modulator model with the mode-locked laser model, the user must ensure that the number of samples per bit and the bit sequence pattern width for both the binary sequence generator and the mode-locked laser model are the same.

The electrical parasitic frequency response for the modulator is written as one of three fitting forms,

( )( )

( )

2

2 3 40 1 2 3 4

fOffset type

1 Coefficients type0 1 ( 2 )

+ + + ( )

powoffset

signal

signal

f fV f CoefV Coef f

a a f a f a f a f dB

+

=+ ⋅

+ Polynomial type

where f is in unit of GHz. Be careful with polynomial fitting, the fitting must be reasonable to frequencies up to the FFT sampling frequency. If an ideal parasitic frequency response is desired, use the Coefficients type with Coef1=1.0 and Coef2=0.0.

After accounting for the parasitic response, the voltage signal is input to the optical modulator and the corresponding intensity response of the modulator is modeled as one of the following forms, depending on the modulation type chosen.

Mach-Zehnder For the Mach-Zehnder type modulator, the following intensity response function is used:

<

≥

−+

= 1 1

1 2

sin

//

/

2

offoni

o

offon

offoni

ooffsetbiassignal

i

o

RII

R

RII

VVVV

II π

π

where Ron off is the extinction ratio of the modulator in linear units (to be input by user as parameter onOffRatio in dB units) and Vsignal represents the electrical signal after being modified by the parasitic frequency response and optionally being level-shifted such that the average level is zero, modeling the behavior of the bias circuitry. The sine function is used instead of the cosine function in the Mach-Zehnder modulator so that the modulated signal will have the same polarity as the original binary sequence. This is important for increased numerical accuracy in simulation. To deactivate the extinction ratio modification to the signal, set the extinction ratio parameter to 0.

Chirp of the modulator is also modeled. The phase change due to chirping is calculated as (see for example, Fumio Koyama and Kenichi Iga, “Frequency chirping in external modulators”, J. of Lightwave Technology, Vol. 6, pp. 87-93, 1988.),

1

2d C dsdt S dtφ =

which gives 00

ln2C S

Sφ φ

= +

where C is the user-specified ChirpFactor, φ is the optical instantaneous phase, and S is the instantaneous intensity of the light.


Ideal For the ideal signal type modulator, the following intensity response function is used:

<

≥

−+

=

offoni

o

offon

offoni

ooffsetbiassignal

i

o

RII

R

RII

VVVV

II

//

/

2

1 1

1 π

where Ron/off is the extinction ratio of the modulator (to be input by user as parameter onOffRatio in dB units), and Vsignal represents the electrical signal after being modified by the parasitic frequency response and optionally being level-shifted such that its average level is zero. To have an ideal parasitic frequency response, use the Coefficients type and set the first coefficient to 1, and the second to 0. To deactivate the extinction ratio modification, set the extinction ratio parameter to 0. Note that intensity is the square of the field; therefore a sinusoidal modulating electrical signal applies a sinusoidal modulation to the electric field of the optical output signal. Also note the behavior when

signal bias offsetV V V+ − < 0 and set the values appropriately to avoid any undesired behavior.

Chirp may be included in the Ideal type modulator, as well. If no chirp is desired, the ChirpFactor parameter should be set to 0. The phase change due to chirping is calculated the same as in the Mach-Zehnder type modulator.

Amplitude This may represent both an ideal amplitude modulator as well as an electroabsorption modulator depending on the parameter settings and whether chirp is specified. For the amplitude type modulator, the following intensity response function is used:

<

≥

−+

=

offoni

o

offon

offoni

ooffsetbiassignal

i

o

RII

R

RII

VVVV

II

//

/

1 1

1 π

where Ron/off is the extinction ratio of the modulator (to be input by user as parameter onOffRatio in dB units), and Vsignal represents the electrical signal after being modified by the parasitic frequency response and optionally being level-shifted such that its average level is zero. To have an ideal parasitic frequency response, use the Coefficients type and set the first coefficient to 1, and the second to 0. To deactivate the extinction ratio modification, set the extinction ratio parameter to 0. In no case is the response allowed to go below 0. If it does, it is clipped at 0.

Chirp may be included in the ideal amplitude type modulator, as well. If no chirp is desired, the ChirpFactor parameter should be set to 0. The phase change due to chirping is calculated the same as in the Mach-Zehnder type modulator.

Phase For the phase type modulator, the following intensity response function is used:

exp signal bias offseto

i

V V VEj

E Vπφ

+ − = ∆

where ∆φ is the specified phase shift and Vsignal represents the electrical signal after being modified by the parasitic frequency response and optionally being level-shifted such that its average level is zero. To have an ideal parasitic frequency response, use the Coefficients type and set the first coefficient to 1, and the second to 0.

Chirp may be included in the phase type modulator, as well. If no chirp is desired, the ChirpFactor parameter should be set to 0. The phase change due to chirping is calculated the same as in the Mach-Zehnder type modulator.


Properties

Inputs #1: Optical signal



Parameter Values Name Type Default Range Units modulationType enumerated MachZehnder Ideal, Amplitude,

Phase, MachZehnder

phaseShift double 0.0 [ -180, 180 ] degrees

vPi double 1.0 [ -1e32, 1e32 ] Volts vBias double 0.0 [ -1e32, 1e32 ] Volts

vOffset double 0.0 [ -1e32, 1e32 ] Volts

onOffRatio double 0 [ 0, 1e6 ] dB

insertionLoss double 0 [ 0, 1e6 ] dB fittingType enumerated Coefficients fOffset, Coefficients,

Polynomial

fOffset double 0.0 [ 0, 1e32 ] GHz

power double 0.0 [ -100, 100 ] none

coef1 double 1 [ -1e32, 1e32 ] none coef2 double 0.0 [ -1e32, 1e32 ] 1/GHz

a0 double 1.0 [ -1e32, 1e32 ] none

a1 double 0.0 [ -1e32, 1e32 ] 1/GHz

a2 double 0.0 [ -1e32, 1e32 ] 1/GHz^2 a3 double 0.0 [ -1e32, 1e32 ] 1/GHz^3

a4 double 0.0 [ -1e32, 1e32 ] 1/GHz^4

ChirpFactor double 0.0 [ -1e32, 1e32 ] none

MatchEnds enumerated No No, Yes LevelShift enumerated No No, Yes

Parameter Descriptions fittingType Modulator electrical parasitics fitting type modulationType Modulation response function type LevelShift Whether to level shift input voltage signal to have an average value of 0 before

applying modulation function to optical signal MatchEnds Whether to alter the signal value in the last two bit periods of the output optical

signal to gradually bring it to the same value as the start of the optical signal vPi vPi of the modulator vBias Bias voltage of the modulator vOffset Offset voltage of the modulator onOffRatio Extinction or on-off ratio


insertionLoss Insertion loss = waveguide + coupling loss foffset Offset for the Frequency Response parasitics fitting type power Exponent for the Frequency Response parasitics fitting type coef1 Coef1 in Coefficients of the parasitics frequency response coef2 Coef2 in Coefficients of the parasitics frequency response ChirpFactor Chirp parameter for modulator a0 a0 of Polynomial Fitting of the parasitics frequency response a1 a1 of Polynomial Fitting of the parasitics frequency response a2 a2 of Polynomial Fitting of the parasitics frequency response a3 a3 of Polynomial Fitting of the parasitics frequency response a4 a4 of Polynomial Fitting of the parasitics frequency response


Electroabsorption Modulator

This model represents an electroabsorption modulator. With voltage scaling enabled, it allows the user to specify the extinction ratio of the output optical signal explicitly, and it scales the input modulating voltage signal as required to obtain the specified extinction ratio at the output. Nonlinear modulation response is also supported with either file-based data interpolation or a 7th order polynomial in terms of the scaled or un-scaled input modulating voltage signal. The model also allows the chirp factor to be specified as a data file or as a 7th order polynomial in terms of the input modulating voltage signal and supports specification of dc chirp with another data file or 7th order polynomial in terms of the input modulating voltage signal. In addition, the user may specify whether or not the output optical signal should be modulated as the inverse of the input electrical signal (only if voltage scaling is enabled), and whether the alpha parameter should be considered positive or negative (to accommodate differing conventions in the literature).

The model has two input ports and one output port. The first input port accepts an optical signal that is modulated to produce the output optical signal, and the second input port accepts an electrical signal that is used to modulate the input optical signal to produce the output optical signal. The input optical signal may be any type of optical signal, but it is recommended to use the CW laser or mode-locked laser sources to provide the input optical signal. The user must ensure that the total number of samples contained within the input electrical and optical signals are the same, as are the sampling rates. An exception to this is when using the CW laser model to produce the PowerValue signal type.

When voltage scaling is enabled (voltage_scaling = Yes), the expression describing the relationship between the input optical signal and the output optical signal is as follows:

( )/10 /1010 1 10L Ro

i

Ix x

I− − = ⋅ − ⋅ +

where Io is the output optical intensity, Ii is the input optical intensity, L is the insertion loss in dB, R is the extinction ratio in dB, and x is as follows:

( ) ( ) ( )

( ) ( ) ( ) ( )

2 30 1 2 3

4 5 6 74 5 6 7

s s s

s s s s

x x V x V x Vx

x V x V x V x V

+ ⋅ + ⋅ + ⋅= + ⋅ + ⋅ + ⋅ + ⋅

where Vs is the input electrical voltage after scaling to range from 0 to 1 and optionally inverting. x may also be interpolated from a data file providing x vs. Vs values.

When voltage scaling is disabled (voltage_scaling = No), the expression describing the relationship between the input optical signal and the output optical signal is as follows:

/1010 Lo

i

Ix

I−= ⋅

where Io is the output optical intensity, Ii is the input optical intensity, L is the insertion loss in dB, and x is as follows:

( ) ( ) ( )

( ) ( ) ( ) ( )

2 30 1 2 3

4 5 6 74 5 6 7

offset offset offset

offset offset offset offset

x x V V x V V x V Vx

x V V x V V x V V x V V

+ ⋅ − + ⋅ − + ⋅ −= + ⋅ − + ⋅ − + ⋅ − + ⋅ −

where V is the un-scaled un-inverted input modulating voltage, and Voffset is a user-specified voltage offset parameter. x may also be interpolated from a data file providing x vs. V-Voffset values.


Expressions describing the frequency chirp and the phase change of the output optical signal due to frequency chirping are as follows (see for example, Fumio Koyama and Kenichi Iga, “Frequency chirping in external modulators”, J. of Lightwave Technology, Vol. 6, pp. 87-93, 1988.):

Frequency Chirp (Hz)1

2ddtφ

π= −

where φ is the phase in units of radians and

1

2dcdd ds

dt S dt dtφφ α = − +

where

( ) ( ) ( )

( ) ( ) ( ) ( )

2 30 1 2 3

4 5 6 74 5 6 7

m m m

m m m m

V V V V V V

V V V V V V V V

α α α αα

α α α α

+ ⋅ − + ⋅ − + ⋅ −= + ⋅ − + ⋅ − + ⋅ − + ⋅ −

and

( ) ( ) ( )

( ) ( ) ( ) ( )

2 30 1 _ 2 _ 3 _

4 5 6 74 _ 5 _ 6 _ 7 _

m dc m dc m dcdc

m dc m dc m dc m dc

dc dc V V dc V V dc V Vddt dc V V dc V V dc V V dc V V

φ + ⋅ − + ⋅ − + ⋅ −= + ⋅ − + ⋅ − + ⋅ − + ⋅ −

and 0α to 7α are the user-specified parameters for the polynomial chirp expression, V is the input modulating voltage, Vm is the user-specified voltage offset parameter, φ is the optical instantaneous phase, S is the instantaneous intensity of the light, dc0 to dc7 are the user-specified parameters for the polynomial dc chirp expression, Vm_dc is the user-specified dc chirp voltage offset parameter, dtd dc /φ is the phase change due to the dc chirp. α and dtd dc /φ may alternatively be interpolated from data files providing their values vs. V-Vm or V-Vm_dc respectively.

There is an option regarding the sign convention for the alpha parameter in the frequency chirp equations. By default in this model, the sign convention is to use the negative sign in front of the alpha parameter in the phase derivative equation (negative alpha). This convention leads to a positive alpha parameter creating a positive frequency chirp for rising output signal powers. To use the opposite convention, change the alpha_sign parameter to “Pos” instead of “Neg”.

Note that the user specified parameters for dc chirp are actually the parameters in the polynomial expansion for the phase derivative. For example, to specify a positive frequency chirp of 1e9 Hz when the input voltage V is 1.0, set dc_chirp1 to 91*)2( eπ− and the rest of the dc_chirp parameters to 0.

File Format For the data files used by this model, the voltage values should be monotonically increasing. The required file format is:

<num_pts>

<voltage 1> <x 1>

<voltage 2> <x 2>

<voltage 3> <x 3>

…

where <num_pts> specifies the number of data points in the file, voltage specifies the voltage (Vs, V-Voffset, V-Vm or V-Vm_dc, depending on the data file), and x specifies the dependent data value (x, α or dtd dc /φ , depending on the data file).


Properties




Parameter Values Name Type Default Range Units voltage_scaling enumerated Yes Yes, No

invertInput enumerated No No, Yes insertionLoss double 0 [ 0, 1e32 ] dB

extinctionRatio double 13.0 [ 0, 1e32 ] dB

Voffset double 0 [ -1e32, 1e32 ] Volts

Vm double 0 [ -1e32, 1e32 ] Volts alpha_sign enumerated Neg Pos, Neg

alpha_spec enumerated polynomial polynomial, file

alpha_file string

alpha0 double 0.0 [ -1e32, 1e32 ] N/A alpha1 double 0.0 [ -1e32, 1e32 ] N/A

alpha2 double 0.0 [ -1e32, 1e32 ] N/A

alpha3 double 0.0 [ -1e32, 1e32 ] N/A alpha4 double 0.0 [ -1e32, 1e32 ] N/A



alpha7 double 0.0 [ -1e32, 1e32 ] N/A Vm_dc double 0 [ -1e32, 1e32 ] Volts

chirp_spec enumerated polynomial polynomial, file

chirp_file string

dc_chirp0 double 0.0 [ -1e32, 1e32 ] N/A dc_chirp1 double 0.0 [ -1e32, 1e32 ] N/A

dc_chirp2 double 0.0 [ -1e32, 1e32 ] N/A


dc_chirp4 double 0.0 [ -1e32, 1e32 ] N/A dc_chirp5 double 0.0 [ -1e32, 1e32 ] N/A



mod_spec enumerated polynomial polynomial, file mod_file string

x0 double 0.0 [ -1e32, 1e32 ] N/A

x1 double 1.0 [ -1e32, 1e32 ] N/A


x2 double 0.0 [ -1e32, 1e32 ] N/A

x3 double 0.0 [ -1e32, 1e32 ] N/A x4 double 0.0 [ -1e32, 1e32 ] N/A

x5 double 0.0 [ -1e32, 1e32 ] N/A

x6 double 0.0 [ -1e32, 1e32 ] N/A

x7 double 0.0 [ -1e32, 1e32 ] N/A

Parameter Descriptions voltage_scaling Whether to scale input voltage for intensity modulation InvertInput Whether to invert input voltage prior to modulation or not InsertionLoss Optical power loss of input optical signal prior to modulation (dB) ExtinctionRatio Ratio of high and low output power levels (dB) Voffset Offset voltage to use when calculating intensity modulation without voltage

scaling alpha_sign Sign prior to the alpha parameter in phase derivative equation alpha_spec Polynomial or file-based specifications for alpha alpha_file File name for file-based alpha specification Vm Offset voltage for chirp parameter polynomial expression (V) alpha0 … alpha7 Chirp parameters for chirp parameter polynomial expression Vm_dc Offset voltage for dc chirp polynomial expression (V) chirp_spec Polynomial of file-based specifications for dc chirp chirp_file Filename for file-based dc chirp specification dc_chirp0 … dc_chirp7 Chirp parameters for dc chirp polynomial expression mod_spec Polynomial or file-based specifications for modulation mod_file Filename for file-based specification of modulation x0 … x7 Modulation relation parameters for nonlinear input voltage to output optical

power modulation response

OptSim Models Reference: Block Mode Chapter 5: Optical Fibers •••• 123

Chapter 5: Optical Fibers

This chapter describes the optical fibers:

• Nonlinear Fiber provide a detailed implementation of propagation of one or more optical channels in a single mode fiber

• Bidirectional Nonlinear Fiber (Raman Amplifier) provide a detailed implementation of a bidirectional fiber with all dispersive, nonlinear, PMD and Raman effects

• Fiber Delay introduce a time delay onto the optical signals

124 •••• Chapter 5: Optical Fibers OptSim Models Reference: Block Mode

Nonlinear Fiber

This model provides a detailed implementation of propagation of one or more optical channels in a single mode fiber. It takes into account attenuation, dispersion, polarization mode dispersion (PMD) and nonlinearities including Raman effects. When the Single-Channel mode of the MUX is used prior to the fiber model, it also takes into account four wave mixing. Bi-directional effects, especially Raman amplification, should be modeled using the Bidirectional Nonlinear Fiber Model, described later in this chapter.

Background A (scalar) optical field ψ in a fiber obeys a differential equation of the general form

( ) ( ) ( ), ˆ ˆ ,z t

D N z tz

ψψ ψ

∂ = + ∂

(1)

where the operators D and N describe the (linear) dispersive and nonlinear effects, respectively. In the simplest approximation the propagation equation is the well-known nonlinear Schrödinger equation:

( ) ( ) ( ) ( )

22

2 2

, ,, ,

z t z tj j z t z t

z tψ ψ

β γ ψ ψ∂ ∂

= − +∂ ∂

(2)

This equation applies to loss-less fibers with quadratic dispersion of strength 2β and a purely instantaneous nonlinearity of strengthγ that introduces self-phase modulation (SPM). In general, it is necessary to include many further terms in order to account for attenuation, higher-order dispersion and delayed-nonlinearity (the Raman response of the fiber), though for most problems only a subset of these effects will play a significant role. We present all these additional terms below, but first discuss the general method of solution.

Fourier Split-step algorithm

As a second-order nonlinear partial-differential equation, Eq. (2) is at first-sight non-trivial to solve. The standard approach with a long validated history [1] is to separate the treatment of the linear and nonlinear parts. Remembering

that D and N are to be treated as operators, the formal solution for a small increment of the solution of Eq. (1) may be written

( ) ( ) ( )

( )

0 0

0

ˆ êxp

ˆ ˆêxp exp exp .2 2

z dz D N dz z

D Ddz Ndz dz z

ψ ψ

ψ

+ = +

≈

(3)

The second approximate solution is simply the lowest order term in a series expansion, and simplifies the problem by allowing the dispersive and nonlinear parts to be integrated separately. This separation inspires the description “split-


step” for the solution method. The choice of the symmetric form of the expansion guarantees that the approximate solution is accurate to second order in the step-size dz .

For the simple example of Eq. (2), the problem is now much easier to solve. Consider first the nonlinear part with 2N jγ ψ= . This has the elementary exact solution

( ) ( ) ( )

( ) ( )0 0

20 0

ˆ, exp ,

exp , , ,

z dz t N dz z t

j z t dz z t

ψ ψ ψ

γ ψ ψ

+ = =

(4)

where we note that the operator N is diagonal in the time domain. Now consider the dispersive part of the problem with 2 2

2ˆ /D i tβ= − ∂ ∂ . The time derivative makes the dispersion operator non-local in the time domain. Writing the optical

field in the Fourier representation

( ) ( )d e j tt ωψ ω ψ ω∞

−

−∞

= ∫

(5)

the time derivative becomes a simple multiplication in the Fourier domain / t jω∂ ∂ → − and the solution to the dispersive problem can be written

( ) ( )

( )0 0

20 2

ˆ, , exp ,

, exp ,

z dz z Ddz

z j dz

ψ ω ψ ω

ψ ω β ω

+ = =

(6)

which is diagonal in the frequency domain.

The general strategy for solution is then to take alternating steps of size dz , solving the dispersive linear part of the problem in the frequency domain, and the nonlinear part of the problem in the time domain. Even when additional terms are added to the propagation equations to describe effects other than quadratic dispersion and SPM, this split-step approach remains effective. The dispersion operator is always exactly solvable. Certain nonlinear terms involving mixed polarization and the Raman response cause the nonlinear operator to become non-diagonal, and in this case the nonlinear part of the problem is integrated using a standard Runge-Kutta algorithm. The step-size dz must thus be small enough to ensure both an accurate integration of the nonlinear part, and the validity of the split-step approach. As discussed below, the step-size may be controlled explicitly by the user, or else determined automatically during the simulation to satisfy accuracy requirements.

The use of the Fourier-based algorithm implies that the simulation uses periodic boundary conditions – pulses leaving one end of the simulation window reenter at the opposite end. Initially, it may appear that this approach introduces unphysical correlations due to the wrapping of signals and noise. In fact, these correlations have a very minor effect and can always be essentially eliminated by using longer bit patterns. Moreover, periodic boundary conditions are important for WDM simulations since initially coincident pulses in different channels quickly separate due to walk-off. If the pulses did not reenter the time window, inter-channel interactions would be grossly underestimated leading to overoptimistic predictions for cross-talk, gain tilt and XPM effects.

Frequency Decomposition Approach It is frequently the case that the incoming field to the fiber consists of a number of narrow channels separated by a fixed or varying channel spacing. Again, a WDM system is a natural example. Neglecting polarization, the incoming field may then be written


( ) ( ) ( )exp c.c.i ii

E t A t j tω= − +∑

(7)

where the amplitude functions )(tAi are slowly varying in time, and c.c. denotes the complex conjugate of the

preceding expression. When such an expression is substituted into the Kerr nonlinearity source term 3E , numerous terms appear. For just two channels at frequencies 1ω and 2ω , for instance, the nonlinear expression is as follows:

( ) ( )( ) ( )( ) ( ) ( )( )

( )

1 2 1 2

1 2 2 1 1 2 2 1

1 2

2 2 2 231 1 2 2 2 1 1 2

2 * 2 * 2 22 2 2 21 2 2 1 1 2 2 1

2 23 31 2

3 e e 6 e e

3 e e 3 e e

e e c.c.

j t j t j t j t

j t j t j t j t

j t j t

E A A A A A A A A

A A A A A A A A

A A

ω ω ω ω

ω ω ω ω ω ω ω ω

ω ω

− − − −

− − − − − + − +

− −

= + + +

+ + + +

+ + +

(8)

The bracketed terms respectively correspond to self-phase modulation (SPM), cross-phase modulation (XPM), difference frequency generation or four-wave mixing (FWM), sum-frequency generation and third harmonic generation. SPM and XPM, which occur only at the source frequencies 1ω and 2ω , are automatically phase-matched and are always present. In optical fibers, the sum-frequency and third-harmonic generation (THG) terms are never phase-matched and are negligible. However, in a low dispersion fiber, or near the zero-dispersion wavelength, the FWM terms transfer significant energy to new frequencies. For widely spaced channels, or far from the zero-dispersion point however, FWM is unimportant. In this case, all the nonlinear response occurs at frequencies in the incoming field and it is then common to simulate the channels separately using a set of coupled differential equations for each amplitude function )(tAi ,

rather than a single equation for the whole field ( )E t . This is known as the “frequency decomposition” approach [1]. Indeed, it is only in the context of such a decomposition that the identification of different effects such as SPM, XPM or THG is justified. The advantage of such an approach is that one need only simulate the behavior in small subsets of the total spectral range of the problem greatly increasing the speed of the simulation. For example, the plots in Fig. 1 show the input and output fields of a three-channel fiber simulation using frequency decomposition. Since one channel is “missing”, a considerable spectral range has been neglected allowing savings in computational time. Moreover, the maximum step-size for accurate simulation of the dispersive part turns out to decrease with the square of the largest channel bandwidth, so by simulating individual channels the simulation speed is increased many times.

lstmpmux spec Optical Frequency Spectrum

x1011

Optical Frequency (Hz)

1929 1930 1931 1932 1933

Pow

er (d

Bm

)

300-

200-

100-

0lstmpfiber spec Optical Frequency Spectrum

x1011


1929 1930 1931 1932 1933

Pow

er (d

Bm

)

300-

200-

100-

0

Figure 1: Input and output spectra for a fiber operating in multi-channel mode. No four wave mixing is visible.


lstmpmux spec Optical Frequency Spectrum

x1011


1927 1928 1929 1930 1931 1932 1933 1934 1935

Pow

er (d

Bm

)

200-

100-

0lstmpfiber spec Optical Frequency Spectrum

x1011


1927 1928 1929 1930 1931 1932 1933 1934 1935

Pow

er (d

Bm

)

140-

120-

100-

80-

60-

40-

20-

0

Figure 2: Input and output spectra for a fiber operating in single-channel mode. Four-wave mixing peaks are observed both between and outside the existing channels

Mode Selection The choice of whether a simulation is performed using frequency decomposition or “multi-channel mode”, or instead as a single field in “single-channel mode” is determined not by controls in the fiber model, but by the value of the Representation parameter setting of any optical multiplexers preceding the fiber in the topology. If the multiplexer is set to MultiBand, each channel is carried as a separate field and the simulation is performed using the coupled equations. If the multiplexer is set to SingleBand, the channels are combined into a single field using Eq. (7) and simulated using a single equation (per polarization).

Four wave mixing The plots in Fig. 2 illustrate an important limitation of the frequency decomposition approach – its inability to simulate four wave mixing effects. These plots show results for the same three-channel problem as in Fig. 1 , but with the whole field ( )E t now simulated using a single channel. It is apparent that FWM has transferred energy to several new frequency bands, and perhaps more importantly it may have degraded the signal at the input channels. Thus if simulations are performed using multi-channel mode, it is wise to check the results using single-channel mode to ensure that FWM is not being neglected without justification.

Complete model We now present the full set of equations used in the fiber model [1]:

( )

( )

2 3

2 32 3

2 3

2 32 3

1 1 ;2 6 2

1 1 ;2 6 2

x x x xx x yi i i i i

i i i x j jxgi

y y y yy x yi i i i i

i i i y j jygi

A A A Aj A N A Az v t t t

A A A Aj A N A Az v t t t

αβ β

αβ β

∂ ∂ ∂ ∂+ + − + =∂ ∂ ∂ ∂

∂ ∂ ∂ ∂+ + − + =∂ ∂ ∂ ∂

K K

K K

(9)

Here, ,x yi iA A are the x and y polarized complex modal amplitudes of the ith channel. For single-channel

mode simulations, only a single amplitude is used for each polarization. Further, giv is the group velocity,


2iβ is the second-order dispersion coefficient, 3iβ is the third-order dispersion coefficient, evaluated at

the center wavelength iλ and iα is the frequency-dependent attenuation coefficient. The nonlinear

response is represented by the terms ( );x yx j jN A AK K and ( );x y

y j jN A AK K . Since these

expressions are very complicated, we reserve stating them in full until the discussion of the nonlinearity below.

We now discuss the use and implementation of the attenuation, dispersion and nonlinearity, in increasing order of complexity.

Attenuation Using the parameter loss_method, the fiber loss may be specified as either a constant value or as an arbitrary user-specified frequency-dependent profile. The constant value is determined by the parameter loss. The user profile is stored in a file of filename loss_filename, with a format described in the appendix. Note that for both methods, the loss is specified in the familiar units of dB/km and is one of the few quantities in OptSim not expressed in SI units. It is strongly recommended that a frequency-dependent profile be used for all broadband or WDM simulations, particularly for simulations involving more than one of the S, C or L bands. Multi-stage fibers with varying attenuation should be produced by concatenation of two or more fiber components.

Dispersion The dispersive part of the model accounts for the variation in phase velocity with frequency as a result of both material and waveguide dispersion. The dispersion is expressed in terms of the standard parameters

( )

( )

( )

1

2

2 2

3

3 3

dddddd

ββ ωω

ββ ωωββ ω

ω

=

=

=

(10)

where ( )β ω is the frequency-dependent propagation constant of the fiber. In each optical channel, a phase shift is applied according to the equation

( ) ( )2 32 3i i

iφ β ω β ω= ∆ + ∆

(11)

In addition, each channel experiences group velocity walkoff from the other channels according to the group index )/( 11 cn ii β= , with 1/i i

gv c n= . Several different models may be used for dispersion, which

determine how the parameters 1n , 2β and 3β are set for each channel. In all cases, the numerical values of the parameters denote the actual values at the center wavelength of the associated optical signal.

These models are selected through the dispersionModel parameter with the following options:

• defined

The dispersive parameters are obtained from the standard fiber dispersion parameter ( )D λ which satisfies the relations


( ) ( )

( ) ( )( )

( )

( ) ( ) ( )0

2

2

3 4

3 22 2

1 1 0

2d

2 d2

d

Dc

DD

c c

Dλ

λ

λβ λ λπ

λλ λβ λ λπ λπ

β λ β λ λ λ

= −

= +

= + ∫

(12)

In the defined model, the dispersion parameter is modeled with the empirical formula

4

0 00( ) 1

4SD Dλ λλ

λ = − +

(13)

in terms of a dispersion slope 0S (dispersionS0), dispersion offset 0D (dispersionOffset),

and a reference wavelength 0λ (dispersionLambda0). These parameters are frequently available from product specification sheets. Note that the conventional units of ps/km/nm for

( )D λ must be converted to SI units of s/m2 for use in OptSim. The component parameters n1, beta2, beta3 are not used in this dispersion model. This is the recommended model for standard link simulation.

• defined2

This model is similar to the defined model, but uses the simpler formula

[ ]0 0 0( )D S Dλ λ λ= − + .

• custom

The user specifies 1n , 2β and 3β values directly through the parameters n1, beta2, beta3. These values are applied to every channel. This cannot correspond to any real dispersion profile and hence for more than one channel, this model is non-physical. It is intended primarily for testing purposes and for theoretical studies of ultrashort-pulse propagation, when it can be useful to have explicit control of the dispersion. The parameters dispersionS0, dispersionOffset, dispersionLambda0 are not used.

• extrapolated

The user specifies 1n , 2β and 3β values at a specific wavelength 0λ (dispersionLambda0). For all other frequencies, the dispersion properties are extrapolated in a polynomial fashion, according to

3 3 0

2 2 0 3 00

2

3 011 2 0

0 0

( ) ( )

2 2( ) ( ) ( )

( )2 2 2 2( ) ( )2

nc

β λ β λ

π πβ λ β λ β λλ λ

β λπ π π πβ λ β λλ λ λ λ

=

= + −

= + − + −

The parameters dispersionS0 and dispersionOffset are not used. This model should be used


if the user has data in terms of the dispersion coefficients iβ rather than ( )D λ , or for novel fibers with unusual dispersion characteristics such as photonic crystal fibers that are not well-fitted by the defined or defined2 models. Note that this model is physical and should normally be preferred to the custom model.

• extrapolated_full

This model extends the extrapolated approach to seventh order dispersion. The user specifies

1n , 2β , 3β , 4β , 5β , 6β and 7β ,values at a specific wavelength 0λ (dispersionLambda0). For all other frequencies, the dispersion properties are extrapolated in a polynomial fashion, according to

7 7 0

6 6 0 7 0

2

5 5 0 6 0 7 0

2 3

4 4 0 5 0 6 0 7 0

2 3 4

3 3 0 4 0 5 0 6 0 7 0

2

2 2 0 3 0 4 0

( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )2

( ) ( ) ( ) ( ) ( )2 3!

( ) ( ) ( ) ( ) ( ) ( )2 3! 4!

( ) ( ) ( ) ( )2

β λ β λβ λ β λ δω β λ

δωβ λ β λ δω β λ β λ

δω δωβ λ β λ δω β λ β λ β λ

δω δω δωβ λ β λ δω β λ β λ β λ β λ

δω δωβ λ β λ δω β λ β λ

== +

= + +

= + + +

= + + + +

= + + +3 4

5 0 6 0

5

7 0

2 3 4

1 1 0 2 0 3 0 4 0 5 0

5 6

6 0 7 0

2 3 41

1 2 0 3 0 4 0 5 0

5 6

6 0 7 0

( ) ( )3! 4!

( )5!

( ) ( ) ( ) ( ) ( ) ( )2 3! 4!

( ) ( )5! 6!

( ) ( ) ( ) ( ) ( )2 3! 4!

( ) ( )5! 6!

nc

δωβ λ β λ

δω β λ

δω δω δωβ λ β λ δω β λ β λ β λ β λ

δω δωβ λ β λ

δω δω δωβ λ δω β λ β λ β λ β λ

δω δωβ λ β λ

+

+

= + + + +

+ +

= + + + +

+ +

where 0

2 2π πδωλ λ

= −

.

The parameters dispersionS0 and dispersionOffset are not used. This model should only be used for problems with a very large bandwidth in which a great deal of information about the fiber dispersion profile is known.

Nonlinearity The general third-order nonlinear polarization response of the fiber can be written

( ) ( ) ( ) ( ) ( )3 20 d

t

P t E t t g t t E tε χ−∞

′ ′ ′= −∫


(14)

where ( )3χ is the (scalar) third-order susceptibility tensor. The response function may be expressed as

( ) ( ) ( ) ( )1 R Rg t f t f h tδ= − +

(15)

so that the nonlinear polarization is composed of an instantaneous part and a delayed response part [2-6] as follows

( ) ( ) ( ) ( ) ( ) ( ) ( )3 20 1 d .

t

R RP t f E t f t h t t E t E tε χ−∞

′ ′ ′= − + −

∫

(16)

The fraction [0,1]Rf ∈ determines the fraction of the nonlinearity assumed to be non-instantaneous. The instantaneous response gives rise to the nonlinear refractive index – the familiar effects of self-phase modulation, cross-phase modulation and four wave mixing. The delayed response, which accounts for Raman effects, results in both additional phase effects and the transfer of energy to longer wavelengths, allowing Raman amplification and interesting effects such as the soliton self-frequency shift [6]. The fiber model permits simulation of all these nonlinear effects.

It may be shown that for the expansion (7), the nonlinear terms in Eqs. (9) have the form

( )

( ) ( ) ( )

( )

( ) ( ) ( )

2 2 2SPM XPM XPM

2 *XPM

2 2 2SPM XPM XPM

2 *XPM

2; 23

e3

2; 23

e3

x x y x x y xi j j i i i j i j i

j i j

y x j x xi i i i j

y x y y y x yi j j i i i j i j i

j i j

x y j y yi i i i j

N A A j A A A A

j A A R A

N A A j A A A A

j A A R A

δ

δ

γ γ γ

γ

γ γ γ

γ

≠

≠

−

= + +

+ +

= + +

+ +

∑ ∑

∑ ∑

K K

K

K K

K

(17)

The final terms ( ), ;x y x yi j jR A AK K describe the delayed Raman response of the ( )3χ nonlinearity and

are discussed below. The preceding terms describe self phase modulation and cross phase modulation due to the instantaneous part of the nonlinear response. The nonlinear coefficients are given by

( )( )2

SPM XPM 1 ii i R

eff

nfcA

ωγ γ= = −

(18)

where Rf is the fraction of the nonlinearity assigned to the Raman response (see below), and ( ) ( )22 3

03 /(4 )Ln Z n χ= is termed the “nonlinear refractive index” with SI units of m2W- 1, where Ln is the

background refractive index and 0 0 0/Z µ ε= is the impedance of free space. Note from Eqs. (17), that for each channel there are two sets of cross-phase modulation terms corresponding to interactions between different channels and different polarizations. (For simulations in single-channel mode or with unpolarized fields, the appropriate terms are dropped.) An additional term with prefactor 1/3 allows coherent coupling


between opposite polarizations and depends on the phase mismatch parameter ( )x y zδ β β= − ∆ . For

simulations including PMD, these coherent terms are dropped due to phase randomization.

It is important to note while we have introduced separate nonlinear coefficients for SPM and XPM, their numerical values are equal and physically they are both always present. They are distinguished in the model in order that their effects may be studied separately if desired (see section Disabling Effects). For any physical model, both effects should be active.

Nonlinear Parameters For the instantaneous part of the nonlinearity, the only internal parameter required by the model is the coefficient γ . This may be specified in terms of the nonlinear refractive index and the effective area of the fiber as shown in Eq. (18) by setting the parameters n2, aEff, diameter and choosing nonlinearity_model=Constant. In this case, the nonlinear coefficient increases linearly with frequency according to Eq. (18) since the effective area of the fiber is assumed independent of frequency. Note that

aEff is a dimensionless number relating the actual effective area of the mode effA to the core diameter:

effA π= aEff (diameter/2)2 .

For wide-bandwidth simulations, it may be more accurate to take account of the frequency variation of the

effective area, as well as of the nonlinear response ( )3χ itself (though it has a relatively weak frequency dependence and is typically difficult to measure). This may be accomplished by setting nonlinearity_model=File and specifying the nonlinear coefficient γ directly as a function of frequency in a data file of filename nonlinearity_filename. While γ is a derived quantity, it is the most easily accessible parameter in experiments. The file format is discussed in the appendix. Note that even with a user-specified nonlinearity, a single value iγ is used for all frequency components within a single channel. Hence, this option is most appropriate for multi-channel simulations.

Raman Effects The most complicated part of the fiber model is the treatment of the delayed nonlinear or Raman response, which gives rise to Raman gain and intra-pulse effects such as the soliton self-frequency shift. Note that if the intention is to simulate Raman amplifiers with CW pump waves, the Bidirectional Nonlinear Fiber model described later should normally be used. That model provides a complete accounting of Rayleigh scattering and spontaneous emission effects which are not included here. All Raman effects in the present model may be disabled with the parameter raman_effects.

The Raman terms in the nonlinear expression of Eq. (17) take the form [4]

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( )

22

*

d d

d ,ki

t tx x x x xi k R i i i R i i k

k i

tj t tx x x

R i k i kk i

R A jf A t t h t t A t jf A t t h t t A t

jf A t t h t t A t A t e

γ γ

γ

≠−∞ −∞

′− Ω −

≠ −∞

′ ′ ′ ′ ′ ′= − + −

′ ′ ′ ′+ −

∑∫ ∫

∑ ∫

K

(19)

where ki k iω ωΩ = − with a similar equation for ( )y yi kR A K . Note that the model does not include

Raman coupling between orthogonal polarizations (effects which are both weak and poorly characterized experimentally). Users interested in such a feature should contact RSoft Design Group. The terms in Eq. (19) respectively represent intra-pulse Raman scattering (or “self-Raman scattering”) which can be considered as a time-delayed contribution to SPM; the “molecular” contribution to cross-phase modulation; and a coherent Raman coupling which gives rise to energy coupling between channels and thus to Raman


gain. In the model implementation, the convolution integrals in Eq. (19) are performed in the frequency domain for efficiency.

As we discussed above for the instantaneous nonlinear effects, it should be emphasized that the identification of these separate effects is only meaningful in the frequency-decomposition picture. For a single-channel optical signal, only the first term appears: if the input optical signal has one broadband component, it experiences a downward frequency shift; if it has several components at different frequencies, the same process transfers energy to lower frequencies but is typically interpreted as Raman amplification of longer wavelength signals by shorter wavelength signals.

In fact, the Raman response spectrum in silica fiber (see Fig. 3) spans a range of over 30 THz, and for typical channel line-widths in WDM systems, the intra-pulse effects described by the first two terms in Eq. (19) are negligible. They are primarily of interest for detailed simulation of ultra-short pulse propagation of analog signals. In the model, these effects are disabled by default, and may be activated with the parameters raman_self_interaction and include_xpm_mol. As indicated in the previous paragraph, however, the first term controlled by raman_self_interaction is required to account for Raman gain if several optical channels are simulated in single-channel mode, since the final gain term only provides coupling between different signals in multi-channel mode.

Raman Response spectrum

Frequency shift (THz)0 10 20 30 40 50

x 1 0 - 4

g a in (m -1 W -1 )

4 - 3 - 2 - 1 - 0 1 2 3 4 5 6 Im[G(Ω)]exp

Re[G(Ω)]exp

Im[G(Ω)]ana

Re[G(Ω)]ana

Figure 3: Analytic and default experimental Raman response functions. The Raman gain is represented by the imaginary parts.

Raman Gain Effects For most multi-channel mode simulations, the only important term in Eq. (19) is the final gain term, which in a WDM simulation accounts for effects such as Raman-induced signal-signal crosstalk and gain tilt. Since the Raman spectrum varies slowly on the scale of the channel line-width, the gain convolution integral can be simplified for pulses longer than a picosecond or so [4]. The optical fields are taken out of the integral to obtain

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

*gain

2

2

d

d

ki

ki

ti t t

i R i k i kk

ti t t

R i k ik

R i k i kik

R jf A t t h t t A t A t e

jf A t A t t h t t e

jf A t A t h

γ

γ

γ

′− Ω −

−∞

′− Ω −

−∞

′ ′ ′ ′= −

′ ′= −

= Ω

∑ ∫

∑ ∫

∑ %

(20)


where ( ) ( ) ( )h h jh′ ′′Ω = Ω + Ω% % % is the Fourier transform of the response function. Finally, if we neglect

the real part ( )h′ Ω% describing the phase response we find

( )

( )

( )

2gain

2

2

2

i R i ki k ik

R i ki k ik

Rik ki

k ik

R jf j h A A

f h A A

gA A

γ

γ

′′= Ω

′′= −Ω

Ω=

∑

∑

∑

%

%

(21)

where we have defined the well-known Raman gain coefficient [1,4,5]

( ) ( )2Rik ki R i kig f hγ ′′Ω = −Ω%

(22)

The expression for the gain coupling in Eq. (20) is the default model used in the fiber simulation. The parameter raman_sigsig_interaction allows the signal interaction to be described by Eq. (20) (Simple), by the full expression in Eq. (19) (Full) or to be disabled altogether (Off).

Note that due to the wavelength dependence of iγ , if a channel at frequency kω induces gain on a channel

at iω with strength ( )Rik kig Ω , the Raman-induced loss at kω is related to the gain coefficient by

( ) ( ).R Rkki ik ik ik

i

g gωω

Ω = − Ω

(23)

This relation is required since a Raman scattering event conserves the number of photons, not the total optical energy. For each Stokes scattering event (down-shifting of frequency), the energy difference between the pump and Stokes photons is lost to the vibrational modes of the fiber.

Response Function and Gain Spectrum

The model allows either the response function ( )h t or gain spectrum ( )Rik kig Ω to be specified in several

ways through the parameter raman_profile. With the Analytic setting, the response function is given an assumed analytic profile

( ) ( ) ( )2 2

1 21 22

1 2

sin / exp / .h t t tσ σ σ σσ σ

+= −

(24)

Taking the values 151 12.2e sσ −= and 15

2 32.0e sσ −= , (set by the parameters raman_analytic_sig1 and raman_analytic_sig2), Eq. (24) predicts a gain spectrum in broad agreement with measurements on silica fibers (see Fig. 3). The strength of the Raman interaction can then be specified in two ways. If raman_strength_model is set to Fractional, the strength is given exactly according to Eq. (19) with the

delayed-response fraction Rf set with the parameter raman_response_fraction. The commonly accepted


value is 0.18Rf = [3]. If raman_strength_model=Absolute, the response fraction Rf is ignored and

the user instead specifies the maximum value of the gain spectrum ( )Rg Ω in a manner explained shortly.

In detail, the gain profile in silica fibers is considerably more complicated than the profile obtained from Eq. (24) and it is usually better to use an experimentally determined gain profile for ( )Rg Ω . This is achieved by setting gain_profile to either Default, in which case a standard literature profile is assumed [3] or to File, in which the profile is specified in a user file of filename raman_profile_filename. Figure. 3 compares the analytic profile with the industry-standard default experimental profile [3] which is used in Default mode. The file format RamanGainFormat1 for the user-specified spectrum is explained in the

appendix. The user specifies a normalized profile ( )2G πν Ω∆ =% with a peak value of 1.0. This accounts

for the gain profile, but as seen earlier, the frequency response function is complex-valued, with the imaginary part representing the gain. In both the Default and File modes, the model uses symmetry and causality requirements [3] to obtain the real part of the response function from the specified gain spectrum

G% , so that the complete function ( )h ν% is known.

The Fractional mode for the Raman strength may also be used with Default and File forms for the gain spectrum. More commonly, however, experiments provide the peak gain, rather than the fractional response

Rf and the Raman strength should be set with raman_strength_model=Absolute. In this case, the gain experienced by channel I due to channel k is given by

( ) ( ) ( ) ( )maxkiR k

ik ki ref keff ref

Gg G

Aν νν ν ν

ν∆

∆ = Γ%

(25)

In this description, it is assumed that a gain profile has been measured using a reference pump wavelength

/ref refcλ ν= . The peak gain ( )maxrefG ν

for that pump wavelength and the reference wavelength itself are set using the parameters raman_max_gain and raman_pump_ref_lambda. The user should beware that in literature sources, the gain is commonly expressed either in terms of

( ) ( ) ( )maxrefG G Gν ν ν∆ = ∆%

with units of m/W, or as ( )Rg ν∆ with units of m- 1W- 1, according to

whether the description is in terms of optical intensity or optical power. Users must enter the value for ( )max

refG ν in m/W for raman_max_gain. The default values are literature standards for silica fiber: raman_max_gain=0.98e- 13 m/W and raman_pump_ref_lambda=1.0e- 6 m.

Equation (25) indicates that by default, the gain coefficient is inversely proportional to wavelength. This is ultimately a result of the frequency dependence of the nonlinear coefficient [see Eq. (18)]. In experiments, however, it is sometimes found that the gain coefficient is not strictly proportional to frequency, due largely to the frequency dependence of the effective area. Typically, the gain at short pump wavelengths is higher than would otherwise be expected [7]. The additional factor ( )kνΓ is a dimensionless scale factor of order unity that can be included by the user. By default it is always unity, but the user may adjust it to specify a departure from the standard linear dependence on the pump frequency. This feature is activated by setting raman_pump_scaling=File and providing a scaling function in a file of filename raman_pump_scaling_filename as described in the appendix. Equation (25) actually assumes that

0ki k iν ν ν∆ = − > , so that the channel at iν experiences gain at the expense of kν For the reverse

situation 0kiν∆ < , Eq. (23) is replaced by

( ) ( )R Ri

ik ki ik kik

g gνν νν

∆ = − ∆


(26)

The additional frequency ratio was explained after Eq. (23).

Normalization Details To handle the two quite different approaches to specifying the strength of the Raman interaction some

additional rules are imposed. As stated earlier the response fraction Rf is only significant if raman_strength_model=Fractional. Moreover, the molecular contribution to XPM, described by the second term in Eq. (19) is only included in Fractional mode. In most cases, the Raman interaction should be specified using the Default or File-based modes with the Absolute strength configuration. We recommend using the Fractional mode largely for studies of analog ultra-short propagation where the details of the molecular XPM are required or where there is an interest in the effects of varying the

response fraction Rf . It is possible to combine a fractional Raman response with a user-specified nonlinear profile γ .

Stimulated Brillouin Scattering With the model parameter include_sbs set to Yes, Stimulated Brillouin Scattering (SBS) [1],[8]-[12] is included in the fiber model as an additional attenuation on the fiber inputs, and is calculated separately from the primary fiber model equations. In other words, before any other effects are taken into account, each optical signal or pump that is input to the fiber has its power reduced by an amount equal to that lost to SBS in the absence of any other phenomena. As part of this calculation, each optical input is modeled as a CW signal with a linewidth that is specified by the user, or calculated from the input’s optical spectrum. Furthermore, the model neglects the following:

• effects due to multiple channels at the same wavelength

• dynamic SBS effects

• interaction between SBS and other nonlinear phenomena, including Raman effects

At the end of a simulation, the fiber model produces plots of the input powers before and after the effects of SBS have been taken into account.

For each input to the fiber, loss from SBS is determined by solving the following differential equations for the pump power (i.e., that of the signal input) and the resulting Stokes wave power:

( ) ( )0

dpf

pp B s p p

II C I f I

zα ξ ξ ξ

∂ ′= − ⋅ − − ⋅ ∂ ∫

(27)

( ) ( )ss B s p s B p

I I C f I I h f C f Iz

α′∂ ′ ′= + ⋅ − ∆ ⋅ − ⋅ ⋅ ∆ ⋅

∂

(28)

where Ip is the pump power at a carrier frequency fp, Is′ is the resulting Stokes wave power density, fs is the Stokes frequency, ∆f = fp – fs, h is Planck’s constant, α is the fiber loss coefficient, and CB(∆f) is the Brillouin gain coefficient defined as

( ) ( )BB

pol eff

g fC f

k A∆

∆ =


(29)

where kpol accounts for random relative polarization between the pump and Stokes wave (typically 1.5 in conventional fibers), and Aeff is the fiber effective area. gB(∆f) accounts for the Lorentzian Brillouin gain shape:

( )( )

2

7 2 212

2320

2 2

2

B

p BB

A B B p BB

fn p f fg f

c V f f f f f f

πρ

∆ ⋅ ⋅ ⋅ ⋅ ∆ ∆ = ⋅ ⋅

⋅ ⋅ ⋅ ∆ ∆ + ∆ ∆ + ∆ −

(30)

where n is the fiber core’s refractive index, p12 is the elasto-optic coefficient, c is the vacuum speed of light, ρo is the material density, VA is the fiber’s acoustic-wave velocity, ∆fB is the Brillouin linewidth, ∆fp is the pump linewidth, and fB is the peak frequency of the Brillouin gain.

Assuming that the SBS gain is constant over some equivalent bandwidth around the peak Brillouin gain frequency, and dealing only with the total Stokes wave power Is, we have simplified the differential equations:

,maxp

p B s p

II C I I

zα

∂= − ⋅ − ⋅

∂

(31)

,max , ,maxs

s B s p s eq SBS B pI I C I I h f B C Iz

α∂ = + ⋅ − ⋅ − ⋅ ⋅ ⋅ ⋅∂

(32)

where 2

,max 2pB B

Bpol eff B p r

fg fCk A f f f

∆= ⋅ ⋅∆ + ∆

(33)

and

, arctan 22 2

B Beq SBS

B

f fBf

π ∆= ⋅ + ∆

(34)

With these equations, the user need only provide values for the following SBS-related parameters:

• gB (g_B), a nominal Brillouin gain value in m/W

• fr (gain_ref_freq), the frequency in THz at which gB is specified

• kpol (k_pol)

• fB (f_B) in GHz

• ∆fB (Delta_fB) in MHz

The pump linewidth ∆fp may either be calculated from the input signal itself, or specified by the user via the model parameter channel_linewidth (in MHz) when force_linewidth is set to Yes.


PMD Due to unavoidable irregularities in fabrication, nominally circular-core fibers actually exhibit weak birefringence with polarization axes that vary randomly along the fiber with a correlation length

corrL (pmd_corlen). As a result, the polarization state of an incoming signal experiences a random walk on the Poincare sphere as it propagates along the fiber. This very complicated process gives rise to the phenomenon of polarization mode dispersion (PMD), in which a pulse becomes distorted due to the different transit times of the orthogonal polarization states.

Rather than explicitly include the micro-variation of the polarization evolution which would be prohibitively expensive computationally, the fiber model uses the “Coarse-Step” method to simulate PMD [13, 14]. In this approach, the signal polarization is changed randomly after each propagation step in such a way that the average dynamics recover the true PMD behavior.

The user may include PMD in the fiber simulation by setting the pmd_method parameter to either Coarse_Step or Modified_Coarse_Step.

The degree of PMD is controlled by the correlation length pmd_corlen, and the polarization mode dispersion coefficient PD (pmd_coef). The Modified_Coarse_Step method implements a more recent version of the algorithm that is believed to give more accurate results and should normally be chosen in preference to the Coarse_Step option [15].

A simulation with PMD clearly requires both polarization members xE and yE of the optical signal. Incoming channels with only a single polarization member are automatically converted to dual polarization signals before the propagation commences.

The user may control the seeding of the random number generator used to implement PMD with the parameter pmd_seed. This satisfies –1e8<pmd_seed<1 with the following behavior:

• pmd_seed<0 The generator is seeded with the actual value of pmd_seed on every run of the simulation. This is useful for obtaining repeatable results.

• pmd_seed=0 The generator is seeded with an integer hashed from the string value of the component’s name. This is a convenient way to obtain repeatable results on subsequent runs within a single fiber, but different PMD sequences from in a series of fiber components which have different names. This would be appropriate in an amplifier chain, for example.

• pmd_seed=1 The generator is seeded with a random number obtained from the system clock. This is essentially unrepeatable.

Noise The fiber model can treat noise in one of two ways. If the incoming signal contains a separate ASE component, the ASE spectrum is attenuated according to the frequency-dependent loss coefficient. No nonlinear or dispersive effects are applied to the noise. This is the appropriate technique if it desired to perform semi-analytic calculations of the bit-error rate at the end of the link. In some problems however, interesting propagation effects that may impact bit error rates occur due to the nonlinear interaction between the noise and signal [16]. For these effects to be captured, the signal and noise must be treated as a single entity. This is accomplished using the Noise Adder model, described elsewhere in this chapter, which has an option to transfer the noise representation from the separate power spectrum description into a stochastic component of the optical signal. Since there is now no distinction between the representation of noise and signal, all nonlinear effects are correctly accounted for. A disadvantage of this approach is that it precludes the semi-analytic calculation of the BER The BER must instead be calculated using a Monte-


Carlo technique. If nonlinear noise effects might be important, the user must experiment to determine which mode of simulation is the most appropriate.

Skew A constant skew may be added to the signal with the use of the skew parameter. This essentially adds the specified time skew to the signal time axis. This is useful if a parallel fiber ribbon is being modeled, and the skew between the different fibers in the ribbon is desired to be accounted for. For such an application, using a value of zero and a nonzero standard deviation in combination with a statistical simulation is often useful.

Interior Property Maps The fiber model is able to monitor several properties along the length of fiber. These are the total power, the pulse width and pulse location. (The latter two are only well-defined for single-pulse simulations.) In conjuction with the Interior Property Map model, plots can be produced showing the evolution of these properties along a series of fibers. The various fibers and the Map communicate through a common file whose name is specified with the parameter physpropFilename the number of measurements is determined by physpropSteps. Further details of this facility are described in the documentation of the Interior Property Map.

Simulation Features and Settings We now discuss the strictly numerical parameters of the model.

Step Size The step size dz may be determined automatically by the model, or set directly by the user. If the user desires to set the step size manually, the zStepSize parameter should be set to a positive value. A negative value of this parameter indicates that the step size should be automatically computed.

The automatically computed step size is found by evaluating the minimum of the characteristic lengths of second order dispersion,third order dispersion, nonlinearityand Raman gain. The actual stepsize is a user-specified fraction (zStepFac) of the minimum characteristic length. If PMD is active, a minimum of 15 steps will be taken, regardless of the characteristic lengths in order to obtain randomization of the polarization. Reasonable values for zStepFac are 0.01 – 0.1.

The current status, including the step size and the distance simulated so far, may be shown in the simulation window and saved to the log file if the showstatus option is set to Yes.

Some recent work [17] has shown that under certain circumstances, the use of a constant step-size may generate spurious four-wave mixing signals in WDMsimulations unless the step-size is extremely small. If the dominant characteristic length is the dispersion length, the automatically calculated step size is constant. Setting the parameter decreasingStepSize=Yes, causes the model to reduce the step size smoothly as the propagation progresses, which helps to alleviate this problem.

Optimizations One of several optimization levels may be chosen by the user to speed up the simulation. The default optimization level is 3. In some simulations, however, if the optimization level is reduced to 1, less artifacts due to the finite nature of the simulation occur.

Disabling effects All physical effects – dispersion, nonlinearity, PMD in the fiber model may be individually disabled. This might be done for one of several reasons –


To increase efficiency by not simulating effects which are known to be negligible. For example in a low power simulation with an optical bandwidth of a few GigaHertz, the impact of the Raman nonlinearity is negligible.

To increase accuracy by not simulating inappropriate effects. For example, in a high power simulation over a few meters of fiber (less than the fiber correlation length), PMD plays no role and should be turned off. Leaving PMD active would introduce spurious polarization scrambling.

To isolate the effects of different physical phenomena. For example, SPM, XPM and dispersion may all be individually turned off so that the user may determine their individual roles in the simulation. It must be stressed that the results of such a simulation are non-physical – in reality, XPM can never occur without SPM for instance – and the results must always be checked against a full simulation with all relevant effects enabled. An example of this technique might be XPM-induced compression of a weak probe pulse by an accompanying strong pump pulse [18]. If the probe pulse has sufficiently weak intensity, SPM will not affect its propagation, but XPM from the pump pulse induces the compression. This physical expectation could be checked by disabling SPM. However, there is a negligible impact on the efficiency of the simulation, and so SPM should never be disabled merely to enhance simulation speed.

By default, PMD and the Raman self-interaction are disabled; all other effects are enabled. For long distance standard fiber propagation, some form of PMD should normally be enabled, while the Raman self-interaction is usually important only for high-power and large bandwidth signals.

All the following parameters allow disabling of effects. Their actions are explained in the Parameter Descriptions section below.

Pmd_method, raman_effects, raman_self_interaction, raman_sigsig_interaction, include_spm, include_xpm_el, include_xpm_mol, include_dispersion.

Attenuation is trivially disabled by choosing a constant loss parameter of 0.

Test functions The Test button provides previews of a number of frequency dependent quantities including loss, dispersion and Raman responses. The frequency_units parameter determines the units used in the test function plots. The quantity displayed is selected through the test_function parameter with the following options:

• loss The current loss profile set by loss_filename is displayed. There is no display if loss_model is set to Constant.

• dispersion The values of 1n , 2β and 3β and the dispersion parameter D are plotted as a function of frequency according to the settings of dispersionModel and the various dispersion parameters.

• nonlinearity The current nonlinear coefficient profile γ set by nonlinearity_filename is displayed. There is no display if nonlinearity_model is set to Constant.

• raman_gain_profile The current Raman response function is displayed according to the various Raman settings. The gain profile assumes a pump wavelength set by test_raman_pump_lambda and takes account of the values of the parameters raman_response_model, raman_profile, raman_max_gain, raman_response_fraction, raman_pump_ref_lambda, raman_analytic_tau1, raman_analytic_tau2 and raman_pump_scaling.


• raman_pump_scaling

The current Raman pump scaling profile ( )νΓ is displayed as a function of frequency.

There is no display if raman_pump_scaling is set to Default.

References [1] G. P. Agrawal, Nonlinear Fiber Optics, 2nd Edition (Academic Press, San Diego, CA, 1995).

[2] K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron 25, 2665 (1989).

[3] R. H. Stolen, J. P. Gordon, W. J. Tomlinson and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159 (1989).

[4] C. Healey III and G. P. Agrawal, “Unified description of ultrafast stimulated Raman scattering in optical fibers,” J. Opt. Soc. Am. B 13, 2170 (1996).

[5] C. Healey III and G. P. Agrawal, “Noise characteristics and statistics of picosecond Stokes pulses generated in optical fibers through stimulated Raman scattering,” IEEE J. Quantum Electron 31, 2058 (1995).

[6] J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11 662 (1986).

[7] A. Berntson, S. Popov, E. Vanin, G. Jacobsen and Jorgen Karlsson, “Polarisation dependence and gain tilt of Raman amplifiers for WDM systems,” paper MI2, Optical Fiber Communication Conference, Anaheim, CA 2001.

[8] R. W. Boyd, Nonlinear Optics (Academic, New York, 1992). [9] L. Chen and X. Bao, “Analytical and numerical solutions for steady state Stimulated Brillouin scattering in a single-mode fiber,” Opt. Commun., 152, pp. 65–70, 1998.

[10] S. Rae, I. Bennion, and M. J. Carswell, “New numerical model of stimulated Brillouin scattering in optical fibers with nonuniformity,” Opt. Commun., 123, pp. 611–616, 1996.

[11] A. Djupsjöbacka, G. Jacobsen, and B. Tromborg, "Dynamic stimulated Brillouin scattering analysis," J. Lightwave Technol, 18, pp. 416-424, March 2000.

[12] R. W. Boyd, K. Rzazewski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A, 42, pp. 5514–5521, 1990.

[13] P. K. A Wai, C. R. Menyuk, and H. H. Chen, Opt. Lett. 16, 1231 (1991).

[14] S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and S. Bergano, J. Lightwave Technol. 10, 28 (1992).

[15] D. Marcuse, C.R. Menyuk, and P.K. A. Wai, J. Lightwave Technol. 15, 1735 (1997).

[16] R. Hui, D. Chowdhury, M. Newhouse, M. O’Sullivan and M. Poettcker, “Nonlinear amplification of noise in fibers with dispersion and its impact in optical amplified systems,” IEEE Photonics Technol. Lett. 9, 392 (1997).

[17] C. Francia, “Constant step-size analysis in numerical simulation for correct four-wave-mixing power evaluation in optical fiber transmission systems,” IEEE Photonics. Technol. Lett. 11, 69 (1999).

[18] J. E. Rothenberg, “Intrafiber visible pulse compression by cross-phase modulation in a birefringent optical fiber,” Opt Lett. 15, 495 (1990).


Properties



Parameter Values Name Type Default Range Units length double 1e3 [ 0, 1e32 ] m

zStepFac double 0.1 [ 0, 1 ] none

zStepSize double -1 [ -1, 1e10 ] m decreasingStepSize

enumerated No No, Yes

diameter double 8.2e-6 [ 0, 1e-2 ] m

aEff double 1.425 [ 0, 100 ] none

loss_model enumerated Constant Constant, File

loss double 0.25 [ 0, 1e32 ] dB/km loss_filename string

dispersionModel enumerated defined defined, defined2, custom, extrapolated

n1 double 1.4682 [ 0, 100 ] none

beta2 double 0.0 [ -1e32, 1e32 ] s^2/m beta3 double 0.0 [ -1e32, 1e32 ] s^3/m

dispersionLambda0

double 1.312e-6 [ -1e32, 1e32 ] m

dispersionS0 double 0.09e3 [ -1e32, 1e32 ] s/m^3

dispersionOffset double 0 [ -1e32, 1e32 ] s/m^2

nonlinearity_model

enumerated Constant Constant, File

n2 double 2.6e-20 [ 0, 100 ] m^2/W nonlinearity_filename

string

pmd_method enumerated None None, Coarse_Step, Modified_Coarse_Step

pmd_coef double 3.16e-15 [ 0, 1e32 ] s/m^0.5 pmd_corlen double 10 [ 1e-10, 1e32 ] m

pmd_seed integer 0 [ -1e8, 1 ] none

raman_effects enumerated On Off, On

raman_strength_model

enumerated Absolute Absolute, Fractional

raman_profile enumerated Default Default, Analytic, File

raman_profile_file string


name raman_max_gain double 0.98e-13 [ 0, 1e32 ] m/W

raman_response_fraction

double 0.18 [ 0, 1 ] none

raman_pump_ref_lambda

double 1.0e-6 [ 0, 1e32 ] m

raman_analytic_sig1

double 12.2e-15 [ 0, 1e32 ] s

raman_analytic_sig2

double 32.0e-15 [ 0, 1e32 ] s

raman_pump_scaling

enumerated Default Default, File

raman_pump_scaling_filename

string

raman_self_interaction

enumerated Off Off, On

raman_sigsig_interaction

enumerated Simple Off, Simple, Full

include_sbs enumerated No No, Yes g_B double 3e-11 [ -1e32, 1e32 ] m/W

k_pol double 1.5 [ 1, 2 ]

gain_ref_freq double 193.0 [ 1e-32, 1e32 ] THz

Delta_fB double 40.0 [ 0, 1e32 ] MHz f_B double 11.0 [ 1e-32, 1e32 ] GHz

force_linewidth enumerated No No, Yes

channel_linewidth double 100.0 [ 0, 1e32 ] MHz

Include_spm enumerated Yes No, Yes include_xpm_el enumerated Yes No, Yes

include_xpm_mol enumerated Yes No, Yes

include_dispersion

enumerated Yes No, Yes

skew double 0 [ -1e32, 1e32 ] s

optimizationlevel integer 3 [ 0, 3 ] none showstatus enumerated Yes No, Yes

logstatus enumerated No No, Yes

physpropFilename

string

physpropSteps integer 20 [ 2, 1e8 ]

frequency_units enumerated um nm, um, m, Hz, GHz, THz, cm^-1, m^-1, rad/s

test_raman_pump_lambda

double 1.4e-6 [ 0, 1e32 ] m

test_output enumerated raman_gain loss, dispersion, n2, raman_gain, raman_pump_scaling


Parameter Descriptions length Length of the fiber

zStepFac Scaling factor for propagation step-size zStepSize Specify user-defined step size or automatic calculation decreasingStepSize Toggle use of monotonically decreasing step-size diameter Core diameter aEff Effective mode area normalized by core area loss_model Select constant or file-based loss loss Constant fiber attenuation parameter loss_filename Data file for attenuation profile dispersionModel Select dispersion model n1 Group index beta2 Second-order group velocity dispersion beta3 Third-order group velocity dispersion dispersionLambda0 Dispersion reference wavelength dispersionS0 Dispersion slope at dispersionLambda0 dispersionOffset Dispersion offset at dispersionLambda0 nonlinearity_model Select nonlinearity model n2 Constant nonlinear refractive index value nonlinearity_filename Data file for nonlinearity coefficient profile pmd_method Select PMD model if any pmd_coef PMD coefficient D pmd_corlen PMD correlation length corrL

pmd_seed Select randomization mode of PMD raman_effects Toggle all Raman effects raman_strength_model Select Raman strength model raman_profile Select Raman response or gain profile model raman_profile_filename Data file for Raman gain profile raman_max_gain Peak Raman gain at reference pump wavelength raman_response_fraction Fraction of nonlinearity that is non-instantaneous raman_pump_ref_lambda Reference pump wavelength for Raman gain raman_analytic_sig1 Coefficient for analytic Raman response function raman_analytic_sig2 Coefficient for analytic Raman response function raman_pump_scaling Toggle additional scaling of Raman gain raman_pump_scaling_filename Data file for user-specified Raman pump scaling raman_self_interaction Toggle intra-pulse Raman scattering raman_sigsig_interaction Select mode for inter-channel Raman interactions include_sbs Toggle stimulated Brillouin scattering g_B Nominal Brillouin gain value k_pol Random polarization factor


gain_ref_freq Reference frequency for g_B Delta_fB Brillouin linewidth f_B Brillouin gain peak frequency force_linewidth Toggle manual setting of channel linewidths for SBS model channel_linewidth Manual channel linewidth for SBS model include_spm Toggle inclusion of SPM include_xpm_el Toggle inclusion of electronic (normal) XPM include_xpm_mol Toggle inclusion of molecular (Raman) XPM include_dispersion Toggle inclusion of dispersion skew Time skew added to the output optical signal optimizationlevel Simulation optimization level showstatus Toggle display of propagation progress logstatus Toggle logging of propagation progress physpropFilename Filename for recording physical properties for interior maps physpropSteps Number of property measurements to make frequency_units Units for display of test functions test_raman_pump_lambda Pump wavelength for Raman gain profile test function test_output Selection of test function display


Appendix: File formats In the following descriptions, absolute frequencies denoted <freq_absolute> may be expressed in a number of units as determined by the value of the field <freq_absolute_units> which may be any one of [nm], [um], [m], [Hz], [MHz], [GHz], [THz], [cm^(-1)], [m^(-1)] or [rad/s].

Frequency shifts, denoted <freq_shift> may only be expressed in the units [Hz], [MHz], [GHz], [THz], [cm^(-1)], [m^(-1)] or [rad/s], as indicated by the value of the field <freq_shift_units>.

The frequency values in all files must be monotonically increasing or decreasing. The field <num_pts> specifies the number of data lines in the file.

Attenuation A user specified attenuation profile is selected by setting loss_method to File and providing a filename loss_filename. The loss values are interpreted in units of dB/km.

Out of range frequency values: If the loss is required at a frequency outside the range specified in the file, the value at the closer endpoint is used.

Format: FiberLossFormat1 <freq_absolute_units>

<num_pts>

<freq_absolute 1> <loss 1>




Example: FiberLossFormat1 [um]

5

1.40 0.25

1.45 0.21

1.50 0.19

0.17

1.60 0.21

Nonlinearity A user specified nonlinearity profile is selected by setting nonlinearity_method to File, and providing a filename nonlinearity_filename. The file contains values for γ in units of W- 1m- 1.

Out of range frequency values: If the nonlinearity is required at a frequency outside the range specified in the file, the value at the closer endpoint is used.


Format: FiberN2Format1 <freq_absolute_units>

<num_pts>

<freq_absolute 1> <gamma 1>




Example: FiberN2Format1 [um]

5

1.40 0.0015

1.45 0.0014

1.50 0.0012

0.001

1.60 0.0009

Raman gain profile

Selection: A user specified Raman gain profile is selected by setting raman_profile=File, and providing a filename raman_profile_filename. OptSim does not at this time support Raman gain between orthogonal polarizations. Users requiring such a feature should contact RSoft Design Group.

The gain profile is dimensionless and should normally have a maximum value of 1.0.

Out of range frequency values: The gain is assumed to vanish at frequencies outside the specified range.

Format: RamanGainFormat1 <freq_shift_units>

<num_pts>

<freq_shift 1> <parallel gain 1>




Example: RamanGainFormat1 [cm^(-1)]

200

0 0.00 0.00

5 0.04 0.01

10 0.20 0.05

20 0.30 0.10


25 0.40 0.12

…

Raman pump scaling

Selection: A user specified Raman pump scaling profile is selected by setting raman_pump_scaling=File, and providing a filename raman_pump_scaling_filename.

The Γ profile is dimensionless and should normally take values of order 1.0.

Out of range frequency values: For frequencies outside the file range, the value at the closer end point is used.

Format: RamanPumpScalingFormat1 <freq_absolute_units>

<num_pts>

<freq_absolute 1> <Gamma value 1>




Example: RamanPumpScalingFormat1 [um]

4

1.2 1.06

1.3 1.02

1.4 1.0

1.5 0.95


Bi-directional Nonlinear Fiber (Raman Amplifier)

This model provides a detailed implementation of a bidirectional fiber with all dispersive, nonlinear, PMD and Raman effects. CW pump lasers may be attached traveling both forwards and backwards to simulate Raman amplification of a forward and backward traveling signals. Spontaneous emission and Rayleigh back-scatter effects are fully described. The problem is solved in two stages: a bi-directional power solution that solves for the pump and noise distributions, followed by an extended implementation of the standard Nonlinear Fiber Model that adds the influence of the pump waves to the dispersive and nonlinear effects experienced by the signals. For strictly uni-directional problems with no pumps and backward traveling signal or noise, the model is identical to the Nonlinear Fiber Model. The documentation for that model should be read before the following.

All the parameters of the standard Nonlinear Fiber Model are also parameters of the present Bidirectional Fiber Model. The shared parameters are documented in full in the discussion of the Nonlinear Fiber Model and their explanations are not repeated here. The Parameter Descriptions section below indicates all parameters that are common to both models.

Introduction A number of fiber propagation applications require the ability to solve for fields traveling in both directions. The most obvious and important is the simulation of Raman amplification of signals by CW pumps. Raman amplification is a bidirectional problem, firstly because the pumps can be launched from either ends, but also because the noise component to the field travels in both directions – a single pump produces spontaneous emission in both directions, and Rayleigh back-scattering couples fields in both directions. These effects are summarized in Fig. 1. The forward traveling (red) signal is amplified by pumps (blue) traveling in both directions. Spontaneous emission is generated by the pumps (green crosses) and back-scattered. The signal also contributes directly to the noise through double Rayleigh back-scattering indicated by the red loop.

Figure 1: Bidirectional fiber effects in Raman amplification. Incoming signal (red) is amplified by co-propagating and counter-propagating pumps (blue). Deleterious effects include spontaneous emission (green), Rayleigh backscatter of

the spontaneous emission (green) and double-Rayleigh backscatter of the signal (red).

In other applications, bidirectionality arises directly due to signals being launched from both ends. We show below that the only significant interaction between the oppositely-directed signals is the Raman coupling. Therefore, the problem can be solved using the same approach as for Raman amplification with one signal replaced by a CW wave of the equivalent average power. Thus this documentation is geared towards the Raman amplification application but applies generally to all bidirectional problems.


Background

Origin of Raman effects Raman effects may be viewed phenomenologically as a result of a delayed third-order nonlinearity, in contrast to effects such as self-phase modulation, which arise from the instantaneous nonlinearity (see documentation for Nonlinear Fiber Model). Microscopically, a Raman scattering event occurs when a pump photon of frequency pω scatters off a phonon (vibrational quantum). A transfer of energy and

momentum between the particles converts the incoming photon to a new frequency s pω ω= + Ω , with

pωΩ << . If 0Ω < , the new photon is referred to as a “Stokes” wave. The opposite case of a positive frequency shift produces an “anti-Stokes” wave. This is very much less likely and can be largely neglected in fiber systems. Raman gain occurs when the Stokes shift corresponds to the separation between a pump and signal wave, so that the incoming signal is amplified. Stokes photons may also appear spontaneously at frequencies with no preexisting signal. Subsequently amplified they contribute to the ASE noise of the system.

Model Approach It is essentially impossible to perform a complete fully-resolved bidirectional simulation over typical distances of tens to hundreds of kilometers. Since boundary conditions exist at both ends of the fiber rather than just the front end, the initial-value problem of the Nonlinear Fiber Model is replaced by a two-point boundary-value problem that must be solved iteratively. On the other hand, modeling coherent effects such as dispersion and self-phase modulation requires a spatial resolution at the sub-millimeter scale. Satisfying both these requirements simultaneously would be numerically intractable in terms of stability, convergence and computational time.

Fortunately we can construct an alternative approach, which is numerically efficient and captures all important effects. The problem is divided into two distinct stages:

The first stage, solved bidirectionally and iteratively, provides the distribution of optical power (including pumps, signals and noise) at coarse spatial resolution – typically tens of meters or more. All dispersive and instantaneous (non-Raman) nonlinear effects are neglected in this stage. Only Raman gain, attenuation, spontaneous emission and Rayleigh scattering are included.

This second stage, is an extension of the standard Nonlinear Fiber Model that incorporates the information about the pump and noise power distribution obtained in the first stage. In this way both the forward and backward traveling coherent optical signal experience the appropriate amount of Raman gain from both forward and backward pumps, and yet still suffers all coherent effects including dispersion, PMD, and nonlinearities (including Raman effects due to intra-pulse or signal-signal interactions as distinct from the CW pump-induced gain). Raman effects due to signal-signal interactions between counter-propagating signals are not currently accounted for in the model, however, the Raman signal-signal interactions between co-propagating signals are included.

Justification This approach clearly neglects certain effects. In most cases these are of little or no consequence:

• No nonlinear effects between signals traveling in opposite directions are included.

So, if the backward traveling wave contains modulated information, rather than just CW pumps, XPM between these oppositely-directed signals is ignored. However, from the point of view of a pulse traveling forwards, intensity variations in the backward-traveling wave walk-through the forward signal essentially at the speed of light – at a rate much faster than any other process in the system. Hence, the XPM interactions between opposite waves are smeared out so rapidly that each signal sees the reverse one as an effective CW wave. Since the action of XPM by a CW wave is merely to impose a uniform phase shift of no significance, it may be ignored. Four wave mixing (FWM) between counter-propagating


signals is negligible due to the phase-mismatch. For some specially chosen signal frequencies, propagation directions, and fiber dispersion profile the phase matching may be fulfilled. These special cases are not considered.

• It is not possible to include nonlinear interactions between the signal and noise generated within the fiber.

In order to correctly account for bidirectional behavior of the noise – bidirectional emission and Rayleigh backscatter, the noise must be represented as a separate power spectrum, rather than as a stochastic component to the signals. Effects such as parametric gain of the noise can only be obtained in the stochastic representation. Generally, if ASE is a significant source of noise, variations due to nonlinear mixing represent a small perturbation. Pre-existing noise produced before entry into the model can of course be converted to the stochastic representation.

• The modeling of the pumps as idealized CW waves neglects the phenomenon of pump-mediated crosstalk.

In a WDM system, amplification of a pulse in one channel can locally deplete the pump wave, reducing the gain seen by other channels. This effect is not captured in the present model since the pump intensity varies only on macroscopic scales. The effect is only important for co-propagating pumping, since for counter-propagating pumps, the variation is averaged out. Pump-mediated crosstalk could be modeled in a unidirectional fashion using the standard Nonlinear Fiber Model (or this model if backward pumping is also present). The forward pump is launched as a genuine optical signal in the form of a very long though finite pulse from one of the laser sources and interacts with the other channels through signal-signal Raman interactions. In this case of course, it is not possible to include bidirectional noise effects. Users concerned about pump-mediated crosstalk would perform simulations in both ways and determine which source of noise has the dominant impact on bit error rates. A similar argument applies to the impact of pump RIN on the amplified signals.

• The model allows incoming signal and noise in both the forward and backward directions.

Model Operation

Input and output ports The Bidirectional Nonlinear Fiber icon has four input and two output nodes (see Fig. 2). The top input and output nodes function identically to the ports of the standard Nonlinear Fiber – the input node accepts an arbitrary set of optical signals, and the output node contains these signals after propagation through the fiber. The lower input and output node have a similar function, but for the backward propagating optical signals. The input nodes in the middle denote incoming CW pump waves and are interpreted in a different manner. The node on the west side of the icon represents pumps launched at the front of the fiber and co-propagating with the forward-traveling signal; the node on the east represents backward co-propagating pumps launched at the rear of the fiber. The pump inputs do not accept arbitrary optical signals. Their inputs must come from a series of CW Laser components whose signalType parameter is set to PowerValue (the default). Using the Optical MUX model, any number of CW Lasers may be connected to each input node. The restriction to CW Laser inputs serves as a reminder to the user that the pumps are completely specified by their wavelength, power and propagation direction.

The example topology in Fig. 2 should help to make this convention clear. A two-channel signal is generated by modulating the output of two CW Lasers with a random binary sequence, and is connected to the first node of the bidirectional fiber. A single CW Laser is connected to the second node representing a forward-traveling pump. A single-channel modulated signal is connected to the backward incoming signal node on the east side. Three additional CW Lasers are multiplexed (in multi-channel mode) and connected to the middle node on the east side to denote backward-going pumps. The output node on the east side holds the modified version of the signals that enter on the first input node. The output node on the west side


holds the modified version of the signal that enters on the backward signal input node. As explained below, in different modes of operation, any three of the four input nodes need not be connected.

Note finally, that no nodes are provided for the outgoing pumps, and at this stage it is not possible to continue the propagation of the pumps beyond the fiber component.

Figure 2: Input node convention for bidirectional fiber model.

Modes of operation Unlike most components in OptSim, the Bidirectional Fiber can be operated in a number of distinct modes. In addition to the complete two-stage simulation as a part of a full link topology, the component can also be used in a quasi-“stand-alone” fashion in order to optimize gain characteristics such as flatness and noise figure. Which of the four modes of operation is used is controlled by the parameter simulation_mode. In three of these modes, internal plots are generated that may be accessed by double-clicking the component icon. These are discussed below in the section Internal Plots. The four modes with the corresponding setting of simulation_mode are as follows:

• Input only (InputOnly)

The preparatory stage of the model is run analyzing the incoming optical signals. Plots are generated displaying the spectral density and power per channel of the incoming signals, pumps and noise field. No signals are emitted at the output nodes. This is used as a quick check that the incoming pumps and signals have the intended wavelengths and powers.

• Power solution only (PowerOnly)

The first stage of the model is solved, obtaining the power distribution with distance of the signals, pumps and noise fields. A large number of internal plots are generated describing this solution. No signals are emitted at the output nodes. This mode is intended for “stand-alone” use so that the gain flatness, range, noise figure and other parameters may be optimized before analyzing the details of true pulsed coherent signal propagation through the fiber.


• Coherent solution only (CoherentOnly)

This mode is similar to the standard Nonlinear Fiber Model with the additional support of backward propagating signals and noise. It is provided as a convenience to avoid having to switch components if bi-directional Raman effects are not desired at some point. As for the standard fiber, there must be at least one incoming optical signal at one of the input signal nodes, and the field at the end of the fiber appears at the corresponding output node. Any inputs at the pump nodes are ignored.

• Power + Coherent solution (Full)

The complete two-stage bidirectional simulation is performed. The power solution is obtained and the resulting pump distribution is used as an additional source of information for the coherent nonlinear fiber model. The resulting optical signal appears at the output nodes.

Signal Inputs In simulations in PowerOnly mode, the user may be more interested in the total gain spectrum and noise profile, as opposed to the detailed response of a single signal channel. Rather than attaching an incoming signal to the incoming optical signal nodes, a flat spectral density representing the average power of many WDM channels may be added by setting signal_spectrum=FixedRange and selecting a spectral density with sig_fixed_spec_dens and a wavelength range with sig_fixed_lambda_lo and sig_fixed_lambda_hi. This arrangement is useful for determining properties such as gain flatness and noise figure over a wide bandwidth. Once the pump powers and wavelengths have been chosen to achieve the desired properties, the simulation can be run in Full mode to determine the detailed effects on a single channel.

It is possible to use the FixedRange spectrum even in Full mode. In this case, the pump power distribution is found using the fixed range spectrum, while the coherent solution acts on the actual signal input at the incoming optical signal nodes. In this way, gain saturation due to the existence of a large number of WDM channels may be modeled without explicitly including all the channels.

User-Specified Profiles A number of physical quantities may be specified in user data files, notably the Raman gain profile and the loss profile. The formats and parameter settings required for these features are all explained in the Nonlinear Fiber Model documentation to which the reader is referred. We emphasize here, however, that in modeling Raman amplifiers it can be especially important to use measured profiles where available rather than choosing constant values. Typically, the pumps and signals can together span 100-150 nm or more and the frequency-dependence of the attenuation is critical to obtaining accurate results. Similarly, if the pumps are widely spaced, the use of the additional Γ scaling substantially improves accuracy if the correct scaling is available.

Power Solution Equations and Calculation The stage-one bidirectional solution solves for the power distribution in the fiber. In order to pass the ASE noise spectrum to subsequent models and to calculate local quantities such as the noise figure, the noise power is modeled as a distinct component from the signal and pumps. The model supports an arbitrary number of pumps from both ends, an arbitrary number of signals from both ends, incoming ASE noise spectra from both ends, temperature-dependent spontaneous emission, and arbitrary-order Rayleigh scattering of signal, pump and noise components. In reality, Rayleigh scattering contributes to the noise in several ways. Spontaneous emission traveling backwards is back-scattered in to the forward ASE spectrum and vice-versa increasing ASE noise. On the other hand, forward and backward traveling signals are “double back-scattered” producing a kind of highly smeared-out “shadow” of the original signals. These two noise sources have somewhat different spectral characteristics. In this model, the two are considered to collectively contribute to the ASE spectrum and are interpreted as such by subsequent components.


Principal equations The model is an extension of a model of Kidorf et al. [1]. It consists of the following coupled equations for the spectral densities of four quantities: the forward signal and pumps ( )fS ν , backward signal and pumps

( )bS ν , forward noise ( )fN ν and backward noise ( )bN ν :

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

dd

d ,

d , 4 1 , ,

dd

d ,

d , 4 1 , ,

d

ff

sf b f b f

pf f b f b ph

bb

sf b f b b

pb f b f b ph

SS

zG S S N N S

G S S S N N h N T

SS

zG S S N N S

G S S S N N h N T

τ ν

τ ν

τ ν

τ ν

να ν

τ τ ν τ τ τ τ τ ν

τ ν ν τ ν τ τ τ τ τ ν τ

να ν

τ τ ν τ τ τ τ τ ν


>

<

>

<

= −

+ − + + +

− − + + + + + −

− = −

+ − + + +

− − + + + + + −

∫∫

∫∫

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

Nd

d , 2 1 ,

d , 4 1 , ,

dNd

d , 2 1

ff b b

sf b f b f ph

pf f b f b ph

bb f f

sf b f b b

N S Nz

G S S N N N h N T

G N S S N N h N T

N S Nz

G S S N N N h N

τ ν

τ ν

τ ν

να ν γ ν ν ν

τ τ ν τ τ τ τ τ ν ν ν τ


να ν γ ν ν ν

τ τ ν τ τ τ τ τ ν ν

>

<

>

= − + +

+ − + + + + + −

− − + + + + + −

− = − + +

+ − + + + + +

∫∫

∫ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

,

d , 4 1 , .

ph

pb f b f b ph

T

G N S S N N h N Tτ ν

ν τ

τ ν ν τ ν τ τ τ τ τ ν τ<

−

− − + + + + + − ∫

(1)

Here ( )α ν is the attenuation, ( )γ ν is the Rayleigh scattering coefficient (discussed below), h is

Planck’s constant and phN represents an enhancement of spontaneous emission associated with the phonon

spectrum of the fiber. The superscripts on the gain factors sG and pG indicate the terms for which the field on the left hand side is acting as a signal (s) (being amplified by higher frequencies) or as a pump (p) (being depleted by lower frequencies). They satisfy

)(|)(||)|,(

)(|)(||)|,(

ττν

νντνετνν

τν

ττνετντ

Γ−=−

Γ−=−

refref

p

refref

s

gG

gG

(2)


where the first argument to ,s pG denotes the pumping frequency, refg is the gain spectrum at the reference

frequency refν and ( )νΓ is an empirical scaling factor. Thus for ( ) 1νΓ = , sG is proportional to the pump frequency which is the traditional assumption. As discussed in the documentation for the Nonlinear Fiber Model, the frequency dependence of the fiber effective area and other factors can introduce departures from this scaling. It was explained there that this can be accounted for by specifying the

( )νΓ function using the raman_pump_scaling parameters. The additional frequency ratio in the

expression for pG arises because the Raman interaction preserves photon number rather than optical

energy. The gain profile, gref(ν) itself is equivalent to the gain function ( )Rg Ω introduced in the standard fiber documentation and it is specified in this model exactly as in the standard fiber.

The factor ε in Eqs. (2) account for the polarization of the pumps. For raman_pump_polarization=Parallel, 1ε = and it is assumed that all signals and pumps have the same polarization throughout the fiber. In most cases, the pumps and signals are scrambled in polarization, due both to properties of the launched fields and PMD. This corresponds to 1/ 2ε = , and is obtained with raman_pump_polarization=Unpolarized.

The Bose-enhancement of the spontaneous emission factor is given by

( ) ( ) 1, exp / 1 .phN T h kT

− Ω = Ω −

(3)

in terms of the temperature T and frequency shift Ω , with k being the Boltzmann constant. At low temperatures, it enhances the degree of spontaneous emission for small frequency shifts. Physically, this is due to the large phonon population in low energy states. At room temperature, its effect is minor.

Note in Eqs. (1), that the Rayleigh scattering terms appear only in the equations for the noise fields. Thus as discussed earlier, all Rayleigh scattered power is regarded as noise. In the first two equations of (1), it is unnecessary to include terms describing scattering out of these fields since this is subsumed into the attenuation coefficient.

Equations (1) are solved by discretizing the fields in space and frequency and solving the differential equations iteratively to a specified tolerance.

Physical Model Parameters

The Raman gain curves ,s pG are specified in shape and strength using the various parameters described for this purpose in the Nonlinear Fiber Model. Their behavior here is unchanged.

Rayleigh scattering and spontaneous emission effects are enabled with the parameters include_rayleigh and include_spontaneous (both enabled by default). The temperature T in Eq. (3) is set by temperature. The parameter noise_adjustment, by default set to 1.0, allows all spontaneous emission terms to be scaled

by a constant factor: all appearances of Planck’s constant h in Eqs. (1) are replaced by *h h= noise_adjustment%

. This may be occasionally useful for fibers with very small noise

capture factors, but should normally not be used. Exact formulas for the Rayleigh coefficient ( )γ ν are

quite involved and depend on careful knowledge of the modal properties and effective index [2]. We have chosen to support several models for the Rayleigh scattering. The default is to use a power-law fit to the Rayleigh scattering. The different models are listed below, each of which defines )(νγ :

• Constant


)(νγ = rayleigh_coeff_k.

• PowerLaw

( ) ( ) ( )/ / 1.0 mcγ ν ν µ =

rayleigh_coeff_mrayleigh_coeff_k/

(4)

• File )(νγ is set by data taken from a file specified with rayleigh_filename. The format is similar

to the documented loss file formats but the dependent parameter is the scattering coefficient, which must be in units of 1/m. For example:

FiberRayleighFormat1 Nanometers 4 1300 1e-8 1400 2e-8 1500 2.5e-8 1600 3e-8

The default values rayleigh_coeff_k=2.16e- 7 m - 1 and rayleigh_coeff_m=2.7, are suitable values for standard single mode fiber. Note that these values differ from the well-known 41/ λ dependence familiar in scattering theory, due to the wavelength dependence of the capture cross-section of the fiber – most of the scattered light escapes from the fiber and is not guided.

The Rayleigh back-scatter of the pump fields contributes to the noise spectrum in the model leaving large spikes in the spectrum at the pump frequencies. In practice these narrow lines would be removed by narrow-band optical filters and this could be done using OptSim components. The parameter filter_noise enables these lines to be removed from the spectrum before it leaves the model as a convenience. Disabling this feature may produce unexpected results in calculations using the ASE spectrum such as in the BER and Eye Diagram components.

Numerical Model Parameters A large number of parameters control aspects of the numerical solution of the algorithm.

Convergence: The criterion for convergence of the bidirectional algorithm is controlled by bd_tol. The algorithm is considered to have converged when the largest relative change in power at any point in the pump/signal or noise spectra is smaller than bd_tol. The default setting is usually appropriate but may be experimented with. The rate at which the solution relaxes towards the correct answer is determined by iterative_damping which is a number between 0.001 and 1.0. In a very strongly-pumped fiber or for very long fibers, the algorithm may occasionally become unstable with the default settings. In this case, increasing iterative_damping towards 1.0 can improve the convergence. Conversely, in problems with weak Raman interactions, reducing iterative_damping reduces the number of iterations required for convergence.

In the most severe pumping problems, convergence may also be improved by enabling the switch progressive_solution, but this parameter should otherwise be left disabled.

Grid parameters: The frequency and spatial grid spacings are controlled by bd_freqStepSize and bd_zStepSize, respectively. If these parameters have negative values, OptSim will determine appropriate values based on the strength of the Raman gain and the input pump and signal powers. For the spatial grid step, the shortest characteristic length in the problem is calculated and multiplied by zStepFac. This number should not be increased above the default value of 0.1, but can be reduced if required until a consistent result is obtained. If positive values are provided for bd_freqStepSize or bd_zStepSize they are


used directly. If stability problems are encountered (an error is issued if this occurs,) the user should experiment with decreasing bd_zStepSize in addition to the parameters mentioned in the previous paragraph.

By default, the model calculates a range for the frequency/wavelength domain that allows for a broad spontaneous emission spectrum. In some cases, this domain may be larger than necessary and the simulation time may be reduced by specifying upper and lower bounds for the frequency grid with the parameters sim_fixed_wavelength_range, sim_lambda_lo and sim_lambda_hi.

Display Parameters Frequently, the user may be interested in viewing the results over a smaller frequency domain than is required for the numerical solution. It may be desired to “zoom in” on the signal gain with the pumps outside the plot domain. To save having to use the WinPlot zoom feature for each new plot, the freq_plot_lo and freq_plot_hi parameters may be used to set viewing limits for the frequency axes in all the internal plots. The values of these parameters are interpreted in terms of the current setting for frequency_units. If either value is negative, the whole domain is shown. The parameters distance_units, power_units, power_density_units determine the units for other quantities in all plots.

As discussed in the section Internal Plots below, a number of generated plots convert the internal spectral density representation into powers over frequency channels of a given width. This width is set with nominal_channel_width. Typical values in a WDM system would be 25GHz, 50GHz or 100GHz.

Coherent Solution Calculation In CoherentOnly mode, incoming signals are propagated through the fiber using the standard Nonlinear Fiber Model with support for signal propagation in both directions. All parameters described in the documentation for that model are present in this model and behave identically.

In Full mode, the signals are again propagated using the standard Nonlinear Fiber Model. However, for each pump laser connected to the model an additional term appears in the expression for the Raman interaction, using the powers ( )f

mP z and ( )bnP z of the forward and backward propagating pumps which

are known from the power solution which has already been obtained. Thus the gain term from the signal field ( )iA z in Eq. (22) of the Nonlinear Fiber documentation becomes

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

2,

,2 2

i k R i k i kik

R Rmi mi ni nif b

m i n im n

R A jf A t A t h

g gP z A t P z A t

γ

ε ε

= Ω

Ω Ω+ +

∑

∑ ∑

%K

(5)

where the gain factors ( )Rmi mig Ω are defined in Eq. (23) of the Fiber Model documentation and miΩ and

niΩ are the frequency separation of the pumps and signal i . The factor ε accounts for the pump

polarization specified by pump_polarization. For Polarized pumps, 1ε = , while for Unpolarized pumps, 1/ 2ε = . Note that signals at higher frequencies than the pumps are depleted rather than amplified as is

required.

If the signal bandwidths are broad compared to the Raman gain spectrum (that is, bandwidths larger than a THz or so), then the assumption of frequency-independent gain coefficients in Eq. (5) is invalid. Setting the parameter raman_pumpsig_interaction=Full instead of the default Simple causes the full frequency dependence to be taken into account. This works in an analogous fashion to the parameter raman_sigsig_interaction in the standard fiber model. The action of the pumps is completely disabled with raman_pumpsig_interaction=Off, in which case the simulation again reverts to the standard fiber


model. The stage 1 power solution is still performed however. Note that the raman interaction between sampled signals traveling in opposite directions is not currently modeled. This may be added later.

The forward-propagating and backward-propagating noise fields obtained from the PowerSolution is directly assigned to the ASE associated with the optical signals and is not altered in the second stage of the problem.

Internal Plots For diagnostic purposes, the first stage power solution generates a number of internal plots, which can be accessed by double-clicking on the component icon after the simulation is completed. These plots are useful for checking the input powers and wavelengths of the various signals and pumps, and for analyzing properties of the gain spectrum. By examining these plots, the pump configuration can be adjusted to produce acceptable gain bandwidth and flatness without requiring a complete coherent propagation of the signal or subsequent components such as the Spectrum Plot. The meaning of the various plots follows, listed by their WinPlot filenames. The units for many of these plots can be adjusted with the parameters frequency_units, distance_units, power_units, power_density_units.

• raman_spectral_density_launch.pcs Spectral density of the launched forward and backward fields and noise. The pumps and signal energies are combined.

• raman_power_launch.pcs Launched pump, signal and noise power averaged over frequency bins of width nominal_channel_width. This presentation may be more natural than the spectral density representation.

• raman_solve_fwd.pcs Contour plot of the spectral density of the forward-propagating solution as a function of distance and frequency.

• raman_solve_back.pcs Contour plot of the spectral density of the backward-propagating solution as a function of distance and frequency.

• raman_fwd_output.pcs Spectral densities of the forward-propagating fields exiting the rear of the fiber.

• raman_back_output.pcs Spectral densities of the backward-propagating fields exiting the front of the fiber.

• raman_forward_channel_power.pcs Power per frequency bin of width nominal_channel_width of the launched and final forward propagating signal and noise.

• raman_backward_channel_power.pcs Power per frequency bin of width nominal_channel_width of the launched and final backward propagating signal and noise.

• raman_signal_power_evol.pcs Distance evolution of total signal and noise powers.

• raman_pump_evol.pcs Distance evolution of individual and total pump powers.

• raman_forward_gain.pcs

Signal gain spectrum with gain ( ) ( )( )

sigfout

10 sigfin

10 logS

GS

νν

ν

=

• raman_backward_gain.pcs


Signal gain spectrum with gain ( ) ( )( )

sigbout

10 sigbin

10 logS

GS

νν

ν

=

• raman_forward_absolute_gain.pcs

Absolute change in forward signal spectral density ( ) ( )sig sigfout finS Sν ν−

• raman_backward_absolute_gain.pcs

Absolute change in backward signal spectral density ( ) ( )sig sigbout binS Sν ν−

• raman_forward_on_off_gain.pcs

On-off gain forward signal gain spectrum. On-off gain is the standard gain normalized by the attenuation of

the fiber: ( ) ( )( ) ( )( )

sigfout

f_on-off 10 sigfin

10 logexpS

GS L

νν

ν α ν

= −

• raman_backward_on_off_gain.pcs

On-off gain forward signal gain spectrum. On-off gain is the standard gain normalized by the attenuation of

the fiber: ( ) ( )( ) ( )( )

sigbout

b_on-off 10 sigbin

10 logexpS

GS L

νν

ν α ν

= −

• raman_forward_noise_figure.pcs

Effective Raman noise figure ( )F ν defined by ( ) ( ) ( )( )ASE f_on-off 1F G hρ ν ν ν ν= − where

( )ASEρ ν is the spectral density of spontaneous emission in both transverse polarizations.

• raman_backward_noise_figure.pcs

Effective Raman noise figure ( )F ν defined by ( ) ( ) ( )( )ASE b_on-off 1F G hρ ν ν ν ν= − where

( )ASEρ ν is the spectral density of spontaneous emission in both transverse polarizations.

• raman_photons.pcs

Total photon flux in forward and backward directions and total photon flow. In a lossless fiber, the photon flow is conserved rather than the power flow.

• raman_Forward_SNR.pcs

Signal-to-noise ratio (SNR) of forward propagating signals.

• raman_Backward_SNR.pcs

Signal-to-noise ratio (SNR of backward propagating signals.

The parameters frequency_plot_lo and frequency_plot_hi can be used to limit the size of the plotting domain as opposed to the simulation domain size. If either parameter is negative, they are ignored and the whole simulation domain is plotted. The values are interpreted with respect to the current frequency_units setting.

To save time and disk space, the number of plots displayed may be controlled with generate_plots. By default, all plots are shown. Set generate_plots=1 to suppress the contour plots, or generate_plots=0 to suppress all plots.


References

[1] H. D. Kidorf, K. Rottwitt, M. Nissov, M. Ma and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photonics Technol. Lett. 11, 530 (1999).

[2] E. Brinkmeyer, “Analysis of the backscattering method for single-mode optical fibers,” JOSA Lett. 70, 1010 (1980).

Properties


#2: CW optical signals representing forward propagating pumps

#3: CW optical signals representing backward propagating pumps


Parameter Values Name Type Default Range Units simulation_mode enumerated Full InputOnly,

PowerOnly, CoherentOnly, Full

length double 1e3 [ 0, 1e32 ] m

zStepFac double 0.1 [ 0, 1 ] none

zStepSize double -1 [ -1, 1e10 ] m decreasingStepSize

enumerated No No, Yes

diameter double 8.2e-6 [ 0, 1e-2 ] m

aEff double 1.425 [ 0, 100 ] none

loss_model enumerated Constant Constant, File

loss double 0.25 [ 0, 1e32 ] dB/km loss_filename string

dispersionModel enumerated defined defined, defined2, custom, extrapolated

n1 double 1.4682 [ 0, 100 ] none

beta2 double 0.0 [ -1e32, 1e32 ] s^2/m

beta3 double 0.0 [ -1e32, 1e32 ] s^3/m dispersionLambda0

double 1.312e-6 [ -1e32, 1e32 ] m

dispersionS0 double 0.09e3 [ -1e32, 1e32 ] s/m^3

dispersionOffset double 0 [ -1e32, 1e32 ] s/m^2

nonlinearity_model

enumerated Constant Constant, File


n2 double 2.6e-20 [ 0, 100 ] m^2/W

nonlinearity_filename

string

pmd_method enumerated None None, Coarse_Step, Modified_Coarse_Step

pmd_coef double 3.16e-15 [ 0, 1e32 ] s/m^0.5

pmd_corlen double 10 [ 1e-10, 1e32 ] m pmd_seed integer 0 [ -1e8, 1 ] none

signal_spectrum enumerated Incoming Incoming, FixedRange

sig_fixed_lambda_lo

double 1.4e-6 [ 1e-10, 1e32 ] m

sig_fixed_lambda_hi

double 1.6e-6 [ 1e-10, 1e32 ] m

sig_fixed_spec_dens

double 1.e-16 [ 0, 1e32 ] W/Hz

raman_effects enumerated On On, Off

raman_strength_model

enumerated Absolute Absolute, Fractional

raman_profile enumerated Default Default, Analytic, File

raman_profile_filename

string

raman_max_gain double 0.98e-13 [ 0, 1e32 ] m/W

raman_response_fraction

double 0.18 [ 0, 1 ] none

raman_pump_ref_lambda

double 1.0e-6 [ 0, 1e32 ] m

raman_analytic_sig1

double 12.2e-15 [ 0, 1e32 ] s

raman_analytic_sig2

double 32.0e-15 [ 0, 1e32 ] s

raman_pump_scaling

enumerated Default Default, File

raman_pump_scaling_filename

string raman_scale.dat

raman_pump_polarization

enumerated Unpolarized Unpolarized, Parallel

raman_self_interaction

enumerated Off Off, On

raman_sigsig_interaction


raman_pumpsig_interaction


include_rayleigh enumerated Yes No, Yes

rayleigh_model enumerated PowerLaw Constant, PowerLaw, File

rayleigh_coeff_k double 2.16e-7 [ 0, 1e32 ] m^-1


rayleigh_coeff_m double 2.7 [ 0, 1e32 ] none

rayleigh_filename string include_spontaneous


temperature double 298.15 [ 0, 1e32 ] K

noise_adjustment double 1.0 [ 0, 1e32 ] none

filter_noise enumerated Yes Yes, No

include_sbs enumerated No No, Yes g_B double 3e-11 [ -1e32, 1e32 ] m/W

k_pol double 1.5 [ 1, 2 ]

gain_ref_freq double 193.0 [ 1e-32, 1e32 ] THz

Delta_fB double 40.0 [ 0, 1e32 ] MHz f_B double 11.0 [ 1e-32, 1e32 ] GHz

force_linewidth enumerated No No, Yes

channel_linewidth double 100.0 [ 0, 1e32 ] MHz

nominal_channel_width

double 100e9 [ 0, 1e32 ] Hz

include_spm enumerated Yes No, Yes include_xpm_el enumerated Yes No, Yes

include_xpm_mol enumerated Yes No, Yes

include_dispersion


sim_fixed_wavelength_range

enumerated No Yes, No

sim_lambda_lo double 1.2e-6 [ 1e-10, 1e32 ] none

sim_lambda_hi double 1.6e-6 [ 1e-10, 1e32 ] none iterative_damping double .7 [ .001, 1 ] none

bd_freqStepSize double -1 [ -1e32, 1e32 ] Hz

bd_zStepSize double -1 [ -1e32, 1e32 ] m

bd_tol double 1e-4 [ 1e-32, 1e32 ] none progressive_solution

enumerated No Yes, No

freq_plot_lo double -1 [ -1e32, 1e32 ] none

freq_plot_hi double -1 [ -1e32, 1e32 ] none

skew double 0 [ -1e32, 1e32 ] s optimizationlevel integer 3 [ 0, 3 ] none

showstatus enumerated Yes Yes, No

logstatus enumerated No No, Yes

physpropFilename

string

physpropSteps integer 20 [ 2, 1e8 ] generate_plots integer 2 [ 0, 2 ]

frequency_units enumerated um nm, um, m, Hz, GHz, THz, cm^-1, m^-1, rad/s

distance_units enumerated m m, km, Mm


power_units enumerated W uW, mW, W, dBm

power_density_units

enumerated dBm/GHz uW/Hz, uW/GHz, uW/THz, mW/Hz, mW/GHz, mW/THz, W/Hz, W/GHz, W/THz, dBm/Hz, dBm/GHz, dBm/THz

test_raman_pump_lambda

double 1.3e-6 [ 0, 1e32 ] m

test_output enumerated raman_gain loss, dispersion, n2, rayleigh, raman_gain, raman_pump_scaling

Parameter Descriptions † denotes a parameter discussed in detail in the Nonlinear Fiber Model documentation simulation_mode Select type of simulation length† Length of the fiber zStepFac† Scaling factor for propagation step-size zStepSize† Specify user-defined step size or automatic calculation decreasingStepSize† Toggle use of monotonically decreasing step-size diameter† Core diameter aEff† Effective mode area normalized by core area loss_model† Select constant or file-based loss loss† Constant fiber attenuation parameter loss_filename† Data file for attenuation profile dispersionModel† Select dispersion model n1† Group index beta2† Second-order group velocity dispersion beta3† Third-order group velocity dispersion dispersionLambda0† Dispersion reference wavelength dispersionS0† Dispersion slope at DispersionLambda0 dispersionOffset† Dispersion offset at DispersionLambda0 nonlinearity_model† Select nonlinearity model n2† Constant nonlinear refractive index value nonlinearity_filename† Data file for nonlinearity coefficient profile pmd_method† Select PMD model if any pmd_coef† PMD coefficient D pmd_corlen†

PMD correlation length corrL

pmd_seed† Select randomization mode of PMD signal_spectrum Toggle source of input signal spectrum sig_fixed_lambda_lo Lower limit of fixed input signal spectrum sig_fixed_lambda_hi Upper limit of fixed input signal spectrum sig_fixed_spec_dens Spectral density of fixed input signal spectrum


raman_effects† Toggle all Raman effects raman_strength_model† Select Raman strength model raman_profile† Select Raman response or gain profile model raman_profile_filename† Data file for Raman gain profile raman_max_gain† Peak Raman gain at reference pump wavelength raman_response_fraction† Fraction of nonlinearity that is non-instantaneous raman_pump_ref_lambda† Reference pump wavelength for Raman gain raman_analytic_sig1† Coefficient for analytic Raman response function raman_analytic_sig2† Coefficient for analytic Raman response function raman_pump_scaling† Toggle additional scaling of Raman gain raman_pump_scaling_filename†

Data file for user-specified Raman pump scaling

raman_pump_polarization Polarization states of Raman pumps raman_self_interaction† Toggle intra-pulse Raman scattering raman_sigsig_interaction† Select mode for inter-channel Raman interactions raman_pumpsig_interaction Select mode for pump-channel Raman interactions include_rayleigh Toggle inclusion of Rayleigh scattering effects rayleigh_model The type of model to use for Rayleigh scattering effects rayleigh_coeff_k Rayleigh scattering coefficient rayleigh_coeff_m Rayleigh scattering coefficient

rayleigh_filename Filename for Rayleigh gamma data for file-based model include_spontaneous Toggle inclusion of spontaneous emission noise temperature Temperature for spontaneous emission strength noise_adjustment Artificial adjustment of noise strength filter_noise Toggle removal of pump lines from outgoing noise spectrum include_sbs† Toggle stimulated Brillouin scattering g_B† Nominal Brillouin gain value k_pol† Random polarization factor gain_ref_freq† Reference frequency for g_B Delta_fB† Brillouin linewidth f_B† Brillouin gain peak frequency force_linewidth† Toggle manual setting of channel linewidths for SBS model channel_linewidth† Manual channel linewidth for SBS model nominal_channel_width Frequency bin width for power-based plots include_spm† Toggle inclusion of SPM include_xpm_el† Toggle inclusion of electronic (normal) XPM include_xpm_mol† Toggle inclusion of molecular (Raman) XPM include_dispersion† Toggle inclusion of dispersion sim_fixed_wavelength_range Toggle fixed or auto-calculated simulation wavelength domain sim_lambda_lo Fixed wavelength domain lower limit sim_lambda_hi Fixed wavelength domain upper limit iterative_damping Relaxation coefficient for iterative solution scheme bd_freqStepSize Frequency grid step for power solution bd_zStepSize Spatial grid step for power solution bd_tol Convergence tolerance for power solution


progressive_solution Toggle progressive iterative scheme for power solution freq_plot_lo Lower frequency limit for internal plots freq_plot_hi Upper frequency limit for internal plots skew† Time skew added to the output optical signal optimizationlevel† Simulation optimization level showstatus† Toggle display of propagation progress logstatus Toggle logging of propagation progress physpropFilename Filename for recording physical properties for interior maps physpropSteps Number of property measurements to make generate_plots Select which internal plots should be displayed frequency_units† Frequency units for display of test and internal plots distance_units Distance units for display of test and internal plots power_units Power units for display of test and internal plots power_density_units Spectral density units for display of test and internal plots test_raman_pump_lambda† Pump wavelength for Raman gain profile test function test_output† Selection of test function display


Fiber Delay

This model delays the input optical signal(s) by the specified amount to model an ideal fiber delay. This model may be used anywhere in the topology where a specified optical signal time delay is desired.

Properties



Parameter Values

Name Type Default Range Units delay double 0 [ 0, 1e32 ] s

Parameter Descriptions

delay Amount to shift the carrier phase of the input optical signal(s) by

OptSim Models Reference: Block Mode Chapter 6: Optical Amplifiers •••• 169

Chapter 6: Optical Amplifiers

This chapter describes the optical amplifiers:

• Black Box Optical Amplifier a simple physical model of an EDFA

• Physical EDFA a full physical model of an EDFA

• Physical EYCDFA a full physical model of an EYCDFA

• Semiconductor Optical Amplifier (SOA) a physical model of a SOA based on travelling wave approximation

• Controlled Semiconductor Optical Amplifier (SOA) a physical model of a controlled SOA with injection current as sum of pump and external control currents

• Optical Noise Adder add spontaneous emission noise without gain

• Linewidth Adder add linewidth to an optical signal as a parameter or as phase noise

• Liekki LAD Interface interface to Liekki Application Designer (LAD)

170 •••• Chapter 6: Optical Amplifiers OptSim Models Reference: Block Mode

Black Box Optical Amplifier

This block models an optical amplifier as a black box, such as an erbium doped fiber amplifier (EDFA). There are two types of optical amplifier gain models in the black box model: the defined model and the custom model. There are also several types of optical amplifier noise models: the Uniform model, the Gaussian model, and the Custom model.

Defined Gain Model In the defined gain model, the gain variations with wavelength of an optical amplifier are not included. This is partially because the gain flatness depends on the level of saturation of the amplifier and this causes added complexities to the model. The gain flatness will be a significant factor when WDM systems are modeled. The gain saturation at high input powers is included in this model, with the power gain specified as,

sat

ave

PPG

GG

0

0

1+=

where G0 (gain) is the small signal power gain, Psat (Psat) is the saturation output power and Pave is the total average power in the fiber.

Both the signal and the preceding generated spontaneous noise are amplified by the optical power gain G.

Custom Gain Model Both gain variations with wavelength and the gain saturation are included in the custom gain model, where measured or theoretical gain spectra at different total input powers are read from the data file. The program will interpolate the optical gain for every channel under the actual input total power. The custom file is a two column format file which specifies the wavelength (nm) and the power gain (dB) all separated by spaces. Multiple gain curves (each corresponds to different input power) are placed one after another within the data file. There is no limit to the number of gain curves that may be included. The very first line of each gain curve starts with ‘0’ and is followed with spaces and the average input power (dBm) of the curve. This ‘0’ tells the program the beginning of a new gain curve and must not be changed. Comments can be added at the end of the line, and each comment is ended with a ‘*’. No comment or ‘*’ is needed in the gain data rows. The data region is ended with a line ‘-1 -1’ at the end of the file. There should be a carriage return at the end of the last line.

The wavelength-gain curves are passed to the black box optical amplifier module. The black box optical amplifier module will calculate the input optical average power and interpolate the gain at the WDM channel wavelengths. With this new power – gain spectra, the black box optical amplifier module then interpolates to obtain gain under the corresponding input power for different channels.

Gain is also applied to the ASE noise spectra according to wavelength in this gain model.

Defined ASE Noise Models Both the Uniform and Gaussian noise models utilize a Noise Figure number to characterize noise [1]. The amplified spontaneous noise power density is simulated as,

νhnGS spsp ⋅⋅−= )1(

where 2/nsp Fn ≈ is the population inversion factor, while Fn (Fn) is the noise figure of the amplifier. νh is the photon energy. The ASE noise spectrum is defined using the Fn parameter, the noiseShape parameter to define the


shape of the defined noise spectrum, BW parameter to define the noise bandwidth, the noiseCenter parameter to define the center wavelength of the defined ASE noise spectra, and the noiseResolution parameter to define the spectral resolution of the generated ASE noise spectra. The resolution should be set to be approximately an order of magnitude smaller than the bandwidth of the narrowest optical filter in the link.

Custom ASE Noise Model The Custom noise model uses a wavelength and power dependent ASE noise file of a format similar to the custom gain model. The noise spectrum is defined in units of dBm at different average input power levels and wavelengths. The name of the custom noise file is given in the noiseFilename parameter. The noiseResolution parameter is used to set the resolution of the generated ASE noise spectra. The resolution should be set to be approximately an order of magnitude smaller than the bandwidth of the narrowest optical filter in the link.

Note: The optical filter is not included in this optical amplifier model, rather it is a separate model.

References [1] N. A. Olsson, “Lightwave systems with optical amplifiers”, J. of Lightwave Technology 7, 1071-1082 (1989).

Properties



Parameter Values Name Type Default Range Unit type enumerated defined defined, custom

spectraFilename string

noiseFilename string

gain double 30 0 ≤ x ≤ 1e32 dB

Psat double 18 0 ≤ x ≤ 1e32 dBm

Fn double 4.0 3 ≤ x ≤ 1e32 dB

BW double 30e-9 0 ≤ x ≤ 1e6 m

noiseCenter double 1550e-9 1e-32 ≤ x ≤ 1e6 m

noiseShape enumerated Uniform Uniform, Gaussian, Custom

NoiseResolution double 0.01e-9 1e-32 ≤ x ≤ 1e6 m

Parameter Descriptions type defined and custom spectraFilename Filename of the amplifier gain spectrum (custom only)


noiseFilename Filename of the amplifier noise spectrum (custom only) gain Optical amplifier small signal amplitude gain (defined only) Psat Optical amplifier saturation power (defined only) Fn Noise Figure of the Amplifier BW Optical amplifier ASE noise bandwidth (defined only) noiseCenter Optical amplifier ASE noise spectra center wavelength (defined only) noiseShape Optical amplifier ASE noise shape (Uniform, Gaussian, or Custom) noiseResolution Optical amplifier ASE noise resolution


Physical EDFA

This block models the operation of an erbium-doped fiber amplifier (EDFA) via a set of well-established physical equations. The model supports component specifications at different levels of complexity, as well as a variety of pump and signal configurations. Figure 1 illustrates an OptSim schematic that utilizes the Physical EDFA model. Forward-propagating optical signals are launched into the EDFA via the first input node, while backward-propagating signals (e.g., counter-propagating pumps) enter via the second input node. OptSim’s multiplexer components can be used to combine signals and pumps at either input. The EDFA output is available at the output node, and includes any signals, pumps, and amplified spontaneous emission (ASE) that are exiting the amplifier. The EDFA may also be used to simulate bidirectional signal propagation, in which case input signals are expected at both input nodes, and an additional backward output appears at the backward output node.

forward input

backward input

pumps backward output

(bidirectional mode only)

Figure 1: Basic OptSim topology depicting an EDFA with forward and backward inputs.

Background The Physical EDFA model is based on a standard set of equations for an EDFA’s steady-state atomic manifold population densities and the evolution of optical powers along the length of the device [1]-[3]. Optical signals propagating along the EDFA interact with the local population densities, resulting in power gain or loss via stimulated emission and absorption. Spontaneous emission and its subsequent amplification also occur.

Generally, the erbium atomic manifolds of interest can be reduced to a three-level atomic system, as illustrated in Fig.2(a), with population densities in each manifold assumed to be Boltzmann-distributed at thermal equilibrium. Optical signals with wavelengths near 980 nm (henceforth referred to as the 980-band) are used to pump from the first (4I15/2) to the third (4I11/2) level, while 1480-nm signals pump from the first to the second (4I13/2) level. In the former case, fast non-radiative decay from the third to the second level effectively eliminates any stimulated emission from the third to first level, allowing us to simplify the model to a two-level atomic system with zero stimulated emission in the 980-band [2]. This reduced arrangement is depicted in Fig.2(b), where R13 is the stimulated absorption rate for 980-band transitions, R12 and R21 are stimulated absorption and emission rates between the 4I15/2 and 4I13/2 levels, respectively, and τ is the spontaneous emission lifetime of the 4I13/2 level. The optical signals being amplified by the EDFA


usually have wavelengths ranging from 1530-1580-nm (the C-band) or 1580-1610-nm (the L-band), and therefore interact with the same atomic populations as 1480-nm pumps. Thus, R12 and R21 account for both signal and pump transitions at wavelengths typically ranging from 1450-1650-nm (henceforth referred to as the 1550-band). Amplified spontaneous emission (ASE) in the EDFA also occurs in this band of wavelengths.

4I11/2

4I13/2

4I15/2

1550 nm

980 nm

1480 nmpumps

level 2

level 1

R13

R21

R121/ τ

(a) (b)

Figure 2: Erbium atomic manifolds. (a) Three manifolds involved in predominant erbium atomic transitions. (b) Simplified two-level model.

In order to describe the interaction between the erbium ions and local signal, pump, and noise powers, the model uses a set of rate equations for erbium ion densities in each atomic level. However, because the sum of these two densities should equal the total erbium doping density N, we can in fact adopt a single rate equation for the level-2 population density N2. Furthermore, in most EDFA applications, the long lifetime τ of the metastable level-2 population acts to eliminate any significant transient changes in the level populations, thereby allowing us to set the time rate-of-change for N2 to zero. In other words, the EDFA’s operating characteristics depend on average optical powers. Thus [2],

τ

2221112221112113

2 0 NNRNRNRNRNRdt

dNASEASE −−+−+=≅

(1)

where N1 = N – N2 is the level-1 population density, RASE12 is the stimulated absorption rate for spontaneous emission, and RASE21 is the corresponding stimulated emission rate. Generally, because there is a continuum of transition frequencies between the erbium atomic manifolds, each of the transition rates R in Eq.(1) are of the form:

νν

ψνρνσ dh

rR ∫= )()()( r

(2)

where ν is the transition frequency, ( )σ ν is the frequency-dependent transition cross section, ( )ρ ν is the local optical spectral density, and ( )rψ v

is the normalized optical mode profile. By assuming homogeneous broadening of the atomic transitions, we have adopted a single frequency-dependent function for the transition cross sections [2]. Following the approach in Ref.[2], we can discretize the above integral over fixed frequency intervals ν∆ , in which case the transition rates take the form

( )i i i

i i

P PRh

σ ψν

+ −+=∑

(3)


where we have replaced the optical spectral density with forward ( iP+ ) and backward ( iP− ) signal powers in each frequency interval.

To complete the model, rate equations are necessary for describing the evolution of signal, pump, and noise powers along the EDFA. Separate equations for both forward and backward propagation are required for 980-band signals, 1550-band signals, and 1550-band ASE. Again following the approach taken in [2], we have adopted the following equations:

dQdz

N r r r dr N r r r dr Qja j p a j p

jj

±±= ± ⋅ ⋅ − ⋅ −

⋅∫∫2

22π σ ψ σ ψαπ, ,( ) ( ) ( ) ( )

(4)

dPdz

N r r r dr N r r r dr Pke k a k s a k s

kk

±±= ± ⋅ + ⋅ − ⋅ −

⋅∫∫222π σ σ ψ σ ψ απ

( ) ( ) ( ) ( ) ( ), , ,

(5)

( )

, , 2 ,

, 2

2 ( ) ( ) ( ) ( ) ( )2

2 ( ) ( ) 2

l le l a l s a l s l

e l s l

dA N r r r dr N r r r dr Adz

N r r r dr h

απ σ σ ψ σ ψπ

π σ ψ ν ν

±± = ± ⋅ + ⋅ − ⋅ − ⋅

± ⋅ ⋅ ⋅ ⋅ ⋅ ∆

∫ ∫

∫

(6)

where 980-band powers are denoted by jQ± , 1550-band powers by kP± , and ASE powers by lA± . Intrinsic background loss is accounted for by α , and locally generated spontaneous emission is included via the last term in Eq.(6).

A number of assumptions are inherent to Eqs.(4)-(6). First, we have assumed that the erbium-doping and mode profiles have no azimuthal dependence. Second, we have assumed that 980-band signals all have the same normalized mode profile ( )p rψ ; similarly, all 1550-band signals and ASE use a common profile

( )s rψ . Third, higher order effects such as excited-state absorption (ESA) [1]-[2] are neglected. However, higher order effects such as upconversion and pair-induced quenching, while not included in the above equations, are included in the model, and will be explained shortly. Finally, fiber effects such as dispersion and nonlinearities are also neglected, due to the relatively short lengths of most EDFA’s. It should also be noted that the model assumes no spectral overlap between separate optical signals. In cases where this is detected, the model will issue an appropriate warning.

Model Implementation

Model Levels The Physical EDFA block includes a number of implementations of the main equations described above. These models can be selected via the simulation_mode parameter, and can take on values of giles_params, constant_overlap, calculated_overlap, cladding_pumped, and spatial.

Level giles_params As shown in [3], the model equations presented above can be significantly simplified by dealing with the average level-population densities and eliminating their explicit spatial dependence via measured gain and loss spectra (as opposed to emission and absorption cross sections) that are weighted by the local doping and mode profiles. This well known implementation is the Giles model, in which the primary model


parameters are reduced to a set of four, namely the measured gain spectra eg , the measured loss spectra

ag , the intrinsic background loss α , and a fiber saturation parameter ζ (fiber_saturation_param), which accounts for the average doping density N0, the fiber transverse area, and the erbium metastable lifetime. Such a model is generally valid for strongly confined erbium-doping profiles. The complete set of model equations become:

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

, , ,

2

0 , , , , ,

a j j j a k k k a l l l

j k lj k l

a j j j a k e k k k a l e l l l

j k lj k l

g Q Q g P P g A Ah h hN

N g Q Q g g P P g g A Ah h h

ν ν ν

ζν ν ν

+ − + − + −

+ − + − + −

⋅ + ⋅ + ⋅ ++ +

=⋅ + + ⋅ + + ⋅ +

+ + +

∑ ∑ ∑

∑ ∑ ∑

(7)

2, ,

0

ja j a j j j

dQ N g g Qdz N

α±

± = ± ⋅ − − ⋅

(8)

( )2, , ,

0

ka k e k a k k k

dP N g g g Pdz N

α±

± = ± ⋅ + − − ⋅

(9)

( ) ( )2 2, , , ,

0 0

2la l e l a l l l e l l

dA N Ng g g A g hdz N N

α υ υ±

± = ± ⋅ + − − ⋅ ± ⋅ ⋅ ⋅ ⋅∆

(10)

Levels constant_overlap, calculated_overlap In terms of computational complexity, the constant_overlap and calculated_overlap models are equivalent to the giles_params model, but support a more thorough parameterized description of the EDFA, including absorption/emission cross sections, doping profiles, and the metastable lifetime. In these versions of the model, any explicit transverse spatial dependence is replaced by factors that account for the spatial overlap between the erbium population densities and optical modes [1]. In the calculated_overlap model, these overlap factors are determined from explicit doping and mode profiles, whereas in the constant_overlap model, the user provides these values directly. For 980-band signals, the overlap parameter is pΓ

(overlap_980), and for 1550-band signals, it is sΓ (overlap_1550). Like the giles_params model, the overlap models are also largely intended for EDFA’s with strongly confined erbium-doping profiles. The complete set of model equations are [1]:

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

, , ,0

2, , , , ,

a j j j a k k k a l l lp s s

j k lj k l

a j j j a k e k k k a l e l l leffp s

j k lj k l

Q Q P P A AN

h h hN

Q Q P P A AAh h h

σ σ σν ν ν

σ σ σ σ στ ν ν ν

+ − + − + −

+ − + − + −

⋅ + ⋅ + ⋅ + Γ + Γ + Γ ⋅ =

⋅ + + ⋅ + + ⋅ ++ Γ + Γ +

∑ ∑ ∑

∑ ∑ ∑ (11)


, 2 , 0j

p a j p a j j j

dQN N Q

dzσ σ α

±± = ± Γ − Γ − ⋅

(12)

( ), , 2 , 0k

s a k e k s a k k kdP N N Pdz

σ σ σ α±

± = ± Γ + − Γ − ⋅

(13)

( ) ( ), , 2 , 0 , 2 2ls a l e l s a l l l s e l l

dA N N A N hdz

σ σ σ α σ υ υ±

± = ± Γ + − Γ − ⋅ ± Γ ⋅ ⋅ ⋅∆

(14) where Aeff is the effective transverse doping area, and N0 is a calculated effective doping density. The user may specify a value for τ via the metastable_lifetime parameter. Note the similarities between these equations and those of the Giles model.

Level cladding_pumped For cladding-pumped devices, the cladding-pumped version of the model should be used. In this case, the user specifies the core area via the parameter Acore, and the inner cladding area via the parameter Aclad. These areas are then used to calculate the pump overlap factor [4], with the rest of the model defaulting to the overlap approach described above.

Level spatial The most complex version of the model is the spatial model, which directly implements Eq.(1) and Eqs.(4)-(6), thereby providing a full spatial description of the interplay between the erbium population densities and optical mode profiles. As such, it requires the largest amount of computational overhead. The simplified models discussed above are sufficient for most situations of interest. However, in some cases, a detailed study of the EDFA design is required, in which case the spatial model would be used.

Higher-Order Effects The above equations ignore higher-order effects such as homogeneous upconversion [5] and pair-induced quenching [6]. Based on the treatments in [1], [5], and [6], we have incorporated these effects into the Physical EDFA model. Homogeneous upconversion is accounted for via the parameter upconversion_coeff (for non-Giles models) or giles_upconversion_coeff (for the Giles model). Pair-induced quenching is incorporated via the parameter pair_fraction, which should be set equal to the fraction of Er ions that appear in pairs within the fiber. This parameter is equal to twice the parameter k from [1] and [6].

EDFA Configurations In specifying the configuration of the EDFA, the user must always specify a fiber length via the length parameter. They may also provide coupling losses at both the input and output nodes via the parameters forward_input_loss, backward_input_loss, and output_loss. Furthermore, they may select to have any pumps (i.e., optical signals with wavelengths near 980 or 1480 nm) excluded from the model output via the output_pumps parameter. Setting this option to no is useful in cases where the pump signal no longer impacts system performance in components that follow the EDFA.

The EDFA model may also be used to simulate bidirectional signal propagation. In this case, the parameter bidirectional should be set to yes. The user may then provide input signals at both the forward and backward input nodes. The backward output appears at the backward output node of the model. The user may specify a backward output coupling loss via the parameter backward_output_loss.


In addition to these basic configuration settings, the Physical EDFA model also supports various EDFA pump/signal recycling schemes [2]. By placing mirrors at either end of an EDFA, pumps and/or signals may be recycled, thereby providing the opportunity for enhanced amplification. Such configurations may be selected through the mirror_configuration parameter. The most basic option is no_mirror, in which no pump/signal reflectors are included in the EDFA. (This configuration is always activated if bidirectional is set to yes.) In cases where forward-propagating optical inputs are to be recycled (typically pumps), the forward_mirror option should be selected. This arrangement is illustrated in Fig.3(a). Alternatively, backward-propagating inputs may be recycled via the backward_mirror option, shown in Fig.3(b). In this case, the user may select to have the reflected signal included in the EDFA output via the output_reflected option. Finally, the user may choose to adopt a signal recycling scheme, such as that depicted in Fig.3(c), wherein the EDFA input and output are actually at the same end of the device, with an optical circulator providing separation between the two. Both signal and pump may be recycled in this manner. This option may be chosen by setting mirror_configuration to signal_mirror.

input

EDFA

output

EDFA

inputoutput

O

reflector

reflector

input

EDFA

output

reflector

input (bwd)

circulator

(a) (b)

(c)

Figure 3: EDFA pump/signal recycling configurations. (a) Co-propagating pump reflector. (b) Counter-propagating pump reflector. (c) Signal/pump recycler.

The spectral characteristics of any mirrors included in the EDFA are set via the mirror_model, mirror_reflectivity, mirror_center, mirror_bandwidth, and mirror_file parameters. Setting mirror_model to rectangular implements a rectangular spectra with the in-band reflectivity specified by mirror_reflectivity, a center wavelength specified by mirror_center, and a bandwidth specified by mirror_bandwidth. The gaussian option for mirror_model uses the same parameters, but of course implements a Gaussian wavelength dependence. Alternatively, a reflectivity spectrum may be read in directly from a file by setting mirror_model to file, and providing a file name in mirror_file. The file format is described in the appendix.

Noise Settings From the ASE power propagation equations, we can see that the amount of ASE power locally injected at any point along the length of the EDFA is typically 2 hυ υ⋅ ⋅ ∆ [2]. The factor of two takes into account ASE injected in both polarizations. In cases where only a single polarization is desired, the parameter ASE_polarization should be set to single (as opposed to both). Local ASE injection may be completely eliminated from the simulation by setting inject_ASE to no.


Absorption/Emission Spectra For optical signals and ASE in the 1550-band, the user must specify the Giles gain/loss spectra (for the Giles model), or absorption/emission cross-section spectra (for the other models). By setting the spectra_1550_model parameter, the user may select either built-in default spectra or device-specific data that is to be read in from a file.

Two sets of default spectra are available. If spectra_1550_model is set to default_Ge, then spectra for a germanosilicate fiber are used. If spectra_1550_model is set to default_GeAl, then spectra for an alumino-germanosilicate fiber are chosen instead. In both cases, the data is based on analytical expressions taken from [2].

If spectra_1550_model is set to user_specified, then spectra must be provided through input files. For the Giles model, the appropriate Giles loss/gain files are identified with the parameters giles_loss_1550_file and giles_gain_1550_file. For the other models, cross-section data files are specified using the parameters absorption_1550_file and emission_1550_file. The file format is described in the appendix.

A loss or absorption spectrum must also be provided for signals in the 980-band. In this case, three choices are available via the parameter spectra_980_model. For either a rectangular profile (rectangular) or Gaussian profile (gaussian), the user must specify a center wavelength (spectra_980_center) and spectral width (spectra_980_width). For the Giles model, the center-wavelength loss is set via giles_loss_980. For the other models, the absorption cross-section center-wavelength value is set via absorption_980.

If spectra_980_model is set to file, then the 980-band spectrum must be provided through an input file. For the Giles model, the Giles loss file is identified with the parameter giles_loss_980_file. For the other models, the 980-band absorption cross-section data file is specified using the parameter absorption_980_file. The file format is described in the appendix.

Depending on the provided data, the model can automatically determine the range of wavelengths that comprise the 980-band and 1550-band based on the range of the gain/loss or emission/absorption spectra. However, by setting spectra_1550_auto_limit to no, the user may directly limit the 1550-band to a specific range of wavelengths via the parameters spectra_1550_low and spectra_1550_high. Similarly, by setting spectra_980_auto_limit to no, the 980-band limits can be set via the parameters spectra_980_low and spectra_980_high.

Thermal Dependence Thermal dependence of either the absorption/emission or loss/gain spectra is modeled based on the work of M. Bolshtyansky et al. in [7]. Given a set of reference spectra αref (absorption or loss) and gref (emission or gain) at a temperature Tref, spectra αnew and gnew at a new temperature Tnew are calculated as:

1 2

1 2

/ /

/ /

( )( ) ( )( )

new new

ref ref

T T T Tref

new ref T T T Tnew

T K e eT K e e

λα λ α λλ

− −

− −

+= ⋅ ⋅+

(15)

1 2

1 2

/ 1/ /

/ /

( )( )( ) ( )( )( )

ref newnew new

ref ref

T TT T T T

ref refnew ref T T T T

new ref

T gK e eg gT K e e

λλλ λα λλ

−− −

− −

+= ⋅ ⋅ ⋅ +

(16)

where K(λ), T1, and T2 are empirical fitting parameters used to fit the thermal dependence to measured data. K(λ) is equal to F1(λ)/F2(λ), where F1(λ) and F2(λ) are defined in [7].

To activate this model, the user must set thermal_model equal to builtin or user, and specify the temperature (reference_temp) at which the reference spectra were measured, and the new temperature (actual_temp) at which the spectra should be modeled. The absorption/emission or loss/gain spectra specified earlier by the user are considered to be the reference spectra.


For the builtin model, the values given in [7] for K(λ), T1, and T2 are used, as these were reported to give good agreement with experimental results for a wide range of silica-based aluminum co-coped EDFAs with varying levels of germanium and aluminum. If the user has their own measured spectra that they would like to use, then they should select the user model. For both the 980- and 1550-band spectra, they must specify a data file for K(λ) (F1F2_ratio_1550_file and F1F2_ratio_980_file) and values for T1 (T1_1550 and T1_980) and T2 (T2_1550 and T2_980). The format for the data files is described in the appendix. Given a set of measured spectra at reference temperature Tref, and additional spectra at temperatures Ta, Tb, Tc, etc., the following procedure is suggested for determining optimal values for the empirical parameters K(λ), T1, and T2:

1. Select values for T1 and T2.

2. At each temperature Ta, Tb, Tc, etc., use equations (15) and (16) to determine K(λ) for each additional gain and loss spectra.

3. To minimize the differences between the different calculations of K(λ), select new values for T1 and T2. Repeat steps 2 and 3, optimizing T1 and T2 until the error between the different calculations of K(λ) is minimized. Select one of the K(λ) functions for use in the model.

Doping Profile A number of choices are available for specifying the EDFA erbium doping profile. These options are available through the doping_model parameter. For both rectangular and gaussian doping profiles, a doping radius and peak density must be provided via the parameters doping_radius and doping_density, respectively. Alternatively, the user may choose to provide a data file containing the doping density profile by setting doping_model to file and specifying a file name through the parameter doping_file. The file format is described in the appendix.

Mode Profiles For the calculated_overlap and spatial models, 980-band and 1550-band mode profiles are required. Different mode shapes may be selected using the parameter mode_model. If mode_model is set to lp, then LP01 mode solutions [2] are determined for a fiber with the specified numerical aperture (fiber_NA) and fiber core radius (fiber_core_radius). For all 1550-band signals, a wavelength of 1550 nm is used in the calculation, whereas a 980-nm wavelength is adopted for all 980-band signals.

If a rectangular or gaussian set of mode shapes is chosen, then relevant mode widths must be specified via the mode_1550_width and mode_980_width parameters. Because the calculated mode shapes are normalized, no additional information is required.

Mode shapes can also be input from a data file by setting mode_model to file. The 1550-band file name must be provided through the mode_1550_file parameter, while the 980-band file name must be entered through the mode_980_file parameter. The file format is described in the appendix.

Background Loss In most simulations, fiber background loss can be neglected [2], in which case the parameter background_loss_model should be set to no_loss. However, other options are available if required.

If background_loss_model is set to constant, then uniform background loss is assumed, the value of which may be specified with the parameter background_loss. Alternatively, different loss values can be provided for signals in the 1550- and 980-bands by setting background_loss_model to two_constant and specifying an additional 980-band loss value through the background_loss_980 parameter. Finally, a background loss spectrum may be read in from a file by setting background_loss_model to file and specifying a file name with the background_loss_file parameter. The file format is described in the appendix.


Encrypted Data Files Note that all of the model characteristics that may be specified via a data file support encrypted file formats. Data files from vendors may be provided in this manner. In this case, an accompanying password file would be included with the data file, and should be placed in the same directory as the link topology, or in the directory containing the RSoft software license files.

Numerical Settings A number of numerical parameters are available to optimize the numerical solution of the bidirectional rate equations for the pump/signal/ASE power evolution. Adjustment of these parameters may help overcome any convergence difficulties encountered during a simulation.

The numerical integration of the power evolution rate equations is handled via a constant-step-size fourth-order Runge-Kutta algorithm. The integration step along the length of the EDFA may be adjusted using the z_step parameter. Similarly, the nominal width for the discretized frequency intervals in the 980- and 1550-bands may be set using the spectral_step parameter. As this parameter is set in units of nanometers, an equivalent spectral step in Hertz is calculated internally by the model. The resolution of transverse calculations may also be adjusted by setting the radial_points parameter to the desired number of points in the radial direction.

The bidirectional nature of the power-evolution rate equations requires an iterative solution scheme, the control of which may be achieved through the iterative_damping and convergence_tolerance parameters. The iterative_damping controls the rate at which the solution is allowed to progress to a final answer. Larger values slow down this process, and may be required when convergence is difficult to achieve. Parameter convergence_tolerance is the convergence criterion. Convergence is achieved when the largest change in the power solution between successive iterations is less than convergence_tolerance. To limit the time spent on a poorly converging solution, the maximum number of iterations may be set with the parameter max_iterations.

Finally, when amplifying or attenuating any incoming optical signals, the model applies its calculated gain spectrum to a Fourier-domain representation of these signals. Normally, the spectral variation in the gain across the frequency band of each signal is fully accounted for by setting the parameter gain_application to continuous. However, given the narrow-band nature of most signals relative to typical spectral variations in EDFA gain, it sometimes may be easier to use a constant gain value for each signal based on its carrier wavelength. In this case, gain_application should be set to carrier. This approach may prove useful when discontinuities in an optical input’s phase at the signal boundaries lead to anomalous output waveforms.

Reference Plots In order to help study the performance of an EDFA within an optical link, a variety of reference plots can be generated by the model in order to study internal power evolution, signal gain, noise figure, and the atomic-manifold population densities. What plots are generated is determined via the parameters power_plots, spectra_plots, density_plots, and gain_nf_plot (by setting them to either yes or no); they may be displayed at the conclusion of a simulation by double-clicking on the EDFA’s icon. The units for the displayed data may be selected via the parameters spectral_units, power_units, length_units, radial_units, density_3d_units, and signal_gain_units (for gain and noise figure data). Below we summarize the different plots that may be generated, listing them by their root WinPlot file names. power_plots

• signal_ase_evolution.pcs:

Displays the evolution of total power in the 1550-band signals (1480-nm pumps excluded) and ASE along the length of the EDFA. Both forward and backward directions of propagation are included.

• pump_evolution.pcs:


Displays the evolution of forward- and backward-propagating optical powers for pump wavelengths near 980 and 1480 nm.

• gain_evolution.pcs:

Displays the evolution of signal gain along the EDFA.

• 1550_forward_solution.pcs:

Displays a contour plot of the complete forward-propagating 1550-band power spectra solution along the length of the EDFA.

• 1550_backward_solution.pcs:

Displays the corresponding contour plot for the backward-propagating 1550-band power spectra solution.

spectra_plots

• 1550_power_spectra.pcs:

Displays the input/output signal and ASE power spectra in the 1550-band.

• 1550_power_spectra_bwd.pcs:

Bidirectional mode only. Displays the backward input/output signal and ASE power spectra in the 1550-band.


Displays the input and output 980-band power spectra.


Bidirectional mode only. Displays the backward input and output 980-band power spectra.

• ase_power_spectra.pcs:

Displays the internal ASE power spectra at both ends of the EDFA.

• gain.pcs:

Displays the overall 1550-band signal gain spectra (1480-nm pumps excluded).

• gain_bwd.pcs:

Bidirectional mode only. Displays the overall backward 1550-band signal gain spectra (1480-nm pumps excluded).

• absolute_gain.pcs:

Displays a spectrum of the absolute change in input-signal power.

• absolute_gain_bwd.pcs:

Bidirectional mode only. Displays a spectrum of the absolute change in backward input-signal power.

• noise_figure.pcs:

Displays the overall 1550-band noise figure, calculated as a function of signal gain ( )G υ

and output ASE spectral density ( )ASEρ ν [1]:

( )1 1

( )ASENF

G hρ ν

ν ν = ⋅ +

• noise_figure_bwd.pcs:


Bidirectional mode only. Displays the overall backward 1550-band noise figure

density_plots

• average_densities.pcs:

Displays the average population densities along the length of the EDFA in both the upper and lower atomic levels.

• n2(r)_vs_z.pcs:

(spatial model only) Displays a contour plot of the transverse population density along the EDFA for the upper atomic level.

gain_nf_plot

• gain_scan.pcs:

Displays the gain of the EDFA at the wavelength specified via the parameter target_wavelength. If the topology is simulated via a parameter scan, then this plot shows the gain as a function of the scanned parameters. This plot can be useful for gain optimization of the EDFA.

• gain_scan_bwd.pcs:

Bidirectional mode only. Displays the backward gain of the EDFA at the wavelength specified via the parameter target_wavelength.

• noise_figure_scan.pcs:

Displays the noise figure of the EDFA at the wavelength specified via the parameter target_wavelength. If the topology is simulated via a parameter scan, then this plot shows the noise figure as a function of the scanned parameters.

• noise_figure_scan_bwd.pcs:

Bidirectional mode only. Displays the backward noise figure of the EDFA at the wavelength specified via the parameter target_wavelength.

Test Functions In selecting the input parameters for a particular EDFA, it may at times be necessary to visualize the various input spectra, doping profiles, and mode shapes. By setting the test_function parameter to the desired output and clicking the Test button in the component-parameter editing window, the user may display the EDFA characteristics summarized below. Furthermore, the default plot ranges for each characteristic may be overridden by setting test_default_settings to no, and specifying values for test_function_x_low, test_function_x_high, and test_function_points (the number of data points to plot). The appropriate units for each characteristics’ overrides are listed in parentheses below. Units for the displayed plots may be specified using the parameters spectral_units, radial_units, cross_section_units, density_3d_units, loss_units, and reflectivity_units.

• 1550_spectra(nm):

Plots the 1550-band gain/loss (for the Giles model), or emission/absorption cross section spectra (for the other models).


Plots the 980-band loss (for the Giles model), or absorption cross section spectrum (for the other models).

• doping(um):

Plots the erbium doping profile as a function of radius.

• modes(um):


Plots the 1550- and 980-band normalized mode profiles as functions of radius.

• background_loss(nm):

Plots the background-loss spectrum.

• mirror(nm):

Plots the mirror reflectivity spectrum (if a mirror is specified).

References [1] P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers: Fundamentals and Technology. (San Diego, Academic Press, 1999).

[2] E. Desurvire, Erbium-Doped Fiber Amplifiers. (New York, Wiley, 1994).

[3] C. R. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” Journal of Lightwave Technology 9, 271-283 (1991).

[4] M. Söderlund, S. Tammela, P. Pöyhönen, M. Leppihalme, and N. Peyghambarian, “Amplified spontaneous emission in cladding-pumped L-band erbium-doped fiber amplifiers,” IEEE Photonics Technology Letters, 13, 22-24 (2001).

[5] P. Blixt, J. Nilsson, T. Carlnas, and B. Jaskorzynska, “Concentration-dependent upconversion in Er3+-doped fiber amplifiers: Experiments and modeling,” IEEE Photonics Technology Letters, 3, 996-998 (1991).

[6] E. Delevaque, T. Georges, M. Monerie, P. Lamouler, and J.-F. Bayon, “Modeling of pair-induced quenching in erbium-doped silicate fibers,” IEEE Photonics Technology Letters, 5, 73-75 (1993).

[7] M. Bolshtyansky, P. Wysocki, and N. Conti, “Model of temperature dependence for gain shape of erbium-doped fiber amplifier,” Journal of Lightwave Technology, 18, 1533-1540 (2000).

Properties

Inputs #1: Forward-propagating optical signals

#2: Backward-propagating optical signals

Outputs #1: Forward-propagating optical signals

#2: Backward-propagating optical signals (bidirectional mode only)

Parameter Values Name Type Default Range Unit simulation_mode enumerated giles_params giles_params,

constant_overlap, calculated_overlap, spatial, cladding_pumped

bidirectional enumerated no yes, no

mirror_configuration enumerated no_mirror no_mirror, forward_mirror, backward_mirror, signal_mirror


output_reflected enumerated yes yes, no

output_pumps enumerated yes yes, no

inject_ASE enumerated yes yes, no

ASE_polarization enumerated both both, single

forward_input_loss double 0 -1e32≤ x ≤ 1e32 dB

backward_input_loss double 0 -1e32≤ x ≤ 1e32 dB

output_loss double 0 -1e32≤ x ≤ 1e32 dB

backward_output_loss double 0 -1e32≤ x ≤ 1e32 dB

length double 20 0 ≤ x ≤ 1e32 m

metastable_lifetime double 10 0 ≤ x ≤ 1e32 ms

fiber_saturation_param double 3e15 0 ≤ x ≤ 1e32 m-1s-1

overlap_1550 double 0.35 0 ≤ x ≤ 1e32


Acore double 3.14e-12 0 ≤ x ≤ 1e32 m2

Aclad double 7.068e-8 0 ≤ x ≤ 1e32 m2

pair_fraction double 0.0 0 ≤ x ≤ 1

upconversion_coeff double 0.0 0 ≤ x ≤ 1e32 m3/s

giles_upconversion_coeff double 0.0 0 ≤ x ≤ 1e32 m-1s-1

spectra_1550_model enumerated default_Ge default_Ge, default_GeAl, user_specified

spectra_1550_auto_limit enumerated yes yes, no

spectra_1550_low double 1400 0 ≤ x ≤ 1e32 nm

spectra_1550_high double 1650 0 ≤ x ≤ 1e32 nm

giles_loss_1550_file string

giles_gain_1550_file string

absorption_1550_file string

emission_1550_file string

spectra_980_model enumerated rectangular rectangular, gaussian, file




giles_loss_980 double 6.2 0 ≤ x ≤ 1e32 dB/m

absorption_980 double 2e-25 0 ≤ x ≤ 1e32 m2

spectra_980_center double 980 0 ≤ x ≤ 1e32 nm

spectra_980_width double 20 0 ≤ x ≤ 1e32 nm



thermal_model enumerated none builtin, user, none

reference_temp double 25 -273.15 ≤ x ≤ 1e32 °C

actual_temp double 25 -273.15 ≤ x ≤ 1e32 °C

F1F2_ratio_1550_file string

T1_1550 double 90 -1e32 ≤ x ≤ 1e32 K


T2_1550 double 650 -1e32 ≤ x ≤ 1e32 K


T1_980 double 90 -1e32 ≤ x ≤ 1e32 K

T2_980 double 650 -1e32 ≤ x ≤ 1e32 K

doping_model enumerated rectangular rectangular, gaussian, file

doping_radius double 1 0 ≤ x ≤ 1e32 µm

doping_density double 1e19 0 ≤ x ≤ 1e32 cm-3

doping_file string

mode_model enumerated lp lp, gaussian, rectangular, file

fiber_NA double 0.3 0 ≤ x ≤ 1e32

fiber_core_radius double 1 0 ≤ x ≤ 1e32 µm

mode_1550_width double 1 0 ≤ x ≤ 1e32 µm


mode_1550_file string


background_loss_model enumerated no_loss no_loss, constant, two_constant, file

background_loss double 0 0 ≤ x ≤ 1e32 dB/km

background_loss_980 double 0 0 ≤ x ≤ 1e32 dB/km

background_loss_file string

mirror_model enumerated rectangular rectangular, gaussian, file

mirror_reflectivity double 1 0 ≤ x ≤ 1e32

mirror_center double 980 0 ≤ x ≤ 1e32 nm

mirror_bandwidth double 10 0 ≤ x ≤ 1e32 nm

mirror_file string

z_step double 0.1 0 ≤ x ≤ 1e32 m

spectral_step double 1 0 ≤ x ≤ 1e32 nm

iterative_damping double 0.7 0 ≤ x ≤ 1

convergence_tolerance double 1e-4 0 ≤ x ≤ 1e32

max_iterations integer 2500 4 ≤ x ≤ 100000

radial_points integer 100 2 ≤ x ≤ 100000

gain_application enumerated continuous continuous, carrier

power_plots enumerated yes yes, no

spectra_plots enumerated yes yes, no

density_plots enumerated yes yes, no

gain_nf_plot enumerated yes yes, no

target_wavelength double 1550 1 ≤ x ≤ 2000 nm

spectral_units enumerated nm nm, um, m, Hz, GHz, THz, cm^-1, m^-1, s^-1

power_units enumerated mW uW, mW, W, dBm

length_units enumerated m um, mm, cm, m, km, Mm


radial_units enumerated um um, mm, cm, m, km, Mm

cross_section_units enumerated m^2 um^2, mm^2, cm^2, m^2, km^2, Mm^2

density_3d_units enumerated cm^-3 um^-3, mm^-3, cm^-3, m^-3, km^-3, Mm^-3

loss_units enumerated dB/km nm^-1, um^-1, cm^-1, m^-1, km^-1, Mm^-1, dB/nm, dB/um, dB/cm, dB/m, dB/km, dB/Mm

signal_gain_units enumerated dB linear, dB, %

reflectivity_units enumerated linear linear, dB, %

test_function enumerated 1550_spectra(nm) 1550_spectra(nm), 980_spectra(nm), doping(um), modes(um), background_loss(nm), mirror(nm)


test_function_x_low double 1400 0 ≤ x ≤ 1e32

test_function_x_high double 1650 0 ≤ x ≤ 1e32

test_function_points integer 201 2 ≤ x ≤ 100000


simulation_mode model-level options bidirectional switch for simulating bidirectional signal propagation mirror_configuration pump/signal recycling options output_reflected switch for including any reflected backward inputs in the output output_pumps switch for including pump signals in the output inject_ASE switch for injecting ASE within the EDFA ASE_polarization switch for injecting ASE into one or two polarizations forward_input_loss forward-propagating input coupling backward_input_loss backward-propagating input coupling output_loss output coupling backward_output_loss output coupling of backward signal when simulating bidirectional operation length EDFA length metastable_lifetime metastable-level lifetime fiber_saturation_param Giles-model fiber-saturation parameter overlap_1550 constant 1550-band overlap factor overlap_980 constant 980-band overlap factor Acore area of core for double-clad devices Aclad area of inner cladding for double-clad devices pair_fraction fraction of dopant ions that are paired upconversion_coeff upconversion coefficient for non-Giles models giles_upconversion_coeff upconversion coefficient for Giles model spectra_1550_model 1550-band signal spectra options


spectra_1550_auto_limit option for automatically determining 1550-band spectral range spectra_1550_low lowest wavelength for 1550-band spectra spectra_1550_high highest wavelength for 1550-band spectra giles_loss_1550_file user-specified 1550-band Giles loss-spectrum data file giles_gain_1550_file user-specified 1550-band Giles gain-spectrum data file absorption_1550_file user-specified 1550-band absorption cross-section data file emission_1550_file user-specified 1550-band emission cross-section data file spectra_980_model 980-band signal spectrum options spectra_980_auto_limit option for automatically determining 980-band spectral range spectra_980_low lowest wavelength for 980-band spectra spectra_980_high highest wavelength for 980-band spectra giles_loss_980 center-value of 980-band Giles loss spectrum absorption_980 center-value of 980-band absorption cross-section spectra__980_center center-wavelength of 980-band spectrum spectra_980_width bandwidth of 980-band spectrum giles_loss_980_file user-specified 980-band Giles loss-spectrum data file absorption_980_file user-specified 980-band absorption cross-section data file thermal_model option for determining thermal dependency of gain/loss or

emission/absorption spectra reference_temp temperature at which gain/loss or emission/absorption spectra were measured actual_temp actual operating temperature of amplifier F1F2_ratio_1550_file empirical F1/F2 thermal data file for 1550-band spectra T1_1550 empirical T1 thermal parameter for 1550-band spectra T2_1550 empirical T2 thermal parameter for 1550-band spectra F1F2_ratio_980_file empirical F1/F2 thermal data file for 980-band spectra T1_980 empirical T1 thermal parameter for 980-band spectra T2_980 empirical T2 thermal parameter for 980-band spectra doping_model erbium-doping density options doping_radius doping radius doping_density center doping density (at r = 0) doping_file user-specified doping-profile data file mode_model optical mode shape options fiber_NA fiber numerical aperture fiber_core_radius fiber core radius mode_1550_width 1550-band optical-mode width mode_980_width 980-band optical-mode width mode_1550_file user-specified 1550-band mode-profile data file mode_980_file user-specified 980-band mode-profile data file background_loss_model background-loss options background_loss 1550-band background-loss value background_loss_980 980-band background-loss value background_loss_file user-specified background-loss data file mirror_model mirror power-reflectivity spectrum options mirror_reflectivity mirror power reflectivity at center wavelength mirror_center mirror-spectrum center wavelength


mirror_bandwidth mirror-spectrum bandwidth mirror_file user-specified mirror-spectrum data file z_step integration step along EDFA length spectral_step spacing between wavelengths in discretized power spectra iterative_damping iterative damping factor for numerical solution convergence_tolerance convergence tolerance for numerical solution max_iterations maximum number of iterations during bidirectional solution radial_points number of radial points gain_application method of applying gain to each input signal power_plots switch for power-solution reference plots spectra_plots switch for power-spectra reference plots density_plots switch for level-density reference plots gain_nf_plot switch for plots of gain and noise-figure scans target_wavelength target wavelength for plots of gain and noise-figure scans spectral_units units for spectral data power_units units for power data length_units units for positional data radial_units units for radial data cross_section_units units for cross-section data density_3d_units units for density data loss_units units for loss data signal_gain_units units for gain and noise figure results reflectivity_units units for mirror data test_function test-function output selection test_default_settings switch for plotting test-function output using default settings test_function_x_low user-specified lowest x-value for test-function output test_function_x_high user-specified highest x-value for test-function output test_function_points user-specified number of points for test-function output


Appendix: File formats For all data files, the X-values should be monotonically increasing or decreasing. Furthermore, the field <num_pts> specifies the number of data lines in the file, while the choice of settings for the various unit fields are as follows:

• <frequency_units>: [nm], [um], [m], [Hz], [GHz], [THz], [cm^-1], [m^-1], [s^-1]

• <loss_units>: [nm^-1], [um^-1], [cm^-1], [m^-1], [km^-1], [Mm^-1], [dB/nm], [dB/um], [dB/cm], [dB/m], [dB/km], [dB/Mm]

• <area_units>: [um^2], [mm^2] , [cm^2] , [m^2] , [km^2] , [Mm^2]

• <density_units>: [um^-3], [mm^-3] , [cm^-3] , [m^-3] , [km^-3] , [Mm^-3]

• <distance_units>: [um], [mm] , [cm] , [m] , [km] , [Mm]

• <reflectivity_units>: [linear], [dB] , [%]

Giles Gain/Loss Spectra Data files with gain and loss spectra for the Giles model are specified through the parameters giles_gain_1550_file, giles_loss_1550_file, and giles_loss_980_file. The X-values are in units of frequency, and the Y-values are in units of loss per distance.

Format: GilesFormat1 <frequency_units> <loss_units>

<num_pts>

<frequency 1> <gain/loss 1>




...

Example: GilesFormat1 [nm] [dB/m]

5

1450 0.3

1500 0.6

1530 2.0

1550 1.2

1600 0.4

Absorption/Emission Cross Sections Data files with absorption/emission cross sections are specified through the parameters absorption_1550_file, emission_1550_file, and absorption_980_file. The X-values are in units of frequency, and the Y-values are in units of area.

Format: CrossSectionFormat1 <frequency_units> <area_units>


<num_pts>

<frequency 1> <cross_section 1>




...

Example: CrossSectionFormat1 [nm] [m^2]

5

1450 0.7e-25

1500 2.4e-25

1530 7.0e-25

1550 4.3e-25

1600 0.2e-25

Thermal K(λλλλ) Profile The data files for the K(λ) data used in the model’s thermal dependence are specified via the parameters F1F2_ratio_1550_file and F1F2_ratio_980_file. The X-values are in units of frequency, and the Y-values are unitless.

Format: SpectralUnitlessFormat1 <frequency_units>

<num_pts>

<frequency 1> <value 1>




...

Example: SpectralUnitlessFormat1 [nm]

5

1500 0.40

1510 0.45

1520 0.50

1530 0.45

1540 0.40

Doping Profile A user-specified doping profile may be specified via the parameter doping_file. The X-values are in units of distance, and the Y-values are in units of cubic density.


Format: DopingFormat1 <distance_units> <density_units>

<num_pts>

<radius 1> <density 1>




...

Example: DopingFormat1 [um] [cm^-3]

5

0 1.00e19

0.25 1.00e19

0.50 1.00e19

0.75 0.50e19

1.00 0.25e19

Mode Profiles Data files with mode shapes may be specified using the parameters mode_1550_file and mode_980_file. The X-values are in units of distance, and the Y-values are considered unitless (the mode profiles are automatically normalized by the model).

Format: ModeProfileFormat1 <distance_units>

<num_pts>

<radius 1> <value 1>




...

Example: ModeProfileFormat1 [um]

5

0 1

1 0.7

2 0.4

3 0.25

5 0.15

Background Loss A data file with the background-loss spectrum may be specified using the background_loss_file parameter. The X-values are in units of frequency, and the Y-values are in units of loss per distance.


Format: FiberLossFormat1 <frequency_units> <loss_units>

<num_pts>

<frequency 1> <loss 1>




...

Example: FiberLossFormat1 [um] [dB/km]

5

1.40 0.25

1.45 0.21

1.50 0.19

1.55 0.17

1.60 0.21

Mirror Reflectivity Spectrum A data file with the mirror reflectivity spectrum may be specified using the parameter mirror_file. The X-values are in units of frequency, and the Y-values are in units of power-reflectivity.

Format: ReflectivityFormat1 <frequency_units> <reflectivity_units>

<num_pts>

<frequency 1> <reflectivity 1>




...

Example: ReflectivityFormat1 [nm] [%]

5

970 10

975 100

980 100

985 100

990 10


Physical EYCDFA

This block models the operation of an erbium-ytterbium co-doped fiber amplifier (EYCDFA). The model supports component specifications at different levels of complexity, as well as a variety of pump and signal configurations. Figure 1 illustrates an OptSim schematic that utilizes the Physical EYCDFA model. Forward-propagating optical signals are launched into the EYCDFA via the first input node, while backward-propagating signals (e.g., counter-propagating pumps) enter via the second input node. OptSim’s multiplexer components can be used to combine signals and pumps at either input. The EYCDFA output is available at the output node, and includes any signals, pumps, and amplified spontaneous emission (ASE) that are exiting the amplifier. The EYCDFA may also be used to simulate bidirectional signal propagation, in which case input signals are expected at both input nodes, and an additional backward output appears at the backward output node.

forward input

backward input

pumps backward output

(bidirectional mode only)

Figure 1: Basic OptSim topology depicting an EYCDFA with forward and backward inputs.

Background The Physical EYCDFA model is based on a standard set of equations for an EYCDFA’s steady-state atomic manifold population densities and the evolution of optical powers along the length of the device [1]-[5]. Optical signals propagating along the EYCDFA interact with the local population densities, resulting in power gain or loss via stimulated emission and absorption. Spontaneous emission and its subsequent amplification also occur. Both the erbium and ytterbium ions are pumped, with the ytterbium transferring its energy to the erbium via cross relaxation.

The erbium and ytterbium atomic manifolds of interest are illustrated in Fig. 2 [4], with population densities in each manifold assumed to be Boltzmann-distributed at thermal equilibrium. On the erbium side, levels 1 through 4 represent atomic levels 4I15/2, 4I13/2, 4I11/2, and 4I9/2, respectively. On the ytterbium side, levels 5 and 6 represent atomic levels 2F7/2 and 2F5/2, respectively. The erbium and ytterbium manifolds are linked via the cross-relaxation rates R61 (R61) and R35 (R35). R56 and R65 are the ytterbium stimulated absorption and emission rates, respectively, between atomic levels 5 and 6 due to signals near 980-nm (henceforth referred to as the 980-band). ASE in this band also contributes to these stimulated rates. τ65 (metastable_lifetime_Yb) is the spontaneous emission lifetime of the upper ytterbium level. In


the case of erbium, R13 represents the simulated absorption rate between the atomic levels 1 and 3. R12 and R21 are the erbium stimulated absorption and emission rates, respectively, between the atomic levels 1 and 2 due to signals and amplified spontaneous emission (ASE) with wavelengths typically ranging from 1450-1650 nm (henceforth referred to as the 1550-band). 2

2CN accounts for homogeneous upconversion in erbium from level 2, where C (upconversion_coeff) is the upconversion coefficient . Finally, τ21 (metastable_lifetime) is the spontaneous emission lifetime of level 2, τ32 (third_level_lifetime) is the decay lifetime of level 3, and τ43 is the decay lifetime of level 4.

Figure 2: Erbium and ytterbium atomic manifolds

In order to describe the interaction of the erbium and ytterbium ions with local signal, pump, and noise powers, the model uses a set of rate equations for erbium and ytterbium ion densities in each atomic level. In most EYCDFA applications, the dopants’ long metastable lifetimes act to eliminate any significant transient changes in the atomic level populations, thereby allowing us to set the time rate-of-change of all the rate equations to zero. In other words, the EYCDFA’s operating characteristics depend on average optical powers. Furthermore, since decay out of the 4I9/2 erbium level is typically extremely fast, we neglect this atomic level in the model [4]. The resulting set of atomic rate equations are [4],[5]:

232 212 1 21 2 2

32 21

0 2NdN NR N R N CNdt τ τ

≅ = + − − −

(1)

23 32 13 1 61 1 6 35 3 5

32

0dN N CN R N R N N R N Ndt τ

≅ = − + + + −

(2)

1 2 3ErN N N N= + +

(3)

6 656 5 65 6 61 1 6 35 3 5

65

0dN NR N R N R N N R N Ndt τ

≅ = − − − +

(4)

5 6YbN N N= +

(5)

where NEr is the erbium doping density, NYb is the ytterbium doping density, and Ni is the population density of atomic level i.


Generally, because there is a continuum of transition frequencies between the erbium atomic manifolds, each of the transition rates R in the above equations are of the form:

νν

ψνρνσ dh

rR ∫= )()()( r

(6)


is the normalized optical mode profile. By assuming homogeneous broadening of the atomic transitions, we have adopted a single frequency-dependent function for the transition cross sections [2]. Following the approach in [2], we can discretize the above integral over fixed frequency intervals ν∆ , in which case the transition rates take the form

( )n n n

n n

P PRh

σ ψν

+ −+=∑

(7)

where we have replaced the optical spectral density with forward ( nP+ ) and backward ( nP− ) signal powers in each frequency interval. In the case of our EYCDFA model, the relevant transition rates are:

, ,13

( ) ( )a j j j p a l l l p

j lj l

P P A AR

h hσ ψ σ ψ

ν ν

+ − + −+ += +∑ ∑

(8)

, ,12,21

( ) ( )a ei i i s a ek k k s

i ki k

P P A AR

h hσ ψ σ ψ

ν ν

+ − + −+ += +∑ ∑

(9)

, ,56,65

( ) ( )Yb Yba ej j j p a el l l p

j lj l

P P A AR

h hσ ψ σ ψ

ν ν

+ − + −+ += +∑ ∑

(10)

where 1550-band signal powers are denoted by iP± , 980-band signal powers by jP± , 1550-band ASE

powers by kA± , and 980-band ASE powers by lA± . Furthermore, aσ represents the erbium absorption

cross section, eσ is the erbium emission cross section, Ybaσ is the ytterbium absorption cross section, Yb

eσ is the ytterbium emission cross section, ψp is the 980-band mode profile, and ψs is the 1550-band mode profile.

To complete the model, rate equations are necessary for describing the evolution of signal, pump, and noise powers along the EYCDFA. Separate equations for both forward and backward propagation are required for signals and ASE in both the 980- and 1550-bands. Following the approach taken in [2],[4]-[5], we have adopted the following equations:

, , 2 ,2 ( ) ( ) ( ) ( ) ( )2

i ie i a i s a i s i

dP N r r r dr N r r r dr Pdz


±± = ± ⋅ + ⋅ − ⋅ − ⋅ ∫ ∫

(11)


, , 6 ,

, 2 ,

2 ( ) ( ) ( ) ( ) ( )2

2 ( ) ( ) ( ) ( )

j jYb Yb Ybe j a j p a j Yb p j

a j p a j Er p j

dPN r r r dr N r r r dr P

dz

N r r r dr N r r r dr P

απ σ σ ψ σ ψ

π

π σ ψ σ ψ

±±

±

= ± ⋅ + ⋅ − ⋅ − ⋅

± ⋅ − ⋅ ⋅

∫ ∫

∫ ∫

(12)

( )

, , 2 ,

, 2

2 ( ) ( ) ( ) ( ) ( )2

2 ( ) ( ) 2

k ke k a k s a k Er s k

e k s k


N r r r dr h


π σ ψ ν ν

±± = ± ⋅ + ⋅ − ⋅ − ⋅

± ⋅ ⋅ ⋅ ⋅ ⋅ ∆

∫ ∫

∫

(13)

( )

, , 6 ,

, 2 ,

, 6

2 ( ) ( ) ( ) ( ) ( )2

2 ( ) ( ) ( ) ( )

2 ( ) ( ) 2

Yb Yb Ybl le l a l p a l Yb p l

a l p a l Er p l

Ybe l p l


N r r r dr N r r r dr A

N r r r dr h


π σ ψ σ ψ

π σ ψ ν ν

±±

±

= ± ⋅ + ⋅ − ⋅ − ⋅ ± ⋅ − ⋅ ⋅ ± ⋅ ⋅ ⋅ ⋅ ⋅∆

∫ ∫

∫ ∫∫

(14)

Intrinsic background loss is accounted for by α , and locally generated spontaneous emission is included via the last terms in Eqs. (13) and (14).

A number of assumptions are inherent to Eqs. (11)-(14). First, we have assumed that the doping and mode profiles have no azimuthal dependence. Second, we have assumed that 980-band signals and ASE all have the same normalized mode profile ( )p rψ ; similarly, all 1550-band signals and ASE use a common profile

( )s rψ . Third, higher order effects such as excited-state absorption (ESA) [1]-[2] are neglected. Finally, fiber effects such as dispersion and nonlinearities are also neglected, due to the relatively short lengths of most EYCDFA’s. It should also be noted that the model assumes no spectral overlap between separate optical signals. In cases where this is detected, the model will issue an appropriate warning.

Model Implementation

Model Levels The Physical EYCDFA block includes a number of implementations of the main equations described above. These models can be selected via the simulation_mode parameter, and can take on values of giles_params, constant_overlap, calculated_overlap, cladding_pumped, and spatial. Note that in all versions, the model can be simplified to treat the erbium atomic manifold as a two-level system by setting the parameter include_third_level to no.

Level giles_params Following the approach described in [3], the model equations presented above can be significantly simplified by dealing with the average level-population densities and eliminating their explicit spatial dependence via measured gain and loss spectra (as opposed to emission and absorption cross sections) that are weighted by the local doping and mode profiles. This well known implementation is the Giles model. For EYCDFA’s, this results in a much smaller set of primary model parameters. In this model, instead of providing data for cross-section spectra, doping profiles, and mode profiles, the user must supply the measured erbium gain spectra eg and loss spectra ag , the measured ytterbium gain spectra Yb

eg and loss


spectra Ybag , the erbium fiber saturation parameter ζ (fiber_saturation_param), and the ytterbium fiber

saturation parameter ζYb (fiber_saturation_param_Yb). Such a model is generally valid for strongly confined doping profiles. The complete set of model equations become:

2

32 2 2112 12 21 12

0 0 32 0

20 ( 1) g

g g g g

C NN Nr r r rN N N

τζ τ

= − + + ⋅ − ⋅ + − ⋅

(15)

2

32 21 213 13 13

0 32 0 0

61 353 6 6 32

0 0 0 0 0

0

1 1

gg g g

g g

Yb Yb

CNN Nr r rN N N

R RN N N NNN N N N N

ττ ζ

ζ ζ

= − ⋅ − + ⋅ + ⋅ +

⋅ − − ⋅ − ⋅ − ⋅

(16)

( ) 61 356 3 6 6 3256 56 65

0 0 0 0 0 0

0 1 1 1g gg g g

Yb Yb Yb Yb Yb

R RN N N N NNr r rN N N N N Nζ ζ

= − + + ⋅ − ⋅ − − ⋅ + ⋅ − ⋅

(17)

, ,13

( ) ( )a j j j a l l lg

j lj l

g P P g A Ar

h hζ ν ζ ν

+ − + −+ += +∑ ∑

(18)

, ,12,21

( ) ( )a ei i i a ek k kg

i ki k

g P P g A Ar

h hζ ν ζ ν

+ − + −+ += +∑ ∑

(19)

, ,56,65

( ) ( )Yb Yba ej j j a el l l

gj lYb j Yb l

g P P g A Ar

h hζ ν ζ ν

+ − + −+ += +∑ ∑

(20)

2, , ,

0

( )ie i a i a i i i

dP Ng g g Pdz N

α±

± = ± + − − ⋅

(21)

6 2, , , , ,

0 0

( )j Yb Yb Ybe j a j a j j j a j a j j

Yb

dP N Ng g g P g g Pdz N N

α±

± ± = ± + − − ⋅ ± − ⋅

(22)

( )2 2, , , ,

0 0

( ) 2ke k a k a k k k e k k

dA N Ng g g A g hdz N N

α ν ν±

± = ± + − − ⋅ ± ⋅ ⋅ ⋅ ∆

(23)


( )6 62, , , , , ,

0 0 0

( ) 2Yb Yb Yb Yble l a l a l l l a l a l l e l l

Yb Yb

dA N NNg g g A g g A g hdz N N N

α ν ν±

± ± = ± + − − ⋅ ± − ⋅ ± ⋅ ⋅ ⋅ ∆

(24)

In these equations, Cg (giles_upconversion_coeff) is the Giles upconversion coefficient, Rg61 (R61_giles) and Rg35 (R35_giles) are the Giles cross-relaxation coefficients, N0 is the average erbium doping density, and NYb0 is the average ytterbium doping density. All other atomic levels densities are considered to be average values as well. When using the above version of the model, instead of specifying the spontaneous lifetimes directly, you only need to specify the ratio τ32/τ21 (giles_lifetime_ratio).

Levels constant_overlap, calculated_overlap In terms of computational complexity, the constant_overlap and calculated_overlap models are equivalent to the giles_params model, but support a more thorough parameterized description of the EYCDFA, including absorption/emission cross sections, doping profiles, and the metastable lifetimes. In these versions of the model, any explicit transverse spatial dependence is replaced by factors that account for the spatial overlap between the dopant population densities and optical modes [1],[4]-[5]. In the calculated_overlap model, these overlap factors are determined from explicit doping and mode profiles, whereas in the constant_overlap model, the user provides these values directly. For 980-band signals, the erbium overlap parameter is Γp (overlap_980), and for 1550-band signals, it is Γs (overlap_1550). The ytterbium overlap parameter for 980-band signals is Γp (overlap_Yb). Like the giles_params model, the overlap models are also largely intended for EYCDFA’s with strongly confined doping profiles. The complete set of model equations are:

0

2 2112 12 21 2 21 2 12 3

32

0 ( ) 2s s s eff eff sr N r r A N CN A r NτττΓ Γ Γ Γ

= Γ − Γ + Γ + ⋅ − + ⋅ − Γ ⋅

(25)

( ) ( )

22113 0 13 2 13 3 21 2

32

21 61 0 2 3 6 21 35 0 6 3

0 p p eff p

Yb

r N r N A r N CN

R N N N N R N N N

τ ττ

τ τ

Γ Γ Γ

= Γ − Γ − ⋅ + Γ ⋅ + +

⋅ − − ⋅ − ⋅ − ⋅

(26)

( )

( ) ( )56 0 56 65 6

65 61 0 2 3 6 65 35 0 6 3

0 YbYb Yb Yb Yb eff

Yb

r N r r A N

R N N N N R N N Nτ τΓ Γ Γ= Γ − Γ + Γ + ⋅ −

⋅ − − ⋅ + ⋅ − ⋅

(27)

21 , 21 ,13

( ) ( )a j j j a l l l

j lj l

P P A Ar

h hτ σ τ σ

ν ν

+ − + −

Γ

+ += +∑ ∑

(28)

21 , 21 ,12,21

( ) ( )a ei i i a ek k k

i ki k

P P A Ar

h hτ σ τ σ

ν ν

+ − + −

Γ

+ += +∑ ∑

(29)


65 , 65 ,56,65

( ) ( )Yb Yba ej j j a el l l

j lj l

P P A Ar

h hτ σ τ σ

ν ν

+ − + −

Γ

+ += +∑ ∑

(30)

, , 2 , 0( )is e i a i s a i i i

dP N N Pdz

σ σ σ α±

± = ± Γ + − Γ − ⋅

(31)

, , 6 , 0 , 2 , 0( )j Yb Yb YbYb e j a j Yb a j Yb j j p a j p a j j

dPN N P N N P

dzσ σ σ α σ σ

±± ± = ± Γ + − Γ − ⋅ ± Γ − Γ ⋅

(32)

( ), , 2 , 0 , 2( ) 2ks e k a k s a k k k s e k k

dA N N A N hdz

σ σ σ α σ ν ν±

± = ± Γ + − Γ − ⋅ ± Γ ⋅ ⋅ ⋅∆

(33)

( ), , 6 , 0

, 2 , 0 , 6

( )

2

Yb Yb YblYb e l a l Yb a l Yb l l

Ybp a l p a l l Yb e l l

dA N N Adz

N N A N h

σ σ σ α

σ σ σ ν ν

±±

±

= ± Γ + − Γ − ⋅

± Γ − Γ ⋅ ± Γ ⋅ ⋅ ⋅ ∆

(34) where Aeff is the effective transverse erbium doping area, Yb

effA is the effective transverse ytterbium doping area, N0 is a calculated effective erbium doping density, and NYb0 is a calculated effective ytterbium doping density. All atomic level densities are considered to be effective values as well.

Level cladding_pumped For cladding-pumped devices, the cladding-pumped version of the model should be used. In this case, the user specifies the core area via the parameter Acore, and the inner cladding area via the parameter Aclad. These areas are then used to calculate the pump overlap factor [4], with the rest of the model defaulting to the overlap approach described above.

Level spatial The most complex version of the model is the spatial model, which directly implements Eqs. (1)-(5), (8)-(10), and (11)-(14), thereby providing a full spatial description of the interplay of the erbium and ytterbium population densities with the optical mode profiles. As such, it requires the largest amount of computational overhead. The simplified models discussed above are sufficient for most situations of interest. However, in some cases, a detailed study of the EYCDFA design is required, in which case the spatial model would be used.

EYCDFA Configurations In specifying the configuration of the EYCDFA, the user must always specify a fiber length via the length parameter. They may also provide coupling losses at both the input and output nodes via the parameters forward_input_loss, backward_input_loss, and output_loss. Furthermore, they may select to have any pumps (i.e., optical signals with wavelengths near 980 or 1480 nm) excluded from the model output via the output_pumps parameter. Setting this option to no is useful in cases where the pump signal no longer impacts system performance in components that follow the EYCDFA.


The EYCDFA model may also be used to simulate bidirectional signal propagation. In this case, the parameter bidirectional should be set to yes. The user may then provide input signals at both the forward and backward input nodes. The backward output appears at the backward output node of the model. The user may specify a backward output coupling loss via the parameter backward_output_loss.

In addition to these basic configuration settings, the Physical EYCDFA model also supports various EYCDFA pump/signal recycling schemes [2]. By placing mirrors at either end of an EYCDFA, pumps and/or signals may be recycled, thereby providing the opportunity for enhanced amplification. Such configurations may be selected through the mirror_configuration parameter. The most basic option is no_mirror, in which no pump/signal reflectors are included in the EYCDFA. (This configuration is always activated if bidirectional is set to yes.) In cases where forward-propagating optical inputs are to be recycled (typically pumps), the forward_mirror option should be selected. This arrangement is illustrated in Fig.3(a). Alternatively, backward-propagating inputs may be recycled via the backward_mirror option, shown in Fig.3(b). In this case, the user may select to have the reflected signal included in the EYCDFA output via the output_reflected option. Finally, the user may choose to adopt a signal recycling scheme, such as that depicted in Fig.3(c), wherein the EYCDFA input and output are actually at the same end of the device, with an optical circulator providing separation between the two. Both signal and pump may be recycled in this manner. This option may be chosen by setting mirror_configuration to signal_mirror.

Figure 3: EYCDFA pump/signal recycling configurations. (a) Co-propagating pump reflector. (b) Counter-propagating pump reflector. (c) Signal/pump recycler.

The spectral characteristics of any mirrors included in the EYCDFA are set via the mirror_model, mirror_reflectivity, mirror_center, mirror_bandwidth, and mirror_file parameters. Setting mirror_model to rectangular implements a rectangular spectra with the in-band reflectivity specified by mirror_reflectivity, a center wavelength specified by mirror_center, and a bandwidth specified by mirror_bandwidth. The gaussian option for mirror_model uses the same parameters, but of course implements a Gaussian wavelength dependence. Alternatively, a reflectivity spectrum may be read in directly from a file by setting mirror_model to file, and providing a file name in mirror_file. The file format is described in the appendix.

Noise Settings From the ASE power propagation equations, we can see that the amount of ASE power locally injected at any point along the length of the EYCDFA is typically 2 hυ υ⋅ ⋅ ∆ [2]. The factor of two takes into account ASE injected into both polarizations. In cases where only a single polarization is desired, the parameter ASE_polarization should be set to single (as opposed to both). Local ASE injection may be completely eliminated from the simulation by setting inject_ASE to no.


Absorption/Emission Spectra For optical signals and ASE in the 1550-band, the user must specify the erbium Giles gain/loss spectra (for the Giles model), or absorption/emission cross-section spectra (for the other models). By setting the spectra_1550_model parameter, the user may select either built-in default spectra or device-specific data that is to be read in from a file.



An erbium loss or absorption spectrum must also be provided for signals in the 980-band. In this case, three choices are available via the parameter spectra_980_model. For either a rectangular profile (rectangular) or Gaussian profile (gaussian), the user must specify a center wavelength (spectra_980_center) and spectral width (spectra_980_width). For the Giles model, the center-wavelength loss is set via giles_loss_980. For the other models, the absorption cross-section center-wavelength value is set via absorption_980.


Finally, 980-band ytterbium loss/gain or absorption/emission cross-section spectra must also be provided. Two choices are available via the parameter spectra_Yb_model. For a rectangular profile (rectangular), the user must specify a center wavelength (spectra_Yb_center) and spectral width (spectra_Yb_width). For the Giles model, the center-wavelength loss and gain are specified via the parameters giles_loss_Yb and giles_gain_Yb, respectively. For the other models, the absorption and emission cross-section center-wavelength values are set via absorption_Yb and emission_Yb, respectively.

If spectra_Yb_model is set to file, then the 980-band ytterbium spectra must be provided through input files. For the Giles model, the loss and gain files are identified with the parameters giles_loss_Yb_file and giles_gain_Yb_file, respectively. For the other models, the absorption and emission cross-section data files are specified using the parameters absorption_Yb_file and emission_Yb_file, respectively.


Doping Profiles A number of choices are available for specifying the EYCDFA erbium doping profile. These options are available through the doping_model parameter. For both rectangular and Gaussian doping profiles, a doping radius and peak density must be provided via the parameters doping_radius and doping_density, respectively. Alternatively, the user may choose to provide a data file containing the doping density profile by setting doping_model to file and specifying a file name through the parameter doping_file. The file format is described in the appendix.

Similar options are available for specifying the ytterbium doping profile. The associated parameters are Yb_doping_model, Yb_doping_radius, Yb_doping_density, and Yb_doping_file.


Mode Profiles For the calculated_overlap and spatial models, 980-band and 1550-band mode profiles are required. Different mode shapes may be selected using the parameter mode_model. If mode_model is set to lp, then LP01 mode solutions [2] are determined for a fiber with the specified numerical aperture (fiber_NA) and fiber core radius (fiber_core_radius). For all 1550-band signals, a wavelength of 1550 nm is used in the calculation, whereas a 980-nm wavelength is adopted for all 980-band signals.

If a rectangular or Gaussian set of mode shapes is chosen, then relevant mode widths must be specified via the mode_1550_width and mode_980_width parameters. Because the calculated mode shapes are normalized, no additional information is required.





Numerical Settings A number of numerical parameters are available to optimize the numerical solution of the bidirectional rate equations for the pump/signal/ASE power evolution. Adjustment of these parameters may help overcome convergence difficulties encountered during a simulation.

The numerical integration of the power evolution rate equations is handled via a constant-step-size fourth-order Runge-Kutta algorithm. The integration step along the length of the EYCDFA may be adjusted using the z_step parameter. Similarly, the nominal width for the discretized frequency intervals in the 980- and 1550-bands may be set using the spectral_step parameter. As this parameter is set in units of nanometers, an equivalent spectral step in Hertz is calculated internally by the model. The resolution of transverse calculations may also be adjusted by setting the radial_points parameter to the desired number of points in the radial direction.

The bidirectional nature of the power-evolution rate equations requires an iterative solution scheme, the control of which may be achieved through the iterative_damping and convergence_tolerance parameters. The iterative_damping controls the rate at which the solution is allowed to progress to a final answer. Larger values slow down this process, and may be required when convergence is difficult to achieve. Parameter convergence_tolerance is the convergence criterion. Convergence is achieved when the largest change in the power solution between successive iterations is less than convergence_tolerance. To limit the time spent on a poorly converging solution, the maximum number of iterations may be set with the parameter max_iterations.


For some simulations, the iterative solution must be calculated more slowly in order to achieve convergence. In these cases, the parameter progressive_solution should be set to yes, and the parameter progressive_steps set to a suitable value, typically between 1 and 10. The larger the number of steps chosen, the more slowly the iterative solution is calculated.

Finally, when amplifying or attenuating any incoming optical signals, the model applies its calculated gain spectrum to a Fourier-domain representation of these signals. Normally, the spectral variation in the gain across the frequency band of each signal is fully accounted for by setting the parameter gain_application to continuous. However, given the narrow-band nature of most signals relative to typical spectral variations in EYCDFA gain, it sometimes may be easier to use a constant gain value for each signal based on its carrier wavelength. In this case, gain_application should be set to carrier. This approach may prove useful when discontinuities in an optical input’s phase at the signal boundaries lead to anomalous output waveforms.

Reference Plots In order to help study the performance of an EYCDFA within an optical link, a variety of reference plots can be generated by the model in order to study internal power evolution, signal gain, noise figure, and the atomic-manifold population densities. What plots are generated is determined via the parameters power_plots, spectra_plots, density_plots, and gain_nf_plot (by setting them to either yes or no); they may be displayed at the conclusion of a simulation by double-clicking on the EYCDFA’s icon. The units for the displayed data may be selected via the parameters spectral_units, power_units, length_units, radial_units, density_3d_units, and signal_gain_units (for gain and noise figure data). Below we summarize the different plots that may be generated, listing them by their root WinPlot file names.

power_plots


Displays the evolution of total power in the 1550-band signals (1480-nm pumps excluded) and ASE along the length of the EYCDFA. Both forward and backward directions of propagation are included.




Displays the evolution of signal gain along the EYCDFA.


Displays a contour plot of the complete forward-propagating 1550-band power spectra solution along the length of the EYCDFA.



spectra_plots





• 980_ase_output_power_spectrum.pcs:


Displays the output 980-band ASE power spectrum.

• 980_ase_output_power_spectrum_bwd.pcs:

Bidirectional mode only. Displays the backward output 980-band ASE power spectrum.






Displays the internal ASE power spectra at both ends of the EYCDFA.

• ase980_power_spectra.pcs:

Displays the internal 980-band ASE power spectra at both ends of the EYCDFA.

• gain.pcs:


• gain_bwd.pcs:







Displays the overall 1550-band noise figure, calculated as a function of signal gain ( )G υ

and output ASE spectral density ( )ASEρ ν [1]:

( )1 1

( )ASENF

G hρ ν

ν ν = ⋅ +


Bidirectional mode only. Displays the overall backward 1550-band noise figure density_plots


Displays the average erbium population densities along the length of the EYCDFA

• n2(r)_vs_z.pcs:

(spatial model only) Displays a contour plot of the transverse population density along the EYCDFA for the second erbium atomic level (4I13/2).

gain_nf_plot

• gain_scan.pcs:

Displays the gain of the EYCDFA at the wavelength specified via the parameter target_wavelength. If the topology is simulated via a parameter scan, then this plot shows the


gain as a function of the scanned parameters. This plot can be useful for gain optimization of the EYCDFA.


Bidirectional mode only. Displays the backward gain of the EYCDFA at the wavelength specified via the parameter target_wavelength.


Displays the noise figure of the EYCDFA at the wavelength specified via the parameter target_wavelength. If the topology is simulated via a parameter scan, then this plot shows the noise figure as a function of the scanned parameters.


Bidirectional mode only. Displays the backward noise figure of the EYCDFA at the wavelength specified via the parameter target_wavelength.

Test Functions In selecting the input parameters for a particular EYCDFA, it may at times be necessary to visualize the various input spectra, doping profiles, and mode shapes. By setting the test_function parameter to the desired output and clicking the Test button in the component-parameter editing window, the user may display the EYCDFA characteristics summarized below. Furthermore, the default plot ranges for each characteristic may be overridden by setting test_default_settings to no, and specifying values for test_function_x_low, test_function_x_high, and test_function_points (the number of data points to plot). The appropriate units for each characteristics’ overrides are listed in parentheses below. Units for the displayed plots may be specified using the parameters spectral_units, radial_units, cross_section_units, density_3d_units, loss_units, and reflectivity_units.


Plots the 1550-band erbium gain/loss (for the Giles model), or emission/absorption cross section spectra (for the other models).


Plots the 980-band erbium loss (for the Giles model), or absorption cross section spectrum (for the other models).

• Yb_spectra(nm):

Plots the 980-band ytterbium gain/loss (for the Giles model), or emission/absorption cross section spectra (for the other models).

• doping(um):

Plots the erbium doping profile as a function of radius.

• Yb_doping(um):

Plots the ytterbium doping profile as a function of radius.

• modes(um):

Plots the 1550- and 980-band normalized mode profiles as functions of radius.



• mirror(nm):






[4] M. Achtenhagen, R. J. Beeson, F. Pan, B. Nyman, and A. Hardy, “Gain and noise in ytterbium-sensitized erbium-doped fiber amplifiers: Measurements and simulations,” Journal of Lightwave Technology, vol. 19, no. 10, pp. 1521-1526, October 2001.

[5] E. Yahel and A. A. Hardy, “Modeling and optimization of short Er3+-Yb3+ codoped fiber lasers,” IEEE Journal of Quantum Electronics, vol. 39, no, 11, pp. 1444-1451, November 2003.

Properties




#2: Backward-propagating optical signals (bidirectional mode only)


constant_overlap, calculated_overlap, spatial, cladding_pumped








metastable_lifetime_Yb double 1 0 ≤ x ≤ 1e32 ms

include_third_level enumerated yes yes, no

third_level_lifetime double 10e-3 0 ≤ x ≤ 1e32 ms

giles_lifetime_ratio double 10e-3 0 ≤ x ≤ 1e32

R61 double 1e-22 0 ≤ x ≤ 1e32 m3/s

R61_giles double 1e17 0 ≤ x ≤ 1e32 m-1s-1

R35 double 1e-22 0 ≤ x ≤ 1e32 m3/s


R35_giles double 1e17 0 ≤ x ≤ 1e32 m-1s-1


fiber_saturation_param_Yb double 3e17 0 ≤ x ≤ 1e32 m-1s-1



overlap_Yb double 0.7 0 ≤ x ≤ 1e32

Acore double 3.14e-12 0 ≤ x ≤ 1e32 m2

Aclad double 7.068e-8 0 ≤ x ≤ 1e32 m2

upconversion_coeff double 0.0 0 ≤ x ≤ 1e32 m3/s
























spectra_Yb_model enumerated rectangular rectangular, file

giles_loss_Yb double 10.0 0 ≤ x ≤ 1e32 dB/m

giles_gain_Yb double 10.0 0 ≤ x ≤ 1e32 dB/m

absorption_Yb double 1e-24 0 ≤ x ≤ 1e32 m2

emission_Yb double 1e-24 0 ≤ x ≤ 1e32 m2

spectra_Yb_center double 1000 0 ≤ x ≤ 1e32 nm

spectra_Yb_width double 200 0 ≤ x ≤ 1e32 nm

giles_loss_Yb_file string


giles_gain_Yb_file string

absorption_Yb_file string

emission_Yb_file string

doping_model enumerated rectangular rectangular, Gaussian, file



doping_file string

mode_model enumerated lp lp, Gaussian, rectangular, file







Yb_doping_model enumerated rectangular rectangular, gaussian, file

Yb_doping_radius double 1 0 ≤ x ≤ 1e32 µm

Yb_doping_density double 1e20 0 ≤ x ≤ 1e32 cm-3

Yb_doping_file string

mirror_configuration enumerated no_mirror no_mirror, forward_mirror, backward_mirror, signal_mirror

mirror_model enumerated rectangular rectangular, gaussian, file




mirror_file string











progressive_solution enumerated no yes, no

progressive_steps integer 1 1 ≤ x ≤ 100

gain_application enumerated continuous Continuous, carrier
















test_function enumerated 1550_spectra(nm) 1550_spectra(nm), 980_spectra(nm), Yb_spectra(nm), doping(um), Yb_doping(um), modes(um), background_loss(nm), mirror(nm)






simulation_mode model-level options bidirectional switch for simulating bidirectional signal propagation forward_input_loss forward-propagating input coupling backward_input_loss backward-propagating input coupling output_loss output coupling backward_output_loss output coupling of backward signal when simulating bidirectional operation length EYCDFA length metastable_lifetime erbium metastable-level lifetime metastable_lifetime_Yb ytterbium metastable-level lifetime include_third_level option for including third erbium level (4I11/2) third_level_lifetime lifetime of third erbium level (4I11/2) giles_lifetime_ratio ratio of the lifetimes for the third and second erbium levels (τ32/τ21) R61 ytterbium-erbium cross-relaxation coefficient R61_giles Giles ytterbium-erbium cross-relaxation coefficient R35 erbium-ytterbium cross-relaxation coefficient


R35_giles Giles erbium-ytterbium cross-relaxation coefficient fiber_saturation_param Giles-model erbium fiber-saturation parameter fiber_saturation_param_Yb Giles-model ytterbium fiber-saturation parameter overlap_1550 constant 1550-band erbium overlap factor overlap_980 constant 980-band erbium overlap factor overlap_Yb constant 980-band ytterbium overlap factor Acore area of core for double-clad devices Aclad area of inner cladding for double-clad devices upconversion_coeff erbium upconversion coefficient for non-Giles models giles_upconversion_coeff erbium upconversion coefficient for Giles model spectra_1550_model 1550-band erbium signal spectra options spectra_1550_auto_limit option for automatically determining 1550-band spectral range spectra_1550_low lowest wavelength for 1550-band spectra spectra_1550_high highest wavelength for 1550-band spectra giles_loss_1550_file user-specified 1550-band Giles loss-spectrum data file for erbium giles_gain_1550_file user-specified 1550-band Giles gain-spectrum data file for erbium absorption_1550_file user-specified 1550-band erbium absorption cross-section data file emission_1550_file user-specified 1550-band erbium emission cross-section data file spectra_980_model 980-band erbium signal spectrum options spectra_980_auto_limit option for automatically determining 980-band spectral range spectra_980_low lowest wavelength for 980-band spectra spectra_980_high highest wavelength for 980-band spectra giles_loss_980 center-value of 980-band Giles loss spectrum for erbium absorption_980 center-value of 980-band erbium absorption cross-section spectra_980_center center-wavelength of 980-band erbium spectrum spectra_980_width bandwidth of 980-band erbium spectrum giles_loss_980_file user-specified 980-band Giles loss-spectrum data file for erbium absorption_980_file user-specified 980-band erbium absorption cross-section data file background_loss_model background-loss options background_loss 1550-band background-loss value background_loss_980 980-band background-loss value background_loss_file user-specified background-loss data file spectra_Yb_model 980-band ytterbium signal spectrum options giles_loss_Yb center-value of 980-band Giles loss spectrum for ytterbium giles_gain_Yb center-value of 980-band Giles gain spectrum for ytterbium absorption_Yb center-value of 980-band ytterbium absorption cross-section emission_Yb center-value of 980-band ytterbium emission cross-section spectra_Yb_center center-wavelength of 980-band ytterbium spectrum spectra_Yb_width bandwidth of 980-band ytterbium spectrum giles_loss_Yb_file user-specified 980-band Giles loss-spectrum data file for ytterbium giles_gain_Yb_file user-specified 980-band Giles gain-spectrum data file for ytterbium absorption_Yb_file user-specified 980-band ytterbium absorption cross-section data file emission_Yb_file user-specified 980-band ytterbium emission cross-section data file doping_model erbium-doping density options doping_radius erbium doping radius


doping_density center erbium doping density (at r = 0) doping_file user-specified erbium doping-profile data file mode_model optical mode shape options fiber_NA fiber numerical aperture fiber_core_radius fiber core radius mode_1550_width 1550-band optical-mode width mode_980_width 980-band optical-mode width mode_1550_file user-specified 1550-band mode-profile data file mode_980_file user-specified 980-band mode-profile data file Yb_doping_model ytterbium-doping density options Yb_doping_radius ytterbium doping radius Yb_doping_density center ytterbium doping density (at r = 0) Yb_doping_file user-specified ytterbium doping-profile data file mirror_configuration pump/signal recycling options mirror_model mirror power-reflectivity spectrum options mirror_reflectivity mirror power reflectivity at center wavelength mirror_center mirror-spectrum center wavelength mirror_bandwidth mirror-spectrum bandwidth mirror_file user-specified mirror-spectrum data file output_reflected switch for including any reflected backward inputs in the output output_pumps switch for including pump signals in the output inject_ASE switch for injecting ASE within the EYCDFA ASE_polarization switch for injecting ASE into one or two polarizations z_step integration step along EYCDFA length spectral_step spacing between wavelengths in discretized power spectra iterative_damping iterative damping factor for numerical solution convergence_tolerance convergence tolerance for numerical solution max_iterations maximum number of iterations during bidirectional solution radial_points number of radial points progressive_solution option for progressive iterative solution progressive_steps number of iterations to use during progressive solution gain_application method of applying gain to each input signal power_plots switch for power-solution reference plots spectra_plots switch for power-spectra reference plots density_plots switch for level-density reference plots gain_nf_plot switch for plots of gain and noise-figure scans target_wavelength target wavelength for plots of gain and noise-figure scans spectral_units units for spectral data power_units units for power data length_units units for positional data radial_units units for radial data cross_section_units units for cross-section data density_3d_units units for density data loss_units units for loss data signal_gain_units units for gain and noise figure results


reflectivity_units units for mirror data test_function test-function output selection test_default_settings switch for plotting test-function output using default settings test_function_x_low user-specified lowest x-value for test-function output test_function_x_high user-specified highest x-value for test-function output test_function_points user-specified number of points for test-function output


Appendix: File formats For all data files, the X-values should be monotonically increasing or decreasing. Furthermore, the field <num_pts> specifies the number of data lines in the file, while the choice of settings for the various unit fields are as follows:







Giles Gain/Loss Spectra Data files with gain and loss spectra for the Giles model are specified through the parameters giles_gain_1550_file, giles_loss_1550_file, giles_loss_980_file, giles_gain_Yb_file, and giles_loss_Yb_file. The X-values are in units of frequency, and the Y-values are in units of loss per distance.


<num_pts>





...


5

1450 0.3

1500 0.6

1530 2.0

1550 1.2

1600 0.4

Absorption/Emission Cross Sections Data files with absorption/emission cross sections are specified through the parameters absorption_1550_file, emission_1550_file, absorption_980_file, absorption_Yb_file, and emission_Yb_file. The X-values are in units of frequency, and the Y-values are in units of area.



<num_pts>





...


5

1450 0.7e-25

1500 2.4e-25

1530 7.0e-25

1550 4.3e-25

1600 0.2e-25

Doping Profile User-specified doping profiles may be specified via the parameters doping_file and Yb_doping_file. The X-values are in units of distance, and the Y-values are in units of cubic density.


<num_pts>





...


5

0 1.00e19

0.25 1.00e19

0.50 1.00e19

0.75 0.50e19

1.00 0.25e19

Mode Profiles Data files with mode shapes may be specified using the parameters mode_1550_file and mode_980_file. The X-values are in units of distance, and the Y-values are considered unitless (the mode profiles are automatically normalized by the model).



<num_pts>





...


5

0 1

1 0.7

2 0.4

3 0.25

5 0.15



<num_pts>





...


5

1.40 0.25

1.45 0.21

1.50 0.19

1.55 0.17

1.60 0.21

Mirror Reflectivity Spectrum A data file with the mirror reflectivity spectrum may be specified using the parameter mirror_file. The X-values are in units of frequency, and the Y-values are in units of power reflectivity.



<num_pts>





...


5

970 10

975 100

980 100

985 100

990 10


Semiconductor Optical Amplifier (SOA)

This module simulates a Semiconductor Optical Amplifier (SOA). The SOA is a highly nonlinear device. It can be used not only for signal amplification but also for other optical signal processing applications, such as wavelength converting, switching and optical time domain demultiplexing [1].

The SOA is modeled as a travelling wave amplifier. It takes into consideration the time dependence of the gain caused by the saturation effect and the time-dependent phase change due to the gain-index coupling. Figure 1 shows a typical SOA geometry.

Figure 1: Typical SOA geometry

Model Description In the active region of amplifier shown in Fig.1, the material gain coefficient g(N) (per unit length) is dependent on the carrier (electrons and holes) density N. Their relation can be described as:

22 )()()()()( ptranptran NNSNNdNdgNg λλκλλκ −−−=−−−=

(1)

where dg/dN is the differential gain which defines the slope of the dependence of g(N) on N assuming linear gain model and will be denoted as S next (). The parameter Ntran (Carrier_Density_Transp) is the carrier density at transparency point. In Eq.(1) the material gain per unit length assumes a parabolic shape for the gain dependence on wavelength. Parameter κ (Curvature_SpectralGain) is the curvature of spectral gain, λ is wavelength, λp (Wavelength_PeakGain) is the peak gain wavelength. The flag SpectralGainShape allows to switch between flat and parabolic spectral gain shape models. If the flag set to Flat or setting κ = 0 (default), the material gain will be independent of wavelength. The parameters S, Ntran, κ and pλ will be input by users.


The total gain coefficient of the amplifier is related to the optical confinement factor Γ defined as a fraction of the mode power within the active layer (Confinement_Factor) and to the waveguide scattering loss αs (Internal_Loss) as,

stot NgNg α−Γ= )()(

(2)

where Γ and αs are input parameters. Then the total gain (in the sense of power) at the location z (along the active region length L), G(N,z), can be calculated as

]))(exp[(])(exp[),( zNgzNgzNG stot α−Γ==

(3)

Note, in Eq.(3) a constant carrier density over the amplifier length was assumed. Using Eq.(3) the average light power Pav(N,t) over the amplifier length can be calculated as

s

sin

L

sin

L

inav

NgLNg

LtP

dzzNgL

tPdzzNGtP

LtNP

αα

α

−Γ−−Γ

⋅=

−Γ== ∫∫

)(1]))(exp[()(

]))(exp[()(

),()(1),(00

(4)

The dynamic rate equation for the carrier density N, which is a function of time t (hence, from now on, N is replaced by N(t)), can be written as

Vhf

LttNPtNgtNR

qVI

dttdN av )),(())((

))(()( Γ−−=

(5)

where I is the injection current (Pump_Current) into the active region, which can be determined by the bias voltage Vf and the material characteristics of the amplifier. Here I is an input parameter of the model, q is the electron charge, V = L×w×d is the volume of the active region, L (Length) is length, w (Width) is width and d (Thickness) is height, respectively. L, w and d are inputs to the model. The last term of Eq.(5) describes a depletion of inversion due to the stimulated emission. This stimulated emission can be caused by the input signals and the entering ASE noise. The depletion of inversion due to the internally generated ASE noise, i.e. the spontaneous emission has not been accounted for in Eq.(5). f denotes the light frequency and h is the Planck constant. The second term of Eq.(5), R(N(t)), recombination rate can be expressed as

32 )()()())(( tCNtBNtANtNR ++=

(6)

where A (Recomb_ConstA), B (Recomb_ConstB), and C (Recomb_ConstC) are constants representing the different type of recombinations. The inverse of linear term coefficnet A is a carrier lifetime τ, the quadratic term represents radiative recombination, and cubic - Auger recombination.

Taking Eqs.(1), (4), and (6) into (5), Eq.(5) can be written as


VhfNtNS

LNtNStPNtNS

tCNtBNtANqVI

dttdN

sptran

sptraninptran αλλκ

αλλκλλκ

−−−−Γ−−−−−Γ

−−−Γ−

−−−=

])())(([1)])())(([exp(

)(])())(([

)()()()(

2

22

32

(7)

In Eq.(7) N(t) is the only unknown quantity, which will be solved numerically with an initial carrier density given by user.

Once N(t) is obtained the optical field output, )(tEout

r is determined by

( ) ( )

+= LNgjtEtE totinout )()1(21exp α

rr,

( ) sptrantot NtNStNg αλλκ −−−−Γ= ))())((()( 2

(8)

where α (Linewidth_Enhance) is the linewidth enhancement factor (input by user), which accounts for the coupling between the gain and the refractive index of the active layer.

In case of parabolic again shape the FWHM bandwidth for the total gain can be estimated as:

LΓ

=∆κ

λ 2ln22

Note that Eq.(7) is only valid for a single channel signal (single frequency component, strictly). For multi-

channel signals, the term f

tPin )( in (7) should be replaced by ∑

k k

kin

ftP )(

, where fk and )(tP kin are the

central frequency and the input signal power of the k-th channel respectively.

Noise Treatment The model takes into account noise in the incoming optical signal. The contribution of the ASE noise in bins (as opposed to the noise contained within the signal, which will be treated as the signal) to Eq.(7) can be accounted for by replacing Pin(t) by the averaged input noise power. Specifically, in analogy with the contribution to Eq.(7) from the signal, the contribution to Eq.(7) from the noise can be expressed as [2]:


dff

fRVhNtNS

dfLf

fRVhNtNS

LNtNS

dff

fRVhNtNS

NtNS

dfLf

fRVhNtNS

LNtNSNtNS

dfVhf

NtNSLNtNS

fRNtNS

ASEp

stran

pASEp

stran

stran

ASE

stran

tran

pASE

stran

strantran

sptran

sptranASEptran

∫

∫

∫

∫

∫

−Γ−−Γ

+

−Γ−−Γ

−−Γ−−Γ

−

−−Γ−Γ

−

−Γ−−−Γ

−−Γ−Γ≈

−−−−Γ−−−−−Γ

−−−Γ

)()()])(([

1

))(exp()()(

)])(([])))((exp[(

)()])(([

)])(([

))(exp()(

)])(([])))((exp[()])(([

])())(([1)])())(([exp(

)(])())(([

2

22

2

2

22

λλκα

λλκλλκ

αα

α

λλκα

α

αλλκαλλκ

λλκ

(9)

where RASE is the spectral density of the ASE entering the amplifier. Hence, in this SOA model, the noise will saturate the amplifier. The output noise can be evaluated as

])(exp[]))(exp[(

])())(([exp2

_

2__

LtSNLSNP

LLNtNSPP

sptraninn

sptraninnoutn

Γ−−Γ−Γ−=

−−−−Γ=

αλλκ

αλλκ

(10)

where Pn_in and Pn_out are input and output noise power respectively and the angle brackets denotes the time averaging.

In this model the gain saturation caused by the entering ASE has been taken into consideration, but the internally generated ASE noise is neglected. To take into account internal noise we added phenomeno-logical rather than physical model where user has to specify Noise Figure of amplifier. To activate this model user has to switch flag includeNoise to Yes and specify value of Noise Figure in dB in parameter noiseFigure. If the incoming optical signal has a noise component then the gain apllied to the noise will be increased by Noise Figure value. However, if the input signal has no noise or noise is contained in signal (i.e. stochastically defined) then the noise figure model will not be applied.

Simulation Techniques To solve rate equation for carrier density (Eqs.(5) or (7)) we aplly Runge-Kutta method. Two Runge-Kutta equation solvers are available: Runge-Kutta 4th-order and Runge-Kutta 5th-order Adaptive. In the Runge-Kutta4th method, the step size equals to the signal step size (in time domain), but this may not be the case for the Runge-Kutta5th with adaptive step size method. For the case when the signal sampling rate is high the Runge-Kutta5th with adaptive step size method may consume less time solving the rate equation compared to the Runge-Kutta4th method. In case of and Runge-Kutta 5th-order Adaptive method user can specify relative expected error (Relative_Error), initial step size (Stepsize_Initial), minimum step size allowed (Stepsize_Min), and maximum number of steps allowed (MAXSTP).

Additional Notes Usually the operation of the SOA’s depends on the polarization state of light. This SOA gain model assumes it is polarization independent. However, if a user knows expected PDG from measurement or


device specifications he can enter its value into parameter PDG in dB units and then the SOA gain applied to x- and y-poalrization components will be accordingly modified.

The input to the SOA model can be a single- or multi-channel signals. If the input is multi-channel signals, the model requires that all the signals have identical time steps and number of data points.

The model could be also used to simulate the nearly travelling wave optical amplifier with low facet reflectivity (R = 10^(-4 ÷ -5)). In this model the carrier density over the length of amplifier is assumed to be constant. .

The saturation power of SOA can be computed from stationary solution fo the rate equation (i.e. left part of Eq.(5) is zero). In the case of flat gain shape, and negligible radiative and Auger recombination components (B, C = 0) the saturation power can estimated using the following equation [3]:

2ln_ SwdEP satout τΓ

=

(11)

Where E is the photon energy, and τ (equal to 1/A) is the carrier lifetime.

References [1] M. J. Connelly, Semiconductor Optical Amplifiers, (Boston, Kluwer Acdemic Publishers, 2002)

[2] L. Gillner, E. Goobar, L. Thylen, M. Gustavsson, “Semiconductor laser amplifier optimization: an analytical and experimental study”, IEEE J of Quantum Electronics 25, 1822-1827 (1989).

[3] M.J.O’Mahony, “Semiconductor laser optical amplifiers for use in future fiber systems”, J. of Lightwave Technology 6, 531-544 (1988).

Properties



Parameter Values Name Type Default Range Unit Pump_Current Double 0.15 0.0~1.0e32 A

SpectralGainShape Enumerated Flat Flat, Parabolic None

Curvature_SpectralGain Double 0.0 0.0~1.0e32 m^(-3)

Wavelength_PeakGain Double 1.52e-6 0.0~1.0e32 m

Linewidth_Enhance Double 5.0 -1.0e32~1.0e32 None

PDG Double 0 -1000~1000 dB

includeNoise Enumerated No Yes, No None noiseFigure Double 6.0 3~1.0e32 dB

Confinement_Factor Double 0.15 0.0~1.0 None

Internal_Loss Double 4000.0 0.0~1.0e32 m^(-1)


Gain_Slope Double 2.78e-20 0.0~1.0e32 m^2

Recomb_ConstA Double 1.43e8 0.0~1.0e32 s^(-1) Recomb_ConstB Double 1.0e-16 0.0~1.0e32 m^(3)s^(-1)

Recomb_ConstC Double 3.0e-41 0.0~1.0e32 m^(6)s^(-1)

Carrier_Density_Start Double 3.0e24 0.0~1.0e32 m^(-3)

Carrier_Density_Transp Double 1.4e24 0.0~1.0e32 m^(-3) Length Double 5.0e-4 0.0~1.0e32 m

Width Double 3.0e-6 0.0~1.0e32 m

Thickness Double 8.0e-8 0.0~1.0e32 m

Equation_Solver Enumerated Runge-Kutta5thAdapt

Runge-Kutta5thAdapt, Runge-Kutta4th

None

Relative_Error Double 1.0e-8 0.0~1.0e32 None

Stepsize_Initial Double 1.0e-12 0.0~1.0e32 s Stepsize_Min Double 0.0 0.0~1.0e32 s

MAXSTP Integer 100000 1~1e8 None

Parameter Descriptions Pump_Current bias current SpectralGainShape switch between Flat and Parabolic gain models Curvature_SpectralGain curvature of spectral gain curve Wavelength_PeakGain wavelength at peak gain Linewidth_Enhance linewidth enhance factor PDG PDG in dB includeNoise swith On/Off internal noise noiseFigure Noise Figure in dB Confinement_Factor confinement factor Internal_Loss internal loss Gain_Slope slope of gain curve Recomb_ConstA recombination coefficient A Recomb_ConstB recombination coefficient B Recomb_ConstC recombination coefficient C Carrier_Density_Transp carrier density at transpency Carrier_Density_Start initial carrier density Length active region length Width active region width Thickness active region thickness Equation_Solver methods to solve the carrier density rate equations Relative_Error relative error expected Stepsize_Initial initial step size tried Stepsize_Min minimum step size allowed MAXSTP maximum number of steps allowed


Controlled SOA

This module simulates a Controlled Semiconductor Optical Amplifier (SOA). The model is identical to the standard SOA model described earlier except it has now an input port for electrical signal. The total injection current is now a sum of internal bias current (Pump_Current) and external current:

I(t) = Ibias + Iext (t) (1)

The rate equation for carier dynamics (Eq.(5) of SOA model description) will be modified with time-dependent injectrion current I(t) instead of constant bias current I.

Properties




Parameter Values Name Type Default Range Unit Pump_Current Double 0.15 0.0~1.0e32 A

SpectralGainShape Enumerated Flat Flat, Parabolic None

Curvature_SpectralGain Double 0.0 0.0~1.0e32 m^(-3)

Wavelength_PeakGain Double 1.52e-6 0.0~1.0e32 m Linewidth_Enhance Double 5.0 -1.0e32~1.0e32 None

PDG Double 0 -1000~1000 dB includeNoise Enumerated No Yes, No None

noiseFigure Double 6.0 3~1.0e32 dB

Confinement_Factor Double 0.15 0.0~1.0 None

Internal_Loss Double 4000.0 0.0~1.0e32 m^(-1) Gain_Slope Double 2.78e-20 0.0~1.0e32 m^2

Recomb_ConstA Double 1.43e8 0.0~1.0e32 s^(-1)

Recomb_ConstB Double 1.0e-16 0.0~1.0e32 m^(3)s^(-1)

Recomb_ConstC Double 3.0e-41 0.0~1.0e32 m^(6)s^(-1) Carrier_Density_Start Double 3.0e24 0.0~1.0e32 m^(-3)

Carrier_Density_Transp Double 1.4e24 0.0~1.0e32 m^(-3)

Length Double 5.0e-4 0.0~1.0e32 m

Width Double 3.0e-6 0.0~1.0e32 m


Thickness Double 8.0e-8 0.0~1.0e32 m

Equation_Solver Enumerated Runge-Kutta5thAdapt

Runge-Kutta5thAdapt, Runge-Kutta4th

None

Relative_Error Double 1.0e-8 0.0~1.0e32 None

Stepsize_Initial Double 1.0e-12 0.0~1.0e32 s Stepsize_Min Double 0.0 0.0~1.0e32 s

MAXSTP Integer 100000 1~1e8 None

Parameter Descriptions Pump_Current bias current SpectralGainShape switch between Flat and Parabolic gain models Curvature_SpectralGain curvature of spectral gain curve Wavelength_PeakGain wavelength at peak gain Linewidth_Enhance linewidth enhance factor PDG PDG in dB includeNoise swith On/Off internal noise noiseFigure Noise Figure in dB Confinement_Factor confinement factor Internal_Loss internal loss Gain_Slope slope of gain curve Recomb_ConstA recombination coefficient A Recomb_ConstB recombination coefficient B Recomb_ConstC recombination coefficient C Carrier_Density_Transp carrier density at transpency Carrier_Density_Start initial carrier density Length active region length Width active region width Thickness active region thickness Equation_Solver methods to solve the carrier density rate equations Relative_Error relative error expected Stepsize_Initial initial step size tried Stepsize_Min minimum step size allowed MAXSTP maximum number of steps allowed


Semiconductor Optical Amplifier (SOA) – Comprehensive Model

This module simulates a Semiconductor Optical Amplifier (SOA), as well as a Reflective SOA (RSOA). These devices are highly nonlinear and can be used for signal amplification, signal modulation, and optical signal processing applications such as wavelength conversion, switching and optical time domain demultiplexing [1].

The user selects whether to model an SOA or an RSOA via the parameter RSOA. With this parameter set to No, the model simulates an SOA. Figure 1 shows a typical device geometry, with optical signals input at one facet and output at the other. In this model, we treat the SOA as a traveling wave amplifier, with negligible facet reflections. The model takes into consideration the time dependence of the gain caused by the saturation effect, the time-dependent phase change due to the gain-index coupling, and the internal generation of amplified spontaneous emission (ASE) noise. With RSOA set to Yes, the model simulates an RSOA. In this case, the input and output optical signals use the same facet, and the opposite facet is made highly reflective. In this device, signals experience two passes through the SOA before exiting. The user specifies the RSOA power reflectivity via the parameter Reflectivity.

The model has two input ports and one output port. The first input accepts optical signals, which can consist of one or more optical channels. The second input accepts an optional electrical modulation current. Finally, the output port generates the resulting output optical signal. The user controls the input and output coupling losses, in dB, via the parameters Input_Loss and Output_Loss. Note that the model does not support cw optical signals, and all signals input to the SOA must have identical time steps and number of data points. If the user connects an electrical modulation current (Imod) to the SOA, then it is added to the specified bias current Ibias (Pump_Current) to form the total injection current:

( ) ( )bias modI t I I t= + (1)

Figure 1. Typical SOA geometry. In an RSOA, light enters and exits via the same facet, and the opposite facet is highly reflective.


Model Description The comprehensive SOA model uses a reservoir approach as described in [2]-[4] and largely equivalent to the models presented in [5]-[7]. Instead of treating the SOA cavity as a distributed element, the model uses a lumped approach that assumes an average carrier density along the length of the cavity. The model is derived from the following equations for the signal power propagation, ASE power propagation, and local carrier density:

( )ii s i

dP g Pdz

α±

±= ± Γ − ⋅ (2)

( ) ,j

j s j j sp j

dAg A R

dzα ω

±±= ± Γ − ⋅ ± h (3)

( ) ( ) ( )2i i i j j j

i ji j

g P P g A AdN I R Ndt qV A A

ηω ω

+ − + −⋅ + ⋅ +Γ Γ= − − −∑ ∑h h

(4)

Equation (2) models the signal power propagation down the length z of the SOA cavity, with iP± representing the forward (+) and backward (-) signal powers at wavelength λi. Γ (Confine_Factor)is the optical confinement factor, while gi is the material gain at λi and αs is the background loss (both of which are functions of carrier density, to be discussed below).

Equation (3) models the propagation of the forward (+) and backward (-) ASE noise power jA± in a bin of width ∆νj centered about λj. It is equivalent to Equation (2), with an additional term Rsp,j for ASE noise generation (also a function of carrier density, to be discussed below). ωj is the optical radial frequency. Note that the inclusion of Rsp,j is controlled via the parameter includeNoise. If it is set to Yes, then the user must also specify a wavelength range in meters for the ASE generation (ASE_min_wavelength to ASE_max_wavelength) along with a bin width (ASE_bin_width).

Finally, Eq. (4) is the rate equation for the local carrier density N along the length of the cavity, where the two summations account for stimulated emission due to optical signals and ASE, respectively. η (Curr_Inj_Efficiency) is the current injection efficiency, q is the electron charge, I(t) is the total injection current, V is the total cavity volume (equal to L×w×d ), L (Length) is the cavity length, w (Width) is the cavity width, d (Thickness) is the cavity thickness, and A is the cavity cross-sectional area (equal to w×d). The carrier recombination R(N) is calculated as

2 3( )R N A N B N C N= ⋅ + ⋅ + ⋅ (5)

where A (Recomb_ConstA) is the linear recombination coefficient, B (Recomb_ConstB) is the quadratic recombination coefficient, and C (Recomb_ConstC) is the Auger recombination coefficient.

These equations can be simplified considerably via the following procedure. First, we substitute Eqs. (2) and (3) into (4). Next, we integrate over the length of the amplifier and divide through by the length. Finally, we assume a constant carrier density along the cavity length in order to properly calculate the output powers as a function of the input, as well as the average powers along the cavity. This results in the following set of SOA model equations.

First, the reservoir rate equation for the average carrier density N is

( )( )( )

( )( )( )

( ) ( )

,

, ,2

11

12 4 1

i s

j s

j s

g Lin i i

i i i s

g Lj in g Lj j sp j

j sj jj j s j s

P e gdN I R Ndt qV V g

A e g g Re g L

V g V g

α

α

α

ω α

αω α α

Γ −

Γ −

Γ −

⋅ − Γ= − − ⋅ Γ −

⋅ − Γ Γ − ⋅ − ⋅ − − Γ − Γ − Γ −

∑

∑ ∑

h

h

(6)

where Pin,i is the time-dependent input signal power at λi and Ain,j is the input ASE power at λj. For simplicity, instead of using the average of R(N) along the cavity length, we treat the recombination as a function of the average carrier density itself. Similar considerations apply to calculation of the average value for Rsp,j.

The output signal and ASE powers are calculated as


( ), ,

i sg Lout i in iP P e αΓ −= ⋅ (7)

( )( )

( ),, , 1j s j sg L g Lj sp j

out j in jj s

RA A e e

gα αω

αΓ − Γ − = ⋅ + ⋅ − Γ −

h (8)

where Pout,i is the time-dependent output signal power at λi, Aout,j is the output ASE power at λj, and the brackets denote time averaging.

The model also accounts for phase changes due to variations in the cavity carrier density, i.e. chirp. In this case, the output phase φout,i as a function of input phase φin,i is

, ,12out i in i ig Lφ φ α= − ⋅Γ (9)

where α (Linewidth_Enhance) is the linewidth enhancement factor.

Taking into account Eqs. (7) and (9), the time-dependent output signal at λi is thus

,, ,

out ijout i out iE P e φ= (10)

Following a similar procedure, the set of equations for an RSOA with a mirror power reflectivity R2 (where the subscript ‘2’ is used to emphasize that the input facet is nonreflective) are

( )( )( ) ( )

( )( ) ( )

( )( ) ( ) ( ) ( )

,

2

,

2

2, 2 222

11 1

12 1

4 1 12 2

i s

i s

j s

j s

j s j s

g Lin i g Li

i i i s

g Lin j g Lj

j j j s

g L gj sp jj s

j j s

P e gdN I R N R edt qV V g

A e gR e

V g

g R R Re R g L e LV g

αα

α

α

α α

ω α

ω α

αα

Γ −

Γ −

Γ −

Γ −

Γ − Γ −

⋅ − Γ = − − ⋅ ⋅ + ⋅ Γ −

⋅ − Γ − ⋅ ⋅ + ⋅ Γ − Γ − ⋅ ⋅ − − + − Γ − + ⋅ Γ −

∑

∑

∑

h

h (11)

( )2, 2 ,

i sg Lout i in iP R P e αΓ −= ⋅ ⋅ (12)

, ,out i in i ig Lφ φ α= − ⋅Γ (13)

( )( )

( ) ( )2 ,, 2 , 21 1j s j s j sg L g L g Lj sp j

out j in jj s

RA R A e e R e

gα α αω

αΓ − Γ − Γ − = ⋅ ⋅ + ⋅ − ⋅ + ⋅ Γ −

h (14)

Equation (10) remains the same as before.

Gain, Loss, and Spontaneous Emission Models In the active region of the amplifier, the material gain coefficient g(N,λ) and the spontaneous emission Rsp(N,λ) are dependent on the carrier density N and the wavelength λ. The SOA model provides a number of alternatives for modeling these features, which can be selected via the parameter SpectralGainShape. In all cases, the background loss is modeled with a linear carrier-density dependence [2],[8]-[9]:

0 1s K K Nα = + Γ (15)

where K0 (Internal_Loss) is the background loss in the absence of carriers and K1 (Internal_Loss_Slope) is proportional to the slope of the background loss versus carrier density.

Flat, Parabolic, and Cubic Gain


The Flat, Parabolic, and Cubic Gain models are all based on the following equations for the material gain and gain peak wavelength [5],[10]-[12], which model the gain’s parabolic and cubic dependence on wavelength:

( ) ( ) ( )2 3

1 3( , ) ( ) ( )o p pg N a N N a N a Nλ λ λ λ λ= ⋅ − − ⋅ − + ⋅ − (16)

( )2( )p po refN a N Nλ λ= − ⋅ − (17)

where a (Gain_Slope) is the linear gain coefficient, No (Carrier_Density_Transp) is the carrier transparency density, a1 (Curvature_SpectralGain) is the quadratic coefficient of the gain’s spectral dependence, a3 (CubicCoeff_SpectralGain) is the cubic coefficient of the gain’s spectral dependence, λpo (Wavelength_PeakGain) is the peak-gain wavelength at the reference density Nref (Wavelength_PeakGain_RefDensity), and a2 (Wavelength_PeakGain_Slope) is the slope of the peak-gain wavelength versus carrier density.

When SpectralGainShape is set to Flat, the coefficients a1 through a3 are set equal to zero and the gain is simply a linear function of the carrier density. When SpectralGainShape is set to Parabolic, only a3is set equal to zero, and the gain then includes both the carrier-density dependence of the peak gain wavelength as well as the quadratic dependence on wavelength; this approach is a common one for dealing with an SOA’s spectral gain dependence [5],[10]. Finally, if SpectralGainShape is set to Cubic, all terms are included and the model accounts for the gain’s cubic dependence on wavelength as well [11]-[12].

In all three cases, the spontaneous emission is modeled via the following expression [4]-[5],[13]:

( , ) ( , )sp spR N n g Nλ λ ν= ⋅Γ ⋅ ∆ (18)

where the spontaneous emission factor nsp is defined as

spo

NnN N

=−

(19)

Parabolic-in-Frequency Gain

By setting SpectralGainShape to Parabolic_Freq, the user may select an alternative equation for modeling the gain, this time using a quadratic frequency dependence [14]-[15]:

( ) ( )

2

( , ) 1 1gapo

o

c c

g N a N NC N Nν

λ λλ

− = ⋅ − ⋅ − − ⋅ −

(20)

where c is the velocity of light in a vacuum, λgap (Wavelength_Bandgap) is the active-region bandgap wavelength, and Cν (Gain_Bandwidth_Slope) is the gain-bandwidth slope.

For this model, the spontaneous emission is modeled using Eqs. (18) and (19) above.

Bulk Quantum Gain

In some cases, the user may want a more detailed description of the gain for bulk devices, in which case the bulk quantum gain model may be appropriate. By setting SpectralGainShape to Bulk, the user invokes the following quantum gain model for bulk active regions [8]:

( ) ( ) ( )3 / 22

3 / 2 21

2( , )

4 2ge hh

rad rad c ve hh

Em m cg N A B N f fm m hn

λλλπ

= ⋅ + ⋅ ⋅ − ⋅ − + h

(21)

where n1 (Active_Region_neff) is the active region effective index, Arad (Recomb_ConstArad) is the linear radiative recombination coefficient, Brad (Recomb_ConstBrad) is the quadratic radiative recombination coefficient, me (Effective_Mass_Electron) is the electron effective mass, mhh (Effective_Mass_HeavyHole) is the heavy-hole effective mass, and mlh (Effective_Mass_LightHole) is the light-hole effective mass.


Eg is the active region bandgap, and it is carrier-density dependent:

1/ 3( )g go gE N E K N= − (22)

where Ego (Bandgap_NoInjection) is the zero-injection bandgap and Kg (Bandgap_Shrinkage_Coeff) is the bandgap shrinkage coefficient.

fc and fv are the Fermi-Dirac distributions for the conduction and valence bands, respectively, and are functions of carrier density, wavelength, and temperature T (Bulk_Temperature):

1

1 expc

a fc

fE E

kT

=−

+

(23)

1

1 expv

b fv

fE E

kT

=−

+

(24)

hha g

e hh

mhcE Em mλ

= − ⋅ + (25)

eb g

e hh

mhcE Em mλ

= − − ⋅ + (26)

( ) 1/ 4ln 64 0.05524 64fcE kTδ δ δ δ

− = + ⋅ + ⋅ ⋅ + ⋅ (27)

( ) 1/ 4ln 64 0.05524 64fvE kTε ε ε ε

− = − + ⋅ + ⋅ ⋅ + ⋅ (28)

/ cN Nδ = (29)

/ vN Nε = (30)

3/ 2

222

ec

m kTN

π = h

(31)

( )

3 / 22 / 33 / 2 3 / 2

222

hh lhv

m m kTN

π

+ =

h (32)

where k is Boltzmann’s constant.

In addition, the spontaneous emission is modeled as [2],[7]-[8]:

( )1

( , ) ( , ) c vsp

c v

f fR N g N

f fλ λ ν

⋅ −= Γ ⋅∆ ⋅

− (33)

Note that the default settings for this gain model are taken from [8].

File-Based Gain

In the most general case, the SOA may be modeled using tabulated values for the spectral and carrier density dependence of the gain and spontaneous emission, such as described in [2] and [16]:

( , ) ( ) ( ( ))og N a N Nλ λ λ= ⋅ − (34)


( , ) ( , )sp emR N g Nλ λ ν= Γ ⋅ ∆ (35)

1( , ) ( ) ( ( ))em emg N a N Nλ λ λ= ⋅ − (36)

where gem(N,λ) is the emission gain coefficient. a(λ) is the wavelength-dependent linear gain coefficient, and the user specifies the file name for this coefficient’s tabulated values via the parameter Gain_Slope_File. No(λ) is the wavelength-dependent transparency carrier density, and the user specifies the file name for this density’s tabulated values via the parameter Carrier_Density_Transp_File. aem(λ) is the wavelength-dependent linear emission gain coefficient, and the user specifies the file name for this coefficient’s tabulated values via the parameter Emission_Gain_Slope_File. Finally, N1(λ) is the wavelength-dependent emission-gain transparency carrier density, and the user specifies the file name for this density’s tabulated values via the parameter EmGain_Density_Transp_File.

For all data files, the x values should be monotonically increasing or decreasing. Furthermore, in the following description, <num_pts> specifies the number of data lines in the file, while the choice of settings for the various unit fields are as follows:




For the data files Gain_Slope_File and Emission_Gain_Slope_File, the x values are in units of frequency and the y values are in units of area. These files should use the following format.

Format: AreaVsWavelength <frequency_units> <area_units>

<num_pts>

<frequency 1> <slope value 1>




etc.

Example: AreaVsWavelength [nm] [m^2]

5

1500 7e-20

1525 8e-20

1550 8.5e-20

1575 8.2e-20

1600 7.5e-20

For the data files Carrier_Density_Transp_File and EmGain_Density_Transp_File, the x values are in units of frequency and the y values are in units of cubic density. These files should use the following format.

Format: DensityVsWavelength <frequency_units> <density_units>

<num_pts>

<frequency 1> <density value 1>





etc.

Example: DensityVsWavelength [nm] [m^-3]

5

1500 1.1e24

1525 1.0e24

1550 8.5e23

1575 7e23

1600 5.5e23

In all cases, the policy for extrapolating values beyond the range of the tabulated data is controlled via the parameter Boundary_Policy. zero sets all values outside of the tabulated data range to zero, while extend uses the endpoint values.

Polarization Effects Usually the operation of an SOA depends on the polarization state of the optical input signals. While the model described above neglects this polarization dependence, if a user knows the expected PDG from measurement or device specifications, he can enter its value (in dB) into the model via the parameter PDG. In this way the SOA gain applied to the x- and y-polarization components will be accordingly modified.

Numerical Settings To solve the carrier rate equation, the model uses one of two Runge-Kutta-based differential equation solvers, selectable via the parameter Equation_Solver: Runge-Kutta 4th-order (Runge-Kutta4th) and Adaptive Runge-Kutta 5th-order (Runge-Kutta5thAdapt). In the 4th-order method, the step size is held constant, whereas this may not be the case for the adaptive 5th-order method. For the case when the signal sampling rate is high, the 5th-order method may consume less time in solving the rate equation. If the user selects this method, then they can specify the relative expected error (Relative_Error) and the minimum allowed step size (Stepsize_Min). For both solvers, the user may specify an initial step size (Stepsize_Initial) and the maximum number of allowed steps (MAXSTP).

Also, please note that while the model internally solves for the initial carrier density, the user can speed up this process by specifying an initial guess for its solution using the parameter Carrier_Density_Start.

Test Functions The SOA model provides a number of test plots for visualizing the expected performance of a device. By setting the Test_function parameter to the desired output and clicking the Test button in the component-parameter editing dialog, the user may display the SOA characteristics described below. Note that the units for the various results may be controlled via the parameters Test_gain_units (for gain values), Test_ASE_units (for ASE values), Test_wavelength_units (for wavelength values), Test_power_units (for optical power values), and Test_current_units (for bias current values).

• gain_vs_wavelength

Plots the device gain as a function of input signal wavelength at different values of bias current. The wavelength range in meters is specified via the parameters Test_wavelength_min and Test_wavelength_max, and the number of wavelengths to use is specified via Test_wavelength_points. The bias current range in Amperes is specified via the parameters Test_Ibias_min and Test_Ibias_max, and the number of bias currents to use is specified via Test_Ibias_points. The input signal power in Watts is specified via Test_Pin.


• gain_vs_pin

Plots the device gain as a function of input signal power at different values of bias current. The input-power range in Watts is specified via the parameters Test_Pin_min and Test_Pin_max, and the number of powers to use is specified via Test_Pin_points. The bias current range in Amperes is specified via the parameters Test_Ibias_min and Test_Ibias_max, and the number of bias currents to use is specified via Test_Ibias_points. The input signal wavelength in meters is specified via Test_input_wavelength.

• ase_vs_wavelength

Plots the device output ASE spectrum (in the presence of an input signal) at different values of bias current. The wavelength range and number of points to plot are controlled via the model’s ASE generation parameters (ASE_min_wavelength, ASE_max_wavelength, and ASE_bin_width). The bias current range in Amperes is specified via the parameters Test_Ibias_min and Test_Ibias_max, and the number of bias currents to use is specified via Test_Ibias_points. The input signal power in Watts is specified via Test_Pin, and the input signal wavelength in meters is specified via Test_input_wavelength.

References [1] M. J. Connelly, Semiconductor Optical Amplifiers, (Boston, Kluwer Academic Publishers, 2002).

[2] W. Mathlouthi, P. Lemieux, M. Salsi, A. Vannucci, A. Bononi, and L. A. Rusch, “Fast and efficient dynamic WDM semiconductor optical amplifier model,” Journal of Lightwave Technology, vol. 24, no. 11, pp. 4353-4365, November 2006.

[3] S. B. Kuntze, L. Pavel, and J. S. Aitchison, “Controlling a semiconductor optical amplifier using a state-space model,” IEEE Journal of Quantum Electronics, vol. 43, no. 2, pp. 123-129, February 2007.

[4] S. B. Kuntze, A. J. Zilkie, L. Pavel, and J. S. Aitchison, “Nonlinear state-space model of semiconductor optical amplifiers with gain compression for system design and analysis,” Journal of Lightwave Technology, vol. 26, no. 14, pp. 2274-2281, July 15, 2008.

[5] L. Gillner, E. Goobar, L. Thylén, and M. Gustavsson, “Semiconductor laser amplifier optimization: An analytical and experimental study”, IEEE Journal of Quantum Electronics, vol. 25, no. 8, pp. 1822-1827, August 1989.

[6] M. J. Adams, J. V. Collins, and I. D. Henning, “Analysis of semiconductor laser optical amplifiers,” IEE Proceedings, Pt. J, vol. 132, no. 1, pp. 58-63, February 1985.

[7] I. D. Henning, M. J. Adams, and J. V. Collins, “Performance predictions from a new optical amplifier model,” IEEE Journal of Quantum Electronics, vol. QE-21, no. 6, pp. 609-613, June 1985.

[8] M. J. Connelly, “Wideband semiconductor optical amplifier steady-state numerical model,” IEEE Journal of Quantum Electronics, vol. 37, no. 3, pp. 439-447, March 2001.

[9] Y. Suematsu and A. R. Adams, Handbook of Semiconductor Lasers and Photonics Integrated Circuits, London, UK: Chapman & Hall, 1994.

[10] M. J. O’Mahony, “Semiconductor laser optical amplifiers for use in future fiber systems,” Journal of Lightwave Technology, vol. 6, no. 4, pp. 531-544, April 1988.

[11] Y. Kim, H. Lee, S. Kim, J. Ko, and J. Jeong, “Analysis of frequency chirping and extinction ratio of optical phase conjugate signals by four-wave mixing in SOA’s,” IEEE Journal of Selected Topics in Quantum Electronics, vol. 5, no. 3, pp. 873-879, May/June 1999.

[12] A. E. Willner and W. Shieh, “Optimal spectral and power parameters for all-optical wavelength shifting: Single stage, fanout, and cascadability,” Journal of Lightwave Technology, vol. 13, no. 5, pp. 771-781, May 1995.

[13] M. A. Summerfield and R. S. Tucker, “Frequency-domain model of multiwave mixing in bulk semiconductor optical amplifiers,” IEEE Journal of Selected Topics in Quantum Electronics, vol. 5, no. 3, pp. 839-850, May/June 1999.


[14] G. Toptchiyski, S. Kindt, K. Petermann, E. Hilliger, S. Diez, and H. G. Weber, “Time-domain modeling of semiconductor optical amplifiers for OTDM applications,” Journal of Lightwave Technology, vol. 17, no. 12, pp. 2577-2583, December 1999.

[15] G. Talli and M. J. Adams, “Amplified spontaneous emission in semiconductor optical amplifiers: Modelling and experiments,” Optics Communications, vol. 218, pp. 161-166, 2003.

[16] K. Obermann, I. Koltchanov, K. Petermann, S. Diez, R. Ludwig, and H. G. Weber, “Noise analysis of frequency converters utilizing semiconductor-laser amplifiers,” IEEE Journal of Quantum Electronics, vol. 33, no. 1, pp. 81-88, January 1997.

Properties




Parameter Values

Name Type Default Range Unit RSOA enumerated No No, Yes

Reflectivity double 1 [ 0, 1 ]

Pump_Current double 0.15 [ 0, 1e32 ] A

Curr_Inj_Efficiency double 1.0 [ 0, 1e32 ]

Input_Loss double 0 [ -1e32, 1e32 ] dB Output_Loss double 0 [ -1e32, 1e32 ] dB

Length double 5.0e-4 ( 0, 1e32 ] m

Width double 3.0e-6 ( 0, 1e32 ] m

Thickness double 8.0e-8 ( 0, 1e32 ] m Confine_Factor double 0.15 [ 0, 1]

Linewidth_Enhance double 5.0 [ -1e32, 1e32 ]

Recomb_ConstA double 1.43e8 [ 0, 1e32 ] s-1

Recomb_ConstB double 1.0e-16 [ 0, 1e32 ] m3/s

Recomb_ConstC double 3.0e-41 [ 0, 1e32 ] m6/s

PDG double 0 [ -1000, 1000 ] dB includeNoise enumerated No No, Yes

ASE_min_wavelength double 1500e-9 ( 0, 1e32 ] m

ASE_max_wavelength double 1600e-9 ( 0, 1e32 ] m

ASE_bin_width double 1e-9 ( 0, 1e32 ] m SpectralGainShape enumerated Flat Flat, Parabolic,

Cubic, Parabolic_Freq, Bulk, File

Gain_Slope double 2.78e-20 [ 0, 1e32 ] m2


Carrier_Density_Transp double 1.4e24 [ 0, 1e32 ] m-3

Curvature_SpectralGain double 0.0 [ 0, 1e32 ] m-3 Wavelength_PeakGain double 1.52e-6 [ 0, 1e32 ] m

Wavelength_PeakGain_Slope double 0 [ 0, 1e32 ] m4

Wavelength_PeakGain_RefDensity double 1.4e24 [ 0, 1e32 ] m-3

CubicCoeff_SpectralGain double 0.0 [ 0, 1e32 ] m-4 Wavelength_Bandgap double 1600e-9 ( 0, 1e32 ] m

Gain_Bandwidth_Slope double 5e-12 ( 0, 1e32 ] m3/s

Effective_Mass_Electron double 4.1e-32 ( 0, 1e32 ] kg

Effective_Mass_HeavyHole double 4.19e-31 ( 0, 1e32 ] kg Effective_Mass_LightHole double 5.06e-32 ( 0, 1e32 ] kg

Bulk_Temperature double 25 [ -273.15, 1e32 ] °C

Active_Region_neff double 3.22 ( 0, 1e32 ] Recomb_ConstArad double 1.0e7 [ 0, 1e32 ] s-1

Recomb_ConstBrad double 5.6e-16 [ 0, 1e32 ] m3/s

Bandgap_NoInjection double 0.777 [ 0, 1e32 ] eV

Bandgap_Shrinkage_Coeff double 0.9e-10 [ 0, 1e32 ] eV·m

Gain_Slope_File file

Carrier_Density_Transp_File file

Emission_Gain_Slope_File file EmGain_Density_Transp_File file

Boundary_Policy enumerated zero zero, extend

Internal_Loss double 4000.0 [ 0, 1e32 ] m-1

Internal_Loss_Slope double 0 [ 0, 1e32 ] m2 Carrier_Density_Start double 3.0e24 [ 0, 1e32 ] m-3

Equation_Solver enumerated Runge-Kutta5thAdapt

Runge-Kutta4th, Runge-Kutta5thAdapt

Stepsize_Initial double 1.0e-12 [ 0, 1e32 ] s

Stepsize_Min double 0.0 [ 0, 1e32 ] s

Relative_Error double 1.0e-8 [ 0, 1e32 ] MAXSTP integer 100000 [ 1, 100000000 ]

Test_function enumerated gain_vs_wavelength gain_vs_wavelength, gain_vs_pin, ase_vs_wavelength

Test_wavelength_min double 1500e-9 ( 0, 1e32 ] m

Test_wavelength_max double 1600e-9 ( 0, 1e32 ] m

Test_wavelength_points integer 101 [ 1, 100000 ] Test_Ibias_min double 100e-3 [ 0, 1e32 ] A

Test_Ibias_max double 100e-3 [ 0, 1e32 ] A

Test_Ibias_points integer 1 [ 1, 20 ]

Test_Pin double 1e-3 [ 0, 1e32 ] W Test_input_wavelength double 1550e-9 ( 0, 1e32 ] m

Test_Pin_min double 1e-3 [ 0, 1e32 ] W

Test_Pin_max double 10e-3 [ 0, 1e32 ] W


Test_Pin_points integer 101 [ 1, 100000 ]

Test_gain_units enumerated dB linear, dB Test_ASE_units enumerated dBm/Hz uW/Hz, mW/Hz, W/Hz,

dBm/Hz

Test_wavelength_units enumerated nm nm, um, m

Test_power_units enumerated dBm uW, mW, W, dBm

Test_current_units enumerated mA nA, uA, mA, A

Parameter Descriptions RSOA Switch between regular SOA and reflective SOA Reflectivity Mirror reflectivity in RSOA Pump_Current Bias current Curr_Inj_Efficiency Current injection efficiency Input_Loss Input coupling loss Output_Loss Output coupling loss Length Active region length Width Active region width Thickness Active region thickness Confine_Factor Optical confinement factor Linewidth_Enhance Linewidth enhancement factor Recomb_ConstA Linear recombination coefficient Recomb_ConstB Quadratic recombination coefficient Recomb_ConstC Auger recombination coefficient PDG PDG in dB includeNoise Switch on/off internal ASE noise generation ASE_min_wavelength Minimum wavelength of internal ASE generation ASE_max_wavelength Maximum wavelength of internal ASE generation ASE_bin_width Bin width of internal ASE generation SpectralGainShape Switch between different gain models Gain_Slope Linear gain coefficient for the Flat, Parabolic, Cubic, and

Parabolic_Freq gain models Carrier_Density_Transp Transparency carrier density for the Flat, Parabolic, Cubic, and

Parabolic_Freq gain models Curvature_SpectralGain Quadratic curvature of the gain for the Parabolic and Cubic gain models Wavelength_PeakGain Peak-gain wavelength at the reference carrier density for the Parabolic

and Cubic gain models Wavelength_PeakGain_Slope Slope of peak-gain wavelength versus carrier density for the Parabolic

and Cubic gain models Wavelength_PeakGain_RefDensity Peak-gain wavelength reference carrier density for the Parabolic and

Cubic gain models CubicCoeff_SpectralGain Cubic curvature of the gain for the Cubic gain model Wavelength_Bandgap Bandgap wavelength for the Parabolic_Freq gain model Gain_Bandwidth_Slope Gain-bandwidth slope for the Parabolic_Freq gain model Effective_Mass_Electron Electron effective mass for the Bulk gain model Effective_Mass_HeavyHole Heavy-hole effective mass for the Bulk gain model Effective_Mass_LightHole Light-hole effective mass for the Bulk gain model


Bulk_Temperature Bulk temperature for the Bulk gain model Active_Region_neff Active-region effective index for the Bulk gain model Recomb_ConstArad Linear radiative recombination coefficient for the Bulk gain model Recomb_ConstBrad Quadratic radiative recombination coefficient for the Bulk gain model Bandgap_NoInjection Bandgap under zero injection for the Bulk gain model Bandgap_Shrinkage_Coeff Bandgap shrinkage coefficient for the Bulk gain model Gain_Slope_File Name of the file containing the linear gain coefficient data for the File

gain model Carrier_Density_Transp_File Name of the file containing the transparency density data for the File gain

model Emission_Gain_Slope_File Name of the file containing the linear emission-gain coefficient data for

the File gain model EmGain_Density_Transp_File Name of the file containing the emission gain’s transparency density data

for the File gain model Boundary_Policy Policy for treating file-based data beyond the data end points Internal_Loss Internal loss Internal_Loss_Slope Slope of internal loss versus carrier density Carrier_Density_Start Initial carrier density Equation_Solver Method for solving the carrier density rate equation Stepsize_Initial Initial step size tried by the differential equation solver Stepsize_Min Minimum step size allowed by the Adaptive 5th-order Runge-Kutta

method Relative_Error Relative error used by the Adaptive 5th-order Runge-Kutta method MAXSTP Maximum number of steps allowed by the differential equation solvers Test_function Test-plot output selection Test_wavelength_min Minimum value when varying the test input wavelength Test_wavelength_max Maximum value when varying the test input wavelength Test_wavelength_points Number of values when varying the test input wavelength Test_Ibias_min Minimum value when varying the test bias current Test_Ibias_max Maximum value when varying the test bias current Test_Ibias_points Number of values when varying the test bias current Test_Pin Single value of test input power Test_input_wavelength Single value of test input wavelength Test_Pin_min Minimum value when varying the test input power Test_Pin_max Maximum value when varying the test input power Test_Pin_points Number of values when varying the test input power Test_gain_units Units when displaying gain data Test_ASE_units Units when displaying ASE data Test_wavelength_units Units when displaying wavelength data Test_power_units Units when displaying power data Test_current_units Units when displaying current data


Optical Noise Adder

This model provides a mechanism for directly manipulating the ASE noise of an optical spectrum. Additional noise can be added to the signal with various profiles. The representation of the noise can also be altered – the noise may be incorporated as a stochastic component to the signal rather than as a power spectral density.

Direct addition of noise Several components in OptSim can generate ASE noise, including the EDFA and Bi-Directional Nonlinear Fiber model (BDNF) (used as a Raman amplifier). The noise generated by these components has important impacts on bit error rates, eye diagrams etc. At times, the user may be interested in the dependence of such quantities on the noise level, but less interested in modeling the origin of the noise sources. In such a case it is inconvenient to have to use the EDFA or BDNF models just to generate the noise, particularly as these models also affect the optical signals through gain and attenuation. The present model acts as a black-box noise generator, that has no impact on the signal outside of the generation and addition of noise.

With the parameter noiseShape, the user may select noise distributions with Uniform, Gaussian, Custom, or ExistingASEOnly profiles. If the incoming signal contains an existing ASE spectrum, the new noise is added incoherently to the existing spectrum. The profiles are defined in terms of the center wavelength 0λ (noiseCenter), wavelength-FWHM B (noiseBW), and peak spectral density S0 as follows:

• Uniform

( )0 0

0

,2

0,2

BSS

B

ν

ν

ν νν

ν ν

− ≤= − >

where λν /0 c= and /( / 2) /( / 2)o oB c B c Bν λ λ= − − + .

• Gaussian

( ) ( ) ( )( )

20

0 2exp log 2/ 2

S SBν

ν νν

− = −

• Custom The Custom noise model uses a wavelength and power dependent ASE noise file with the same format as the custom noise model in the optical amplifier (EDFA) model. The noise spectra is defined in units of dBm at different average input power levels and wavelengths. The program will interpolate the ASE noise for every channel under the actual input total power. The name of the custom noise file is given in the noiseFilename parameter. The noiseResolution parameter is used to set the resolution of the generated ASE noise spectra. The resolution should be set to be approximately an order of magnitude smaller than the bandwidth of the narrowest optical filter in the link.

The Custom file is a two column format file which specifies the wavelength (nm) and the ASE noise (dBm) all separated by spaces. Multiple noise curves (each corresponds to different input power) are placed one after another within the data file. There is no limit to the number of noise curves that may be included. The very first line of each curve starts with ‘0’ and is followed with spaces and the average input power (dBm) that corresponds to the curve. This ‘0’ tells the program the beginning of a new curve and must not


be changed. Comments can be added at the end of the line, and each comment is ended with a ‘*’. No comment or ‘*’ is needed in the curve data rows. The data region is ended with a line ‘-1 -1’ at the end of the file. There should be a carriage return at the end of the last line.

• ExistingASEOnly

The ExistingASEOnly noise model simply takes the ASE noise already present in the signal as part of the ASE noise bins representation and adds it into the signal waveform. It does not add new ASE noise to the signal, but merely changes its representation from a power spectral density to a stochastic component in the signal waveform.

Normalization If noiseNormalization=Absolute, the peak spectral density 0S is specified directly by noisePeak in units of W/Hz. If noiseNormalization=SNR, the spectral density of the signal is sampled in bins of width noiseSNR_BW to find the maximum value max

sS . The peak noise density is then chosen as max0 sS S= −noiseSNR .

The sampling rate of the noise is specified in Hz with the noiseResolution parameter and should be chosen to adequately resolve the features of the noise.

Figure 1: Optical spectrum before and after addition of uniform noise profile

Figure 1 shows the optical spectrum before and after the addition of a uniform noise profile with bandwidth 5 nm.

Stochastic representation of noise Generally, OptSim represents optical signals and ASE noise as separate entities – signals as time dependent complex waveforms, and ASE noise as a frequency-dependent power spectral density. For most purposes, this is a useful separation since it allows quasi-analytical estimation of bit-error rates, and for the noise to span much larger bandwidths than individual optical signals.

One disadvantage of this separation, however, is that it requires that a number of fiber propagation effects are not applied to the noise. That is, the noise does not experience either dispersion or nonlinear interactions with the signal. In problems in which ASE noise dominates over receiver noise sources, this approximation may not be justified. For example, four wave mixing can significantly amplify noise close to a signal channel. To enable modeling of such effects, the ASE noise can be transferred from the power spectral density representation, into a stochastic component of the optical signal. This is accomplished by setting the noiseRepn parameter to InSignal, instead of the default value InNoiseBins. This is implemented by Fourier transforming the optical signal, and adding a complex random number related to the local spectral noise density. In the ensemble average, the same noise power is present in the signal as was originally present in the noise power spectrum. If the power spectrum is broader than the bandwidth of the signal, parts of the power spectrum outside the signal bandwidth are unchanged.


Figure 2 shows the optical spectrum and signal from Fig.1 with the noise transferred to the signal. If the incoming signal is a multi-channel signal, the noise is transferred to each signal according to the local spectral density of the noise. Note that if the different signal bandwidths overlap, there is no sensible way to divide the noise amongst the competing channels. The result is an overestimate of the noise power in that region. OptSim will issue a warning in this situation, and the user should either alter the signal bandwidths so that they do not overlap, or convert to single-channel mode, which may be more efficient anyway.

In transferring noise to the signal, it is important to consider the way in which the polarization of the noise is treated in OptSim. The numerical value of a noise bin represents the sum of the noise powers in each polarization at that frequency. Thus when the noise is transferred to the signal, half of the noise bin power is assigned to each polarization. This fact is especially significant if the incoming signal has only an Ex component and no Ey component. In this case, half the noise is transferred to the x component, and the other half is discarded. The whole noise energy is only transferred when both polarization components are present. This can always be accomplished, if required, by preceding the Noise Adder with the Polarization Controller set to provide a zero rotation, which creates an empty Ey component.

Figure 2: Spectrum for Figure 1 with stochastic representation of noise: a) spectrum, b) time series.

Seed for stochastic noise The user may control the seeding of the random number generator used to implement the stochastic representation. This satisfies –1e8 <= noise_seed <= 1 with the following behavior:

• noise_seed < 0

The generator is seeded with the actual value of noise_seed on every run of the simulation. This is useful for obtaining repeatable results.

• noise_seed = 0

The generator is seeded with an integer hashed from the string value of the component’s name. This is a convenient way to obtain repeatable results on subsequent runs within a single fiber, but different noise fields in a series of fiber components which have different names. This would be appropriate in a set of parallel WDM inputs to a fiber, for instance.

• noise_seed = 1


Properties




Parameter Values Name Type Default Range Unit noiseNormalization enumerated Absolute Absolute, SNR,

Photons/SpatialMode

noiseShape enumerated Uniform Uniform, Gaussian, Custom, ExistingASEOnly

noiseFilename string

noisePeak double 1e-15 0 ≤ x ≤ 1e32 W/Hz

noiseS double 20 0 ≤ x ≤ 1e32 dB

noiseSNR_BW double 50e9 0 ≤ x ≤ 1e32 m

noiseCenter double 1.425 1e-32 ≤ x ≤ 1e6 m

noiseBW double 30e-9 0 x ≤ 1e6 m

noiseResolution double 0.01e-9 1e-32≤ x ≤ 1e32 m

noiseRepn enumerated InNoiseBins InNoiseBins, InSignal

noiseSeed int 0 -1000000 ≤ x ≤ 1

Parameter Descriptions noiseNormalization Select absolute or relative (SNR) noise intensity noiseShape Select shape of noise profile noiseFilename Filename for custom noise profile noisePeak Peak spectral density for absolute noise intensities noiseSNR Peak SNR for relative noise intensities noiseSNR_BW Sampling bin width for signal spectral density noiseCenter Center of added noise spectrum noiseBW Full bandwidth of added noise spectrum noiseResolution Bin spacing for noise bin representation of noise spectrum noiseRepn Toggle representation of noise in bins or as stochastic signal noiseSeed Random number seed for stochastic representation


Linewidth Adder

This model inserts source linewidth into any optical signals at its input. The linewidth is inserted as either a constant parameter, or via the introduction of phase noise into the signal. The model does not check whether the incoming optical signal already accounts for any source linewidth.

Inclusion of a source linewidth in the optical output is controlled via the parameter linewidth_model, which by default is set to phase_ noise. In this case, if the optical signal is not a power-value cw signal, then linewidth is added to the output via random phase variations, which are seeded via the randomSeed parameter. This results in a Lorentzian output power spectrum under cw conditions [1]. If linewidth_model=value, then the constant linewidth value is attached to each optical signal. linewidth_model=none deactivates the model.



Properties

Inputs #1-N: Optical signal

Outputs #1-N: Optical signal

Parameter Values Name Type Default Range Units linewidth_model enumerated phase_noise none, phase_noise,

value




Parameter Descriptions linewidth_model Select linewidth representation linewidth_units Select linewidth units linewidth Linewidth value randomSeed Random number seed for phase randomization. (Standard OptSim seed

i )


convention).


Liekki LAD Interface

The Liekki LAD Interface provides the user with an interface to Liekki Application Designer (LAD), Liekki’s fiber amplifier and laser simulation tool. This interface allows users to embed LAD designs within their OptSim schematic. At simulation time, the Interface calls LAD to simulate the LAD design. The Interface is used as follows. After placing the Liekki LAD Interface block in a schematic, open its Properties dialog. To load an LAD design, select one from the pull-down selection box at the top of the dialog and click on the Load… button. To appear in the pull-down selection box, an LAD project file should be located in the Workgroup directory, User directory or products/optsim/block_mode/usermodels directory of the OptSim installation. Alternatively, if you want to load an LAD project file from disk, select “From Disk…” in the selection box and click on the Load… button. Once you have loaded the design, the project_file parameter will be updated to display the path to the selected file, and input and output ports will be added to the Interface block, reflecting the definition of input and output ports in the LAD design file. You may now connect the Interface block to other components in your design and run a simulation. Note that at each input port of the interface, ASE is ignored and each input signal is treated as a single optical channel. Furthermore, the input ports in the LAD schematic determine which inputs are considered signals within LAD, and which are considered pumps. Finally, at the output ports, ASE is included, and both LAD signals and pumps are treated as OptSim optical signals (i.e., OptSim does not maintain a distinction between the two). Currently, all output signals will be cw.

Properties

Inputs: Optical signals, one for each input port in the LAD project file.

Outputs: Optical signals, one for each output port in the LAD project file.

Parameter Values Name Type Default Range Unit project_file file


project_file Name of LAD project file.

OptSim Models Reference: Block Mode Chapter 7: Optical Components •••• 247

Chapter 7: Optical Components

This chapter describes following optical components:

• Optical Attenuator

attenuate an optical signal by a specified level

• Optical Power Normalizer

normalize input optical signal(s) to the specified average output power level

• Optical Phase Shift

shift the carrier phase of the input optical signal(s)

• Optical Phase Conjugator

generate an optical signal having phase opposite to that of the input optical signal

• Polarization Transformer

transform the polarization of the input optical signal(s)

• Ideal Frequency Converter

shift the carrier frequency of the input optical signal(s)

248 •••• Chapter 7: Optical Components OptSim Models Reference: Block Mode

Optical Attenuator

This model attenuates the input optical signal by the specified level of attenuation. This model may be used anywhere in the topology where a specified level of optical attenuation is desired. It has two parameters. The first and primary parameter is the attenuation value in units of dB. This attenuation is applied to the x polarization portion of the signal. If the signal contains a y polarization component as well, then the second parameter, xy_differential, is used to set the attenuation of the y polarization component. The attenuation of the y polarization component (y_attenuation) is expressed as follows:

y_attenuation = attenuation – xy_differential

Properties



Parameter Values Name Type Default Range Units attenuation double 0.0 [ -1e16, 0 ] dB

xy_differential double 0.0 [ -1e16, 1e16 ] dB


attenuation Amount to attenuate the x polarization component of the input optical signal by xy_differential Differential amount to attenuate the y polarization component compared to the x polarization component

attenuation


Optical Power Normalizer

This model normalizes the optical signal power by attenuating the input optical signal(s) to the specified average output power level. This model is most often used to control the input optical power at the receiver when preparing a BER vs. received optical power curve plot. The model may be used to attenuate all input optical signals to the same average output power regardless of their different average input powers, or it may be used to attenuate all input optical signals by the same amount such that the signal with the largest average input power has the specified average output power.

Properties



Parameter Values

Name Type Default Range Units AttenuationType enumerated Uniform Uniform, Nonuniform

AvePowerOut double -20 [ -1e16, 1e16 ] dBm

Parameter Descriptions AvePowerOut Average output optical power of the optical power normalizer AttenuationType Uniform attenuates all input optical signals by the same amount such that the optical signal with the

largest input average power has the specified average output power; Nonuniform attenuates all input optical signals by different amounts such that each output optical signal has the specified average output power


Optical Phase Shift

This model shifts the carrier phase of the input optical signal(s) by the specified amount. This model may be used anywhere in the topology where a specified optical phase shift is desired.

Properties



Parameter Values Name Type Default Range Units PhaseShift double 0 [ -180, 180 ] degrees

Parameter Descriptions PhaseShift Amount to shift the carrier phase of the input optical signal(s) by


Optical Phase Conjugator

This model simulates an ideal phase conjugator, i.e. a device that ideally generates an optical output signal whose phase is opposite with respect to the one of the input signal. An efficiency parameter η and a constant phase_shift shiftφ are taken into account. The usual frequency-shift introduced by real phase-conjugator device is not taken into account.

))(exp()()( tjtEtE inin φ=

)exp())(exp()()( shiftinout jtjtEtE φφη −=

Properties



Parameter Values Name Type Default Range Units

efficiency double 1 [ 0, 1 ] none

phase_shift double 0 [ 0, 360 ] degrees

Parameter Descriptions efficiency Efficiency of the component referred to the signal amplitude phase_shift Constant phase shift applied to the output optical signal


Polarization Transformer

This model transforms the polarization of the input optical signal(s) according to the specified parameters. This model may be used anywhere in the topology where a specified polarization transformation is desired.

There are several modes of transformations supported. The modes are described below:

• Custom

In the custom mode, the input optical signal’s polarization is transformed according to the specified complex matrix:

=

i

i

o

o

YX

cccc

YX

2221

1211

• Rotation

In the Rotation mode, the polarization is rotated by the specified angle.

• LinearPolarizer

In the LinearPolarizer mode, the polarization is made to be linear at the specified angle.

• CircularLeft

In the CircularLeft mode, the polarization is made to be circular in the left direction.

• CircularRight

In the CircularRight mode, the polarization is made to be circular in the right direction.

• PolarizationController

In the PolarizationController mode, the polarization states of the output signals are controlled by the parameters “azimuth”, “ellipticity”, “symmetryFactor_x” and “symmetryFactor_y” to simulate an ideal polarization controller. The polarization states of the outputs do not depend on the input polarization while the optical power is preserved.

Assume the input signal is expressed as

=

)](exp[|)(|)](exp[|)(|

)(tjtEtjtE

tE inyy

inxx

in δδ

v

vv

,

and the output signal is expressed as

22 ||||)](exp[

)](exp[1)( yxout

y

outx

out EEtjk

tjktE

vvv+

−=

δδ

where k is the power splitting parameter. The phases of the input and output signals can be related as

out outy xδ δ δ− = −


y

xiny

outy

inx

outx

αα

δδδδ

=−−

where

k

kk21

cos)1(2)2tan(

−−

=δ

η

δε sin)1(2)2sin( kk −=

xa is called symmetryFactor_x, ya is called symmetryFactor_y, h is called azimuth and e is called ellipticity. These parameters are specified by the user.

An optical loss may also be included in the transformation via the loss parameter.

Properties



Parameter Values Name Type Default Range Units mode enumerated Custom Custom, Rotation,

LinearPolarizer, CircularLeft, CircularRight, PolarizationController

loss double 0 [ 0, 1e32 ] dB

angle double 0 [ -180, 180 ] degree

azimuth double 0 [ -90, 90 ] degree

ellipticity double 0 [ -45, 45 ] degree symmetryFactor_x double 0 [ -1e32, 1e32 ] none

symmetryFactor_y double 1 [ -1e32, 1e32 ] none

coef11r double 1 [ -1e32, 1e32 ] none

coef11i double 0 [ -1e32, 1e32 ] none coef12r double 0 [ -1e32, 1e32 ] none

coef12i double 0 [ -1e32, 1e32 ] none


coef21i double 0 [ -1e32, 1e32 ] none coef22r double 1 [ -1e32, 1e32 ] none



Parameter Descriptions mode Parameterized mode or Custom mode loss Optical loss at input ports angle Amount to adjust the polarization state by coef11r Real portion of c11 coef11i Imaginary portion of c11 coef12r Real portion of c12 coef12i Imaginary portion of c12 coef21r Real portion of c21 coef21i Imaginary portion of c21 coef22r Real portion of c22 coef22i Imaginary portion of c22 azimuth Azimuth of the signal polarization ellipse ellipticity Ellipticity of the signal polarization ellipse symmetryFactor_x Related to the phase change for the x polarization component symmetryFactor_y Related to the phase change for the y polarization component


Ideal Frequency Converter

This model shifts the carrier frequency of the input optical signal(s) by the specified amount, which may be in either frequency units of Hz or wavelength units of meters. This model may be used anywhere in the topology where a specified optical carrier frequency conversion is desired.

Properties



Parameter Values

Name Type Default Range Units specMode enumerated Wavelength Frequency, Wavelength shift double 0 [ -1e32, 1e32 ] Hz or m

Parameter Descriptions shift Amount to shift the carrier frequency of the input optical signal(s) by specMode Whether shift amount is specified in frequency or wavelength units


Sagnac Effect model for Interferometric Fiber Optic Gyroscope (I-FOG)

This block models the differential phase shift belonging to the Sagnac effect between the clockwise and counter-clockwise propagating optical signals in a fiber optic loop.

The Sagnac effect is an interferometric phenomenon that manifests itself in a ring setup undergoing rotation. A beam of light is split and the two beams are made to follow trajectories in opposite directions, enclosing a given area and producing interferometry upon recombination. The light transmission can be guided through fiber optics.

The phase difference belonging to the Sagnac effect Sϕ is proportional to the dot product of the rotation rate vector

Ωr

by the area vector Ar

enclosed by the optical path:

Ω⋅=rr

AcS 24ωϕ

whereω is the angular frequency of the light source and c is the light velocity in vacuum.

In multi-turn fiber coils the sensitivity is enhanced multiplying the enclosed area A by a certain number of turns N. The Sagnac effect phase difference can then be rewritten as:

//2 Ω=

cLD

S λπϕ

where L is the total coil length, D is the coil diameter, λ is the light source wavelength, and //Ω is the rotation rate component parallel to the rotation axis.

In the Sagnac effect OptSim model the rotation can be constant during the simulation time or instantaneous. The block property parameter rotation_rate controls the constant rotation rate, while the electrical input enables the user to provide an electrical signal representing the instantaneous rotation rate in degrees/second.

References [1] H. Lefevre: The Fiber-Optic Gyroscope (Artech House, Boston, 1993)

Properties

Inputs #1: Optical Signal propagating clockwise in the I-FOG

#2: Optical Signal propagating counter-clockwise in the I-FOG

#3: Electrical Signal representing the instantaneous I-FOG rotation in degrees/second


Outputs #1: Optical Signal propagating clockwise in the I-FOG with Sagnac effect phase shift

#2: Optical Signal propagating counter-clockwise in the I-FOG with Sagnac effect phase shift

Parameter Values Name Type Default Range Units fiber_length Double 200 ( 0, 1e32 ] m

coil_diameter Double 3 ( 0, 1e32 ] cm rotation_rate Double 0 Deg/s

Parameter Descriptions fiber_length Total length of the fiber used for the I-FOG multiturn coil

coil_diameter Diameter of the I-FOG multiturn coil rotation_rate I-FOG rotation rate, constant during the simulation time

OptSim Models Reference: Block Mode Chapter 8: WDM Components •••• 259

Chapter 8: WDM Components

This chapter describes following WDM components:

• Optical Splitter (1××××N)

ideally split the optical signal

• Optical Coupler (2××××2)

couple the input optical signals

• Controlled Optical Coupler

couple the input optical signals according to control signal

• Optical Filter

model a various types of optical filters

• Fiber Bragg Grating Filter

model a fiber Bragg grating filter

• Optical Multiplexer (N××××1 MUX)

multiplex N optical signals

• Optical DeMultiplexer (1××××N DEMUX)

demultiplex incoming optical signal

• Optical Add Multiplexer

add an optical signal to the WDM signal

• Optical Drop Multiplexer

drop an optical signal from the WDM signal

• Optical Add/Drop Multiplexer

add and drop optical signal to/from the WDM signal

• General Multiport Optical Device (N××××M and WDM)

represent a general N×M multiport optical WDM device via a data-driven transfer matrix

• Jones Matrix Transfer Function

represent an optical device via a data-driven (experiment or theory) linear transfer function model based on the Jones matrix

260 •••• Chapter 8: WDM Components OptSim Models Reference: Block Mode

Optical Splitter (1xN)

This model represents an ideal optical splitter. It takes a single input signal, and divides it equally among N output ports with 1/N splitting loss, plus excess loss determined by the transmission model parameter.

Properties



Parameter Values Name Type Default Range Units transmission double 1.0 [ -1e30, 1e30 ] none

Parameter Descriptions transmission overall transmission coefficient to account for excess loss


Optical Coupler (2x2)

This model represents an optical coupler. It takes an optical input signal on each port, and uses one of two ways to couple the optical signals together. If the mode is set to Parameterized, it uses the following complex matrix to couple the signals, where A represents the complex optical field amplitude:

−−=

2

1

2

1

11

i

i

o

o

AA

jj

AA

αααα

If the mode is set to Custom, the following complex matrix is used to couple the signals:

=

2

1

2221

1211

2

1

i

i

o

o

AA

cccc

AA

There is also a loss factor that may be applied to both input signals

Properties


#2: Optical signal


#2: Optical signal

Parameter Values Name Type Default Range Units mode enumerated Parametric Parametric, Custom

loss double 0 [ 0, 1e32 ] dB alpha double 0.5 [ 0, 1 ] none



coef12r double 0 [ -1e32, 1e32 ] none coef12i double 0 [ -1e32, 1e32 ] none



coef22r double 1 [ -1e32, 1e32 ] none coef22i double 0 [ -1e32, 1e32 ] none


Parameter Descriptions mode Parameterized mode or Custom mode loss Optical loss at input ports alpha Parameter for parameterized mode coef11r real portion of c11 coef11i imaginary portion of c11 coef12r real portion of c12 coef12i imaginary portion of c12 coef21r real portion of c21 coef21i imaginary portion of c21 coef22r real portion of c22 coef22i imaginary portion of c22


Controlled Optical Coupler (2x2)

This model represents an optical coupler controlled by a separate control signal. It takes an optical input signal on each port, and uses the following complex matrix to couple the signals where A represents the complex optical field amplitude and α is alpha_on if the control signal is 1, and alpha_off if the control signal is 0:

−−=

2

1

2

1

11

i

i

o

o

AA

jj

AA

αααα

There is also a loss factor that may be applied to both input signals

Properties


#2: Optical signal

#3: Binary signal


#2: Optical signal

Parameter Values Name Type Default Range Units loss double 0.0 [ 0, 1e32 ] dB

alpha_off double 0.0 [ 0, 1 ] none

alpha_on double 1.0 [ 0, 1 ] none

Parameter Descriptions loss Optical loss at input ports alpha_on Parameter for 1 state of control signal alpha_off Parameter for 0 state of control signal


Optical Filter

This model represents one of the following types of optical filters: Fabry Perot, Gaussian, RaisedCosine, Lorentzian, Trapezoidal, Ideal, Custom1, Custom2, and Custom3. Each filter type except for the custom types may also be inverted. A wavelength signal whose filtered peak power does not exceed the user-specified drop threshold will not be passed by the filter.

The Ideal filter type is an ideal filter in which the response is 1 from 2/0 Bw−λ to 2/0 Bw+λ and zero outside this range.

The Trapezoidal filter type uses a Trapezoidal filter response to filter the optical signal amplitude. The user specifies the bandwidth in the wavelength domain (units of m) for the flat portion of the response as BW0dB. The user also specifies the 3dB bandwidth in the wavelength domain.

The Gaussian filter type uses a Gaussian filter response to filter the optical signal amplitude. The user specifies the 3dB bandwidth in the wavelength domain, and the order of the filter.

The Fabry-Perot option models a Fabry-Perot filter. The optical filter will affect both the optical signal and the noise level and bandwidth. The transfer function for the Fabry-Perot resonant cavity model is computed as,

LeR

eRT i

i 11

)1()(2/

⋅⋅−⋅−= δ

δ

ν

where L is the insertion loss of the filter, R is the power reflectivity of the facets (assuming equal reflection at both facets), δ is a phase delay,

F

nlνπν

λθπδ 2cos4 ==

where, n is the refractive index of the medium, l is the separation between the facets, θ is the incident angle, λ is the optical signal wavelength, v is the optical frequency, Fν is the free spectral range (FSR).

2/1νν ∆⋅= FF

F is the finesse factor,

RRF

−=

1π

2/1ν∆ is the 3 dB bandwidth in frequency, and is calculated as,

)2//()2//( 002/1 BwcBwc +−−=∆ λλν

where 0λ is the filter center wavelength, Bw the user input 3 dB optical bandwidth in wavelength.

The free spectral range may be input by the user in units of wavelength (m), or it may be calculated from the reflectivity. If the reflectivity is desired to be entered directly, it should be set to a nonzero value. If the FSR is desired to be entered directly, the reflectivity should be set to 0.

The RaisedCosine filter applies the following transfer function to the optical signal amplitude:


( )

( )

( ) ( )

( )

0 0 1/ 2

0 1/ 2 0 1/ 2 0 0 1/ 20 1/ 2

0 0 1/ 2

1, 1

1 1 sin , 1 122

0, 1

R

T R RR

R

α

α

ν ν ν

πν ν ν ν ν ν ν νν

ν ν ν

− < − ∆ = − − − ∆ − ∆ < − < + ∆ ∆ − > + ∆

The Lorentzian filter type uses a multiple-stage Lorentzian filter response to filter the optical signal amplitude. The transfer function for the Lorentzian filter type is computed as,

N

o

T

∆

−+

= 2

2/1

2

)(21

1)(

ννν

ν

where user specify N as order of the filter (or number of stages), center wavelength as a wavelength and 3-dB bandwidth of a filter in wavelength domain as BW. Values of 0ν and 2/1ν∆ will be calculated the same way as for Fabry-Perot filter.

There are three custom filter types supported. These are specified as Custom1, Custom2, and Custom3. Each of these custom filter types uses the data file, as per the format of the NxM multiport optical device, to specify the filter response in the wavelength domain. The Custom1 filter’s response at each wavelength is taken directly from the data file.

The Custom2 type treats the data file as a normalized optical filter response, in which the 3dB bandwidth is 1 and the center wavelength is zero. The user specifies the center wavelength as the wavelength parameter of the filter. In practice, whatever the center wavelength of the filter data in the data file is, the wavelength parameter is added to it. The bandwidth of the filter is also multiplied by the BW parameter of the model. The overall effect of this filter type is to shift the center wavelength of the filter specified in the data file by the center wavelength specified in the model parameter, and scale the bandwidth of the filter specified in the data file by the BW parameter specified in the model.

The Custom3 type treats the data file as a regular optical filter response which the user wishes to shift and scale to have a different center wavelength and a different bandwidth. The user specifies the desired center wavelength of the filter as the wavelength parameter. The user specifies the desired wavelength range which the filter data should be scaled to occupy in the BW parameter. The overall effect of this filter type is to map the filter response specified in the data file to a new center wavelength (wavelength instead of Lmax+Lmin/2.0) and covering a different wavelength range (wavelength-BW/2.0 to wavelength+BW/2.0 instead of Lmin to Lmax).

Carrier shifting The model may be used to shift the carrier frequency of the optical signal(s) output from the filter. When the signals being filtered are of the multiple channel representation, in which each channel’s signal is represented in its own frequency band centered about a specified carrier frequency, this option would not be desired. When the optical filter is being used to filter a particular channel out of a single channel representation, in which all channels are represented together in one frequency band, this option can be used to reset the center frequency of the filtered frequency band to match the center of the filter. If the desired channel is centered at the center wavelength of the filter, using this option will ensure that the center wavelength of the signal output from the filter is the desired channel’s center wavelength.


Test Parameters The wavelength and frequency response of the optical filters may be viewed graphically by using the Test function button on the parameter editing window.

Properties





type enumerated Gaussian Ideal, Trapezoidal, Gaussian, RaisedCosine, Lorentzian, FabryPerot, Custom1, Custom2, Custom3

filename string

wavelength double 1550e-9 [ 1e-32, 1 ] meters

reflectivity double 0 [ 0, 1 ] none

FSR double 100e-9 [ 0, 1e32 ] meters

BW double 0.1e-9 [ 1e-32, 1e32 ] meters

BW0dB double 0.0 [ 0, 1e32 ] meters

order integer 1 [ 1, 128 ] none

rolloff double 0.1 [0, 1] none

alpha double 1 [0, 1e32] none

inversion enumerated None None, Power

shiftcarrier enumerated No No, Yes

dropThreshold double 0 [ 0, 1e32 ] Watt


type Type of filter operation: several filter types are supported

filename File name for the custom filter types inversion Whether to invert the power response of optical filter (except custom)


dropThreshold Signals whose peak value does not exceed threshold will be dropped wavelength Optical filter center wavelength BW 3dB filter bandwidth in wavelength BW0dB Zero dB bandwidth for Trapezoidal filter type FSR Free spectral range of Fabry Perot optical filter reflectivity Reflectivity of Fabry Perot optical filter (set to 0 when FSR is specified) loss Optical Filter Insertion Loss order Order of Gaussian or Lorentzian optical filter rolloff Rolloff parameter for RaisedCosine filter alpha alpha parameter for RaisedCosine filter shiftcarrier Whether to shift the carrier wavelength specification of the signal to match filter


Fiber Bragg Grating Filter

This model simulates a fiber Bragg grating filter. It can be described in the scattering matrix S or the transfer matrix T as in the following diagram.

a1

b1

a2

b2

SS, T

where

=

2

1

aa


=

2

1

bb

b represents the reflection waves. ,Sab =

=

2221

1211

SSSS

S . On the other hand, the relationship of the inputs and the outputs can be expressed as

=

1

1

2

2

ba

Tba

,

=

2221

1211

TTTT

T . The transfer matrix was used because it is convenient to deal with

the cascaded blocks, i.e., the transfer matrix of the cascaded blocks equals to the multiplication of the transfer matrix of each block. The transformation from the transfer matrix to the scattering matrix is as

,22

2111 T

TS −= ,1

221221 T

SS == 22

1222 T

TS = .

This model is implemented as shown in the following diagram:

a1

a2=0

b2

b1

SS,T


where 2b (from the first output port) is the transmitted wave and 1

b (from the second output port) is the

reflected wave of the grating.

For uniform fiber Bragg gratings, the effective refractive index of the core at a location z can be expressed as

)2cos()( znnzn eff0Λ

⋅∆+= π

where effn is the effective background refractive index of the host fiber, n∆ is the amplitude of the

effective refractive index modulation (index modulation depth) and 0Λ is the grating period. For this uniform fiber Bragg grating the transfer matrix T can be obtained from the solution of the coupled-mode equations (Raman Kashyap, Fiber Bragg Gratings, Academic Press,1999 and Turan Erdogan, “Fiber Grating Spectra”, J of Lightwave Tech. 15(8), pp1277-1294, August, 1997), i.e.,

α

αδα )sinh()cosh(11LjLT −=

α

αδα )sinh()cosh(22LjLT +=

α

ακ )sinh(12

LjT ac−=

α

ακ )sinh(21

LjT ac=

where L is the length of the grating, acκ represents the coupling between the counter-propagation fields and can be calculated as

nneff

ac ∆Λ

=02πκ

and

00

2Λ

−=Λ

−= πλ

ππβδ effn

22 δκα −= ac

where λ is the wavelength.

For a linearly chirped fiber Bragg grating with an apodised refractive index modulation profile, its effective refractive index can be expressed as

))(

2cos()()()( zz

znznzn eff Λ⋅∆+= π

where the grating period, )(zΛ can be written as


L

zz ⋅∆Λ+Λ=Λ 0)(

where ∆Λ is the linear chirp coefficient and will be given by the user. This amount of linear chirp will produce a primary reflection over a spectral range ∆Λ=∆ effn2λ .

)(zn∆ is the apodised refractive index modulation amplitude and can be expressed as

)()( zAnzn ⋅∆=∆

where )(zA , the apodisation profile, represents the refractive index modulation envelope along z . In the current model, following apodisation patterns are supported, in which a generalised parameter η , apodisation coefficient, η is used to specify the details of each apodisation profile.

• Uniform Grating

LzLzA 5.05.01)( ≤≤−=

• Quarter-Cosine Apodisation

)5.0(||1)( η−<= LzzA

LzLL

LzzA 5.0||)5.0(])5.0(||2

cos[)( ≤≤−−+⋅= ηη

ηπ

• Raised-Sinusoid Apodisation

)5.0(||1)( η−<= LzzA

LzLL

LzzA 5.0||)5.0(])5.0(||2

cos[15.0)( ≤≤−−+⋅+= ηη

ηπ

• Tanh Apodisation

LzLzzA 5.05.0)tanh(

|)]|21(tanh[)( ≤≤−−=η

η

• Blackman Apodisation

LzLzzzA 5.05.0)1(2

)4cos()2cos()1(1)( ≤≤−+

+++=η

πηπη

To calculate the transfer matrix for the chirped fiber Bragg grating with an apodised refractive index modulation, one can discretize the grating into short sections which can be approximated to uniform gratings. The transfer matrix of the grating will be the multiplication of the transfer matrix of each section.

For analyzing the characteristics of the grating, a text file named as “fbgcurve.txt” will be generated after the simulation is done. In this text file, there are five columns. The first column is the wavelength (nm). The second and third columns are the reflectivity of the grating without unit and in dB respectively. The fourth column is the time delay (ps) introduced by the reflection of the grating, and the fifth column is the phase of reflection coefficient (radian). This characteristic is useful for simulating the dispersion compensation effect of fiber Bragg gratings.


Properties



Parameter Values Name Type Default Range Units period double 5.35355e-7 [ 0, 1e32 ] m

length double 1.0e-2 [ 0, 1e32 ] m ModulationDepth

double 1.0e-4 [ -1e32, 1e32 ] none

RefractiveIndex double 1.45 [ 0, 1e32 ] none

LinearChirp double 0 [ -1e32, 1e32 ] nm

ApodisationPattern

enumerated UniformGrating UniformGrating, Cosine, RaisedSinusoid, Tanh, Blackman

ApodisationCoef

double 0 [ -1e32, 1e32 ] none

temperature double 20.0 [ 0, 1e32 ] C

Parameter Descriptions ApodisationCoef Apodisation coefficient. ApodisationPattern Apodisation pattern (Uniform Grating, Cosine, RaisedSinusoid, Tanh, Blackman). LinearChirp Linear chirp coefficient. ModulationDepth The effective refractive index modulation depth. RefractiveIndex The effective background refractive index of the host fiber. Length The length of grating. Period The period of the grating. Temperature The operation temperature of the grating.


Optical Multiplexer (Nx1 MUX)

This model represents an optical WDM multiplexer (see also the General Multiport Optical Device described below). It accepts multiple optical signals at its input ports and produces a WDM optical signal at its output port which includes all the input WDM optical signals.

There are two signal representations that may be used at the output, depending on whether four wave mixing is desired for use in the fiber model. The multiple-band mode will put each optical signal band in its own signal representation, thereby decreasing the overall memory load of the simulation by not including the unused frequency bands between the bands. This approximation can only be made when the bands do not overlap significantly. If there is significant overlap, or it is desired to include the effects of four wave mixing in the fiber simulation, the single-band mode must be used.

To use the single-band mode, the samples per bit (pointsPerBit) in the signal generator for all the bands must be set high enough to include all the frequency components of each of the bands in the frequency domain representation of the signal. A good rule of thumb is that the simulation bandwidth be chosen to be about 3 times the total signal bandwidth (number of bands times band spacing). The simulation bandwidth is given by the following:

sPerBitpos

sPerBitpo

ssim

bitRateT

bitRateT

BW

int

int

21

21

•=

•==

where BWsim is the simulation bandwidth, Ts is the time step between data points, bitRate is the bit rate specified in the PRBS generator, and pointsPerBit is the sampling rate specified in the electrical signal generator. The pattern length, samples per bit and bit rate must be equal for all the bands. These may be set conveniently through user defined variables in the symbol table.

To represent a more realistic optical MUX, an optical filter may be specified for the input channels. For details on the optical filter types supported, see the documentation on the Optical Filter model. When the filter is used, the spacing between the filters may be uniform in either frequency or wavelength, depending on the setting of the filterSpecMode parameter. If this parameter is set to Frequency, then all filter bandwidths and centers are specified in units of Hz, along with the spacing between the filter centers. When the custom filter types are used, different custom filter responses may be specified for each of the input ports by using the file format of the General Multiport Optical Device and specifying that file in the filename parameter.

Properties



Parameter Values

Name Type Default Range Units


Representation enumerated MultiBand MultiBand, SingleBand


filterType enumerated None None, Ideal, Trapezoidal, Gaussian, FabryPerot, Custom1, Custom2, Custom3

filename string filterSpecMode enumerated Wavelength Frequency,

Wavelength

filterBW double 0.1e-9 [ 1e-32, 1e32 ] Hz or meters

filterBW0dB double 0.0 [ 0, 1e32 ] Hz or meters

filterOrder integer 1 [ 1, 128 ] none

filterSpacing double 0.8e-9 [ 1e-32, 1e32 ] Hz or meters firstFilterCenter double 1550e-9 [ 1e-32, 1e32 ] Hz or meters

filterFSR double 100e-9 [ 1e-32, 1e32 ] Hz or meters

Parameter Descriptions Representation Whether to use multiple-band or single-band representation for WDM optical signals filename File name for the custom filter types filterType Type of filter operation on input ports: several filter types are supported filterSpecMode Whether filter specs are in frequency or wavelength units firstFilterCenter First input port’s optical filter center frequency or wavelength filterSpacing Spacing between filter centers from one input port to the next filterBW 3dB filter bandwidth in wavelength filterBW0dB Zero dB bandwidth for Trapezoidal filter type filterFSR Free spectral range of Fabry Perot optical filter loss Optical Filter Insertion Loss filterOrder Order of Gaussian optical filter


Optical DeMultiplexer (1xN DEMUX)

This model represents an optical WDM demultiplexer (see also the General Multiport Optical Device described below). It accepts a WDM optical signal at its input port and produces N single channel optical signals at its output ports, one channel per port. This is accomplished by applying the specified filter to the input signal for each of the output ports. For details on the optical filter types supported, see the documentation on the Optical Filter model. The spacing between the filters may be uniform in either frequency or wavelength, depending on the setting of the filterSpecMode parameter. If this parameter is set to Frequency, then all filter bandwidths and centers are specified in units of Hz, along with the spacing between the filter centers. When the custom filter types are used, different custom filter responses may be specified for each of the output ports by using the file format of the General Multiport Optical Device and specifying that file in the filename parameter.

The carrier frequency of the output optical signal(s) may be shifted to match the center frequency of the filters if desired. The user may choose to let the program decide automatically whether to do this, or to turn this feature on or off. The automatic mode will shift the carrier frequency if the signal is not a multiple frequency band WDM signal.

Properties



Parameter Values

Name Type Default Range Units loss double 0 [ 0, 1e9 ] dB

filterType enumerated Gaussian Ideal, Trapezoidal, Gaussian, FabryPerot, Custom1, Custom2, Custom3

filename string

filterSpecMode enumerated Wavelength Frequency, Wavelength

filterBW double 0.1e-9 [ 1e-32, 1e32 ] Hz or meters

filterBW0dB double 0.0 [ 0, 1e32 ] Hz or meters

filterOrder integer 1 [ 1, 128 ] none filterSpacing double 0.8e-9 [ 1e-32, 1e32 ] Hz or meters

firstFilterCenter double 1550e-9 [ 1e-32, 1e32 ] Hz or meters

filterFSR double 100e-9 [ 1e-32, 1e32 ] Hz or meters

shiftcarrier enumerated Auto No, Yes, Auto


dropThreshold double 0 [ 0, 1e32 ] Watt

Parameter Descriptions filterType Type of filter operation: several filter types are supported filename File name for the custom filter types filterSpecMode Either Frequency or Wavelength specifications firstFilterCenter Optical filter center frequency or wavelength of first output channel filterSpacing Either uniform frequency or wavelength spacing between output channels filterBW 3dB filter bandwidth in either frequency or wavelength filterBW0dB Zero dB bandwidth for Trapezoidal filter type, either frequency or wavelength filterFSR Free spectral range of Fabry Perot optical filter, either frequency or wavelength loss Optical Filter insertion loss filterOrder Order of Gaussian optical filter shiftcarrier Whether to shift carrier frequency of signal (Auto, Yes, No) dropThreshold Signals whose peak value does not exceed threshold will be dropped


Optical Add Multiplexer

This model represents an optical WDM add multiplexer (see also the general multiport optical device described below). It accepts a WDM optical signal at its first input port, and a single-band optical signal at second input port (WDM signals may be used due to the filtering operation performed on them). The signals from the first input port pass through a notch filter to remove the channel being added from the second input port. The signals from the second input port are filtered with a bandpass filter to separate out the desired channel to add to the WDM signal at the output. The output is a WDM optical signal with the channel from the second input port added to the WDM signal provided at the first input port.

The filter used to filter out the desired channel from the first input port’s WDM signal is an inverse of the filter used to select the desired channel from the second input port’s optical signal. For details on the optical filter types supported, see the documentation on the optical filter model. The filter specifications are provided in units of wavelength. Specific custom defined filters may be specified for these by using one of the custom filter types. In this case, each of the two input filters would be specified independently using the N×M multiport optical device file format.

There are two signal representations that may be used at the output, depending on whether four wave mixing is desired for use in the fiber model. The multiple-band mode will put each optical signal band in its own signal representation, thereby decreasing the overall memory load of the simulation by not including the unused frequency bands between the channels. This approximation can only be made when the channels do not overlap significantly. If there is significant overlap, or it is desired to include the effects of four wave mixing in the fiber simulation, the single-band mode must be used.

To use the single-band mode, the samples per bit (pointsPerBit) in the signal generator for all the bands must be set high enough to include all the frequency components of each of the channels in the frequency domain representation of the signal. A good rule of thumb is that the simulation bandwidth be chosen to be about 3 times the total signal bandwidth (number of bands times band spacing). The simulation bandwidth is given by the following:

sPerBitpos

sPerBitpo

ssim

bitRateT

bitRateT

BW

int

int

21

21

•=

•==

where BWsim is the simulation bandwidth, Ts is the time step between data points, bitRate is the bit rate specified in the PRBS generator, and pointsPerBit is the sampling rate specified in the electrical signal generator. The pattern length, samples per bit and bit rate must be equal for all the channels. These may be set conveniently through user defined variables in the symbol table.

The carrier frequency of the optical signal being added may be shifted to match the center frequency of the filter if desired. The user may choose to let the program decide automatically whether to do this, or to turn this feature on or off. The automatic mode will shift the carrier frequency if the signal is not a multiple frequency band WDM signal.

Properties



#2: Optical signal


Parameter Values

Name Type Default Range Units Representation enumerated MultiBand MultiBand,

SingleBand



filename string

filterBW double 0.1e-9 [ 1e-32, 1e32 ] meters

filterBW0dB double 0.0 [ 0, 1e32 ] meters filterOrder integer 1 [ 1, 128 ] none

filterCenter double 1550e-9 [ 1e-32, 1e32 ] meters

filterFSR double 100e-9 [ 0, 1e32 ] meters


Parameter Descriptions Representation Whether to use multiple-band or single-band representation for WDM optical

signals filterType Type of filter operation: several filter types are supported filename File name for the custom filter types filterCenter Optical filter center wavelength of channel being added filterBW 3dB filter bandwidth in wavelength filterBW0dB Zero dB bandwidth for Trapezoidal filter type, wavelength filterFSR Free spectral range of Fabry Perot optical filter, wavelength loss Optical Filter insertion loss filterOrder Order of Gaussian optical filter shiftcarrier Whether to shift center frequency of added optical signal (carrier frequency)


Optical Drop Multiplexer

This model represents an optical WDM drop multiplexer (see also the general multiport optical device described below). It accepts a WDM optical signal at its input port, and outputs the WDM signal minus the dropped channel at the first output port, and the dropped channel itself at the second output port. The signals from the input port pass through a notch filter to remove the channel being dropped from the first output port. The signals from the input port are filtered with a bandpass filter to separate out the desired drop channel to the second output port.

The filter used to filter out the dropped channel from the first output port’s WDM signal is an inverse of the filter used to select the desired channel for the second output port’s optical signal. For details on the optical filter types supported, see the documentation on the optical filter model. The filter specifications are provided in units of wavelength. Specific custom defined filters may be specified for these by using one of the custom filter types. In this case, each of the two filters would be specified independently using the N×M multiport optical device file format.

The carrier frequency of the dropped output optical signal may be shifted to match the center frequency of the filter if desired. The user may choose to let the program decide automatically whether to do this, or to turn this feature on or off. The automatic mode will shift the carrier frequency if the signal is not a multiple frequency band WDM signal.

Properties



#2: Optical signal

Parameter Values

Name Type Default Range Units loss double 0 [ 0, 1e9 ] dB filterType enumerated Gaussian Ideal, Trapezoidal,

Gaussian, FabryPerot, Custom1, Custom2, Custom3

filename string

filterBW double 0.1e-9 [ 1e-32, 1e32 ] meters filterBW0dB double 0.0 [ 0, 1e32 ] meters


filterCenter double 1550e-9 [ 1e-32, 1e32 ] meters

filterFSR double 100e-9 [ 0, 1e32 ] meters shiftcarrier enumerated Auto No, Yes, Auto


Parameter Descriptions filterType Type of filter operation: several filter types are supported filename File name for the custom filter types filterCenter Optical filter center wavelength of channel being added filterBW 3dB filter bandwidth in wavelength filterBW0dB Zero dB bandwidth for Trapezoidal filter type, wavelength filterFSR Free spectral range of Fabry Perot optical filter, wavelength loss Optical Filter insertion loss filterOrder: Order of Gaussian optical filter shiftcarrier Whether to shift center frequency of dropped channel signal


Optical Add/Drop Multiplexer

This model represents an optical WDM add/drop multiplexer (see also the general multiport optical device described below). It accepts a WDM optical signal at its first input port, and a single-band optical signal at second input port (WDM signals may be used due to the filtering operation performed on them). The block performs a combined function of the add multiplexer and the drop multiplexer. The specified signal is first dropped from the first input port’s WDM signal and provided at the second output port, and then the second input port’s signal is added to the resulting WDM signal minus the dropped channel and provided at the first output port. The first output port’s signal is a WDM optical signal with the channel from the second input port added to the WDM signal provided at the first input port. The second output port’s signal is a single-band optical signal which represents the dropped channel. The block diagram representing this component’s function follows:

The filter used to filter out the desired channel from the first input port’s WDM signal is an inverse of the filter used to select the desired channel from the second input port’s optical signal. For details on the optical filter types supported, see the documentation on the optical filter model. The filter specifications are provided in units of wavelength. Specific custom defined filters may be specified for these by using one of the custom filter types. In this case, each of the two input filters would be specified independently using the N×M multiport optical device file format.

There are two signal representations that may be used at the output, depending on whether four wave mixing is desired for use in the fiber model. The multiple-band mode will put each optical signal band in its own signal representation, thereby decreasing the overall memory load of the simulation by not including the unused frequency bands between the signal bands. This approximation can only be made when the bands do not overlap significantly. If there is significant overlap, or it is desired to include the effects of four wave mixing in the fiber simulation, the single-band mode must be used.

To use the single-band mode, the samples per bit (pointsPerBit) in the signal generator for all the bands must be set high enough to include all the frequency components of each of the bands in the frequency domain representation of the signal. A good rule of thumb is that the simulation bandwidth be chosen to be about 3 times the total signal bandwidth (number of bands times band spacing). The simulation bandwidth is given by the following:

sPerBitpos

sPerBitpo

ssim

bitRateT

bitRateT

BW

int

int

21

21

•=

•==

where BWsim is the simulation bandwidth, Ts is the time step between data points, bitRate is the bit rate specified in the PRBS generator, and pointsPerBit is the sampling rate specified in the electrical signal generator. The pattern length, samples per bit and bit rate must be equal for all the channels. These may be set conveniently through user defined variables in the symbol table.


The carrier frequency of the added input and dropped output optical signal(s) may be shifted to match the center frequency of the filters if desired. The user may choose to let the program decide automatically whether to do this, or to turn this feature on or off. The automatic mode will shift the carrier frequency if the signal is not a multiple frequency band WDM signal.

Properties


#2: Optical signal


#2: Optical signal

Parameter Values

Name Type Default Range Units Representation enumerated MultiBand MultiBand,

SingleBand



filename string filterBW double 0.1e-9 [ 1e-32, 1e32 ] meters

filterBW0dB double 0.0 [ 0, 1e32 ] meters


filterCenter double 1550e-9 [ 1e-32, 1e32 ] meters filterFSR double 100e-9 [ 0, 1e32 ] meters


Parameter Descriptions Representation Whether to use multiple-band or single-band representation for WDM optical signals filterType Type of filter operation: several filter types are supported filename File name for the custom filter types filterCenter Optical filter center wavelength of channel being added filterBW 3dB filter bandwidth in wavelength filterBW0dB Zero dB bandwidth for Trapezoidal filter type, wavelength filterFSR Free spectral range of Fabry Perot optical filter, wavelength loss Optical Filter insertion loss filterOrder Order of Gaussian optical filter


shiftcarrier Whether to shift center frequency of filtered signals to match filter center


General Multiport Optical Device (NxM and WDM)

This model represents a general N×M multiport optical WDM device via a data-driven (experiment or theory) transfer matrix model. It can be used to model devices such as arrayed waveguide grating (AWG) multiplexers, add, drop, or add/drop multiplexers, and other optical filters and components whose response is determined by a wavelength dependent transfer matrix. The key parameter to the model is the name of a data file containing the wavelength dependent transfer matrix (format documented below). Additional parameters include the CommandLine, which allows integration of OptSim with device level tools or measuring instruments which can generate the required transfer matrix (e.g. RSoft Design Group’s BeamPROP), and SignalBandwidth, which allows the model to treat the signal as having a narrow bandwidth compared to the filter function for optimization, or to treat a fully broadband signal without approximation.

The transfer matrix data file is an ASCII file, which begins with header information as follows: TransferMatrixFormat1 <SubFormat> <FactorFormat>

<# inputs, N> <# outputs, M> <# of wavelength points, K> <Min λ> <Max λ>

The <SubFormat> can be either POWER, REAL, REAL_IMAG, AMP_PHASE, AMP_PHASE_RAD, POWER_PHASE, POWER_DELAY, POWER_DELAY_PS, AMP_DELAY, or AMP_DELAY_PS. This format indicates the type of transfer matrix coefficient as will be described shortly. The <FactorFormat> may be left out, or it may be specified as “DB”. If it is left out, the transfer matrix coefficients are assumed to be unitless, ranging from 0 to 1. If it is set to DB, the transfer matrix coefficients are assumed to be in units of dB. The other items have the obvious meanings. The header is followed by blocks containing the wavelength dependent transfer matrix coefficients. Each block looks like

<λ 1> <coefficient 11> <coefficient 12> … <coefficient 1M>

<λ 2> <coefficient 21> <coefficient 22> … <coefficient 2M>

<λ K> <coefficient K1> <coefficient K2> … <coefficient KM>

The coefficient depends upon the <SubFormat>:

• POWER

The coefficient should be the (real) power transfer coefficient.

• REAL

The coefficient should be the (real) amplitude transfer coefficient.

• REAL_IMAG

The coefficient should be a pair of numbers, the first of which is the real value of the amplitude transfer coefficient, and the second of which is the imaginary value. Both of these numbers should be on the same line, separated by a space.

• AMP_PHASE

The coefficient should be a pair of numbers, the first of which is the magnitude of the amplitude transfer coefficient, and the second of which is the phase of the amplitude transfer coefficient in units of degrees. Both of these numbers should be on the same line, separated by a space.

• AMP_PHASE_RAD

The coefficient should be a pair of numbers, the first of which is the magnitude of the amplitude transfer coefficient, and the second of which is the phase of the amplitude transfer coefficient in units of radians. Both of these numbers should be on the same line, separated by a space.

• POWER_PHASE


The coefficient should be a pair of numbers, the first of which is the magnitude of the power transfer coefficient, and the second of which is the phase of the power transfer coefficient in units of degrees. Both of these numbers should be on the same line, separated by a space.

• POWER_DELAY

The coefficient should be a pair of numbers, the first of which is the magnitude of the power transfer coefficient, and the second of which is the group delay in units of seconds. Both of these numbers should be on the same line, separated by a space.

• POWER_DELAY_PS

The coefficient should be a pair of numbers, the first of which is the magnitude of the power transfer coefficient, and the second of which is the group delay in units of picoseconds. Both of these numbers should be on the same line, separated by a space.

• AMP_DELAY

The coefficient should be a pair of numbers, the first of which is the magnitude of the amplitude transfer coefficient, and the second of which is the group delay in units of seconds. Both of these numbers should be on the same line, separated by a space.

• AMP_DELAY_PS

The coefficient should be a pair of numbers, the first of which is the magnitude of the amplitude transfer coefficient, and the second of which is the group delay in units of picoseconds. Both of these numbers should be on the same line, separated by a space.

All wavelengths should be specified in units of meters. Currently, the wavelength column is ignored and the wavelength assumed to vary linearly from the minimum to maximum values specified in the header. The order of the blocks should be as follows:

<block for input port 1>

<block for input port 2>

<block for input port N>

where each block contains a series of coefficients for each of the M output ports. Each wavelength line in the block has a series of coefficients for each of the M output ports listed in order, as follows:

<λ 1> <coefficient(s) for output port 1> <coefficient(s) for output port 2> …

<λ 2> <coefficient(s) for output port 1> <coefficient(s) for output port 2> …

…

<λ K> <coefficient(s) for output port 1> <coefficient(s) for output port 2> …

If there are any questions regarding this format, please contact RSoft Design Group.

Properties


Outputs #1-M: Optical signal

Parameter Values


Name Type Default Range Units SignalBandwidth enumerated Wide Narrow, Wide

CommandLine string FileName string

Parameter Descriptions CommandLine Optional command line which will be executed to generate the transfer matrix

data (for integration with RSoft Design Group’s BeamPROP or other device tools) FileName Name of the file containing the transfer matrix data SignalBandwidth Bandwidth assumption used in the calculation (Narrow or Wide)


Jones Matrix Transfer Function

This model represents an optical device via a data-driven (experiment or theory) linear transfer function model based on the Jones matrix. It can be used to model devices such as arrayed waveguide grating (AWG) multiplexers, add, drop, or add/drop multiplexers, and other optical filters and components whose response is characterized by a wavelength- and polarization-dependent linear transfer function based on the Jones matrix.

Theoretical Background The Jones matrix is a 2×2 matrix of numbers that represent the modification of the state of polarization (SOP) of a device on an input signal. It is shown below:

=

)()(

)()(

tEtE

DCBA

tEtE

yi

xi

yo

xo

where Exi(t) and Eyi(t) are the input optical signal’s instantaneous electric field scalar values corresponding to the x and y polarization states, Exo(t) and Eyo(t) are the output optical signal’s instantaneous electric field scalar values corresponding to the x and y polarization states, and A, B, C, and D are the elements of the Jones matrix (also referred to as J11, J12, J21, and J22, respectively).

The linear transfer function goes beyond the Jones matrix in including the magnitude and phase response through the use of a complex number for each of A, B, C and D, as well as providing values for these elements at each wavelength or frequency data point in the frequency or wavelength spectrum of the component. The transfer function is specified in the frequency domain, and is shown below:

=

)()(

)()()()(

)()(

ωω

ωωωω

ωω

yi

xi

yo

xo

EE

DCBA

EE

where Exi(ω) and Eyi(ω) are the input optical signal’s complex frequency domain representation corresponding to the x and y polarization states, Exo(ω) and Eyo(ω) are the output optical signal’s complex frequency domain representation corresponding to the x and y polarization states, and A, B, C, and D are the complex elements of the linear transfer function.

The linear transfer function represents the linear transformation of an input optical signal’s amplitude, phase, and polarization state by an optical device in producing the output signal. This representation provides a complete characterization of the linear input/output relationship of an optical device. From this data, magnitude and phase response, group delay (GD), differential group delay (DGD), and polarization dependent loss (PDL) can be determined. When actual linear transfer function measurements of a device are used with this block to model it in a system simulation, its system-level performance can be analyzed and system designs utilizing the device optimized.

As discussed in the OptSim User Manual, an optical signal is represented using a slowly varying envelope approximation as a set of magnitude and phase vectors vs. time. This is transformed into the frequency domain for linear transformations using the linear transfer function in this model. The group delay (GD) property of an optical component is defined as the first derivative of optical phase with respect to optical frequency. The chromatic dispersion (CD) is the second derivative of optical phase with respect to optical frequency. The formulas for these quantities are shown below:

ωϕ

∂∂=GD 2

2

ωϕ

∂∂=CD

Where φ is the optical phase and ω is the optical frequency.


Using the Model A linear transfer function can be imported from optical test equipment provided by leading manufacturers, including Agilent Technologies, Luna Technologies, and FiberWork Optical Communications. Alternatively, users can define their own data or import it from other tools and software using the RSoft data format specified in this documentation.

The key parameter for the model is FileName, the name of the data file containing the Jones matrix data. Additional parameters include the following: FileType, which specifies the file format of the data file; CommandLine, which allows integration of OptSim with other software tools or measuring instruments which can generate or operate on the required Jones matrix; DataColumn which specifies which data column to use in a multicolumn format input file; Threshold, which specifies the threshold below which the optical signal or spectrum is considered to have zero power; and SignConvention which specifies whether the group delay should be computed as a positive or negative value in the Test function. Note that the SignConvention parameter only affects the sign of the GD and CD outputs of the Test function, not the simulation results.

Test Features This model block includes test functions used to ascertain properties of the selected DataColumn in the provided data file including the following:

• Wavelength response (attenuation and phase vs. wavelength) for each Jones matrix element

• Frequency response (attenuation and phase vs. frequency) for each Jones matrix element

• Differential group delay (DGD) vs. wavelength

• Differential group delay (DGD) vs. frequency

• Group delay (GD) vs. wavelength

• Group delay (GD) vs. frequency

• Chromatic dispersion (CD) vs. wavelength

Note that the group delay and chromatic dispersion are reported as averages of the x polarization and y polarization values. The differential group delay is reported as the difference between the x polarization and y polarization group delays. The SignConvention parameter may be modified to specify the sign of the group delay output in the Test function. Note that the SignConvention parameter only affects the sign of the GD and CD outputs of the Test function, not the simulation results.

File Formats

Luna File Format The Luna format is based on Luna Technologies’ Binary measurement data file versions 3.0 through 3.3, generated by the Luna Optical Vector Analyzer (OVA) products. Contact Luna Technologies for specifics on this file format.

Agilent File Format The Agilent format Jones matrix data file is an ASCII file. This file format is generated by the Agilent 81910A Photonic All-Parameter Analyzer. Following is a description of this file format. Contact Agilent for further details on the file format.

The first line begins with '%' and is a comment line whose contents are not important for the simulation. The second line begins with '%' and is a comment line for the user describing the contents of each column of the comma separated values in the main body of the file. Again, this line is not important for the simulation.


The third line begins with '%' and contains a list of the sets of data columns in the main body of the file. Each set of data columns is identified by a channel number written in the format "Channel_x" where x is the channel number, followed by a space and either "Rx" or "Tx" depending on whether the data corresponds to reflection or transmission data, followed by a space and then a description of the data, e.g. "Measurement". It is essential that each set of data columns have a corresponding "Channel_x" entry for the simulation. The fourth and subsequent lines are the data columns. Each line represents a single wavelength. On that line, there are 9 columns of comma separated values for each set of data identified in the third comment line of the file. These columns are wavelength (m) followed by the 8 numerical values for the Jones matrix in the following format and order: Amp(J11), Phase(J11), Amp(J12), Phase(J12), Amp(J21), Phase(J21), Amp(J22), Phase(J22)

The Port parameter can be set to choose which set of data columns in the data file to use for the output port of the model block (e.g., “Channel_1 Tx” data, “Channel_2 Tx” data, etc.)

RSoft File Format The RSoft format Jones matrix data file is an ASCII file, which begins with header information as follows (on two lines):

<Format> <SubFormat> <FactorFormat>

<# inputs, N> <# outputs, M> <# of wavelength or frequency points, K> <Min λ or f> <Max λ or f>

Presently, the model supports one input port and one output port. However, the data file may provide data for one input port and multiple output ports. <N> therefore must be 1, but <M> may be any number representing the number of output ports for which data are provided in the data file. Since only one output port is supported by the model block, the DataColumn parameter is used to specify which output port (or column) from the data file is to be used for the single output port of the model block. The values for <K>, <Min λ or f>, and <Max λ or f> must also be specified in this line. The data in this line overrides the wavelength or frequency data column in the following data lines.

The <Format> can be either JonesMatrixFormat1 or JonesMatrixFormat2. JonesMatrixFormat1 indicates that the data in the file is evenly spaced in wavelength. JonesMatrixFormat2 indicates that the data in the file is evenly spaced in frequency.

The <SubFormat> can be either REAL_IMAG, AMP_PHASE_RAD, POWER_PHASE_RAD, AMP_DISP, AMP_DELAY_PS, or POWER_DELAY_PS. This format indicates the type of Jones matrix coefficient as will be described shortly. The <FactorFormat> may be left out, or it may be specified as “DB”. If it is left out, the Jones matrix coefficients are assumed to be unitless, ranging from 0 to 1. If it is set to DB, the Jones matrix (amplitude or power) coefficients are assumed to be in units of dB. The header is followed by blocks containing the wavelength- and polarization-dependent Jones matrix coefficients. Each block looks like

<λ 1> <Jones matrix 11> <Jones matrix 12> … <Jones matrix 1M>

<λ 2> <Jones matrix 21> <Jones matrix 22> … <Jones matrix 2M>

<λ K> <Jones matrix K1> <Jones matrix K2> … <Jones matrix KM>

The coefficient depends upon the <SubFormat>:

• REAL_IMAG

The Jones matrix is a set of eight numbers representing the Jones matrix for the given wavelength or frequency, depending on whether the format is JonesMatrixFormat1 or JonesMatrixFormat2: the real value of A, the imaginary value of A, the real value of B, the imaginary value of B, the real value of C, the imaginary value of C, the real value of D, and the imaginary value of D. All of these numbers should be on the same line, separated by a space, comma, colon, semicolon, or tab. There are M Jones matrices on each line, representing the M possible output ports of the modeled device. The Jones matrix used to model the component in the simulation is chosen by the setting of the parameter DataColumn.

• AMP_PHASE_RAD


The Jones matrix is a set of eight numbers representing the Jones matrix for the given wavelength or frequency, depending on whether the format is JonesMatrixFormat1 or JonesMatrixFormat2: the magnitude of the amplitude of A, the phase (in radians) of A, the magnitude of the amplitude of B, the phase (in radians) of B, the magnitude of the amplitude of C, the phase (in radians) of C, the magnitude of the amplitude of D, and the phase (in radians) of D. All of these numbers should be on the same line, separated by a space, comma, colon, semicolon, or tab. There are M Jones matrices on each line, representing the M possible output ports of the modeled device. The Jones matrix used to model the component in the simulation is chosen by the setting of the parameter DataColumn.

• POWER_PHASE_RAD

The Jones matrix is a set of eight numbers representing the Jones matrix for the given wavelength or frequency, depending on whether the format is JonesMatrixFormat1 or JonesMatrixFormat2: the magnitude of the power of A, the phase (in radians) of A, the magnitude of the power of B, the phase (in radians) of B, the magnitude of the power of C, the phase (in radians) of C, the magnitude of the power of D, and the phase (in radians) of D. The power values in this format are equivalent to the square of the amplitude values in the AMP_PHASE_RAD format. All of these numbers should be on the same line, separated by a space, comma, colon, semicolon, or tab. There are M Jones matrices on each line, representing the M possible output ports of the modeled device. The Jones matrix used to model the component in the simulation is chosen by the setting of the parameter DataColumn.

• AMP_DISP

The Jones matrix is a set of eight numbers representing the Jones matrix for the given wavelength or frequency, depending on whether the format is JonesMatrixFormat1 or JonesMatrixFormat2: the magnitude of the amplitude of A, the dispersion of A, the magnitude of the amplitude of B, the dispersion of B, the magnitude of the amplitude of C, the dispersion of C, the magnitude of the amplitude of D, and the dispersion of D. All of these numbers should be on the same line, separated by a space, comma, colon, semicolon, or tab. There are M Jones matrices on each line, representing the M possible output ports of the modeled device. The Jones matrix used to model the component in the simulation is chosen by the setting of the parameter DataColumn.

• AMP_DELAY_PS

The Jones matrix is a set of eight numbers representing the Jones matrix for the given wavelength or frequency, depending on whether the format is JonesMatrixFormat1 or JonesMatrixFormat2: the magnitude of the amplitude of A, the group delay (in units of picoseconds) of A, the magnitude of the amplitude of B, the group delay (in units of picoseconds) of B, the magnitude of the amplitude of C, the group delay (in units of picoseconds) of C, the magnitude of the amplitude of D, and the group delay (in units of picoseconds) of D. All of these numbers should be on the same line, separated by a space, comma, colon, semicolon, or tab. There are M Jones matrices on each line, representing the M possible output ports of the modeled device. The Jones matrix used to model the component in the simulation is chosen by the setting of the parameter DataColumn.

• POWER_DELAY_PS

The Jones matrix is a set of eight numbers representing the Jones matrix for the given wavelength or frequency, depending on whether the format is JonesMatrixFormat1 or JonesMatrixFormat2: the magnitude of the power of A, the group delay (in units of picoseconds) of A, the magnitude of the power of B, the group delay (in units of picoseconds) of B, the magnitude of the power of C, the group delay (in units of picoseconds) of C, the magnitude of the power of D, and the group delay (in units of picoseconds) of D. Note that the power values are the square of the amplitude values in the AMP_DELAY_PS format. All of these numbers should be on the same line, separated by a space, comma, colon, semicolon, or tab. There are M Jones matrices on each line, representing the M possible output ports of the modeled device. The Jones matrix used to model the component in the simulation is chosen by the setting of the parameter DataColumn.

All wavelengths should be specified in units of meters and frequencies in units of Hz. Currently, the wavelength/frequency column is ignored and the value assumed to vary linearly from the minimum to maximum values specified in the header.


If there are any questions regarding this format, please contact RSoft Design Group.

Properties



Parameter Values

Name Type Default Range Units FileName file None

FileType enumerated RSoft Agilent, Luna, RSoft None

SignConvention Enumerated Positive Positive, Negative None CommandLine string None

DataColumn int 1 1 ≤ x ≤ 128 none

Threshold double 0.0 0 ≤ x ≤ 1e32 None

Parameter Descriptions FileName Name of the file containing the Jones matrix data FileType Whether it is in format of Agilent, Luna, or RSoft SignConvention What sign convention is used for group delay computation in Test function

CommandLine Optional command line which will be executed prior to use of the Jones matrix data file; can be used to generate or operate on the data by other tools

DataColumn The Jones matrix column or port in the data file used by model Threshold The transmission threshold below which the transmitted power is assumed to be

zero

OptSim Models Reference: Block Mode Chapter 9: Optical Receivers •••• 291

Chapter 9: Optical Receivers

• Compound optical receiver detect an optical signal and produce an amplified electrical signal

• Photodetector model just the PIN or APD photodetector at the start of a receiver unit

292 •••• Chapter 9: Optical Receivers OptSim Models Reference: Block Mode

Compound Optical Receiver

This models an optical receiver and all its standard parts. The OptSim photoreceiver model is composed of several individual building blocks: the photodetector, the preamplifier, and the postamplifier/filter complex:

Figure 1: Basic components of an optical receiver

Each block is a separate entity complete with its own input parameters and options. The photodetector model converts an optical input signal to an electrical current. This photocurrent is then passed to the preamplifier model which converts it to a voltage. Finally, the postamplifier model contains a set of baseband filters that shape the output waveforms. The model also computes the photoreceiver noise components.

In fact the receiver model is implemented directly in terms of the three stand-alone models the PIN/APD Photodetector, Electrical Amplifier and Electrical Filter model described elsewhere in this manual and all the parameters of each of those models are also parameters of the monolithic receiver model. In other words the two configurations shown in the topology in Figure 2 serve equivalent functions.

There are two motivations for providing this additional receiver model. As a matter of convenience and topology compactness, some users may find it useful to represent the receiver configuration with a single block. However, this function could also have been achieved by creating a super-block containing the three individual models. Rather, the principal reason for providing a combined model is to allow a “Quasi-Analytic” (QA) treatment of the receiver noise. Both the photodetector and electrical amplifier model a number of noise sources – shot noise, dark current noise, signal-spontaneous emission beta noise, thermal noise in the pre-amp transistor etc. In these two models, the noise is added directly as a stochastic contribution to the electrical signal – the so-called Monte-Carlo (MC) picture.

Figure 2: Two configurations for modeling an optical receiver.


However, as explained in the chapter on signal representations, electrical noise in OptSim may also be represented as a time-dependent vector of the standard deviation of a Gaussian white noise source. This representation is especially useful in bit error rate calculations, since it allows the contribution of each bit to the BER to be evaluated separately and trivially accounts correctly for intersymbol interference (“pattern-dependent”) effects. This is referred to as the QA method for BER calculation. As discussed in the BERTester documentation, extracting the BER from a signal with Monte-Carlo noise is considerably more involved, particularly if the signal exhibits strong pattern dependence. Thus, provided the receiver components are the dominant source of noise in the system, the QA approach is preferred due to increased accuracy and speed. On the other hand, if ASE is the dominant source of noise, and the propagation through the fiber is strongly nonlinear, then the noise properties are non-Gaussian and the MC approach is unavoidable. Which of these two cases is true is determined by experimentation with simulations.

So the representation of noise as a separate standard deviation is necessary to allow BER calculations with the QA method, but as we explore below in the section QA noise expressions, this is only possible if all stages of the receiver from photodetection from filtering can be considered together because the variances for each noise source depend on parameters from at least two of the sub-models.

Model Parameters Since the Receiver is constructed from the photodector, electrical amplifier and electrical filter models, it includes all the physics and essentially all the parameters of those three models. For the sake of space and clarity, we will not repeat the explanations of the deterministic parts of the models –definition of photodetector quantum efficiency, amplifier spectral response, filter bandwidth, etc – which are identical for this model. Moreover, we will not repeat the discussions of these many parameters here and the reader should consult the specific documentation for each of the other models. The parameters have the same names in the monolithic receiver as in the individual models with the addition of prefixes denoting which part of the system the parameter belongs to. The prefixes are pd_ for the photodetector parameters, fe_ (front end) for the amplifier parameters and flt_ for the filter parameters. So for example, the dark current in the photodetector is set by pd_darkCurrent, the transimpedance of the amplifier by fe_tZ and the filter bandwidth by flt_bandwidth. While not discussed in detail, all the parameters are listed in the tables at the conclusion of the receiver documentation.

Since the calculation of noise effects involve several models, certain of the noise parameters do not have any prefixes. So the spectrum of thermal noise is specified by n_a0, n_a2, n_a4, n_a6 exactly as in the electrical amplifier model.

Noise Representation and Effects The choice between the quasi-analytic and Monte-Carlo treatments of noise is set with the parameter n_representation. With n_representation=MC, the Monte-Carlo approach is used, and the model is exactly equivalent to the concatentation of the three component models.

The quasi-analytic treatment (n_representation=QA) is identical with respect to the deterministic parts of the component models. That is, the generated electrical signal is identical to the signal that would be generated by concatenating the three models and disabling all noise terms. The noise terms are calculated quite differently as we see shortly. It is important to note that the QA representation of noise incurs approximations in the spectral features of the noise. Any noise source which has a non-flat spectrum such as the thermal noise or the spontaneous-spontaneous beat noise which typically has a triangular shape is replaced by a flat spectrum of an equivalent noise density. This is an unavoidable consequence of treating the noise strictly as variances in the time domain.

We now discuss the expressions for the various noise sources and how they are combined. Each noise source may be separately disabled using the parameters beginning include_. Note that when the Monte-Carlo noise treatment is used, the spontaneous emission noise is controlled via the parameter include_SE_noise, whereas when the quasi-analytic treatment is used, the signal-spontaneous and spontaneous-spontaneous ASE beat noise are included separately via the parameters include_sigspon and include_sponspon, respectively.

Preamplifier Noise Parameters OptSim accounts for the following types of noise in the receiver model: circuit or thermal noise, shot noise due to both the detector dark current and the received signal, signal-spontaneous beat noise (when ASE noise is present in the


received signal), spontaneous-spontaneous beat noise, APD excess noise (for APD receivers), and relative intensity noise (when RIN is specified in the transmitter). In the literature, noise is described mathematically as the variance (σn

2 or <in

2>) of the photogenerated current. In the receiver, the noise is amplified along with the signal; therefore, the noise contained in the electrical signal output from OptSim’s receiver model is represented as the standard deviation of the amplified voltage signal coming out of the receiver. The standard deviation is simply:

><σ 2or nn i

The spectral density is also a commonly used noise representation. This quantity describes the amount of noise power per unit frequency (A2/Hz) and is related to the noise variance by:

dffSi in )(0

2 ∫∞=

Many noise sources are described in the literature in terms of their spectral density. It is common practice to denote the effective bandwidth of the photoreceiver by Beff (as computed from the receiver front end amplifier and filter response)

( ) 2

0

deffB H f f∞

= ∫ ,

where the combined filter response ( )H f is assumed to vanish at sufficiently high frequencies. In numerical

modeling, there is a necessary cutoff to the response at the sampling frequency, so for a flat response 1/ 2 .effB t= ∆

The current variance is then

effiin BfSffSi )()(2 =∆=

The photocurrent in the detector is described by:

Phc

PqI ℜ=

ηλ= 0

where q is the charge of an electron, η is the quantum efficiency of the photodetector, λ is the wavelength of the received optical signal, P is the optical power of the received optical signal, h is Planck’s constant, c is the speed of light, and ℜ is the responsivity. The following noise expressions are relative to the photocurrent, so they are called input-referred noise expressions. For APD photodetectors, the noise expressions are relative to the photogenerated current after multiplication in the avalanche region, or MI. OptSim represents the circuit or thermal noise as a power series expansion of frequency. The total noise power per Hz bandwidth at the input can be expressed as:

66

44

220, )( fafafaafS circuiti +++=

This is a more general form of the commonly accepted expression:

22

,)2(

44

)( fgC

kTRkT

fSm

T

fcircuiti

πΓ+=

which describes the thermal contribution of the feedback resistor in the transimpedance amplifier and the thermal channel noise in the preamplifier input transistor. In this expression, k is Boltzmann’s constant, T is the temperature, Rf is the amplifier feedback resistance, gm is the transconductance of the preamplifier input transistor, Γ is the excess channel noise factor, and CT is the total input capacitance. The generalized polynomial representation is chosen to allow the user to tailor the noise spectral density as he sees fit. It is also useful when actual noise spectra are available since it allows the noise to be represented by simply fitting the polynomial coefficients to measured data. To model white noise, simply set the coefficients a2, a4, and a6 to zero.


The photoreceiver is characterized by a transimpedance response; in other words, the transfer function of the preamplifier converts an input current to an output voltage. The relationship between the output noise voltage and the input noise current (or equivalently the input spectral density) is:

∫∞=

0

22 )()( dffSfHv iout

Random fluctuations in current are characterized by shot noise which is described in general by:

qIfSqIBi ieffshot 2)(22 =⇔=

Here q is the electronic charge and I is the current under consideration. In the OptSim photoreceiver model, shot noise due to the photoreceiver dark current and the photogenerated signal current are considered. Consequently, the shot noise due to the dark current is represented by:

effdarkdarkshot BFIqMi 22, 2=

where M is the gain of the APD (1 for PIN devices), Id is the dark current, Beff is the effective receiver bandwidth), and F is defined as follows:

)12)(1( MkkMF −−+=

where k is the APD ionization coefficient, and the other terms are as previously defined. Similarly, the shot noise due to the photogenerated signal itself is represented by:

effsignalshot FIBqMi 22, 2=

The signal-spontaneous and spontaneous-spontaneous beat noise sources are derived from standard approaches in the literature and describe the effects of amplified spontaneous emission (ASE) on the noise performance of the photoreceiver. These analyses have their roots in the fact that the optical field can be analytically decomposed into two parts, the signal Es and the noise En:

E(t) = Es(t) + En(t)

Since the optical power is defined to be the intensity of the electric field, when light hits a detector, the resulting photocurrent can be described loosely by:

++ℜ=+ℜ=ℜ= 2222 )()()(2)()()()()( tEtEtEtEtEtEtEtI nnssnsph

As seen by the last term on the right, the square-law detection process performed by the detector causes the various components of the optical field to multiply, or “beat,” with each other. The first term on the right represents the photocurrent signal itself sPℜ . The second term models the effect of the signal field beating with the noise field while the third term represents the beating of the noise with itself.

The variance due to signal-spontaneous ASE beat noise is modeled in the literature essentially by multiplying the spontaneous emission at every frequency by the optical signal:

( ) )(42222 fPPBFMi n

f

ff

seffsps

end

start

∑=

− ℜ=

Here, fstart and fend represent the beginning and end frequencies of the ASE noise spectrum. In general, the frequency range fend - fstart will be much larger than the photoreceiver bandwidth; thus, the signal-spontaneous beat noise is effectively truncated when it is passed through the receiver transfer function.

Similarly, the variance due to spontaneous-spontaneous ASE beat noise is:


( ) )()(4 212222

1 2

fPfPBFMi n

f

ff

f

ff

neffspsp

end

start

end

start

∑ ∑= =

− ℜ=

Here, a double summation is required because the ASE noise at each frequency beats with the ASE noise that exists at every other frequency in the noise bandwidth. Again, the ASE spectral density is captured from the power spectrum portion of the spectrum analyzer measurement tool and will generally be truncated by the finite receiver bandwidth.

The RIN noise is represented by the following expression:

effRINRIN BNFIMi 222 )(=

where NRIN is the RIN noise parameter defined in the transmitter and the other terms are as previously defined.

The total input-referred noise expression in the receiver is then:

222222RINspspspsshotcircuitn iiiiii ++++= −−

Seed for stochastic noise The user may control the seeding of the random number generator used to implement the stochastic representation. This satisfies –1e8 <= monte_seed <= 1 with the following behavior:

• monte_seed < 0

The generator is seeded with the actual value of monte_seed on every run of the simulation. This is useful for obtaining repeatable results.

• monte_seed = 0

The generator is seeded with an integer hashed from the string value of the component’s name. This is a convenient way to obtain repeatable results on subsequent runs within a receiver, but different noise fields in bank of receivers which have different names. This would be appropriate in a set of parallel WDM receivers, for instance.

• monte_seed = 1


Calibrating Receiver Sensitivity The bit error rate for Gaussian systems can be roughly described by:

1BER erfc2 2

Q =

20

21

01

nn ii

IIQ

+

−=

where I1 and I0 are the photocurrents in the one and zero states and the standard deviation terms represent the noise in the respective states. The complementary error function is represented by erfc. Thus, for a desired BER, the Q factor can be calculated by inverting the complementary error function; although this cannot be performed analytically, it is a simple matter to either perform it numerically or look it up in tables.


Using the equations given in this documentation, it is thus possible to set the receiver parameters to model a specific device. It is important to model the receiver’s frequency response as accurately as possible using the photodetector, front end amplifier, and filter parameters; then using the receiver’s sensitivity based on a given BER with a given average input optical power and difference between a 1 level and a 0 level, set the noise parameters so that the model’s sensitivity matches the device being modeled. Note that it is very important that the noise parameters be set properly to represent the receiver you are using if you wish the simulation results to correlate with a particular device. Also, note that simply changing the default parameters for the APD multiplier and ionization coefficient will not convert an optimal PIN receiver into an optimal APD receiver, as there are other parameters which will vary between PIN receivers and APD receivers since their designs generally differ from one another.

The documentation for the amplifier describes how to set the frequency response of the preamplifier to match a 3-dB frequency that is either measured or found on a data sheet. As a further example, suppose the desired BER for a particular receiver is 10-12; the corresponding value of Q can be easily determined from tables to be approximately 7. Once the value of Q is known, some further design guidelines can be established. One parameter that is often found on photoreceiver data sheets is the sensitivity or, more accurately, the minimum average received power. If we neglect noise terms that depend explicitly on the signal level (usually a good assumption), this quantity can be approximated as:

211

nirrQ

P

−+

ℜ=

Here, r is the extinction ratio of the signal, namely I0/I1. Thus, if the BER (and hence Q), the detector responsivity, the signal extinction ratio, and the receiver sensitivity are known, it is possible to estimate the standard deviation of the noise that is needed to model such a device.

Test Function A number of spectral responses are available as test functions, selected with test_output. The exact format for the display of the response functions is controlled with test_display. The available settings for test_output and their meanings are elec_filter_spectrum (electrical filter response function), photodetector_resp (photodetector response function), front_end_resp (amplifier response function), cumulative_resp (combined response of the detector, amplifier and filter) and noise_spectrum (thermal noise spectrum of the amplifier).

References [1] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics. New York, NY: John Wiley & Sons, Inc., 1991.

[2] M. E. Van Valkenberg, Analog Filter Design. New York, NY: Oxford University Press, 1982.

[3] M. Jeruchim, P. Balaban, and K. Shanmugan, Simulation of Communication Systems. New York, NY: Plenum Press, 1994.

[4] Optical Fiber Telecommunications II., ed. S. E. Miller and I. P. Kaminow, chapters 14 and 18. Academic Press, 1988.

[5] N. A. Olsson, “Lightwave systems with optical amplifiers,” IEEE Journal of Lightwave Technology, vol. 7, no. 7, pp. 1071-1082, 1989.

Properties




Parameter Values Name Type Default Range Units n_representation enumerated QA QA, MC pd_APD_Multiplier double 1.0 [ 1, 1e32 ] none

pd_ionizationCoef double 1.0 [ 0, 1 ] none

pd_QEmethod enumerated Defined Defined, Computed

pd_quantumEff double 0.8 [ 0, 1 ] none pd_absorptionCoeff double 0.68e6 [ 0, 1e32 ] 1/m

pd_layerThickness double 0.5e-6 [ 0, 1e32 ] m

pd_reflectivity double 0.04 [ 0, 1 ] none pd_detect enumerated false false, true

pd_modeltype enumerated empirical empirical, intrinsic

pd_loadResistance double 50.0 [ 0, 1e32 ] Ohms

pd_seriesResistance

double 5.0 [ 0, 1e32 ] Ohms

pd_deviceCapacitance

double 50e-15 [ 0, 1e32 ] F

pd_electronVelocity double 6.5e6 [ 0, 1e32 ] m/s pd_holeVelocity double 4.8e6 [ 0, 1e32 ] m/s

pd_lossGain double 0 [ -1e32, 1e32 ] dB

pd_respfp double 52.259e9 [ 0, 1e32 ] Hz

pd_respfo double 25.679e9 [ 0, 1e32 ] Hz pd_respg double 0.5874 [ 0, 1e32 ] none

pd_darkCurrent double 1e-6 [ 0, 1e32 ] A

fe_modeltype enumerated defined defined, custom

fe_filename string fe_tZ double 1.0 [ 0, 1e18 ] Ohms

fe_zero double 0.0 [ 0, 1e18 ] Hz

fe_pole double 1e18 [ 0, 1e18 ] Hz

fe_lo_trunc enumerated extend extend, zero fe_hi_trunc enumerated extend extend, zero

n_a0 double 5.4617e-23 [ 0, 1e32 ] A^2/Hz

n_a2 double 2.924e-43 [ 0, 1e32 ] A^2/Hz^3

n_a4 double 1.1118e-63 [ 0, 1e32 ] A^2/Hz^5 n_a6 double 0 [ 0, 1e32 ] A^2/Hz^7

flt_type enumerated LPbessel LPbessel, LPbutterworth, LPchebyshev, LPideal, HPbessel, HPbutterworth, HPchebyshev, HPideal, BPbessel, BPbutterworth, BPchebyshev, BPideal

flt_preserve_alignm enumerated YES NO, YES


ent flt_bandwidth double 10e9 [ 0, 1e18 ] Hz

flt_order integer 4 [ 0, 128 ] none

flt_lossGain double 0.0 [ -1e32, 1e32 ] dB flt_passbandRipple double 1.0 [ 1e-15, 1e32 ] dB

flt_geometricCenter double 0.0 [ 0, 1e32 ] Hz

include_thermal enumerated YES NO, YES

include_sigshot enumerated YES NO, YES include_darkshot enumerated YES NO, YES

include_sigspon enumerated YES NO, YES

include_sponspon enumerated YES NO, YES

include_SE_noise enumerated YES NO, YES include_rin enumerated YES NO, YES

monte_seed integer 0 [ -1e8, 1 ] none


test_output enumerated elec_filter_spectrum elec_filter_spectrum, photodetector_resp, front_end_resp, cumulative_resp, noise_spectrum


General Receiver Parameters n_representation Select quasi-analytic or Monte-Carlo treatment of noise include_thermal Toggle inclusion of thermal noise include_sigshot Toggle incluson of signal shot noise include_darkshot Toggle inclusion of dark current shot noise include_sigspon Toggle inclusion of signal-spontaneous beat noise in quasi-analytic treatment of noise include_sponspon Toggle inclusion of spontaneous-spontaneous beat noise in quasi-analytic treatment of noise include_SE_noise Toggle inclusion of ASE beat noise in Monte-Carlo treatment of noise include_rin Toggle inclusion of relative intensity noise monte_seed Random number seed for Monte-Carlo noise test_display Format for display of response functions test_output Select response function for display

Photodetector Parameters pd_APD_Multiplier APD multiplier value (1.0 for PIN detector)


pd_ionizationCoef APD ionization coefficient (1.0 for PIN detector) pd_QEmethod If Computed, the quantum efficiency is computed by the model. If Defined, the entered

value is used. pd_quantumEff Quantum efficiency. pd_layerThickness Thickness of the active region pd_absorptionCoeff Absorption coefficient (α) pd_reflectivity Reflectivity at the photodiode pd_detect If true then use a PD frequency response model, else assume it is included in front end

response pd_modeltype ‘intrinsic’ or ‘empirical’ models for frequency response pd_loadResistance Load resistance pd_seriesResistance Series resistance pd_deviceCapacitance Device capacitance pd_electronVelocity Electron saturated velocity pd_holeVelocity Hole saturated velocity pd_lossGain Gain or loss of the photodetector response (only used if pd_detect is set true) pd_respfp Parasitic frequency of the empirical frequency response pd_respfo Resonance frequency of the empirical frequency response pd_respg Gamma of the empirical frequency response fit pd_darkCurrent Dark current for dark current noise computation

Preamplifier Parameters fe_modeltype Select parameterized or user-defined transfer function fe_filename Filename for user-defined transfer function fe_tZ Transimpedance coefficient fe_zero Single zero low frequency rolloff fe_pole Single zero high frequency rolloff fe_lo_trunc Low frequency truncation behavior for user-defined transfer function fe_hi_trunc High frequency truncation behavior for user-defined transfer function n_a0 Thermal noise coefficient n_a2 Thermal noise coefficient n_a4 Thermal noise coefficient n_a6 Thermal noise coefficient

Filter Parameters flt_type Filter type flt_preserve_alignment Preserve alignment of incoming bits flt_bandwidth Filter 3dB bandwidth flt_order Geometric center frequency for bandpass filters flt_lossGain Order of the filter flt_passbandRipple Filter gain or loss


flt_geometricCenter Passband ripple for Chebyshev filter


Photodetector

This component models standard photodetectors – either a PIN photodiode or an avalanche photodiode – converting an optical signal into an electrical current. Noise in this model is treated stochastically. For an analytic treatment of the noise effects, refer to the monolithic receiver model.

Detection Process This model acts as the first stage of an optical receiver assembly to convert optical signals to an electrical current. Both PIN and avalanche photodiode (APD) type detectors can be modeled. Typically the output of the photodiode would be connected to an electrical amplifier and then to a filter. If a BER analysis is to be performed, at the very least an amplifier must be placed somewhere between the photodetector and BER Tester to convert the current output of the photodetector to the voltage source expected by the BER Tester.

The quantum efficiency is a measure of the percentage of incident photons that are successfully converted to electrons. This quantity may be input directly by the user as a parameter quantumEff; alternatively, the user can allow OptSim to calculate the quantum efficiency internally from the user-specified absorption layer thickness L (layerThickness), absorption coefficient (absorptionCoeff), and reflectivity of the photosensitive area of the detector Rf (reflectivity) according to:

( ) ( )1 1 e LQ fR αη −= − −

To input the quantum efficiency directly, set the QEmethod parameter to Defined and enter the desired value in the quantumEff field. To have the model compute it for you, set the QEmethod parameter to Computed and assign values to layerThickness, absorptionCoeff, and reflectivity.

Regardless of the method chosen to specify the quantum efficiency ηQ, the responsivity is computed as:

hc

e Qλη=ℜ

where e is the electron charge, λ is the wavelength of the input optical signal, h is Planck’s constant, and c is the speed of light. The responsivity has units of A/W and gives the quantity of electrical current generated for each watt of incident optical power P . That is, the generated photocurrent obeys the square-law relation

( ) ( ) ( )( ) ( )

( ) ( ) ( )

0

2 2

, , ,x j y jj

I t I t N t

P t N t

E t E t N t

= +

= ℜ +

= ℜ + + ∑

where the sum is over all optical channels in the incoming signal. (Recall that in OptSim, the square of the electric field for an optical signal has units of Watts). The term ( )N t includes all noise effects and is discussed below.

In an avalanche photodiode, the photocurrent is multiplied by successive ionization and the final current is modified to


( ) ( ) ( )

( ) ( )0

,

I t MI t N t

M P t N t

= +

= ℜ +

where the avalanche factor M is specified with the parameter APD_Multiplier. The behavior of APD detectors also differs in the noise response (see below).

There is no specific switch to convert from a PIN model to an APD model. Simply setting APD_Multiplier>1, and the noise-related parameter ionizationCoefficient (see below), produces APD behavior. Note however, that simply changing the default parameters for the APD multiplier and ionization coefficient will not convert an optimal PIN receiver into an optimal APD receiver, as there are other parameters which will vary between PIN receivers and APD receivers since their designs generally differ from one another.

Frequency Response If desired, the photodetector may be given a frequency response that acts by filtering the photocurrent

( )I t . In most cases it may be more convenient to model the total electrical response of the receiver assembly together in the subsequent electrical amplifier or electrical filter models. For that approach, set the parameter detect=false.

To control the photodetector response directly, set detect=true, and configure the modeltype parameter to use one of the following frequency responses. This may be useful, for example, if the photodetector is simply connected to a 50 Ω amplifier.

• Intrinsic Response (modeltype=intrinsic) The frequency response is described by:

( )( ) ( )

e h

/ 20

-j -j

1 1100 1 1 e

1 e e e 1 1-e 1 e ee ee h

GL

d s p

j jL LL L

e e h h

ii j R R C

j L j j L j

α

ωτ ωτ ωτ ωτα αα α

ωω

ωτ α ωτ ωτ α ωτ

−

− −−− −

= + + − − − −+ + + + −

In this formulation, the electron and hole transit times are given by τe = L/ve and τh = L/vh, Cp is the parasitic capacitance of the photodetector, Rd is the load resistance, Rs is the series resistance, and the other symbols are as previously defined. The user should enter values for seriesResistance, loadResistance, deviceCapacitance, electronVelocity, holeVelocity, layerThickness and absorptionCoeff and lossGain (G).

• Empirical Response (modeltype=empirical) The response is described by:

( )( ) ( )

/ 20

2 2 22 2 2 2 20 0

10 10 1 / 1 / 4 /

G

p

ii f f f f f f

ω

γ

= + − +

where G , pf , 0f and γ are empirically determined by the user and supplied to the model in the form of the parameters lossGain, respfp, respfo and respg, respectively. The response can be multiplied by a gain or loss constant by setting the lossGain parameter to a nonzero value. This representation is particularly useful when the inductance of the bond wire dominates the photodetector response. In this situation, the frequency response is often modeled as the second order response depicted above.


Noise Response The photodetector model allows modeling of the following noise effects: shot noise of the signal, shot noise associated with the dark current of the device, RIN noise and both signal-spontaneous and spontaneous-spontaneous beat noises. Each of these effects may be separately disabled or enabled with the parameters include_sigshot, include_darkshot, include_rin and include_SE_noise respectively.

In this model, all these noise sources appear as stochastic terms added directly to the sampled electrical signal. The monolithic receiver model should be used if an analytic treatment of noise is desired, (typically to use the Quasi-Analytic approach to bit error estimation).

Spontaneous Emission Noises The two spontaneous emission noise effects – signal-spontaneous (sig-spon) and spontaneous-spontaneous (spon-spon) beat noise arise from the action of the square law detection mixing the deterministic signal and random noise. When include_SE_noise=YES, before applying the square-law detection to generate the photocurrent, the model first converts any power in ASE noise bins in the optical signal into a stochastic time series which is added to the sampled optical signal. (See the NoiseAdder documentation for details on this procedure). The spontaneous emission noises thus arise naturally in the detection calculation. Note that any ASE power outside the simulation bandwidth of the sampled signal is dropped. Since an electrical filter with a bandwidth narrower than the simulation bandwidth will normally be included later in the topology, this is not a significant restriction.

Due to the direct inclusion of ASE as stochastic noise, it is not possible to separately control the inclusion of sig-spon and spon-spon noise.

Other Noise Sources The remaining noise sources (shot noise, dark current shot noise and RIN), are treated by calculating noise variances for each effect and then adding a Gaussian random variable to each sample of the photocurrent:

( ) 2 2 2shot dark RINˆ

i iN t i i iξ= + + ,

where iξ is a Gaussian random variable of zero mean and unit variance. The seed for the random variable is set with random_seed. This variable obeys the standard OptSim convention for the meanings of negative, zero and unity seeds. Since the spectrum of each noise source is assumed to be white (a reasonable approximation for electrical bandwidths), it is necessary to choose an effective bandwidth for the noises. In each case the effective bandwidth is taken to be the numerical one-sided bandwidth

( )eff 1/ 2B t= ∆ , where t∆ is the sampling rate. Below we give the expressions for each noise variance

and the associated spectral density functions ( )S f . The two are related through the expression

eff2

0( )

B

n ni S f df= ∫

• Shot noise

The shot noise derives from the random distribution in arrival times of photons at the photodetector. The expressions are

( ) ( ) ( )2shot eff2 2i qI t B S f qI t= ⇔ =

where q is the charge on the electron. Note that the noise variance depends on the photocurrent, but that for electrical frequencies the noise can be considered “locally” white.

• Dark current noise


The dark current noise is shot noise associated with leakage currents in the active region of the photodetector which flow even in the absence of incident optical poser, and is described by

( )2 2 2dark dark eff dark2 2i qI M FB S f qI M F= ⇔ = .

Here the (time-independent) dark current darkI is entered as the parameter dark_current. For

PIN diodes, both M and F are unity. For an APD, M is again the multiplier factor, while

)12)(1( MkkMF −−+=

is an excess noise factor associated with the ratio /k α β= of the electron and hole ionization coefficients. The ratio k is determined by ionizationCoef.

• RIN

The RIN variance is given by

( )2 2 2 2 2RIN RIN eff RIN RIN( ) ( )i M FI N B S f M FI N= ⇔ =

where RINN is the RIN parameter specified in the source models.

Test function The model's test functions displays the current frequency response of the detector if detect=true. The format for display of the complex response can be controlled through test_display.





[5] N. A. Olsson, “Lightwave systems with optical amplifiers” IEEE Journal of Lightwave Technology, vol. 7, no. 7, pp. 1071-1082, 1989.

Properties


Outputs #1: Electrical signal (current)

Parameter Values



APD_Multiplier double 1.0 [ 1, 1e32 ] none

ionizationCoef double 1.0 [ 0, 1 ] none QEmethod enumerated Defined Defined, Computed

quantumEff double 0.8 [ 0, 1 ] none

absorptionCoeff double 0.68e6 [ 0, 1e32 ] 1/m

layerThickness double 0.5e-6 [ 0, 1e32 ] m reflectivity double 0.04 [ 0, 1 ] none

detect enumerated false false, true

modeltype enumerated empirical empirical, intrinsic

loadResistance double 50.0 [ 0, 1e32 ] Ohms seriesResistance double 5.0 [ 0, 1e32 ] Ohms

deviceCapacitance

double 50e-15 [ 0, 1e32 ] F

electronVelocity double 6.5e6 [ 0, 1e32 ] m/s

holeVelocity double 4.8e6 [ 0, 1e32 ] m/s

lossGain double 0 [ -1e32, 1e32 ] dB respfp double 52.259e9 [ 0, 1e32 ] Hz

respfo double 25.679e9 [ 0, 1e32 ] Hz

respg double 0.5874 [ 0, 1e32 ] none

darkCurrent double 1e-6 [ 0, 1e32 ] A include_sigshot enumerated YES NO, YES

include_darkshot enumerated YES NO, YES

include_SE_noise enumerated YES NO, YES

include_rin enumerated YES NO, YES random_seed integer 0 [ -1e8, 1 ] none


Parameter Descriptions APD_Multiplier APD multiplier value (1.0 for PIN detector) ionizationCoef APD ionization coefficient (1.0 for PIN detector) QEmethod If Computed, the quantum efficiency is computed by the model. If Defined, the

quantum efficiency defined above is used instead. quantumEff Quantum efficiency. layerThickness Thickness of the active region absorptionCoeff Absorption coefficient (α) reflectivity Reflectivity at the photodiode detect If true then use a PD frequency response model, else assume it is included in front end

response modeltype intrinsic or empirical models for frequency response loadResistance Load resistance seriesResistance Series resistance


deviceCapacitance Device capacitance electronVelocity Electron saturated velocity holeVelocity Hole saturated velocity lossGain Gain or loss of the photodetector response (only used if pd_detect is set true) respfp Parasitic frequency of the empirical frequency response respfo Resonance frequency of the empirical frequency response respg Gamma of the empirical frequency response fit darkCurrent Dark current for dark current noise computation include_sigshot Enable/disable optical shot noise include_darkshot Enable/disable dark current shot noise include_SE_noise Enable/disable spontaneous emission noise from ASE spectrum include_rin Enable/disable RIN noise random_seed Seed for random number generation for noise sources test_display Display format for test function of photodetector response

OptSim Models Reference: Block Mode Chapter 10: Analyzers •••• 309

Chapter 10: Analyzers

This chapter describes a variety of optical and electrical signal analyzers.

• Interior Property Map

plot the evolution of a number of quantities inside a sequence of fibers

• Property Map

plot dispersion and power maps along a fiber link

• Optical Monitor

plot a number of standard properties of an optical signal

• Gain/NF Analyzer

measure gain and noise figure for a pair of optical signal

• Polarization Monitor

measure a number of polarization state related properties of an optical signal

• Optical Eye Analyzer

plot a number of parameters related to noise, waveform and eye diagram of an optical signal

• Bit Error Rate Tester

compute BER, Q-factor and eye properties of an electrical signal

• X-Y Plotter

produce X-Y plots by combining outputs from scanning-aware model blocks

• Signal Analyzer

display signal waveforms

• Constellation Diagram Analyzer

plot constellation diagram of a signal

• Eye Diagram Analyzer

plot eye diagram of a signal

• Signal Spectrum Analyzer

plot spectrum of a signal

• Optical Frequency/Wavelength Chirp Analyzer

plot chirp of an optical signal

310 •••• Chapter 10: Analyzers OptSim Models Reference: Block Mode

• Optical Autocorrelator Analyzer

characterize ultra-short pulses

• Multiplot create and recreate multiple plots with different settings

• Transfer Function plot tranfer function (frequency response) of component(s) under test


Interior Property Map

This model produces plots of the evolution of a number of quantities inside a sequence of fibers. The model is closely related to the Property Map. That model measures the accumulated dispersion and/or optical power at the output of a series of fibers or other components and assembles them into a map of dispersion or power as a function of distance. However, as the Property Map only has access to the optical signals that are output from each component, it is unable to resolve the evolution of any properties within a particular fiber. Since in many systems, interesting pulse evolution can occur within a single fiber span it is important to be able to visualize this evolution. This is the function of the Interior Property Map.

Model Usage Figure 1 shows a topology employing an Interior Property Map to generate the power map in Figure 2. This topology is identical to the example shown in the documentation for the Property Map but with the Interior Property Map replacing the Property Map icon. Note the unusual feature that the Interior Property Map has no input or output nodes and is not connected to any other components. In fact the fibers in the link and the Interior Property Map communicate through access to a common data file specified by the user. This is necessary since as stated above, the optical signals emerging from the fibers do not themselves contain sufficient information to construct the map within each fiber. In this case there are two input signals and the generated maps contain traces for each signal. The output in Figure 2 should be compared to the corresponding figure in the standard Property Map documentation. The present map shows the same periodic amplification and gradual transfer of energy from the short wavelength channel to the long wavelength channel, but also resolves the exponential decay of optical power within each fiber.

Figure 1: Topology demonstrating use of Interior Property Map


Figure 2: Interior power map generated from Figure 1

To force a fiber to generate the output needed to produce an interior map, the user specifies a filename for the fiber parameter physpropFilename, which is available in both the Nonlinear Fiber and Bidirectional Fiber models. The same filename is entered for the physpropFilename parameter of the Interior Property Map. The fibers write to this file during the simulation, and the file is then read and analyzed by the Interior Property Map at the conclusion of the simulation. Since the parameter name physpropFilename is the same for both the fibers and the Interior Property Map, it can be conveniently set for all the components at once, by using the mouse to select all the fibers and the map and using the Selective Multi-Component Parameter Edit feature under the Edit menu. If the topology contains several separate links, multiple interior maps can be generated by adding additional Interior Map icons and using separate filenames for each set of fibers and map.

Available Maps The model can produce maps of the optical power, pulse position, pulse width, and bandwidth as determined by the parameters PowerMap, PositionMap, WidthMap, WidthFWHMMap, and BandwidthMap. There is no option to produce dispersion maps since, as the dispersion is constant in any fiber, the standard Property Map model handles this case completely.

The optical power option measures the total optical power in each signal, neglecting any ASE noise carried in noise bins. The pulse position 0x and pulse width τ are defined by the following equations:

( )( )( ) ( )

( )

22

00 22

0

22 2002

22

0

1/ 22

dt

dt

dt

dt

T

x y

T

x y

T

x y

T

x y

t E Et

E E

t t E Et

E E

tτ γ

+=

+

− +=

+

=

∫∫

∫∫

where γ is a scale factor explained in a moment. These definitions are really only useful for simulations involving a single pulse – for instance studying the evolution of a pulse produced by the ModeLocked Laser in single-pulse (pattern=Single) mode.


The scale factor γ (parameter widthScaling) allows the second moment width 1/ 22t to a number of other

width measures which may be more useful. For example, if the pulse is known to be Gaussian in shape, then with widthScaling=gaussianFWHM, the model plots the FWHM width, while the standard pulse

width 0τ for the pulse amplitude ( )( )220exp / 2t τ− is obtained with widthScaling=gaussianTau.

In full, the conversion factors are as follows:

• second moment width (widthScaling=unity)

1γ =

• Gaussian FWHM (widthScaling=gaussianFWHM)

2 2 log 2γ =

• Gaussian width (widthScaling=gaussianTau)

2γ =

• sech FWHM (widthScaling=sechFWHM)

( ) ( )4 3 / log 1 2γ π= +

• sech width (widthScaling=sechTau)

2 3 /γ π=

If the pulse is neither Gaussian nor sech-like in shape, the setting widthScaling=unity may be most appropriate.

Scanning If the Interior Property Map is used in a scanning simulation then it must be attached to the output of the last fiber being monitored. This causes the map to be called at the end of each scan rather than once at the end of the simulation. If this is not done, the output from all the iterations will be combined in a single plot and appear very confusing.

Display Parameters The parameters DistanceUnits, PowerUnits and TimeUnits control the units used for the display of the various quantities. The output file prefix can be altered with the optional value FilenameRoot, while Plot controls whether plots are suppressed, saved or automatically displayed.

Properties

Inputs None

Outputs None


Parameter Values Name Type Default Range Units physpropFilename

string

FilenameRoot string PowerMap enumerated Yes No, Yes

PositionMap enumerated No No, Yes

WidthMap enumerated No No, Yes

WidthFWHMMap enumerated No No, Yes BandwidthMap enumerated No No, Yes

widthScaling enumerated gaussianFWHM unity, gaussianFWHM, gaussianTau, sechFWHM, sechTau

DistanceUnits enumerated m m, km, Mm

PowerUnits enumerated dBm uW, mW, W, dBm

TimeUnits enumerated s fs, ns, ps, us, ms, s

Plot enumerated Save No, Save, Display


physpropFilename Shared file for data output by fiber models

FilenameRoot Filename root for output files

PowerMap Toggle calculation of power map

PositionMap Toggle calculation of positionmap

WidthMap Toggle calculation of width map

WidthFWHMMap Toggle calculation of width FWHM map

BandwidthMap Toggle calculation of bandwidth map

widthScaling Scaling factor for width measure

DistanceUnits Distance units in output plots

PowerUnits Power units in output plots

TimeUnits Time units in output plots (for position and width)

Plot Suppress, save or automatically display plots


Property Map

This model produces maps of dispersion and power along a fiber link.

Frequently we construct links consisting of a series of fibers and amplifiers and it is useful to monitor the power evolution along the link. In addition, the dispersion in the fibers comprising the link often varies from span to span in order to provide dispersion compensation or dispersion management. This model allows the current power and dispersion at the output of any component to be recorded and output as a map along the link.

Model Usage Figure 1 shows a topology containing two sources – direct modulated lasers driven by a PRBS source – multiplexed into a link containing a series of fibers and amplifiers. The link includes a repetition loop.

A number of the optical components along the link are connected to the Property Map block at the top right. After running the simulation, the Property Map produces the plots shown in Figure 2 – maps of accumulated dispersion and optical power as a function of distance. The accumulated dispersion in a fiber is defined as the product ( )2 0 Lβ λ , where ( )2 0β λ is the second order dispersion at the center

frequency of the optical signal and L is the fiber length.

In each map, a separate trace is produced for each optical channel. (Though see the section on Limitations below for restrictions related to Single-Band mode). For this example, the dispersion map indicates a sequence of normal and anomalous dispersion spans. The accumulated dispersion of each channel differs due to cubic dispersion. The power map shows the attenuation in each fiber followed by gain in the amplifiers. In this example, the long wavelength channel slowly increases in power at the expense of the short wavelength channel due to Raman interactions.

Note that there is no significance to the order of inputs to the Property Map block – the first component in a link need not be connected to the first input port of the block. The Property Map automatically sorts the incoming signals on the basis of propagation distance.

Model Parameters The creation of the maps may be controlled separately through the parameters DispersionMap, PowerMap, WidthMap, WidthFWHMMap, and BandwidthMap, and the units for display of each quantity are controlled with DispersionUnits, PowerUnits, TimeUnits, DistanceUnits, and FrequencyUnits. As for other plot blocks, the root filename is set with FilenameRoot, and the plots may be displayed automatically or suppressed altogether with Plot.


Figure 1. Topology demonstrating use of power and dispersion maps.

Figure 2. Dispersion and power maps generated from Figure 3.

Scanning Behavior For simulations using inner, outer or statistical scanning, separate traces are generated for each trace if iteration meta-prefixes are entered in the main scan dialog. No plots are produced if the meta-prefixes are empty.

Limitations Currently, the maps contain one trace for each optical signal in the link, as opposed to one for each optical channel. The dispersion is measured at the center wavelength of the optical signal, and the power is the total power of the whole optical signal. This gives the expected behavior for several channels in Multi-Band representation. For a series of channels represented in Single-Band mode however, only a single trace is shown representing the dispersion at the center wavelength or the total power in all the channels. For the dispersion map, this limitation can be overcome by converting the multiplexers to multi-band mode and repeating the simulation (with a low sampling rate if desired for speed). However, since four wave mixing influences the power evolution, it is not possible to generate correct individual power traces for each channel in Single-Band mode.

There are complications in connecting components within a repetition loop to a property map block. The map block should receive inputs from components inside at most one repetition loop, and should not receive inputs from any block after a repetition loop. Such a configuration may produce a map that is clearly incorrect. This is because the block is never called until all its input components have been called once.

These limitations will be removed in future versions of OptSim.

Properties

Inputs #1 - #512: Optical Signal


Outputs None

Parameter Values Name Type Default Range Units FilenameRoot string

DispersionMap enumerated No No, Yes

PowerMap enumerated Yes No, Yes WidthMap enumerated No No, Yes

WidthFWHMMap enumerated No No, Yes

BandwidthMap enumerated No No, Yes

DistanceUnits enumerated m m, km, Mm DispersionUnits enumerated ps/nm ps/nm, ps^2

PowerUnits enumerated dBm uW, mW, W, dBm

TimeUnits enumerated s fs, ps, ns, us, ms, s

FrequencyUnits enumerated Hz Hz, GHz, THz Plot enumerated Save No, Save, Display

Parameter Descriptions FilenameRoot Filename root for output files DispersionMap Toggle calculation of dispersion map PowerMap Toggle calculation of power map WidthMap Toggle calculation of pulse width map WidthFWHMMap Toggle calculation of pulse FWHM map BandwidthMap Toggle calculation of bandwidth map DistanceUnits Units for display of accumulated distance DispersionUnits Units for display of accumulated dispersion map PowerUnits Units for display of power map TimeUnits Units for display of time FrequencyUnits Units for display of frequency Plot Suppress, save or automatically display plots


Electrical Monitor

This model provides the facility to measure the average power of an electrical signal. The results may be displayed in several forms – as a detailed table containing full statistical information, or as plots showing the dependence of the power on scanned parameters. Moreover, if the model is given an output port, the collected information can be passed to the XY-Plotter model to be combined with data from another point in the topology. In this way, it is possible to easily generate plots showing the dependence of any quantity on any other.

Model Use The Average Power measurement is activated with the parameter AverageElectricalPower, whose options are None, Local, NonLocal and Both. When this parameter is set to Local, the power is calculated and reported both in a combined table (with filename ending “table.txt”) and in plot form (with filename ending “.txt”). When the parameter is set to NonLocal and the model has an output port connected to an XY-Plotter, data is not presented locally but is passed to the XY-Plotter for constructing plots with data from elsewhere in the topology. The user should consult the XY-Plotter documentation and Chapter 5 of the User Manual – “Scanning Facilities” for further details. The setting Both both exports the data to the plotter and generates the local table and plot.

Support for Scanning This model fully supports OptSim's scanning capabilities. When a simulation is launched using the scan dialog, the Electrical Monitor automatically produces plots showing the measured quantities for the entire scan. On the local plots, the inner scan variable appears as the x-axis, and if an outer scan variable is used, a series of curves is produced, one for each value of the outer scan variable. Using the XY-Plotter the user can produce plots in which any measurable quantity appears on the x-axis.

The table output lists the results for every iteration. For statistical simulations the table output also reports the mean, maximum, minimum and standard deviation of each quantity.

The parameter StatisticalPlotMode controls whether plots show any or all of the mean (Average), maximum (Maximum), minimum (Minimum), standard-deviation (Std.Dev.), or most recent (Latest) results. This parameter can also be set to StatRun in order to display the detailed statistical variation of the results. Setting this parameter to All requests that all statistical plots be created.

Measurement Definitions We now provide the precise definition of the Average Electrical Power.

• Average Electrical Power (parameter AverageElectricalPower)

2

0

1 ( )T

aveP s t dtT

= ∫

where the electrical signal s(t) is either in units of volts or amperes, and the power is calculated across a unity load. The units for display are controlled with the parameter PowerUnits.


Properties


Outputs #1: Measurand Signal

Parameter Values Name Type Default Range Units AverageElectricalPower enumerated None None, Local,

NonLocal, Both

PowerUnits enumerated W W, dBm

SaveOrDisplayPlots enumerated Save Save, Display

StatisticalPlotMode enumerated Latest Average, Minimum, Maximum, Std.Dev., Latest, StatRun, All

Parameter Descriptions AverageElectricalPower Toggle calculation of average electrical power

PowerUnits Units for power measurement SaveOrDisplayPlots Save plots to file or display automatically

StatisticalPlotMode Select statistic for display in plots


Optical Monitor

This model provides the facility to measure a number of standard properties of an optical signal, specifically Average Optical Power (a Power Meter), Total NoisePower, Center Frequency, Averaged Stokes Parameters, Optical Signal to Noise Ratio, and Pulse Position.

The results may be displayed in several forms – as a detailed table containing full statistical information, or as plots showing the dependence of the properties on scanned parameters. Moreover, if the model is given an output port, the collected information can be passed to the XY-Plotter model to be combined with data from another point in the topology, either another Optical Monitor or else BERTester or OpticalEyeAnalyzer data. In this way, it is possible to easily generate plots showing the dependence of any quantity on any other.

The Optical Monitor only accepts single-channel optical inputs contained within a single optical signal. Thus, multi-channel signals should be represented in Single-Band mode, and must first be separated using a demultiplexing component.

Model Use The physical properties available are activated with the parameters OpticalPower, NoisePower, CenterFreq, FreqShift, Stokes, OpticalSNR, and Position. Each of these parameters has the settings None, Local, NonLocal and Both. When a parameter is set to Local, that quantity is calculated and reported both in a combined table (with filename ending “table.txt”) and in plot form (with filename ending “.txt”). When a parameter is set to NonLocal and the model has an output port connected to an XY-Plotter, data is not presented locally but is passed to the XY-Plotter for constructing plots with data from elswhere in the topology. The user should consult the XY-Plotter documentation and Chapter 5 of the User Manual – “Scanning Facilities” for further details. The setting Both both exports the data to the plotter and generates the local table and plot.

Support for Scanning This model fully supports OptSim's scanning capabilities. When a simulation is launched using the scan dialog, the Optical Monitor automatically produces plots showing the physical properties for the entire scan. On the local plots, the inner scan variable appears as the x-axis, and if an outer scan variable is used, a series of curves is produced, one for each value of the outer scan variable. Using the XY-Plotter the user can produce plots in which any measurable quantity appears on the x-axis.



Property Definitions We now provide the precise definitions of each the quantities available.

• Average Optical Power (parameter OpticalPower)

( ) ( ) 22opt 0

dT

x yP E t E t t= +∫


The units for display are controlled with the parameter PowerUnits.

• Total Noise Power (parameter NoisePower)

This is the summed noise spectral density in all noise bins:

( )ASE ASE jj

P Sδν ν= ∑ ,

where ( )ASE jS ν is the spectral density in the noise bin at frequency jν .

• Center Frequency (parameter CenterFreq)

This is the weighted frequency average of the optical signal:

( ) ( )( )

( ) ( )

/ 2 22

/ 2/ 2 22

0/ 2

d

d

c

c

c

c

x y

x y

E Ec

E E

ν

νν

ν

ν ν ν νν

λ ν ν ν−

−

+= +

+

∫∫

,

where 0λ is the center wavelength of the optical signal and 1/c tν = ∆ is the sampling frequency.

• Frequency Shift (parameter FreqShift)

This is simply the frequency average relative to the center frequency:

( ) ( )( )

( ) ( )

/ 2 22

/ 2/ 2 22

/ 2

d

d

c

c

c

c

x y

x y

E E

E E

ν

νν

ν

ν ν ν νν

ν ν ν−

−

+∆ =

+

∫∫

.

• Time-Averaged Stokes Parameters (parameter Stokes)

This is a three-component vector quantity of the time-averaged Stokes Parameters

( )1 2 3, ,s s s .

The instantaneous Stokes Parameters ( )0 1 2 3, , ,S S S S describe the polarization state of an optical signal and are defined as:

( ) ( )( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )

220

221

* *2

* *3

,

,

,

.

x y

x y

x y x y

x y x y

S E t E t

S E t E t

S E t E t E t E t

S i E t E t E t E t

= +

= −

= +

= −

These quantities describe the location of the polarization state on the Poincare Sphere and

satisfy a relation 2 2 20 1 2 3S S S S= + + .

The polarization of the sampled optical field may vary with time. Thus it can be useful to measure the averaged Stokes parameters defined as


( )

( )

( )

( )

0 00

1 100

2 200

3 300

S t d ,

1 S t d ,

1 S t d ,

1 S t d .

T

T

T

T

s t

s ts

s ts

s ts

=

=

=

=

∫

∫

∫

∫

For these parameters we have the relation 2 2 21 2 3 1,s s s+ + ≤ with equality only when the

polarization state is constant in time. Thus the averaged Stokes vectors provide a measure both of the character of the polarization state and its constancy.

• Optical Signal to Noise Ratio (parameter OpticalSNR)

The Optical SNR is defined as

( )

( ) ( ) ( ) ( )

0

0

0 0

0 0

/ 2

sig/ 2

/ 2 / 2

sig ASE sig ASE/ 2 / 2

d

1 d d2

sig

sig

sig sig nse

sig nse sig

POSNR

P P P P

ν ν

ν ν

ν ν ν ν

ν ν ν ν

ν ν

ν ν ν ν ν ν

+∆

−∆

−∆ +∆ +∆

−∆ −∆ +∆

= + + +

∫

∫ ∫,

where 0ν (OSNR_f0) is the center frequency of the signal, sigν∆(OSNR_DeltafSig) is the

nominal bandwidth of the signal, nseν∆ (OSNR_DeltafNse) is the sampling bandwidth of the

noise, while ( )sigP ν and ( )ASEP ν are the power spectral density of the signal and noise bins respectively. A negative value for OSNR_f0 causes the center frequency of the channel to be used for 0ν .

This definition is better understood with reference to the spectrum Figure 1. The signal power is measured over a central bandwidth sigν∆ , and the noise is integrated over the adjacent

frequency bands of width nseν∆ .


Figure 1. Optical SNR integration bandwidths

The user may also control aspects of the calculation of the signal power spectral density ( )sigP ν . It turns

out that the simple definition ( ) ( ) 2

sig j jP Eν ν= , where ( )jE ν is the Discrete Fourier Transform of

the sampled electrical field is not an ideal estimator for the power spectral density [1]. In particular, with this definition the variance in the spectral density at any given frequency does not decrease with an increase in the number or rate of samples. Improvements in the variance can be made by both undersampling the frequency data and applying a “window function” to the sampled data before the DFT is performed. The maximum number of frequency components in the calculated spectral density is given by M=2OSNR_SubSamples. If the original DFT contains additional components, the spectral density is obtained with an overlap-add averaging method [1]. The window function is selected with OSNR_FFTWindow. See Ref. [1] for the definitions of the window functions. The default settings of these parameters should normally be appropriate.

• Pulse Position (Parameter Position)

The pulse position is defined as the mean position of the integrated optical energy in the signal:

2 .22

02 .

22

0

( ) ( )

( ) ( )

n

n

T

x y

T

x y

t E t E t dt

t

E t E t dt

+

= +

∫

∫

where 2n is number of bits in the bit sequence, and T is the bit duration.

References [1] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C, Chapter 13, (Cambridge University Press, Cambridge UK, 1995).

Properties



Parameter Values Name Type Default Range Units OpticalPower enumerated None None, Local,

NonLocal, Both

PowerUnits enumerated dBm dBm, W, mW, uW

NoisePower enumerated None None, Local, NonLocal, Both


CenterFreq enumerated None None, Local, NonLocal, Both

FreqShift enumerated None None, Local, NonLocal, Both

Stokes enumerated None None, Local, NonLocal, Both

OpticalSNR enumerated None None, Local, NonLocal, Both

Position enumerated None None, Local, NonLocal, Both

OSNR_f0 double -1 [ -1e32, 1e32 ] Hz

OSNR_DeltafSig double 50e9 [ 0, 1e32 ] Hz OSNR_DeltafNse double 25e9 [ 0, 1e32 ] Hz

OSNR_SubSamples integer 9 [ 1, 18 ] 2^n

OSNR_FFTWindow enumerated Bartlett None, Bartlett, Hann, Welch



Parameter Descriptions OpticalPower Toggle calculation of Optical Power PowerUnits Units for Optical Power NoisePower Toggle calculation of Noise Power CenterFreq Toggle calculation of Center Frequency FreqShift Toggle calculation of Frequency Shift Stokes Toggle calculation of Stoke parameters OSNR Toggle calculation of Optical SNR OSNR_f0 Signal frequency for OSNR. OSNR_DeltafSig Signal bandwidth for OSNR OSNR_DeltafNse Noise integration bandwidth for OSNR. OSNR_SubSamples Number of samples in power spectral density OSNR_FFTWindow Window function for power spectral density SaveOrDisplayPlots Save plots to file or display automatically StatisticalPlotMode Select statistic for display in plots


Gain/NF Analyzer

Given a pair of optical signals, this model provides the facility to measure the Gain and Noise Figure. The model’s top input port corresponds to an output signal, and the bottom input port corresponds to an input signal. For example, the input signal could be the signal launched down a chain of optical amplifiers, and the output signal could be the amplified result. The measurements may be displayed in several forms – as a detailed table containing full statistical information, or as plots showing the dependence of the properties on scanned parameters. Moreover, if the model is given an output port, the collected information can be passed to the XY-Plotter model to be combined with data from another point in the topology. In this way, it is possible to easily generate plots showing the dependence of any quantity on any other. The Gain/NF Analyzer only accepts single-channel optical signals at identical center frequencies. Multi-channel signals should be represented in Single-Band mode, and must first be separated using a demultiplexing component.

Model Use The Gain and Noise Figure measurements are activated with the parameters Gain and NF. Each of these parameters has the settings None, Local, NonLocal and Both. When a parameter is set to Local, that quantity is calculated and reported both in a combined table (with filename ending “table.txt”) and in plot form (with filename ending “.plc”). When a parameter is set to NonLocal and the model has an output port connected to an XY-Plotter, data is not presented locally but is passed to the XY-Plotter for constructing plots with data from elsewhere in the topology. The user should consult the XY-Plotter documentation and Chapter 5 of the User Manual – “Scanning Facilities” for further details. The setting Both both exports the data to the plotter and generates the local table and plot.

Note that the units of the measured data can be selected via the parameter Gain_NF_Units.

Support for Scanning This model fully supports OptSim's scanning capabilities. When a simulation is launched using the scan dialog, the Gain/NF Analyzer automatically produces plots showing the measured quantities for the entire scan. On the local plots, the inner scan variable appears as the x-axis, and if an outer scan variable is used, a series of curves is produced, one for each value of the outer scan variable. Using the XY-Plotter the user can produce plots in which any measurable quantity appears on the x-axis.



Gain/NF Definitions We now provide the precise definitions of the Gain and Noise Figure.

• Gain (parameter Gain)

The total powers of the input (Pin) and output (Pout) signals are measured, and the resulting gain is calculated as Pout/Pin.

• Noise Figure (parameter NF)


The Noise Figure (NF) is calculated based on the calculated Gain (G), and the input (ρASE,in) and output (ρASE,out) ASE spectral densities at the signal center frequency ν [1]:

,

,

1 1

1

ASE out

ASE in

G hNF

h

ρν

ρν

⋅ + = +


Properties


#2: Optical Signal


Parameter Values Name Type Default Range Units Gain enumerated None None, Local,

NonLocal, Both

NF enumerated None None, Local, NonLocal, Both

Gain_NF_Units enumerated dB dB, linear



Parameter Descriptions Gain Toggle calculation of Gain

NF Toggle calculation of Noise Figure

Gain_NF_Units Units for Gain/NF measurements SaveOrDisplayPlots Save plots to file or display automatically

StatisticalPlotMode Select statistic for display in plots


Polarization Monitor

This model provides the facility to measure a number of polarization state related properties of an optical signal, specifically Differential Group Delay (DGD), Degree Of Polarization, Averaged Stokes Parameters and Instantaneous Stokes Parameters.

The results may be displayed in several forms – as a detailed table containing full statistical information, or as plots showing the dependence of the properties on scanned parameters. Moreover, if the model is given an output port, the collected information can be passed to the XY-Plotter model to be combined with data from another point in the topology, either another Polarization Monitor or else Optical Monitor, BERTester or OpticalEyeAnalyzer data. In this way, it is possible to easily generate plots showing the dependence of any quantity on any other.

The model also allows plotting Stokes parameters of an optical signal on Poincare Sphere.

The Polarization Monitor only accepts single-channel optical inputs. Multi-Channel signals must first be separated using a demultiplexing component.

Model Use The physical properties available are activated with the parameters DGD, DegreeOfPolarization, and AverageStokes. Each of these parameters has the settings None, Local, NonLocal and Both. When a parameter is set to Local, that quantity is calculated and reported both in a combined table (with filename ending “table.txt”) and in plot form (with filename ending “.txt”). When a parameter is set to NonLocal and the model has an output port connected to an XY-Plotter, data is not presented locally but is passed to the XY-Plotter for constructing plots with data from elswhere in the topology. The user should consult the XY-Plotter documentation and Chapter 5 of the User Manual – “Scanning Facilities” for further details. The setting Both both exports the data to the plotter and generates the local table and plot.

The plot for instantaneous Stokes parameters can be activated (disabled) by setting the parameter InstantStokes to YES (NO).

Poincare Sphere can be activated with by setting the parameter PlotPoincareSphere. This parameter has following settings: None, AverageStokes, InstantStokes, and Both. To plot average Stokes parameters on Poincare Sphere one has to set PlotPoincareSphere to AverageStokes and AverageStokes to Local or Both. To plot instantaneous Stokes parameters on Poincare Sphere one has to set PlotPoincareSphere to InstantStokes and InstantStokes to YES.

Support for Scanning This model fully supports OptSim's scanning capabilities for the parameters defined as time-averaged. When a simulation is launched using the scan dialog, the Polarization Monitor automatically produces plots showing the physical properties for the entire scan. On the local plots, the inner scan variable appears as the x-axis, and if an outer scan variable is used, a series of curves is produced, one for each value of the outer scan variable. Using the XY-Plotter the user can produce plots in which any measurable quantity appears on the x-axis.


The parameter StatisticalPlotMode controls whether plots show any or all of the mean (Average), maximum (Maximum), minimum (Minimum), standard-deviation (Std.Dev.), or most recent (Latest) results.


This parameter can also be set to StatRun in order to display the detailed statistical variation of the results. Setting this parameter to All requests that all statistical plots be created.

Property Definitions We now provide the precise definitions of each the quantities available.

• Differential Group Delay (parameter DGD)

Differential Group Delay (DGD) shows the group delay between x- and y-polarization components of the signal due to PMD effect and is derived as:

yx tt −=∆τ

where pulse positions for x- and y-polarizations components yxt , are defined as:

∫

∫= T

yx

T

yx

yx

dttE

dttEtt

0

2

,

0

2

,

,

)(

)(

Here T is total signal duration, i.e. (2n x To) – bit duration times number of bits.

• Time-Averaged Stokes Parameters (parameter AverageStokes)

This is a three-component vector quantity of the time-averaged Stokes Parameters ( )1 2 3, ,s s s .

The instantaneous Stokes Parameters ( )0 1 2 3, , ,S S S S describe the polarization state of an optical signal and are defined as:

( ) ( )( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )

220

221

* *2

* *3

,

,

,

.

x y

x y

x y x y

x y x y

S E t E t

S E t E t

S E t E t E t E t

S i E t E t E t E t

= +

= −

= +

= −

These quantities describe the location of the polarization state on the Poincare Sphere and

satisfy a relation 2 2 20 1 2 3S S S S= + + .

The polarization of the sampled optical field may vary with time. Thus it can be useful to measure the averaged Stokes parameters defined as


( )

( )

( )

( )

0 00

1 100

2 200

3 300

S t d ,

1 S t d ,

1 S t d ,

1 S t d .

T

T

T

T

s t

s ts

s ts

s ts

=

=

=

=

∫

∫

∫

∫

For these parameters we have the relation 2 2 21 2 3 1,s s s+ + ≤ with equality only when the

polarization state is constant in time. Thus the averaged Stokes vectors provide a measure both of the character of the polarization state and its constancy.

• Degree of Polarization (parameter DegreeOfPolarization)

Degree of Polarization (DOP) is a ratio of polarized portion of the signal power to total signal power:

dunpolarizepolarized

polarized

PPP

DOP+

=

or in terms of normalized Stokes parameters:

23

22

21 sssDOP ++=

where time-averaged Stokes Si parameters are defined in previous section for AverageStokes.

• Instant Stokes Parameters (parameter InstantStokes)

This option will plot the normalized instantaneous Stokes Parameters ),,( 321 sss as a function of time for each sample of optical filed. These parameters defined as

)(/)()( 0 tStSts ii = (i = 1,2,3)

where definition for iS is given in section for AverageStokes.

Poincaré Sphere Plot This option produces a Poincare Sphere plot of the optical signal as a means of visualizing in three dimensions the signal’s polarization state.

In case of the InstantStokes option, the instantaneous Stokes vector is plotted on the Poincare Sphere. Figure 1(a) shows the example of a Poincare Sphere plot for the instantaneous Stokes vector of an optical signal after traveling through 100 km of dispersive fiber with PMD. Each point on the surface corresponds to the Stokes vector ))(),(),(( 321 nnn tststs at time t = tn..

In case of the AverageStokes option, only one point is depicted on the Poincare Sphere corresponding to the time-averaged Stokes vector. However, in case of simulations with a parameter scan or statistical run each iteration result for the averaged Stokes vector will be plotted on the Poincare Sphere. Figure 1(b) shows the


example of a Poincare Sphere plot for the average Stokes vector after a parameter scan of 100 values for the PMD seed in a 100-km fiber span with PMD on.

(a) (b)

Figure 1. Examples of the Poincare Plot for (a) instantaneous Stokes parameters, and (b) time-averaged Stokes parameters after parameter scan.

The Poincare Sphere Plot can be manipulated with the mouse. Left mouse click-and-hold allows the user to rotate the Poincare Sphere, and right mouse click-and-hold allows the user to zoom in/out by moving up/down.

Properties



Parameter Values Name Type Default Range Units FilenameRoot String

DGD enumerated None None, Local, NonLocal, Both

DegreeOfPolarization enumerated None None, Local, NonLocal, Both

AverageStokes enumerated None None, Local, NonLocal, Both

InstantStokes enumerated NO NO, YES

PlotPoincareSphere enumerated None None, AverageStokes, InstantStokes, Both




Parameter Descriptions FilenameRoot Root of the filename store the plot data DGD Toggle calculation of Differential Group Delay DegreeOfPolarization Toggle calculation of Degree of Polarization AverageStokes Toggle calculation of time-average Stokes parameters InstantStokes Whether to plot instant Stokes parameters or skip it PlotPoincareSphere Whether to plot Poincare Sphere for average Stokes parameters, instant Stokes

parameters, or both SaveOrDisplayPlots Save plots to file or display automatically StatisticalPlotMode Select statistic for display in plots


Optical Eye Analyzer

This model computes a number of useful parameters related to the noise, signal waveform, and eye diagram of the input optical signal. These may be plotted vs. the scanned variables by this block.

There are several modes of operation in regards to the use of an input reference binary signal. When a reference binary signal is provided at the input, the model synchronizes the input optical signal to the corresponding original binary signal, generates eye data and performs the parameter computations. In this mode, the analysis block accepts two inputs (one input pair) and provides no outputs to other blocks. These two inputs are the optical signal and the reference binary signal (the transmitted binary signal for the simulation).

If no reference input binary signal is provided, the model makes an assumption on the value of the bit at the eye center by whether it is above or below the eye center. This mode is useful when a reference binary signal is not available. It can be less accurate in cases where the signal is degraded to the point that the assumption is no longer valid.

This block can be used to display a number of different plots. The plot function is used in conjunction with the sweep dialog. If sweeping a variable, the variable name which is used as the value of the component parameter to be swept is also used as the x axis of the plot. The user sets the start value, step value, and stop value for the sweep range of the swept variable in the scan dialog. The block is executed in each sweep of the simulation and builds the plots as it is swept. If two variables are being swept, then a family of curves is plotted. If a simulation is performed without sweeping, then the selected data such as extinction ratio is calculated and output as a single value.

There are several plot options available. The PlotAveHigh plots the average high level of the optical signal. The PlotAveHighSteady plots the average high level of runs of high level bits. The PlotAveLow plots the average low level of the optical signal. The PlotAveLowSteady plots the average low level of runs of low level bits. PlotAveSig plots the total average signal value. PlotExtRatio plots the extinction ratio of the optical signal. PlotEyeArea plots the area of the eye opening. PlotEyeCenter plots the vertical center of the eye opening. PlotEyeHeight plots the height of the eye opening. PlotEyeWidth plots the width of the eye opening. PlotPeakNoise plots the peak ASE noise value. PlotEyeLids option will plot the upper and the lower lids used in finding the eye as well as the eye diagram of the signal in a separate plot. The eye center and level are indicated by the zero values of the axes in both these plots.

For most models and simulations, the optical and electrical signals remain stored in positions in their respective data arrays to directly correspond to the original pulse positions, and consequently to the original bit pattern. In some models and simulations, most notably ones in which the single channel mode in the optical multiplexer is used to simulate four wave mixing effects, these data arrays may become out of alignment with the original bit positions. This misalignment could cause problems with the normal operation of this model, including an inability to find an open eye in the data. To compensate for this, this block provides the AlignBinaries option. This option aligns the binary sequences with their corresponding incoming optical signals to maximize their correlation before beginning the computations. If a simulation provides a poorer result than expected, and binary and optical signal plot comparisons shows a misalignment between the optical and binary input signals, set the AlignBinaries option to compensate.

For eyes in which a 1 level falls below a 0 level for another bit, the eye is technically closed and the normal eye detection algorithm is replaced with a modified algorithm called the average eye method. This algorithm uses the average 1 level and average 0 level of each of the bits to determine the eye opening. When this happens, a message is given to warn the user that there is a closure in the eye somewhere in the bit stream. When the eye is too completely closed, the routine may report that the eye diagram could not be found. When this occurs, the bit error rate is reported as 1. To diagnose these problems, the user may select the options to plot the eye lids and plot the signals.


Even when there is no open eye, or the input signals are not binary, results which do not depend on binary sequences such as the PlotPeakNoise and PlotAveSig will still produce good results.

Properties


#2: Binary signal

Outputs None


AlignBinaries enumerated NO NO, YES preBits integer 2 [ 0, 1e8 ] none

postBits integer 3 [ 0, 1e8 ] none

decisionLevelSelection

enumerated Automatic Automatic, Defined

decisionLevel double 0 [ -1000, 1000 ] V

showOffsets enumerated NO NO, YES ISI_Bits integer 0 [ 0, 8 ] none

bitsPerLine integer 0 [ 0, 10000000 ] none

StatisticalPlotMode

enumerated All Average, Minimum, Maximum, Std.Dev., Latest, StatRun, All

SaveOrDisplayPlots

enumerated Save Save, Display

SaveAllData enumerated YES NO, YES

PlotEyeHeight enumerated None None, Local, NonLocal, Both

PlotEyeWidth enumerated None None, Local, NonLocal, Both

PlotEyeCenter enumerated None None, Local, NonLocal, Both

PlotEyeArea enumerated None None, Local, NonLocal, Both

PlotEyeClosure enumerated None None, Local, NonLocal, Both

PlotAveHigh enumerated None None, Local, NonLocal, Both

PlotAveLow enumerated None None, Local, NonLocal, Both

PlotAveSig enumerated None None, Local, NonLocal, Both


PlotPeakNoise enumerated None None, Local, NonLocal, Both

PlotAveHighSteady

enumerated None None, Local, NonLocal, Both

PlotAveLowSteady

enumerated None None, Local, NonLocal, Both

PlotExtRatio enumerated None None, Local, NonLocal, Both

PlotEyeLids enumerated NO NO, YES

ContourEyes enumerated NO NO, YES

Parameter Descriptions AlignBinaries Whether to align the input binary signals with the optical signals such that their

correlation is maximized before performing computations FilenameRoot Base of the filename used to store the results StatisticalPlotMode Which value to plot during statistical simulation iterations: average, minimum,

maximum, or latest of all statistical simulation runs PlotExtRatio Whether to plot the extinction ratio curves or not PlotPeakNoise Whether to plot the peak ASE noise value curves or not PlotAveHigh Whether to plot the average high level or not PlotAveHighSteady Whether to plot the average high level for runs of high bit values or not PlotAveLow Whether to plot the average low level or not PlotAveLowSteady Whether to plot the average low level for runs of low bit values or not PlotEyeArea Whether to plot the area of the eye opening or not PlotEyeWidth Whether to plot the width of the eye opening or not PlotEyeHeight Whether to plot the height of the eye opening or not PlotEyeCenter Whether to plot the vertical center or decision level of the eye opening or not PlotEyeLids Whether to plot the eye diagram and eye lids with the center time and level indicated or

not SaveOrDisplayPlots Whether to save plots to files, or to also display them automatically


Bit Error Rate Tester

This model computes the Bit Error Rate (BER) for the input electrical signal(s) as well as a number of useful parameters such as the Q factor and electrical eye properties such as the height, width, area and extinction ratio. The BER may be calculated using either a Quasi-Analytical or Monte-Carlo algorithm depending on the nature of the dominant noise sources in the simulation. The module is fully integrated with the generalized scanning facilities described in Chapter 7 of the OptSim User Manual and so can produce important scan plots such as power penalty diagrams for the BER or other quantities.

We begin the description of the model with a discussion of the different techniques of numerical BER estimation. Following this exposition, we enumerate the steps in the BER implementation and finally describe the non-BER features of the model.

BER Estimation Techniques The Bit Error Rate (BER) of a fiber link is the most important measure of the faithfulness of the link in transporting the binary data from transmitter to receiver. From time to time, due to signal degradations from dispersion, nonlinearities and noise, a signal is so distorted that the detector makes a mistake – a binary one or “mark” is recorded where a binary zero or “space” was transmitted. Clearly, if the link is to be of any use, the frequency of such errors must be as small as possible. The BER quantifies the rate of errors and is defined as the probability of an error occurring per transported bit. Typical benchmarks for the BER are rates of 10-9 and 10-12, though with the use of Forward Error Correcting (FEC) codes, much lower rates can be acceptable.

In the laboratory, the BER is measured directly. A pseudo-random bit source modulating the source is also connected to the BER testing apparatus, and the binary value of every transmitted bit is compared against the value of the same bit at the receiver. The apparatus literally counts the events in which the comparison fails. For a 10 Gbps system, a BER near 10- 9 can be measured in a few seconds, while rates near 10-12 can be measured in hours. With numerical simulation, the situation is completely different. A typical OptSim simulation involves hundreds to thousands or perhaps tens of thousands of bits, and if the link is long or the simulation challenging a single run may take minutes or even hours. To establish a bit error rate of 10-12 by direct counting of errors – a technique sometimes known as Monte-Carlo simulation – would require the simulation of trillions of bits, which is clearly out of the question. Instead of this direct approach, numerical tools attempt to extrapolate an estimate of the BER from simulation of just hundreds or thousands of bits. If the BER is acceptable, then it is very unlikely that an “error bit” will actually occur within the simulation time window, but by measuring the shape of the received electrical eye, the simulator can extract basic parameters describing the distribution of bits and produce a BER estimate with confidence limits.

OptSim provides two algorithms for calculating the BER, referred to as the Quasi-Analytic (QA) and Monte-Carlo (MC) techniques. (Despite the shared name, the Monte-Carlo technique we will describe has nothing in common with the “Monte-Carlo” error counting method mentioned above. On occasion, users may encounter or read about other simulation tools which do calculate the BER by propagation and detection of billions of bits with direct counting of errors. These simulations require great quantities of time and can include only the most cursory and inadequate physical treatment of the components within the link.) Rather in OptSim, QA and MC refer to the representation in which noise in the electrical signal is stored. In Chapter 8 of the User Manual, we explain that electrical noise may be represented in two complementary fashions. We summarize the pictures here, but the reader is encouraged to read the full discussion in Chapter 8.

In the MC picture, there is no real distinction between the signal and noise – noise is included directly by adding Gaussian random variates to the signal samples kV . Viewed with a SignalAnalyzer or


SpectrumAnalyzer, the resultant signal genuinely appears “noisy”, and if the simulation is repeated, the noisy signal appears different in successive runs (More accurately, the repeatibility of the noise from run to run depends on the setting of the seed value for the random number generator in the model generating the noise. Most OptSim models that generate noise allow the user to specify whether the seed changes or remains the same from run to run). In the QA picture, the noise and signal are stored separately – one vector of length N stores the signal, and a second vector of the same length stores standard deviation samples for the noise. So the complete electrical signal may be written

, 1,kk

k

vE k N

σ

= =

K

in terms of the signal samples kv and the noise samples kσ which represent the instantaneous standard deviation of a Gaussian noise source. As discussed in Chapter 8 of the User Manual, the MC approach is very general as it allows for noise of any distribution or correlation function. The QA approach is more restrictive since it assumes a Gaussian noise sources. The primary motivation for its use is that it permits a very efficient BER estimation algorithm as we now see.

Quasi-Analytic BER estimation This technique is the simpler of the two approaches and is the default BER implementation in OptSim.

To use the QA approach, the BERTester block must be preceded by a monolithic Receiver block which incorporates the photodectector, transimpedance amplifier and electrical filter. As indicated above, these sub-components contribute various noise sources which all contribute to a single Gaussian variance that is a function of time, parallel to the outgoing voltage signal. Representing the noise in this fashion discards any spectral features of the incoming ASE, and also only partially respects filtering operations within the receiver – the total noise power is reduced by the filtering but the spectral weighting is lost since the noise is represented by a strictly real number in time domain. Expressions for each of the noise variances are described in the Monolithic Receiver model documentation.

Figure 1: Eye diagram and distribution functions for spaces (zeros).

The basis of the technique is illustrated in Figure 1. In this example, the eye diagram has a very simple structure with unambiguous levels for the marks and spaces. Due to thermal noise, there is a significant spread in values around each of the levels. We are concerned with the distribution of values at the decision point indicated by the vertical line at the center of the eye. Since the noise is known to be Gaussian, the

vlevel

f0(v)

f1(v)


two distributions 0 ( )f v and 1( )f v which govern the spread of the upper and lower bars of the eye can be

specified just by mean values 0v and 1v , and widths 0σ and 1σ :

( ) ( )

( ) ( )

22 20

0 0 0

22 21

1 1 1

2 exp / 2 ,

2 exp / 2 .

f v v v

f v v v

σ σπσ σπ

= − −

= − −

(1)

To find the probability of an error given a specified threshold decision level levelv (indicated by the dotted horizontal line in Fig. 6), we integrate that part of each distribution function falling on the “wrong” side of the decision level. So the probability 0 1P → of mistakenly measuring a space as a mark is given by

( ) ( )level

0 1 0 0 level 0

0 level

0

, , d

1 erfc ,2 2

vP v v f v v

v v

σ

σ

∞

→ =

−= −

∫

(2)

where erfc denotes the complementary error function. The probability 1 0P→ of measuring a mark as a space is given by

( ) ( )level

1 0 1 1 level 1

1 level

1

, , d

1 erfc .2 2

vP v v f v v

v v

σ

σ

→ −∞=

−=

∫

(3)

The total bit error rate is then just

0 10 1 1 0

0 1 0 1

BERn nP P

n n n n→ →= ++ +

.

(4)

where 0n and 1n are the numbers of spaces and marks in the signal. If 0 1n n≈ , we have

1 level 0 level

1 0

1BER erfc erfc4 2 2

v v v vσ σ

− −≈ + −

.

(5)

The error rate in Eq. (5) is minimized by choosing

0 1 1 0level

0 1

v vv σ σσ σ

+=+

.

(6)


Finally, defining the “Q-factor”

1 0

1 0

v vQσ σ

−=+

,

(7)

the bit error rate takes the simple form

1BER erfc2 2

Q =

.

(8)

The Q-factor is a very useful parameter since Eq. (7) can easily be evaluated from measurements of the eye diagram, and because the Q typically takes values in the range Q ≈5÷15, which can be more convenient than the many orders of magnitude spanned by the BER. It is also common in communication community to refer to Q-factor in dB units using following relationship

Q dB = 20 log10 (Q linear) Then, for example, BER = 10-13 according to Eq. (8) corresponds to Q-factor equal 7.34 in linear units and equal 17.32 in dB units. User can choose linear or dB units for Q-factor by switching parameter unitsForQ to linear or Q^2(dB).

Pattern dependence and the QA approach In fact, the formulae just presented represent an oversimplification, since in real systems the received eye diagram is rarely as simple as that in Fig. 1. More commonly, eye diagrams show signs of “pattern dependence” or “inter-symbol interference” (ISI). Due to laser properties, dispersion, nonlinearity or certain noise effects, the shape of a received bit is influenced by the bit values in its neighborhood. So for instance, in an NRZ system, the central bit in the pattern 11111 would be less distorted by dispersion than the central bit in the pattern 00100. Pattern dependence gives rise to eyes that might look something like that in Fig. 2 which was obtained by driving the Direct Modulated Laser model with a PRBS signal at 10 Gbps. In Fig. 2, there are two relatively smooth bars corresponding to sequences of successive ones and zeros, but sequences involving a transition between a one and zero in either direction produce overshooting.

The problem with an eye exhibiting pattern dependence is that it is impossible to fit a simple Gaussian distribution to either the mark or space level. At the decision time in Fig. 2, the marks are arranged in a double-peaked distribution and it is obvious that an error is much more likely to arise from a sample in the lower of these two peaks. Thus blindly fitting a Gaussian distribution to the whole ensemble of marks would overestimate the width 1σ , and therefore overestimate the BER.

The Quasi-Analytic representation provides a solution to the pattern dependence problem. Recall that the QA picture provides an explicit value for the standard deviation of the noise at the decision time for each bit in the simulated sequence. Moreover, since the PRBS_Generator produces maximal length sequences, every possible sequence of marks and spaces up to a given length appears in the simulated bit train. Therefore, we can use Eqs. (2) and (3) to separately evaluate the probability of an error for each of the N bits:

( )( )

0 1QA

1 1 0

, , bit k is a zero1BER, , bit k is a one

Nk k

k k k

P vN P v

σσ

→

= →

=

∑

(9)

where kv and kσ are the signal and standard deviation values at the decision point for bit k . The maximal length properties of the bit sequence guarantee that the various patterns receive an equal


weighting in the sum in Eq. (9), and we can successfully predict the BER for eyes with strong ISI such as that in Fig. 2.

Figure 2: 10 Gbps eye diagram showing pattern dependence.

Monte-Carlo BER Estimation The QA approach solves the problem of pattern dependence, but as mentioned above and discussed in detail in Chapter 8 of the OptSim User Manual, the QA representation is limited in the types of noise and operations it can handle. In particular, it cannot describe colored noise or noise filtering, or the nonlinear amplification of noise as occurs for ASE in a nonlinear fiber. For systems, in which these kinds of noise are present, the Monte-Carlo noise representation, in which the noise is directly included as a stochastic contribution to the signal vector, must be used.

In contrast to the QA representation, when the electrical signal enters the BERTester, we no longer have access to explicit values to the noise standard deviations, and therefore, for the width of the bars in the electrical eye. Instead, the algorithm must construct histograms for the upper and lower parts of the eye, and so explicitly measure the means and widths of the mark and space bars to obtain estimates for 1v , 0v ,

1σ and 0σ . For the simple eye in Figure 1, this is relatively straightforward and the BER is obtained directly from Eq. (5). However, for eyes displaying pattern dependence such as that in Fig. 2, the situation is more complex. With the QA representation, we solved this problem by examining every bit separately. But since in the MC approach we must measure the statistical quantities 1v , 0v , 1σ and 0σ from a histogram composed of many samples we cannot treat the bits individually.

The solution is to split up the eye diagram into a set of “partial eyes” by selecting a maximum number of neighbors considered to affect a bit. In the model, this number is set by the parameter ISI_Bits. For


instance, with ISI_Bits = 1, the model considers only one adjacent bit on either side, and construct different eyes for the patterns 000, 001, 100, 101, 010, 011, 110 and 111. For the original eye in Fig. 2, we would obtain the eight partial eyes displayed in Fig. 3. It is easy to see that in general, the number of partial eyes is given by 2ISI_Bits*2+1.

Figure 3: Partial eyes for ISI_bits=1.

Now, for each of these eyes, we separately measure any the mean and standard deviations of the pattern

,j jv σ at the chosen decision time t and calculate the individual error rates ( ) ( )levelpat , ,j jjB v vσ using

Eqs. (2) or (3) as appropriate for each of the eight patterns pat( )j : 000, 001, 010, etc .


The total error rate is then

( )patBER( ) jlevel j

j tot

nv B

n=∑ ,

(10)

where the sum is over the ISI_Bits*2 1ISI 2N += patterns, nj is the number of occurrences of each pattern, and

ntot is the total number of bits. The value for ISI_Bits must be chosen with some care. If it is too small, the different patterns will not be properly separated in the partial eyes, and the fitting of the distribution will be inaccurate. However, setting ISI_Bits increases the amount of work required, since the bit stream must be sufficiently long to generate a reasonable distribution for every pattern. The model will refuse to perform an MC analysis if the partial eyes contain less than 8 instances each and accurate results they should contain many more than this. We examine the convergence properties of the QA and MC methods later.

Confidence Limits for the BER and Q In BER and Q estimation, it is important to have confidence limits available since with an insufficient number of bits or a difficult signal, the BER can easily be uncertain by several orders of magnitude. In OptSim, we obtain confidence limits in different ways for the two BER estimation approaches.

For the QA approach, we use basic results of Gaussian statistics to obtain estimates for the uncertainty in the moments 1v , 0v , 1σ and 0σ , and so evaluate the standard deviation of the Q. The standard errors for the moments are given by

SE ,

SE .2

jj

jj

vN

N

σ

σσ

=

=

(11)

Using these values with standard rules applied to Eq. (7), the standard error in the Q is given by

[ ]( ) ( ) ( )2 2

2 22

1,2

2

SE SE SE

.2

j jj j j

dQ dQQ mdm d

QN

σσ=

= +

=

∑

(12)

Choosing one-deviation confidence limits for the Q-factor: [ ] [ ]lo hiSE , SEQ Q Q Q Q Q = − = + ,

we invert Eq. (8) to find confidence limits for the BER as

( ) ( )lo lo hi hiBER BER ,BER BERQ Q = = .

For the MC method, for which the Q is less well-defined, it is simpler to base error limits directly on calculation of the BER. For each bit pattern, standard errors are estimated for the expressions

( ) ( )1 1 level 1/ 2v vµ σ= − and ( ) ( )0 level 0 0/ 2v vµ σ= − .


Confidence limits are chosen as

( ) ( )1 1lo hi2 2BER erfc SE / 2,BER erfc SE / 2j j

j j j jµ µ µ µ = − = +

and averaged over all bit patterns. The limits for the Q-factor are then found by converting the BER limits to Q values with Eq. (8).

Complete BER Estimation Algorithm We are now ready to describe all the steps of the BERTester implementation, of which the above theory is just a part, and introduce the relevant model parameters.

Incoming Signals - AlignBinaries: NO

x10-9

Time (s)

0 1 2 3 4 5

Bina

ry V

alue

0.0

1.0

Signal M

agnitude (V)

0.001

Incoming Signals - AlignBinaries: YES

x10-9

Time (s)

0 1 2 3 4 5

Bina

ry V

alue

0.0

1.0

Signal M

agnitude (V)

0.001

Figure 4: Incoming binary and electrical signals with a misalignment.

In the sections below, we introduce many output and plotting facilities. As with all output/plot blocks, the FilenameRoot parameter allows the user to pick a prefix for all generated files. If FilenameRoot is empty, a prefix is generated from the component’s name.

The algorithm is implemented in 5 stages as follows.

1. Inputs The BERTester uses input nodes in sets of two that accept pairs of electrical and binary signals. The BERTester can function with only an electrical input, but if the original binary signal is available, it is of assistance to the algorithm. Typically, one connects the PRBS_Generator at the start of the link to the BERTester at the end. In WDM simulations where there may be numerous binary signals generated, it is important to match the binary signals to the correct BERTester. A plot of the incoming binary and electrical signals is generated if the parameter PlotSignals=YES (see Fig. 4 for examples). If more than one pair of electrical and binary signals are connected, the BERTester calculates “Word Error Rates”. This is discussed later.

The parameters preBits and postBits determine the number of bits at the start and end of the signal that the BERTester ignores in calculating the BER. These should be normally be set to the same values of the corresponding parameters in the PRBS. The purpose of these ignored slots is to deal with any phase noise introduced through the periodic boundary conditions of the FFT operation. The signature of this is unusually noisy or oscillatory signal patterns at the start or end of the time window. The MC method is particularly sensitive to this kind of noise, and on occasion it may be necessary to ignore 5 or more bits at either end.

2. Creation of Provisional Bitstream In the first stage of the algorithm, the BERTester generates a provisional set of binary values for the incoming signals. This bitstream is used later to construct “eyelids” for the eye diagram. If a binary signal


has been attached, then the contents of that signal are used as the provisional binary value. If no binary signal is available, the BERTester crudely estimates the decision threshold as the mean value of the electrical signal, and then assigns binary values based on the instantaneous value of the electrical signal in each bit slot. If the parameter bitsPerLine is nonzero, the model writes the provisional bit stream to a file with the suffix “provisional_bitstream”.

If a binary signal input is not provided, the BERTester is still usually able to construct eyelids, but the model has no way of determining if a detection error has occurred. When a binary signal is available and an error has occurred, the BERTester detects that the eye is formally closed and refuses to calculate a BER (since an error is essentially certain in a real system).

In some simulations involving delays or filters, it is possible for the electrical and binary signals to become misaligned, producing a signal plot like that in Fig. 4(a). If the misalignment is severe (larger than 1.5 bit periods,) the algorithm may subsequently have trouble constructing an eye diagram. To avoid this problem, the setting AlignBinaries = YES instructs the model to realign the signals to obtain a maximum correlation. Fig. 4(b) shows the same signals after realignment.

3. Construction of Eye Diagram and Eyelids The model now constructs an electrical eye and “eyelids” from the incoming electrical signal and the provisional bit stream as indicated in Fig. 5. Plots are generated for the user if PlotEyeLids = YES. The eye itself is only generated for the user’s convenience and is disabled if PlotEyeLids = NO. Using bitstreams of longer than 210 or setting ContourEye = YES causes the eye to be plotted as a contour plot rather than as a line plot. The former type is sometimes more suited to comparison with laboratory instruments.

The eye lids are created by finding the minimum values of all marks, and the maximum values of all spaces, where the provisional bit streams generated previously are used to determine whether a bit slot is to be considered a mark or space. Thus the eye lids represent the largest curve that can be inscribed within the eye diagram.

Figure 5: Eye diagrams and eye lids for 10 Gbps problem.

The crosshairs in the eye and eye lid plots indicate provisional values for the decision level and decision time. In Fig. 5, the plots have been shifted to place the decision points at 0 sec and 0 V. The actual values for the decision points are shown if showOffsets = YES.

The provisional decision time is chosen to be the vertical line dividing the eye opening into two equal areas, and the provisional decision level is the average of the upper and lower eye lid values at the decision time. Below, we show how the decision level and time can be determined more accurately or specified directly by the user.

Several other eye properties are calculated from the eye lids. These properties, with their definitions and the parameters controlling their output, are:


• eye height (V, PlotEyeHeight) – the largest vertical distance between the upper and lower lids (which does not necessarily occur at the decision time).

• eye width (s, PlotEyeWidth) – the horizontal distance between the points at which the eye lids cross.

• eye area (Vs, PlotEyeArea) – the integrated area inside the eye lids.

• eye closure (PlotEyeClosure) – 10eye height eye width10log

eye area ×

If the eye is closed, that is to say, a mark or space is found on the wrong side of the threshold, then the model produces an “Average Eye” found by taking the mean values of all marks and all spaces. This produces eye lids which roughly correspond to the darkest parts of the eye diagram itself. The model outputs these plots, posts a warning and exits without proceeding to calculate an error rate.

4. BER Calculation The BER is now finally calculated using the preferred method as set by n_representation. We emphasize again that to use the QA method, the BER must be preceded by a monolithic receiver model also set to n_representation = QA. The MC method may be preceded by either the monolithic receiver, or a chain consisting of photodetector, amplifier and filter, but if the receiver is used, the values of n_representation must again match.

To this point, the decision time and level have been set provisionally by examination of the eye. For the QA method, this frequently works very well, but the provisional estimate is less reliable for the MC method. Therefore, the model has several features for optimizing the values of these quantities. If decisionLevelSelection = Automatic (the default), the model calculates the BER for a range of decision levels and searches for the optimum. Alternatively, if decisionLevelSelection = Defined, the user directly sets the decision level with decisionLevel. The user may also influence the decision time, but rather than setting it directly, the user specifies an offset from the model’s automatic selection, in units of the bit period with decisionPointShift. The most accurate but most time-consuming way to set these quantities is with the setting jointOptimization = YES, in which the model uses a two-dimensional iterative algorithm to simultaneously find the optimum values for the decision level and time. Generally, this option should only be used if necessary. For most work, the default settings perform well – the decision time is taken from the eye analysis, and the decision level is found by one-dimensional optimization. Once the optimum levels have been found, the bit error rate is calculated once more to produce the final optimum answer.

Regardless of whether the actual BER is found by automatic optimization or from user specified values, scans of the BER against decision level and time are produced if PlotDecisionLevelScan and PlotDecisionTimeScan are activated. Two-dimensional plots of the BER and Q-factor as a function of the level and time are activated with PlotBERContours and PlotQContours. The 2D plots are very useful in MC mode, since our experience indicates that a multi-peaked contour is often a good indication of a bitstream of insufficient length or an insufficient ISI_Bits setting. The actual decision level selected is plotted if PlotEyeCenter = YES.

The model issues a warning if the decision level or time scans fail to find minimum values. In such a case, the user should investigate these curves and the appearance of the eye to determine why the decision points are not being found correctly. Usually, this occurs if the eye is so noisy that it is essentially closed. For such a case, the BER is so close to unity, that an accurate determination is not justified anyway. The system should be improved until the eye quality is reasonable. A common cause of errors from the model is to use the Defined mode for decision level selection but provide a value that does not fall inside the eye at all. This can happen particularly easily in scans, when the correct decision level may change from run to run.

5. Additional Output For the MC method, the model can plot all the partial eyes (PlotPartialEyes = All), or just the eight partial eyes (PlotPartialEyes = Critical) that give the greatest contribution to the BER.


If bitsPerLine is nonzero, a second bitstream is written to a file with extension the “measured_bitstream”. This sequence is found by measuring the electrical signal with the final values of the decision level and time.

Additional BER Features

Simulated Detection Jitter In real systems, the BER can be affected by jitter in either the decision level or time due to imperfections in the detector circuitry. Using the parameters timingJitter and decisionLevelJitter the user may specify standard deviations for the decision level and time. For each bit, the model adds a random shift to both quantities selected from a Gaussian distribution truncated at two standard deviations. The BER and Q are calculated as an average over numberAve repetitions.

Word Error Rates If more than one electrical input is connected to the BERTester, the model calculates a “word error rate”. The procedure is identical to that described above with the following exceptions. The eye diagram and eye lids are constructed by superimposing all the signals into a single eye. Thus single decision points are selected for all the signals. The BER is calculated for each signal and the worst case is taken as the actual error rate. In QA mode, this is done on a bit by bit basis; in MC mode, it is done for each partial eye.

Other Distribution Functions Some researchers have estimated the BER using distributions that are non-Gaussian, such as the chi-squared distribution. Selecting a different distribution intelligently requires detailed knowledge of the nature of the noise in the problem. Currently OptSim does not support additional distribution functions, but users requiring such a feature should contact RSoft Design Group.

Forward Error Correction Forward Error Correction (FEC) coding techniques help optical links to achieve higher performance by detecting and correcting errors on the link. FEC allows a predetermined amount of error to occur during transmission, and detects and corrects them at the receiving end. These techniques, previously used in wireless systems and data storage applications, now are being widely used in optical telecommunications systems, especially long-haul applications. Almost without exception, all modern transoceanic systems starting with TPC-5 (1996) use FEC. Two ITU specifications - ITU-T G.709 and ITU-T G.975 - recommend FEC in transmission systems [1].

Performance improvements due to FEC can be used to increase inter-amplifier spacing and/or to increase system capacity, to relax the specifications on the optical components or fiber (hence lower cost), and so on. In a WDM system these capacity improvements can be achieved by increasing the bit rate of each WDM channel or by decreasing the channel spacing, allowing more WDM channels for a given amplifier bandwidth.

The disadvantage of using FEC is that the inserted check symbols consume bandwidth within the communications channel and a system using FEC requires a slightly higher bit rate to support this additional correction data.

There are a large number of error-correction codes, each with different properties that are related to how the codes are generated and consequently how they perform. Some examples of these are the linear and cyclic Hamming codes, the cyclic Bose-Chaudhuri-Hocquenghem (BCH) codes, the cyclic Golay and Fire codes, and the Turbo convolutional and product codes (TCC, TPC). The codes that are most attractive at present for application in high bit-rate communication systems are a set of cyclic, non-binary, block codes known as Reed-Solomon (RS) codes [2].


Reed-Solomon codes are described as (N, K) where N is the total number of symbols (bits) per code-word, K is the number of information symbols (data bits), and R is the number of check symbols (N-K). The overhead of the code is simply the ratio of the check symbols to code-word symbols. For example, the Reed-Solomon codes used in ITU-T G.709 and G.975 are both (255,239) so will consist of a 239 information symbols and 16 check symbols with about 6.7% overhead.

Efficiency of FEC is measured by coding gain (in units of Q-factor in dB). To achieve a BER = 1.0E-13 (corresponding Q = 17.3dB) after FEC with RS(255,239) code, the raw BER before FEC has to be 1.42E-4 (and corresponding Q=11.20 dB). Here, if we compare two identical systems, one with and the other without FEC, then gross FEC coding gain is about 6 dB, whereas net coding gain has to take into account the bit rate overhead and is system dependent (system length, bit rate, etc. will define system impairments). So, the transatlantic WDM system showed about 5 dB net coding gain at 6.7% redundancy rate [3]. RS(255,239) is standardized by ITU recommendations G.709/G.975 and also can be referred to in literature as a standard FEC or generation one FEC.

FEC Gross Coding Gain

input Q2 (dB)4 6 8 10 12 14 16 18

outp

ut B

ER

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100No FEC

FEC RS(255,239) Code

FEC RS Concat. Code

Figure 6. FEC gross coding gain for RS(255,239) and RS Concatenated codes.

To achieve higher coding gain different combinations of concatenated RS codes were tried, providing proper interleaving of outer and inner codes. These type of codes can generate higher gross coding gain of 8-10 dB compared to RS(255,239) but require higher overhead rate, 14-25%. [4,5]. These types of concatenated codes have been deployed commercially in several recent undersea systems and are referred to as super FEC or generation two FEC. One particular concatenated scheme considered here employs RS(223,207) Outer and RS(255,223) Inner interleaved codes, which provides about 9 dB gross coding gain for BER requirement of 1.0E-13 at the cost of 23% bit rate overhead.

Theoretical conversion formula for FEC encoding/decoding can be derived if the errors are assumed to be from additive white Gaussian noise [6]. The OptSim BER Tester model block has implemented FEC conversion formula for RS(255,239) and Concatenated RS(223,207)/RS(255,223) codes. Fig. 6 shows the gross coding gains for both FEC schemes. In addition, the third scheme option is available for user-defined FEC conversion table.

Note: The model does not actually decode the data but rather applies FEC coding gain to input BER according to the selected FEC scheme. Also note that the coding gain can be optimistic in cases with high PMD, where long periods of high error rates may degrade the error correcting capability of the code [7].


Implementation To include FEC coding during BER estimation the parameter includeFEC should be set to YES. Next parameter FEC_CodingScheme allows user to choose one of three available coding schemes: RS(255,239) (RS(255,239)_Code), RS Concatenated Code (RS_Concat_Code), and User Defined Code (UserDefined). If a UserDefined FEC coding option is selected, then the user must specify a file name in parameter FEC_Filename, where the file contains a data table for the BER before and after FEC. The file format is given below:

File Format: FECFormat1

<BER_input 1> <BER_output 1>

<BER_input 2> <BER_output 2>

…

Example: FECFormat1

1.00E-02 1.81E-02

7.50E-03 1.35E-02

…

It is important that the user adjust the bit-rate at the transmitting end of a link in accordance with the overhead incurred due to the selected coding scheme. The case of RS(255,239)_Code requires 6.7% overhead (e.g. 10 Gbps bit-rate has to be changed to 10.67 Gbps), RS_Concat_Code requires 23%, and in case of UserDefined scheme has to be defined by the user.

Plotting functions are supplied to plot BER and Q after applying FEC. Also a test function will plot FEC coding gain for selected FEC scheme.

Performance Budget The performance budget is an important element of the optical link designs at all its stages starting from the system bidding following by a desktop system design, laboratory trials, manufacturing, installation, and commissioning and acceptance on installed system [8]. It is also a valuable tool to study upgrade paths on existing links to their maximum capacities.

The purpose of performance budget is to evaluate design viability for selected system configuration at given requirements of system BER performance. The difference of system performance (measured or simulated) and system performance requirement is a measure of system viability and called a system margin. Traditionally system margin is being calculated in units of Q-factor in dB. For example, if link performance requirement from a system provider is BER no worse than 1.0E-13 (corresponding Q-factor is 17.3 dB) and simulation BER for given link configuration is 1.42E-20 (corresponding Q=19.3 dB), and then system margin will be 2 dB. Typical design objectives for system margin are to be between 3 and 6 dB to absorb numerous performance impairments due to the deviation of real system from ideal design. To name few of these impairments there are manufacturing and environmental impairments (due to temperature fluctuations, fluctuations in fiber length, loss, and dispersions in different spans, etc.), time variation penalties (fading, PMD, etc.), and time related penalties (fiber aging, components aging/fails, repairs). Some of these impairments can be estimated by user himself based on available manufacturing or measured data, and some of them can be directly included in simulations (e.g. PMD, statistical distribution of fiber parameters, etc.).


Implementation OptSim implements the performance budget described above. It calculates the system margin by subtracting required Q-factor (given by user in system BER requirement) from simulation Q. In a case when FEC coding\decoding is included, the required system Q will be recalculated by applying FEC coding gain. Tables 1 and 2 show examples of generated performance budget with and without FEC, respectively.

To generate performance budget the parameter genPerfBudget should be set to YES. The next parameter requirementForBER has to specify the BER requirement for system performance. Performance budget will be stored in file “topname”_BERTest_perfbudget.txt. To open it user has to double-click on BERTester model and pick the abovementioned file from a plot list.

Table 1. A sample of performance budget generated by OptSim for case without FEC.

1. Simulation Q (dB) 18.89

2. Simulation BER 7.06E-19

3. BER Requirement 1.00E-09

4. Required Q (dB) 15.56

5. System Margin (dB) 3.33

Table 2. A sample of performance budget generated by OptSim for case with FEC.

1. Simulation Q (dB) 14.28

2. Simulation BER 1.15E-07

3. BER Requirement 1.00E-13

4. Required Q (dB) 17.32

5. Required Q before FEC (dB) 11.2

6. Required BER before FEC 1.42E-04

7. System Margin (dB) 3.08

NON-BER facilities A number of other quantities related to the optical eye are calculated during the BER computation. These quantities and the parameters enabling their output are as follows:

• Average mark level (PlotAveHigh)

This is the mean value of the voltage at the decision point for all marks.

• Average space level (PlotAveLow)

This is the mean value of the voltage at the decision point for all spaces.

• Width of the mark level (PlotAveHighNoise)

This is the mean value of the noise standard deviation at the decision point for all marks.

• Width of the space level (PlotAveLowNoise)


This is the mean value of the noise standard deviation at the decision point for all spaces.

• Mark level for successive ones (PlotAveHighSteady)

This is the mean voltage at the decision point for all marks which are adjacent to at least one other mark. It is a measure of the “one” bar at the top of the eye.

• Mark level for successive zeros (PlotAveLowSteady)

This is the mean voltage at the decision point for all spaces which are adjacent to at least one other space. It is a measure of the “zero” bar at the bottom of the eye.

• Extinction Ratio (PlotExtRatio)

This is defined as 10Average one level10logAverage zero level

• Rectangular Eye Opening (PlotRectOpening)

This constructs the largest rectangle inside the eye according to the following algorithm. The rectangle has a horizontal length defined as a fraction of a bit period by rectOpeningFrac. The code looks for the tallest rectangle of that length that fits inside a region where the BER<minBER_Crit.

Eye Mask BERTester model also allows to plot normalized eye diagram with eye mask specified by user. To activate Eye Mask feature one has to switch parameter PlotEyeMask to YES and define the eye mask coordinates in parameter eye_mask_array, which is double array. Figure 7 shows an example of eye mask definition used in IEEE 802.3ae Standard [9]. Eye mask is plotted as a polygon on a normalized time and voltage (amplitude) scale. In this case eye mask is a hexagon with 6 vertices (for some other standards it can be rectangular mask with 4 sides or decagon mask with 10 sides).

Figure 7. Example of eye mask definition from IEEE 802.3ae standard (Ref [9], Figure 53-8).


Figure 8 shows how to construct eye mask array. Each vertex of the mask polygon should be given by its (X, Y) coordinate and then all vertices coordinates should be arranged in counter-clockwise sequence to form a closed polygon. So, in case of Figure 8 the double array for eye mask should be defined as

eye_mask_array = X1, Y1, X2, Y2, X3, Y3, X4, Y4, X5, Y5, X6, Y6

Figure 8. Construction of eye mask points.

The default values used are for eye mask are shown at Figure 8, i.e. 0.2, 0.5, 0.35, 0.25, 0.65, 0.25, 0.8, 0.5, 0.65, 0.75, 0.35, 0.75. Finally, Figure 9 shows an example for OptSim simulation output plot with eye diagram with eye mask.

Decision Eye with Eye Mask

Normalized Time-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Nor

mal

ized

Am

plitu

de

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Figure 9. Example of eye diagram with eye mask in OptSim.

Scanning Facilities In most problems, one wishes to know the BER as a function of one or more system parameters, so the ability to scan results is important. The BERTester model is fully integrated into the OptSim’s generalized scanning facility described in Chapter 7, Section 7.1.2 of the User Manual. Using these facilities, the BER, Q and a large number of other quantities may be automatically plotted as functions of either the inner or outer scan variables. By attaching the BERTester to an XY-Plotter, plots may be produced as a function of any other measured parameter. This is particularly useful for creating power-penalty plots. The scanned properties are also reported in a text file with the extension “table.txt” which includes complete information on the values, confidence limits and statistical properties. Instructions for using these features


are included in Chapter 7 of the User Manual and are not repeated here. The facilities are available for all properties whose parameters take the values None, Local, NonLocal and Both. Specifically, these parameters are PlotBER, PlotQ, PlotBER_FEC, PlotQ_FEC, PlotEyeHeight, PlotEyeWidth, PlotEyeCenter, PlotEyeArea, PlotEyeClosure, PlotAveHigh, PlotAveLow, PlotAveHighNoise, PlotAveLowNoise, PlotAveHighSteady, PlotAveLowSteady, PlotExtRatio and PlotRectOpening.

Validation: 10 Gbps Example The remainder of this discussion compares the use of the two QA and MC noise treatments for the 10 Gbps example in the tutorial section of the user manual.

We start with convergence properties of the quasi-analytic approach as the length of the bit stream increases. For convenience in scales, we use the Q factor for comparisons. Similar results are found for the BER. Recall from Eq. (12) that we have defined confidence limits that decrease as the sequence length N increases. These confidence limits are shown in the plot in Fig. 10.

Figure 10: Convergence of QA method with increasing bit sequence length.

The different colors correspond to the different standard drive currents of the laser and the line styles denote the number bits in each simulation. Above 29 bits, there is only a small change in the values for each current. Note that the error bars indeed give a reasonable estimate of the uncertainty in the curves.

We now compare how the MC results converge to the QA results as the sequence length and the ISI window are increased. In all plots, the dashed lines are the QA results for 215 bits, and the solid lines are the MC results. With no ISI, (Fig. 11) the results are completely invalid. The MC method grossly underestimates the Q, since the noise estimates are too large.


Figure 11: Comparison of MC and QA analysis. Sequence length = 29 bits, ISI_Bits=0

In Figs. 12-14, we use increasing values for ISI_Bits and show curves for sequence lengths of 29, 211, 213

and 215 bits. Figs. 12 and 13 show an improvement in agreement over Figure 11, but surprisingly the

agreement does not improve as the sequence length increases. An improvement might have been expected, since as indicated by the error bars on the MC curves, the confidence limits in theory decrease with longer sequences. In fact, only when ISI_Bits = 3 in Fig. 14, does the increase in pattern length improve the convergence. For that case with 215 bits, there is excellent agreement for all but the I0 = 0.01 A curve. As we show later, the weak current causes difficulties due to very strong pattern dependence.

In general then, we conclude that it is necessary to check the MC results at a number of values of ISI_Bits until the results converge. With insufficient ISI settings, as in Figs. 12 and 13, there is little advantage to using longer bit sequences.

The error bars can be misleading in this sense. They indicate how reliable the results are, assuming that the current ISI value is sufficient. It is always necessary to check higher ISI values before accepting the results. Note also that for the MC method, the Q is a somewhat artificial concept, since each ISI bit pattern has an error rate calculated separately. Nevertheless, Eq. (8) still serves as a useful definition for the Q, allowing us to display error rates on a linear scale.



Figure 12: Comparison of QA and MC methods for increasing bit sequence length with ISI_Bits=1.







Now we return to the problems seen in Fig. 14 for the blue curve corresponding to a laser drive current of

0I =0.01A. Even with ISI_Bits=3 and very long bit sequences, the MC method underestimates the Q. This is explained by the partial eyes in Fig. 15.


Figure 15: Partial eyes for 0I =0.01A, showing residual pattern dependence.

Despite accounting for 3 bits of ISI, there is still a visible pattern dependence at the peaks of the curves. On the right hand side for pattern 0001000, this leads to a spread in the signal which is still quite large and inflates the BER. This is still the case with ISI_Bits = 4. Moving to larger values becomes difficult since to maintain the same number of samples in each partial distribution, the pattern length must be multiplied by 4 for each increment in ISI_Bits. There will probably always be noisy signals like this that are difficult to treat. But if the ISI impacts the error rate this strongly, then subtle noise effects in the fiber are probably insignificant and the QA treatment is appropriate.

References [1] ITU-T Recommendations: G.709 Interfaces for the Optical Transport Network (02/2001), G.975

Forward error correction for submarine systems, (10/2000).

[2] G.C. Clark, J. B. Cain, Error-Correction Coding for Digital Communications, (Plenum Pub, 1981).

[3] S. Yamamoto, et al, “5 Gb/s optical transmission terminal equipment using forward error correction code and optical amplifier,” Electronics Letters 30, 254 (1994).


[4] H. Taga, et al, “Performance improvement of highly nonlinear long-distance optical fiber transmission system using novel high gain forward error correcting code”, paper TuF3, Technical Digest of OFC 2001.

[5] O. Ait Sab: FEC techniques in submarine transmission, paper TuF1, Technical Digest of OFC 2001.

[6] H. Kidorf et al, “Performance improvement in high capacity ultra-long distance, WDM systems using forward-error correction codes”, paper ThS3, Technical Digest of OFC 2000.

[7] M. Tamizawa et al, “FEC performance in PMD-limited high speed optical transmission systems“, Technical Digest of ECOC 2000, vol.2, pp.97-99, 2000.

[8] E.Golovchenko, “Tutorial: the challenges of designing long-haul WDM systems”, p.79, Technical Digest of OFC 2002.

[9] IEEE Std 802.3ae™-2002 (Amendment to IEEE Std 802.3, 2002 Edition).

Properties


#2: Binary signal

#3-N: Repetitions of electrical and binary signal pairs

Outputs None or #1: MeasurandSig

Paramater Values Name Type Default Range Units FilenameRoot string

AlignBinaries enumerated NO NO, YES

n_representation enumerated QA MC, QA

unitsForQ enumerated linear linear, Q^2(dB) preBits integer 2 [ 0, 1e8 ] none

postBits integer 3 [ 0, 1e8 ] none

jointOptimization enumerated NO NO, YES

decisionLevelSelection enumerated Automatic Automatic, Defined decisionLevel double 0 [ -1000, 1000 ] V

timingJitter double 0 [ 0, 1 ] s

decisionLevelJitter double 0 [ 0, 1e3 ] V

decisionPointShift double 0 [ -1, 1 ] bit_period numberAve integer 1 [ 1, 1e8 ] none


ISI_Bits integer 0 [ 0, 8 ] none

showOffsets enumerated NO NO, YES bitsPerLine integer 8 [ 0, 10000000 ] none

minBER_Crit double 1e-9 [ 1e-300, .5 ] none

rectOpeningFrac double 0.2 [ 0, 1 ] none

genPerfBudget enumerated NO NO, YES requirementForBER double 1.e-9 [ 0, 1 ] none

includeFEC enumerated NO NO, YES

FEC_CodingScheme enumerated RS(255,239)_Code RS(255,239)_Code, RS_Concat_Code, UserDefined

FEC_Filename string

PlotEyeMask enumerated NO NO, YES

eye_mask_array double array [0.2, 0.5, 0.35, 0.25, 0.65, 0.25, 0.8, 0.5, 0.65, 0.75, 0.35, 0.75]

StatisticalPlotMode enumerated All Average, Minimum, Maximum, Std.Dev., Latest, StatRun, All


PlotBER enumerated Local None, Local, NonLocal, Both

PlotQ enumerated None None, Local, NonLocal, Both

PlotBER_FEC enumerated None None, Local, NonLocal, Both

PlotQ_FEC enumerated None None, Local, NonLocal, Both

PlotEyeHeight enumerated None None, Local, NonLocal, Both

PlotEyeWidth enumerated None None, Local, NonLocal, Both

PlotEyeCenter enumerated None None, Local, NonLocal, Both

PlotEyeArea enumerated None None, Local, NonLocal, Both

PlotEyeClosure enumerated None None, Local, NonLocal, Both

PlotAveHigh enumerated None None, Local, NonLocal, Both

PlotAveLow enumerated None None, Local, NonLocal, Both

PlotAveHighNoise enumerated None None, Local, NonLocal, Both

PlotAveLowNoise enumerated None None, Local, NonLocal, Both

PlotAveHighSteady enumerated None None, Local, NonLocal, Both


PlotAveLowSteady enumerated None None, Local, NonLocal, Both

PlotExtRatio enumerated None None, Local, NonLocal, Both

PlotRectOpening enumerated None None, Local, NonLocal, Both

PlotEyeLids enumerated NO NO, YES

PlotPartialEyes enumerated None None, All, Critical PlotSignals enumerated NO NO, YES

PlotDecisionLevelScan enumerated NO NO, YES

PlotDecisionTimeScan enumerated NO NO, YES

PlotBERContours enumerated NO NO, YES PlotQContours enumerated NO NO, YES

ContourEyes enumerated NO NO, YES

Parameter Descriptions FilenameRoot Filename prefix for data files. AlignBinaries Toggle realignment of binary and electrical signals. n_representation Toggle QA and MC treatments of noise. preBits Number of initial bits to ignore. postBits Number of final bits to ignore. jointOptimization Toggle joint optimization of decision level and time. decisionLevelSelection Toggle automatic calculation of decision level. decisionLevel User specified decision level if selection is user-defined. timingJitter Standard deviation of jitter in the decision time. decisionLevelJitter Standard deviation of jitter in the decision level. decisionPointShift Offset of decision time in fractions of a bit period. numberAve Number of repetitions to average if jitter is present. ISI_Bits Number of adjacent bits (either side) to include in MC mode. showOffsets Toggle absolute or relative times and voltages in eye diagrams. bitsPerLine Activate output of incoming binary sequence. minBER_Crit Maximum BER for rectangular eye opening. rectOpeningFrac Horizontal width of rectangular eye opening in bit periods. genPerfBudget Activate output of Performance Budget requirementForBER BER requirement for Performance Budget includeFEC Toggle inclusion of FEC FEC_CodingScheme Type of FEC coding scheme: several types are supported FEC_Filename File name for UserDefined FEC coding data PlotEyeMask Toggle output of eye diagram with eye mask eye_mask_array Eye mask definition StatisticalPlotMode Toggle statistical quantity to plot in scanning simulations. SaveOrDisplayPlots Toggle automatic or manual display of plots, or disable plots. PlotBER Toggle output of BER plot and data.


PlotQ Toggle output of Q plot and data. PlotBER_FEC Toggle output of BER after FEC plot and data. PlotQ_FEC Toggle output of Q after FEC plot and data. PlotEyeHeight Toggle output of eye height plot and data. PlotEyeWidth Toggle output of eye width plot and data. PlotEyeCenter Toggle output of decision level plot and data. PlotEyeArea Toggle output of eye area plot and data. PlotEyeClosure Toggle output of eye closure and data. PlotAveHigh Toggle output of average mark level plot and data. PlotAveLow Toggle output of average space level plot and data. PlotAveHighNoise Toggle output of average mark noise plot and data. PlotAveLowNoise Toggle output of average space noise plot and data. PlotAveHighSteady Toggle output of average successive marks plot and data. PlotAveLowSteady Toggle output of average successive spaces plot and data. PlotExtRatio Toggle output of extinction ratio plot and data. PlotRectOpening Toggle output of rectangular eye opening plot and data. PlotEyeLids Toggle output of eye diagram and eye lid diagram. PlotPartialEyes Control output of partial eyes in MC mode. PlotSignals Toggle display of incoming signals. PlotDecisionLevelScan Toggle plot of decision level scan. PlotDecisionTimeScan Toggle plot of decision time scan. PlotBERContours Toggle contour plot of BER. PlotQContours Toggle contour plot of Q. ContourEyes Toggle contour/line plot mode for eye diagram. unitsForQ Toggle linear or Q^2(dB) units for the Q plot


Karhunen-Loeve BER Estimator

The purpose of this model is to implement an efficient semi-analytical technique, based on the results described in [1]-[3], for the estimation of the Bit-Error Rate (BER) performance of direct-detection receivers (see Fig. 1) or differential receivers based on the use of an Asymmetric Mach-Zehnder Filter (AMZ) (see Fig. 2).

The model can for instance be used for performance evaluation of the following modulation formats:

• Intensity-Modulation Direct-Detection (IMDD)

• Duobinary

• Differential Phase-Shift Keying (DPSK)

• Differential Polarization-Shift Keying (DPolSK)

• Differential Quadrature Phase-Shift Keying (DQPSK)

The results may be displayed in several forms – as a detailed table containing full statistical information, or as plots showing the dependence of the properties on scanned parameters. Moreover, if the model is given an output port, the collected information can be passed to the XY-Plotter model to be combined with data from another point in the topology, for example Optical Monitor data. In this way, it is possible to easily generate plots showing the dependence of any quantity on any other.

Model Use The model takes as input a single-band optical signal (multi-channel signals should be represented in Single-Band mode, and must first be separated using a demultiplexing component) and, if Synchronization_flag is set equal to Yes, a corresponding bit sequence with the same number of bits as the optical signal. Furthermore, if a bit sequence is provided, it should at least be as long as the fundamental pattern from which it is generated (for example, 511 bits for a 29-1 PRBS sequence). The fundamental pattern length is specified via the parameter Fundamental_Pattern_Length. In all cases, the input signal should contain at least 11 bits, since the first and last 5 bits are ignored by the BER estimation algorithm (a fact which should be kept in mind when specifying pre- and post-bits in any binary signals).

The results produced by the model are activated with the parameters BER_optimal_threshold, Q_optimal_threshold, Optimal_Threshold, BER_average_threshold, Q_average_threshold, Average_Threshold, and Sampling_Instant. Each of these parameters has the settings None, Local, NonLocal and Both. When a parameter is set to Local, that quantity is calculated and reported both in a combined table (with filename ending “table.txt”) and in plot form (with filename ending “.txt”). When a parameter is set to NonLocal and the model has an output port connected to an XY-Plotter, data is not presented locally but is passed to the XY-Plotter for constructing plots with data from elsewhere in the topology. The user should consult the XY-Plotter documentation and Chapter 5 of the User Manual – “Scanning Facilities” for further details. The setting Both both exports the data to the plotter and generates the local table and plot.

Support for Scanning This model fully supports OptSim's scanning capabilities. When a simulation is launched using the scan dialog, the model automatically produces plots showing the results for the entire scan. On the local plots, the inner scan variable appears as the x-axis, and if an outer scan variable is used, a series of curves is produced, one for each value of the outer scan variable. Using the XY-Plotter the user can produce plots in which any measurable quantity appears on the x-axis.




Model of the Direct-Detection Optical Receiver With Modulation_format set equal to IMDD, the block models a complete direct-detection optical receiver, including a pre-detection optical filter, one photodetector of type PIN or APD, an electrical amplifier and a post-detection electrical filter. The electric filter may be followed by a FIR filter equalizer with the following transfer function:

2( ) eqN

j fnTeq n

n NH f c e π−

=−

= ∑ (1)

where 2N+1 is the total number of taps and Teq is the time delay between the taps.

Figure 1: Direct-Detection receiver.

The noise sources taken into account are the following:

• ASE noise nASE(t), introduced by the optical amplifiers (EDFAs). It is modeled as additive white Gaussian noise, characterized by its one-sided power spectral density N0.

• Thermal noise nsh(t) introduced by the electrical amplifier. It is modeled as additive white Gaussian noise. The variance of noise [mV2] on the decision variable is equal to:

22 2 0 ( )2th t L

NR H f dfσ

+∞

−∞= ∫ (2)

where HL(f) is the transfer function of the post-detection filter, N0/2 is the “input-referred” bilateral power spectral density [(pA)2/Hz] and Rt is the overall RX transfer function, which is generally a transimpedance [Ohm] that transforms the input current [mA] into an output voltage [mV] 1.

• Shot noise nsh(t) introduced by photodetectors, modeled as a non stationary additive white Gaussian random process with zero mean and signal-dependent variance. The variance of shot noise on the decision variable is equal to [4]:

1 Since the overall RX performance does not depend on the value of Rt, in the simulation program it is assumed to be equal to 1 and neglected.


( ) 22 2 2 ( ) ( )xsh t e Rt G qRR h P t dσ τ τ τ+= −∫ (3)

where q is the electron charge, R is the responsivity of the photodetector, G is the avalanche gain of the APD, hL(t) is the impulse response of the overall electrical filtering (post-detection filter and equalizer) and PR(t) is the power of the filtered optical signal at the input of the photodetector.

Model of Optical Binary Differential Receiver With Modulation_format set equal to DPSK, the block models the receiver shown in Fig. 2. The optical filter is followed by an AMZ filter, with the following optical field transfer function:

( )21( ) 1 e2

AMZj fTAMZH f π δϕγ +± = ± (4)

where TAMZ is the optical (macroscopic) path difference within the interferometer (ideally equal to the inverse of the symbol rate), δϕ is a phase-mismatch due to an additional microscopic path difference inside the interferometer and γ <1 represents a finite extinction ratio. The ± sign reflects the two interferometer output ports. The AMZ is then followed by a balanced photo-detector (BPD) device.

Figure 2: Binary differential receiver.

The implemented algorithm considers the following possible imperfections of the receiver components (see [3-5]):

• AMZ frequency offset ∆f, corresponding to the detuning between the laser frequency and the Asymmetric Mach-Zehnder central frequency. It is related to the phase mismatch δϕ through the relationship:

2 Sf Rδϕπ

∆ = (5)

where Rs is the symbol rate (equal to the bit-rate in case of binary modulation).

• AMZ extinction ratio, related to the parameter γ through the following relationship:

( )( )

2

10 2

110 log

1dB

γε

γ

+= ⋅

− (6)

• AMZ delay error, defined as the difference between the actual AMZ delay TAMZ and the symbol time T=1/RS.

• BPD phase imbalance, defined as the propagation delay difference ∆τ between the two arms of the BPD.


• BPD amplitude imbalance β, due to the difference between the responsivity values of the two photodetectors:

1 1 2 2

1 1 2 2

R G R GR G R G

β −=

+ (7)

The considered noise sources are the same as for the direct-detection receiver.

Model of DQPSK Receiver With Modulation_format set equal to DQPSK_upper_arm or DQPSK_lower_arm, the block models the corresponding arm of the receiver shown in Fig. 3. It is composed of two differential receivers (like the one shown in Fig. 2), with a differential optical phase between the AMZ arms equal to ±π/4.

When DQPSK modulation is used, a differential pre-coder [5] must be inserted at the transmitter in order to properly use the synchronization feature of the BER estimating block.

Figure 3: DQPSK receiver.

.

Brief Description of the Semi-Analytical Technique The semi-analytical technique used to evaluate the BER performance of the system is based on the following main steps:

1. The noiseless optical signal at the input of the receiver is generated through simulation;

2. The probability density function (PDF) associated with each detected bit is determined using the KLT technique described in the following (since beat noise between signal and ASE is inherently non-stationary, the PDFs are different for each bit);

3. The overall BER at a given decision threshold sth and sampling instant ts is evaluated as:

( )0,1,

1

1( , ) ,N

th s e k th sk

BER s t P s t kTN =

= +∑ (8)

where T is the inverse of the symbol rate and ( )0,1, ,e k th sP s t kT+ denotes the error probability

associated with the decision of the k-th bit (either a 0 or a 1):


( ) ( )0 1, ,( , ) , , ( , ) ,

th

th

s

e k th k e k th ks

P s t PDF s t ds P s t PDF s t ds∞

−∞

= =∫ ∫ (9)

4. Joint minimization of (8) over the couple (sth,ts) is performed to obtain the BER at the optimum threshold and sampling instant.

Karhunen-Loeve Technique (KLT) Neglecting thermal and shot noise, the noisy electrical signal at the output of the receiver can be written as:

21 1( ) ( )* ( ) * ( )o Lv t R G e t h t h t= (10)

for the direct detection receiver, and as:

2 2

1 1 2 2( ) ( )* ( )* ( ) ( )* ( )* ( ) * ( )o AMZ o AMZ Lv t R G e t h t h t R G e t h t h t h t+ − = − (11)

for the differential receiver, where e(t)=s(t)+nASE(t).

The exact PDFs of the random variable v(t) can be efficiently evaluated by using the method described in [2], based on the Karhunen-Loève decomposition of signal and noise in the frequency domain. In the frame of this method, the signals in (10) and (11) are written as inverse Fourier transforms [3]:

( ) [ ]

1 * *1 1

*1 1 2 2 1 2 1 2

( ) ( ) ( )* ( ) ( ) ( )

( ) , ( ) exp 2 ( )

o o L

DD

v t F R G E f H f E f H f H f

E f K f f E f j f f t df dfπ

− = − − =

= −∫∫ (12)

for the direct detection receiver, and as:

( ) [ ]

1 * * *1 1

* * *2 2

*1 1 2 2 1 2 1 2

( ) ( ) ( ) ( )* ( ) ( ) ( ) ( )

( ) ( ) ( )* ( ) ( ) ( ) ( )

( ) , ( ) exp 2 ( )

o AMZ o AMZ L

o AMZ o AMZ L

diff

v t F R G E f H f H f E f H f H f H f

R G E f H f H f E f H f H f H f

E f K f f E f j f f t df dfπ

− + +

− −

= − − −

− − − − =

= −∫∫

(13)

for the differential receiver, where:

( ) *1 2 1 1 1 1 2 2, ( ) ( ) ( )DD o L oK f f R G H f H f f H f= − − (14)

( ) * *

1 2 1 1 1 1 1 2 2 2

* *2 2 1 1 1 2 2 2

, ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )diff o AMZ L o AMZ

o AMZ L o AMZ

K f f R G H f H f H f f H f H f

R G H f H f H f f H f H f

+ +

− −

= − − −

− − − − (15)

E(f) is the Fourier transform of the unfiltered optical field corrupted by ASE noise.

It can be shown (see [2]-[3]) that the signals (12) and (13) can be written in the form:

2( ) ( ) ( )i i ii

v t b t n t τ= +∑ (16)

where bi(t) and vi(t) are the coefficients of the series expansion of the Fourier transform S(f) and NASE(f) of the noiseless received signal s(t) and of the ASE noise random process nASE(t), respectively :

[ ]

[ ]

*

*

( ) ( ) exp 2 ( )

( ) ( )exp 2 ( )

n n

n ASE n

b t S f j ft f df

v t N f j ft f df

π φ

π φ

=

=

∫∫

(17)

φm(t) is a set of orthonormal functions satisfying the following eigenvalue integral equation:

( )2 1 2 2 1( ) , ( )m m mf K f f df fφ τ φ=∫ (18)


where K(f1,f2) is as in (14) and (15) for direct detection and differential receivers respectively.

Using numerical integration algorithms (see [8], Chapter 4), the solution of equation (18) can be reduced to the eigenvalue and eigenvector problem for an Hermitian matrix:

( ) ( ) ( )2

2 1 2 2 2 1 2 2 10

( ) , ( ) , ( ) ,MAX

MAX

f k

m m m i iif

f K f f df f K f f df x K f xφ φ φ=−

≈ ≈∑∫ ∫ (19)

where i MAXi kx f

k−= . The integration interval [-fMAX, fMAX] must be chosen in a way that the integrand

evaluated in f >fMAX is sufficiently small and does not affect the result anymore2. The choice of the number of integration points (2k+1) is a compromise between accuracy and computational complexity3.

Equation (18) can thus be rewritten as:

( )2

0( ) , ( ) , 0,..., 2

k

m i n i m m ni

x K x x x n m kφ τ φ=

= =∑ (20)

i.e., in matrix form:

⋅ = ⋅K φ τ φ (21)

with Ki,j=K(xi,xj), ϕϕϕϕij=ϕj(xi), ττττij=τiδi,j. Since K is Hermitian, the problem (22) is straightforward to solve using the Jacobi routines in [8], Chapter 11.

Since nASE(t) is a white random process, it can be shown that the noise expansion coefficients ni(t) are statistically independent Gaussian random variables with zero mean and variance N0τi. Thus (16) represents a weighted sum of nonzero-mean squared Gaussian random variables, whose moment generating function (MGF) is equal to [5]:

( )

2

10

0 0

exp1

( , )1

val

i i

ni

Mi i

z bz N

h t zz N

ττ

τ

−

=

−

+ =+

∏ (22)

with M=1 or 2 for single or double polarization representation, respectively. Ideally, the product in (22) should span from 0 to ∞ . In practice, only a finite number nval of eigenvalues give a non-negligible contribution to the MGF evaluation4.

If also additive electrical noise sources are present (like shot or thermal noise), which are statistically independent from the optical ASE noise, the overall moment generating function of the decision variable can be evaluated by multiplying (22) by the MGF of a Gaussian random process with zero mean and variance σ2(t)=(σsh(t))2+(σth)2, i.e.:

( )2 21( , ) exp2elh t z t zσ = −

(23)

Using the steepest descent approximation method [7], the probability of error can be evaluated as:

2 The criterion used in the simulation program consists of choosing fMAX such that both following conditions are satisfied: |Ho(±fMAX)|<max|Ho(f)/100 (i.e. the extreme values in the matrix K are sufficiently small) and fMAX < 0.25 k/TAMZ (i.e. one period of the AMZ transfer function is represented by at least 5 frequency points). 3 In the simulation program, a value of k equal to 50 has been adopted. It has been verified that, choosing k as high as 100, would yield almost coincident results in most practical cases. 4 It has been verified that choosing nval as high as 30 is more than sufficient in all practical cases, thus we adopted it in the simulation program.


( ) [ ]

( ) [ ]

00,

0

1,

1

exp ( , )( , ) ,

2 "( , )

exp ( , )( , ) ,

2 "( , )

th

th

e k th ks

si

e k th k

t zP s t PDF s t ds

t z

t zP s t PDF s t ds

t z

θπθ

θπθ

∞

−∞

= ≈

= ≈

∫

∫ (24)

with θ (t,z) defined as:

[ ] 2 21exp ( , ) exp ( )

2( , ) lnthzs h t z t z

t zz

σθ

− − − =

(25)

θ ” is the second derivative of θ with respect to z and z0 and z1 are respectively the positive and negative roots of the equation:

( , )'( , ) 0t zt zz

θϑ ∂= =∂

(26)

Equation (26) can be efficiently solved using the bisection method described in [8], Chapter 9.

Filter Data File Formats The syntax to be used for filter data files must match the following rules. Note that the optical filter data file should contain a double-sided spectrum, whereas the electrical filter data file should contain positive frequencies only.

• Files must be written in plain ASCII text.

• The first part of the file may contain comment lines. A line that has two '#' characters in the first two columns is mandatory and it ends the comment section.

• All comment lines must have a '#' character in the first column.

• Comment lines are allowed only after the first line and above the '##' line.

• The data must be written in lines, separated by blank space, each containing:

o the frequency as the first field [THz for optical filters, GHz for electrical filters]

o the filter’s square amplitude in linear scale

o the filter’s phase in radians

• Abscissas (either frequency or wavelength) should be given in ascending order while abscissas are not required to be strictly equispaced. Two or more points with the same frequency (or wavelength) are not allowed: if this situation happens only the first line containing the repeated value is considered and a warning message is shown. The maximum number of allowed points is 4095; if the file contains more than 4095 points, only the first 4095 points are considered and a warning message is shown.

References [1] G. Bosco and R. Gaudino, “Towards new semi-analytical techniques for BER estimation in optical

system simulation”, National Fiber Optics Engineers Conference NFOEC 2000, Denver (CO), Tuesday Sess. E1, USA, Aug. 2000.

[2] J. S. Lee and C. S. Shim, “Bit error rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain”, IEEE Journal of Lightwave Technology, vol. 12, pp. 1224-1229, 1994.


[3] A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” IEEE Journal of Lightwave Technology, vol. 23, n. 1, pp. 115-130, Jan. 2005.

[4] L. Kazovsky, S. Benedetto and A. Willner, Optical Fiber Communication Systems, Artech House, 1996.

[5] G. Bosco and P. Poggiolini, “On the joint effect of receiver impairments on direct-detection DQPSK systems,” IEEE Journal of Lightwave Technology, vol. 24, pp. 1323-1333, Mar. 2006.

[6] A. M. Mathai, S. B. Prevost, Quadratic forms in random variables, New York, Marcel Dekker, 1992, Chapter 3.

[7] C. W. Helstrom, “Distribution of the filtered output of a quadratic rectifier computed by numerical contour integration”, IEEE Trans. on Inform. Theory, vol. IT-32, no. 4, Jul. 1986, pp. 450-463.

[8] W. H. Press, S. A. Teukolsky and W. T. Vetterling, Numerical recipes in C, Cambridge University Press, 1992.

Properties


#2: Binary Signal


Parameter Values Name Type Default Range Units Modulation_format enumerated DPSK IMDD, DPSK,

DQPSK_upper_arm, DQPSK_lower_arm

Synchronization_flag enumerated No No, Yes

Fundamental_Pattern_Length integer 1 1 ≤ x ≤ 1000000

Sampling_Instant_Shift integer 0 -1000000 ≤ x ≤ 1000000

ASE_source enumerated Signal Signal, Specified ASE_Power_Spectral_Density double 0 -1e32 ≤ x ≤ 1e32 dB[mW/GHz]

Polarizer_Flag enumerated No No, Yes

Thermal_Noise_Flag enumerated No No, Power_Spectral_Density, Variance

Thermal_Noise_PSD double 0 0 ≤ x ≤ 1e32 pA^2/Hz

Thermal_Noise_Variance double 0 0 ≤ x ≤ 1e32 mA^2

Shot_Noise_Flag enumerated No No, Yes Photodiode1_Responsivity double 1 0 < x ≤ 1e32 A/W

Photodiode1_Avalanche_Gain double 1 1 ≤ x ≤ 1e32

Photodiode1_Excess_Noise_Factor double 0 0 ≤ x ≤ 1e32

Photodiode2_Responsivity double 1 0 < x ≤ 1e32 A/W

Photodiode2_Avalanche_Gain double 1 1 ≤ x ≤ 1e32

Photodiode2_Excess_Noise_Factor double 0 0 ≤ x ≤ 1e32


Optical_Filter enumerated Super Gaussian_Order2

File, Matched, None, SuperGaussian_Order1, SuperGaussian_Order2, SuperGaussian_Order3, SuperGaussian_Order4, SuperGaussian_Order5, SuperGaussian_Order6, SuperGaussian_Order7, SuperGaussian_Order8, Lorentzian, RaisedCosine

Optical_Filter_File string

Optical_Filter_Bandwidth double 30 0 < x ≤ 1e32 GHz

Optical_Filter_Rolloff double 0.5 0 ≤ x ≤ 1

Electrical_Filter enumerated Bessel _Order5

File, Matched, None, Lorentzian, Bessel_Order2, Bessel_Order3, Bessel_Order4, Bessel_Order5, Bessel_Order6

Electrical_Filter_File string Electrical_Filter_Bandwidth double 10 0 < x ≤ 1e32 GHz

MZ_frequency_offset double 0 -50 ≤ x ≤ 50 % Bit Rate

MZ_extinction_ratio double 1000 0 < x ≤ 1e32 dB

BPD_phase_imbalance double 0 -1e32 ≤ x ≤ 1e32 % Symbol Time

MZ_delay_mismatch double 0 -1e32 ≤ x ≤ 1e32 % Symbol Time

Flag_equalizer enumerated No No, Yes

Taps_delay double 100 0 < x ≤ 1e32 % Symbol Time

Equalizer_Coefficient_m7 double 0 -1e32 ≤ x ≤ 1e32







Equalizer_Coefficient_0 double 1 -1e32 ≤ x ≤ 1e32

Equalizer_Coefficient_p1 double 0 -1e32 ≤ x ≤ 1e32







BER_optimal_threshold enumerated None None, Local, NonLocal, Both

Q_optimal_threshold enumerated None None, Local, NonLocal, Both


Optimal_Threshold enumerated None None, Local, NonLocal, Both

BER_average_threshold enumerated None None, Local, NonLocal, Both

Q_average_threshold enumerated None None, Local, NonLocal, Both

Average_Threshold enumerated None None, Local, NonLocal, Both

Sampling_Instant enumerated None None, Local, NonLocal, Both



Parameter Descriptions Modulation_format Select IMDD (or duobinary), DPSK (or other binary differential

modulation format), DQPSK – upper arm, or DQPSK – lower arm Synchronization_flag If this flag is set to Yes, the received signal is aligned with the

(deterministic) input bit sequence Fundamental_Pattern_Length Fundamental pattern length of input bit sequence (actual bit sequence

at input may be longer) Sampling_Instant_Shift Difference (in number of samples) between the actual sampling

instant used for the BER evaluation and its optimum estimated value. If Synchronization_flag is set to Yes, the optimum sampling instant is evaluated through the alignment of the received signal with the known input bit sequence, otherwise it is estimated by maximizing the eye opening.

ASE_source Select source of ASE value – at the input, or from a user-specified value. Note that for ASE at the input, if the input signal has no Y polarization, then only one polarization of the ASE is used.

ASE_Power_Spectral_Density One-sided single-polarization ASE noise power spectral density in dB[mW/GHz]

Polarizer_Flag When this flag is set to Yes, the model takes into account only the X polarization for signal and ASE noise. This parameter is useful for POLMUX systems where the channels are separated with a polarizer and at the same time the ASE noise is filtered out.

Thermal_Noise_Flag This flag selects how the thermal noise is specified (either it is neglected, defined through its power spectral density, or defined through its variance after the electrical filter)

Thermal_Noise_PSD “Input-referred” thermal noise bilateral power spectral density in pA2/Hz (as usually found in datasheets). The variance of noise on the decision variable is then equal to:

22 0 ( )2th L

N H f dfσ+∞

−∞= ∫

where HL(f) is the overall transfer function of electrical filtering. Thermal_Noise_Variance Variance of thermal noise on the decision variable

Shot_Noise_Flag Select whether or not to include shot noise

Photodiode1_Responsivity Responsivity of either the only photodiode for IMDD or the upper arm photodiode for differential receivers

Photodiode1_Avalanche_Gain Avalanche gain of APD (linear). If a PIN is used, it must be set to 1.

Photodiode1_Excess_Noise_Factor Excess noise factor of APD


Photodiode2_Responsivity Responsivity of the lower arm photodiode

Photodiode2_Avalanche_Gain Avalanche gain of APD (linear) in the lower arm. If a PIN is used, it must be set to 1

Photodiode2_Excess_Noise_Factor Excess noise factor of APD Optical_Filter Specify optical filter type. The options are File, Matched (rectangular

in time), None, Super-Gaussian of order n, Lorentzian, and Raised Cosine

Optical_Filter_File Optical filter data file

Optical_Filter_Bandwidth For file-based filter data, this parameter is equal to the scaling factor, i.e. the real number that specifies the expansion/stretch factor in the frequency scale of the transfer function stored in the data file. This parameter should be set to 1 when scaling is not required. For Matched filters, this parameter is the inverse of the time duration of the rectangular pulse (in GHz). For all other filters, this parameter is the -3dB optical filter bandwidth in GHz.

Optical_Filter_Rolloff Roll-off factor of the raised-cosine optical filter.

Electrical_Filter Specifies electrical filter type. The options are File, Matched (rectangular in time), None, Lorentzian, and Bessel of order n.

Electrical_Filter_File Electrical filter data file Electrical_Filter_Bandwidth For file-based filter data, this parameter is equal to the scaling factor,

i.e. the real number that specifies the expansion/stretch factor in the frequency scale of the transfer function stored in the data file. This parameter should be set to 1 when scaling is not required. For Matched filters, this parameter is the inverse of the time duration of the rectangular pulse (in GHz). For all other filters, this parameter is the -3dB optical filter bandwidth in GHz.

MZ_frequency_offset Detuning ∆f between the laser frequency and the Asymmetric Mach-Zehnder central frequency (expressed in % of the bit rate)

MZ_extinction_ratio Extinction ratio ε in dB of AMZ BPD_phase_imbalance Propagation delay difference ∆τ between the two arms of the BPD

(expressed in %T, with T=1/RS, where RS is the symbol rate) MZ_delay_mismatch AMZ delay error, defined as:

100 100AMZT T TT T

δ −⋅ = ⋅

where TAMZ is the delay introduced by the AMZ and T is the inverse of the symbol rate

Flag_equalizer If it is set to Yes, an FIR filter equalizer is inserted after the post-detection filter. The transfer function of the filter is equal to:

2( ) eqN

j fnTN

n NH f c e π−

=−

= ∑

Taps_delay Time delay Teq between the equalizer taps (expressed in percentage of symbol time T)

Equalizer_Coefficient_m7 FIR filter tap coefficient c-7



Equalizer_Coefficient_m4 FIR filter tap coefficient c-4 Equalizer_Coefficient_m3 FIR filter tap coefficient c-3



Equalizer_Coefficient_0 FIR filter tap coefficient c0


Equalizer_Coefficient_p1 FIR filter tap coefficient c1


Equalizer_Coefficient_p3 FIR filter tap coefficient c3 Equalizer_Coefficient_p4 FIR filter tap coefficient c4



Equalizer_Coefficient_p7 FIR filter tap coefficient c7 BER_optimal_threshold Toggle reporting of BER at optimal threshold

Q_optimal_threshold Toggle reporting of Q at optimal threshold

Optimal_Threshold Toggle reporting of optimal threshold

BER_average_threshold Toggle reporting of BER at average threshold Q_average_threshold Toggle reporting of Q at average threshold

Average_Threshold Toggle reporting of average threshold

Sampling_Instant Toggle reporting of sampling instant

SaveOrDisplayPlots Save plots to file or display automatically StatisticalPlotMode Select statistic for display in plots


Monte Carlo DPSK BER Estimator

This model uses a state-of-the-art Monte-Carlo (MC) technique to compute the Bit Error Rate (BER) and Q factor for a received DPSK signal. The algorithm for calculating the noise statistics of a received DPSK signal is based on work done originally at the University of Central Florida (UCF). It is applicable to both RZ- and NRZ-DPSK systems with bit rates as high as 10 Gbps, and which are limited by both linear and nonlinear noise (as would be the case in a practical transmission link) and which exhibit reasonably small transceiver imperfections (or none at all). (Note that since the algorithm treats SPM as the dominant source of nonlinearity, results may not be as accurate for 40-Gbps systems in which FWM plays a significant role.) The module is fully integrated with the generalized scanning facilities described in Chapter 7 of the OptSim User Guide and so can produce important scan plots such as power penalty diagrams for the BER or Q.

MC DPSK BER Estimation Technique The Monte-Carlo technique we will describe has nothing in common with the “Monte-Carlo” error counting method. Rather, MC refers to the representation in which noise in the electrical signal is stored. In Chapter 8 of the User Guide, we explain that electrical noise may be represented in two complementary fashions. We summarize the MC picture here, but the reader is encouraged to read the full discussion in Chapter 8.

In the MC picture, there is no real distinction between the signal and noise – noise is included directly by adding random variates to the signal samples Vk. Viewed with a SignalAnalyzer or SpectrumAnalyzer, the resultant signal genuinely appears “noisy”, and if the simulation is repeated, the noisy signal appears different in successive runs. (More accurately, the repeatability of the noise from run to run depends on the setting of the seed value for the random number generator in the model generating the noise. Most OptSim models that generate noise allow the user to specify whether the seed changes or remains the same from run to run.) As discussed in Chapter 8 of the User Guide, the MC approach is very general as it allows for noise of any distribution or correlation function.

Given that noise is directly included as a stochastic contribution to the signal vector, the MC BER Estimation algorithm must construct histograms for the upper and lower parts of the received electrical eye, and so explicitly measure the statistics of the probability distribution functions for marks and spaces. These distributions govern the spread of the upper and lower bars of the eye, respectively. Unlike the BER Tester model (described elsewhere in this manual), which assumes Gaussian probability distribution functions in determining the noise statistics, the MC DPSK BER Estimator uses an algorithm developed at UCF for determining a more accurate probability distribution function that is applicable to DPSK systems. This alternate approach is necessary since the assumption of Gaussian noise statistics in DPSK systems is not accurate [1]. Figure 1 illustrates the accurate calculation of BER in DPSK systems using this algorithm in comparison with an error counting method.

Once the probability functions have been calculated for a given electrical signal, we can find the probability of an error given a specified threshold decision level by integrating that part of each distribution function falling on the “wrong” side of the decision level. We denote as 0 1P → the probability of mistakenly measuring a space as a mark, and as 1 0P→ the probability of measuring a mark as a space.

The total bit error rate is then just

0 10 1 1 0

0 1 0 1BER

n nP Pn n n n→ →= +

+ + (1)

where n0 and n1 are the numbers of spaces and marks, respectively, in the signal.


1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

-1 0 1 2 3 4 5Power (dBm)

BER

Error CountingUCF MC DPSK Algorithm

Figure 1: Comparison of BERs in an RZ-DPSK system of 72 fiber spans, calculated using the UCF algorithm and error counting [2].

Finally, we define the “Q-factor” via the relationship

1BER erfc2 2

Q =

(2)

Strictly speaking, this equation is only valid for systems with Gaussian noise statistics, but as Q is a common metric in optical system analysis, we adopt Eq. (2) as a definition for relating the BER to Q, rather than as a strict theoretical relationship. Note that it is common in the communication community to refer to Q in dB units using the following relationship

QdB = 20 log10 (Qlinear) (3)

Then, for example, a BER of 10-13 corresponds, according to Eqs. (2) and (3), to a Q-factor equal to 7.34 in linear units and 17.32 in dB units. Users can choose linear or dB units for Q by setting the parameter unitsForQ to linear or Q^2(dB).

Model Inputs The BER Estimator has three input nodes, two for electrical signals and one for a binary signal. Figure 2 depicts OptSim topologies illustrating the signal connections to each of these input nodes in the context of typical DPSK receivers. The BER Estimator requires that the DPSK receiver consist of an optical filter, optical demodulator, and a balanced receiver. In Fig. 2(a), the optical filter is the component labeled OptFilt, the optical demodulator is the delay interferometer MZDI, and the balanced receiver consists of two single-ended receivers and an expression block to calculate the difference between their outputs. In addition, the model requires that an additional single-ended receiver be connected to the signal node directly before the optical demodulator. This receiver is necessary for accurately calculating the contribution of linear noise to the DPSK BER, and must be included in your simulations using the same parameters as the individual receivers used to form the balanced receiver. Figure 2(b) depicts an alternate configuration in which the balanced receiver is broken up into its constituent components of photodetector, amplifier, and filter, while the additional single-ended receiver is similarly subdivided.

As shown in Fig. 2, the first input node of the BER Estimator should be connected to the electrical signal output from the balanced receiver. The second node should be connected to the output of the additional single-ended receiver. Finally, a reference binary signal should be connected to the model’s binary-signal input node, as shown in Fig. 2. Typically, one connects the PRBS Generator at the start of the link to the BER Estimator at the end. In WDM simulations where there may be numerous binary signals generated, it is important to match the binary signals to the correct BER Estimator.


In order to properly calculate a BER, the model requires some additional information about the input signals. First, opticalFilterBW should be set equal to the bandwidth in Hz of the optical filter used in the DPSK receiver. This is necessary since this filtering information is absent from the receiver’s electrical signal output. Second, ASE_polarization is used to indicate to the model what was the polarization of ASE at the optical filter input. Single corresponds to single-polarization ASE, and Dual corresponds to dual-polarization ASE. Finally, the parameters preBits and postBits determine the number of bits at the start and end of the signal that the BER Estimator ignores in calculating the BER. These should normally be set to the same values of the corresponding parameters in the PRBS Generator. The purpose of these ignored slots is to deal with any phase noise introduced through the periodic boundary conditions of the FFT operation. The signature of this is unusually noisy or oscillatory signal patterns at the start or end of the time window. The MC method is particularly sensitive to this kind of noise, and on occasion it may be necessary to ignore 5 or more bits at either end.

(a)

(b)

Reference Bitstream

DPSK Receiver

Receiver for Linear Noise Estimation

DPSK Input

DPSK Receiver

DPSKInput

Reference Bitstream

Receiver for Linear Noise Estimation

Figure 2: OptSim topologies depicting proper usage of the MC DPSK BER Estimator. (a) Topology based on the Compound Optical Receiver model. (b) Topology based on individual receiver subcomponents.


Scanning Facilities In most problems, one wishes to know the BER as a function of one or more system parameters, so the ability to scan results is important. The BER Estimator model is fully integrated into OptSim’s generalized scanning facility described in Chapter 7, Section 7.1.2 of the User Guide. Using these facilities, the BER and Q may be automatically plotted as functions of either the inner or outer scan variables. By attaching the BER Estimator to an XY-Plotter, plots may be produced as a function of any other measured parameter. This is particularly useful for creating power-penalty plots. The scanned properties are also reported in a text file with the extension “table.txt”. Instructions for using these features are included in Chapter 7 of the User Guide and are not repeated here. The facilities are available for all properties whose parameters take the values None, Local, NonLocal and Both. Specifically, these parameters are PlotBER and PlotQ.

References [1] P. J. Winzer, S. Chandrasekhar, and H. Kim, “Impact of filtering on RZ-DPSK reception,” IEEE Photonics Technology Letters, vol. 15, no. 6, pp. 840-842, June 2003.

[2] Y. Han (private communication), 2007.

Properties

Inputs #1: Electrical signal (detected DPSK signal)

#2: Electrical signal (pre-demodulation DPSK signal)

#3: Binary signal

Outputs None or #1: MeasurandSig

Parameter Values Name Type Default Range Units FilenameRoot string opticalFilterBW double 10e9 ( 0, 1e32 ] Hz

ASE_polarization enumerated Dual Single, Dual

unitsForQ enumerated linear linear, Q^2(dB)

preBits integer 2 [ 0, 1e8 ] none postBits integer 3 [ 0, 1e8 ] none

StatisticalPlotMode enumerated All Average, Minimum, Maximum, Std.Dev., Latest, StatRun, All


PlotBER enumerated Local None, Local, NonLocal, Both

PlotQ enumerated None None, Local, NonLocal, Both


Parameter Descriptions FilenameRoot Filename prefix for data files. opticalFilterBW Bandwidth of optical filter used in DPSK receiver. ASE_polarization Specify whether ASE is single- or dual-polarization. unitsForQ Toggle linear or Q^2(dB) units for the Q plot. preBits Number of initial bits to ignore. postBits Number of final bits to ignore. StatisticalPlotMode Toggle statistical quantity to plot in scanning simulations. SaveOrDisplayPlots Toggle automatic or manual display of plots, or disable plots. PlotBER Toggle output of BER plot and data. PlotQ Toggle output of Q plot and data.


XY-Plotter

This is a utility model that combines the output from scanning-aware blocks, to produce X-Y plots. Its use is demonstrated in the section of the manual discussing scanning techniques, which should be read in conjunction with this passage.

Several blocks – currently the Optical Monitor, Optical Eye Analysis block and the BERTester – support special facilities for automatically generating plots during scans. In these plots, the independent variable (x-axis quantity) is always the inner scan variable, and therefore must be an input parameter. However, it is often necessary to use measured quantities as the x-axis variable. For example, a bit error rate could be plotted as a function of output laser power or received optical power. In this case, the optical power would be measured by an Optical Monitor block and the bit error rate by a BERTester. The XY-Plotter combines these quantities into a single plot.

Figure 1 and Figure 2 illustrate the use of the model. Output ports have been added to the Optical Monitor and BERTester. The OpticalPower and PlotBER parameters of the Optical Monitor and BERTester respectively have been set to NonLocal or Both. This causes these quantities to be exported to the XY-Plotter. When a scan is run, the output appears as in Figure 2. On the left, is the BER plot produced internally by the BERTester. The x-axis show the drive_current of the direct modulated laser which was the inner scan variable. The right-hand plot, produced by the XY-Plotter, replaces the drive_current with the optical power obtained from the Optical Monitor to produce a power-penalty plot.

Figure 1: Use of XY-Plotter for combining quantities.

Figure 2: BER plots generated locally by BERTester (left), and by combining with Optical Monitor data in the XY-Plotter.


Combining Quantities Each of the scanning-aware blocks can output a number of quantities simultaneously. For example, the Optical Monitor can output optical power, noise power, center frequency and several other quantities. All these quantities are exported through the single output port of the Optical Monitor, and the same is true of the various quantities calculated by the Optical Eye Analysis block and the BERTester. As a result, it is possible for the x- and y-input ports of the XY-Plotter to accept several quantities at once. The most common case is to have a single x-input quantity (say optical power) and multiple y-inputs. In this case, the XY-Plotter produces one plot for each y quantity. If there are also several x-inputs, the parameter CombineMode determines how they are combined with the y-inputs. With CombineMode=OneToOne, the XY-Plotter expects the same number of x- and y-inputs and combines them in sequence. That is, the nth x-quantity is plotted against the nth y-quantity. With CombineMode=AllCombs, the number of inputs on each port need not be equal, and every x-quantity is plotted against every y-quantity.

Penalty Plots The XY-Plotter can also generate penalty plots such as the power penalty for a given BER as a system parameter is changed. To activate this feature, a comma-separated series of values is specified for the parameter PenaltyLevels. For example, continuing the earlier example, if we set PenaltyLevels=1e-3, 1e-5, 1e-7, the model generates plot shown in Figure 3. This plot is constructed by finding the intersections of the curves in the above BER scan at each of the penalty levels, and can be considered as an axis rotation of the standard scan. As such, it contains no additional information, but arranges the axes in such a way that power penalties can be easily extracted as vertical distances between the curves.

Figure 3: Penalty plot for the BER scan in Figure 2.


Properties

Inputs #1-2: MeasurandSig

Outputs None

Parameter Values Name Type Default Range Units CombineMode enumerated OneToOne OneToOne, AllCombs

FilenameRoot string Plot enumerated Save No, Save, Display

PenaltyLevels string

Parameter Descriptions CombineMode Control combinations of multiple x and y inputs FilenameRoot Filename root for output files Plot Suppress, save or automatically display plots PenaltyLevels Set levels for interpolating penalty plot.


Signal Analyzer

The Signal Analyzer block is used to display the signal waveform(s) of a signal at the node connected to its input port(s). It has various settings which allow the user to customize the signal waveform display. The user may set the timing offset of the start of the plot, set the title of the plot and the filename it is to be written to, and determine whether to display the electrical signal noise or optical ASE noise on the plotted signal value or not.

In addition, the user may choose among several signal representation options: the signal may be displayed as its magnitude only, combined magnitude and phase plots with separate y axes, combined optical signal magnitude and frequency chirp plots with separate y axes, combined electrical signal magnitude and noise term plots with separate y axes, or separate real and imaginary components. By default, an optical signal is displayed in magnitude representation and an electrical signal is displayed in real representation. The optical noise is not displayed by the signal plot routine, so the magnitude and noise plot mode is treated the same as the magnitude only plot mode for optical signals

When displaying optical signals, the polarization state to be displayed should be chosen by the user. When the optical signal plot representation is set to magnitude and phase, magnitude and chirp, real, or real and imaginary plots, the combined x and y polarization mode is not supported. If these combinations are attempted, the x polarization state alone will be plotted. For magnitude plots, the combined x and y polarization states may be displayed to view the total power in the signal as a function of time.

When displaying optical signal magnitude with frequency chirp, the powerThreshold parameter may be used to limit the displayed chirp to only the region of the signal which is greater than the threshold value.

The block may be set to save the plot to a file for later retrieval (either from the graph view function, or by double clicking on the plot icon), save and display the plot as soon as it is generated, or not to plot. These options may be overridden globally for all plot icons in the simulation start dialogs.

Properties

Inputs #1-N: Electrical, Optical, or Binary signal (all inputs must be of same type)

Outputs None

Parameter Values

Name Type Default Range Units FilenameRoot string

Representation enumerated Default Magnitude, Mag/Phase, Mag/Chirp, Mag/Noise, Real, Real/Imag, Default

Polarization enumerated comb_x_and_y x_only, y_only, sep_x_and_y, comb_x_and_y, all


ShowNoise enumerated TRUE FALSE, TRUE

Toffset double 0 [ -1e15, 1e15 ] s powerThreshold double 0 [ 0, 1e32 ] W

PowerUnits enumerated Watt Watt, dBm


Parameter Descriptions Toffset Time offset of the start of the plot showNoise Whether to show the noise in the plot or not Plot Whether to create plot data, display the plot, or skip it Polarization Which polarization states should be included in plot FilenameRoot Root of the filename used to store the plot data Representation Whether to display magnitude, magnitude and phase, magnitude and noise separately,

real portion only, real and imaginary separately, or use default according to input signal type

PowerUnits Units for the magnitude power of optical signal: Watt or dBm powerThreshold Optical signal power threshold above which the phase and frequency chirp may be

plotted


Constellation Diagram Analyzer

The Constellation Diagram block is used to display the constellation diagram of an electrical or optical signal at the node connected to its input port. It displays the imaginary part of optical or electrical field as a function of real part of optical or electrical field - i.e. in-phase and quadrature-phase signals are given on x- and y-axes, correspondingly.

The Constellation Diagram Analyzer only accepts single-channel optical inputs contained within a single optical signal. In the case when the input optical signal has both x- and y-polarizations then a separate cosntellation diagram will be plotted for each polarization. In addition constellation diagram has settings which allow the user to set the title of the plot and the filename it is to be written to, and determine whether to display the optical signal ASE or electrical signal noise on the plotted diagram or not. The block may be set to save the plot to a file for later retrieval (either from the graph view function, or by double clicking on the plot icon), save and display the plot as soon as it is generated, or not to plot

Properties

Inputs #1: Electrical or Optical signal

Outputs None

Parameter Values Name Type Default Range Unit FilenameRoot string

ShowNoise enumerated No No, Yes

seed integer 1 [ -1e8, 1] none


Parameter Descriptions FilenameRoot Root of the filename used to store the plot data ShowNoise Whether to show the optical signal ASE or electrical signal noise in the plot

or not seed Random number generator seed for stochastically added noise. Follows

OptSim seed convention. Plot Whether to create plot data, display the plot, or skip it


Eye Diagram Analyzer

The Eye Diagram block is used to display the eye diagram of a signal at the node connected to its input port(s). By default, it displays the magnitude of optical signals and the real value of electrical signals. It has settings which allow the user to shift the eye diagram’s center in the plot window, set the title of the plot and the filename it is to be written to, and determine whether to display the optical signal ASE or electrical signal noise on the plotted eye diagram or not.


Properties

Inputs #1-N: Electrical or Optical signal

Outputs None


Type enumerated Default Default, Line, Contour, ContourBW

Representation enumerated Default Magnitude, Real, Default

Polarization enumerated comb_x_and_y x_only, y_only, comb_x_and_y

ShowNoise enumerated TRUE FALSE, TRUE

Toffset double 0 [ -1e15, 1e15 ] s PowerUnits enumerated Watt Watt, dBm


Parameter Descriptions Toffset Time offset of the start of the plot ShowNoise Whether to show the optical signal ASE or electrical signal noise in the plot or not Plot Whether to create plot data, display the plot, or skip it Polarization Which polarization states to include in plot Representation Whether to display the magnitude of an electrical signal or the real portion only; Optical

signals are always displayed as magnitude, while Default sets electrical signals to display real only


FilenameRoot Root of the filename used to store the plot data PowerUnits Units of optical signal magnitude power: Watt or dBm


Signal Spectrum Analyzer

The Signal Spectrum block is used to display the spectrum of a signal at the node connected to its input port(s). In addition to plotting the magnitude of the spectrum (in forms of power, absolute value of field amplitude, real and imaginary parts of the amplitude), the phase and group delays may also be plotted. A parameter may be set to only plot the phase and group delays over regions where the power exceeds the specified power threshold. It has options to plot the baseband signal spectrum, or in the case of an optical signal, the optical spectrum in either wavelength domain or frequency domain. By default, it plots the wavelength spectrum of an optical signal and the baseband spectrum of an electrical signal. For an optical signal, the ASE noise may be either plotted separately from the signal, with the signal in the same plot (deterministically), both of the preceding, stochastically combined with the signal, or not at all. In case of stochastic noise the parameter seed will define random number generation following OptSim seed convention. However, there are some restrictions on noise application to a signal spectrum. First, noise cannot be added if the spectrum is plotted in baseband domain. Second, if the spectrum analyzer block has more than one input, then noise can be either added stochastically or ASE noise bins for each input will be plotted separately, but not in combined mode. The power spectrum plot may optionally be normalized when specified in dBm units. The dc offset may optionally be ignored or included in the spectrum plot. To save on disk space and time to save data files, the user may select a reduced resolution, or may optionally save the plot data in the full simulated resolution. The user may also set the title of the plot and the filename it is to be written to.


When viewing optical signal spectra (in Power spectrum representation only) in frequency or wavelength format, the user may view the full optical spectrum data contained in OptSim’s signal representation, or model the effect of a real-world optical spectrum analyzer (OSA) on the optical signal spectrum before it is displayed. Simply put, an optical spectrum analyzer in a laboratory uses a window over a portion of the spectrum to measure the optical power contained within that window. This window can be modeled as an optical filter In addition, it has limitations on how small a detected power can be. This is specified as the sensitivity of the OSA. By using the filtering and sensitivity limiting options of OptSim’s optical spectrum plot block, simulation results can be displayed in a form which more closely resembles the experimental results obtained by real-world optical spectrum analyzers in the laboratory. To make use of these capabilities, specify the filter parameters which include the filter type and its parameters in either wavelength or frequency units. The special filterThreshold parameter is used to improve the computational efficiency of the block. To maximize the computational efficiency while not significantly impacting the plot results, the threshold should be set to a value which is small enough that it makes no significant difference if the filter function is treated as zero at that point in the filter response.

Properties

Inputs #1-N: Electrical or Optical signal

Outputs None

Parameter Values


Name Type Default Range Units FilenameRoot string

Spec_Representation enumerated Power Amp/Phase, Pow/Phase, Amp/Delay, Pow/Delay, Real/Imag, Power

Domain enumerated Default Baseband, Frequency, Wavelength, Default

Plot enumerated Save No, Save, Display Polarization enumerated comb_x_and_y x_only, y_only,

comb_x_and_y

PowerThreshold double 0 [ 0, 1e32 ] W

PowerUnits enumerated dBm Watt, dBm

IncludeDCoffset enumerated Include Ignore, Include

Normalize enumerated No No, Yes Resolution enumerated Limited Limited, Full

ResolutionLimit integer 2049 [ 2, 1000000 ] points

filterType enumerated None None, Gaussian, Trapezoidal

filterSpecMode enumerated Frequency Frequency, Wavelength

filterBW double 10e9 [ 1e-32, 1e32 ] Hz or meters

filterBW0dB double 0.1e9 [ 0, 1e32 ] Hz or meters filterThreshold double -100 [ -1000, 0 ] dB


OSAsensitivityMode enumerated Unlimited Unlimited, Limited

OSAsensitivity double -65 [ -1000, 0 ] dBm Spec_ShowNoise enumerated Combined No, Separately,

Combined, Both, Stochastic

seed integer 1 [ -1e8, 1 ] none

Parameter Descriptions FilenameRoot Root of the filename used to store the plot data Spec_Representation Which combinations of signal spectrum amplitude, power, phase, group delay, or

real/imaginary to plot Domain The type of spectrum plot Polarization Which polarization states to include in optical spectrum plot Normalize Whether to normalize the plot to dB or not Resolution Whether to use the Full simulated resolution in the plot, or a reduced resolution for a

smaller plot file ResolutionLimit The maximum number of data points to be included in plot data file if resolution is

reduced IncludeDCoffset Whether to include or ignore the DC offset of the signal when computing the


spectrum PowerThreshold Optical spectrum power threshold above which to plot the phase or group delay PowerUnits Units for the magnitude power of optical signal: Watt or dBm Plot Whether to create plot data, display the plot, or skip it Spec_ShowNoise Whether to plot the ASE noise of an optical signal in the same plot as the signal itself,

in a separate plot, both, stochastically combined, or neither. Valid only in wavelength or frequency domain

seed Random number generator seed for stochastically added noise. Follows OptSim seed convention.

filterSpecMode Whether OSA optical filter specs are specified in frequency units of Hz or wavelength units of m

filterType Whether a Gaussian, trapezoidal, or no OSA optical filter is used. Can be applied only when spectrum is in Power representation

filterBW Bandwidth of OSA optical filter in units determined by filterSpecMode filterBW0dB For trapezoidal OSA optical filter, the bandwidth of the flat top of the filter response

with 0dB loss filterOrder For Gaussian OSA optical filter, the order of the filter function filterThreshold The filter loss at which all portions of filter response with a lower loss are set to

infinite loss OSAsensitivityMode Whether to display full OSA optical spectrum data or limit it to power which is above

the specified sensitivity. Can be applied when signal is in Power representation, in dBm units, and either wavelength or frequency domain.

OSAsensitivity The minimum optical power which the OSA registers; behaves as an optical noise floor


Optical Frequency/Wavelength Chirp Analyzer

The Plot Chirp block is used to display the frequency chirp of an optical signal at the node connected to its input port(s). It may be plotted in units of frequency (Hz) or wavelength (m). It has various settings which allow the user to customize the signal waveform display. The user may set the timing offset of the start of the plot, set the title of the plot and the filename it is to be written to, and set which of the two polarization states should have their chirp displayed.


The frequency chirp is computed directly from the input optical signal using the following relation:

Frequency Chirp (Hz)dtdφ

π21−=

where φ is the phase in units of radians.

Properties


Outputs None


Representation enumerated Frequency Frequency, Wavelength

Polarization enumerated sep_x_and_y x_only, y_only, sep_x_and_y

Toffset double 0 [ -1e15, 1e15 ] s

powerThreshold double 0 [ 0, 1e32 ] W


Parameter Descriptions Toffset Time offset of the start of the plot Plot Whether to crate plot data, display the plot, or skip it FilenameRoot Root of the filename used to store the plot data Polarization Determines which polarization states should be shown


Representation Determines whether to plot in units of Frequency (Hz) or Wavelength (m). PowerThreshold Optical signal power level above which the chirp should be plotted.


Optical Autocorrelator Analyzer

This model implements an optical autocorrelator for characterization of ultra-short pulses.

For pulses shorter than approximately 5-10 ps, real world photodetectors and other receivers are not able to resolve the pulse profile. For ultra-short pulse work, autocorrelators may be used to characterize pulse shapes and phase profiles. Of course in simulations, the PlotSignal model presents all time information for pulses of any length. However, for comparison with experiments it may be useful to represent the pulse as an autocorrelation.

For each field component ( ),x yE t , the model calculates the complex autocorrelation function

( ) ( ) ( )*, , ,dx y x y x yA t E t E tτ τ

∞

−∞

= +∫

The plot is normalized to a maximum of 1.0 at 0τ = .

If there is an ASE noise spectrum associated with the signal, the noise may be transferred to the stochastic signal representation with the ShowNoise parameter.

Properties


Outputs None


Representation enumerated Power/Phase Power, Power/Phase, Power/WrappedPhase, Real/Imag

ShowNoise enumerated No No, Yes PowerThreshold double 1e-10 [ 0, 1e32 ] W


Parameter Descriptions Representation Display mode for plot PowerThreshold Minimum power to display phase


ShowNoise Add ASE noise to signal before calculation


Multiplot

This model combines the features of five of the previous analyzer blocks: the Signal Analyzer, EyeDiagram Analyzer, Spectrum Analyzer, Chirp Analyzer and the Optical Autocorrelator. Signals received by this block may be displayed using any of these five analyzers and redisplayed with different settings at any time. The Multiplot is thus far more flexible than the other plot blocks.

Usage In setting up a topology, the Multiplot is used rather like any of the five standard analyzer blocks, in that its input node is connected to any desired output source. Figure 1, for example shows the standard 10 Gigabit per second example from Tutorial 1, with the various analyzers replaced by Multiplot blocks. Note that since a Multiplot serves as all of the other blocks and since its settings can be changed and plots regenerated without repeating the simulation, there is never any need to attach more than Multiplot to a particular block. This leads to simpler topologies as well as more flexible post-processing.

Figure 1: Tutorial 1 topology using Multiplot blocks.

After the simulation is run, the output of the Multiplot is not activated by double clicking on the block as for a standard analyzer block, but by opening the parameter dialog using a right-click. The parameter dialog for a Multiplot (see Fig. 2) is a little different to the standard appearance for all other blocks. Note the five buttons along the top. These are used to generate any of the corresponding plots at any time simply by clicking the desired icon. (The icons match those for the stanard plot blocks). The parameters of the dialog are divided into six tabs. The first General tab contains properties that apply to all or several of the different output modes. The remaining five tabs contain parameters that only relate to a particular output mode, say the Spectrum plot. The meanings of all these parameters are identical to the corresponding parameters of the same names which are documented under the topics for each individual Analyzer block and we will not repeat them here.


Figure 2: Multiplot parameter dialog showing plot icons for an electrical input.

The very first parameter datafile on the General tab is rather special. It is empty until a simulation is run, after which it contains a list of available data files. These binary-format files contain a complete representation of the incoming signal. The type of signal is indicated by the extension: .lsb (binary signal), .lse (electrical signal) or .lso (optical signal). Since the data files contain a complete description of the signal, we can use them to generate any form of plot with any settings we like as follows.

Display Procedure • The first step is to select a file from the datafile list. When this is done, OptSim will enable the appropriate

set of plot icons along the top row for that signal type. So if the selected datafile holds a binary signal, only the Signal Analyzer icon is enabled and the remainder are greyed out. For an electrical signal, the EyeDiagram and Spectrum icons are also enabled, while for an optical signal, all the icons are enabled. (see Figs. 2 and 3).

• To display a plot simply hit the icon of your choice.

• On viewing the plot, you may decide that it would be more useful or more attractive with different settings. You can adjust any of the properties in the parameter dialog you like and then regenerate the plot by hitting the icon button again.


• Each time an icon button is hit to generate a plot, a WinPLOT control file and one or more ascii data files are created to represent the plot just as with the standard Analyzer blocks. By changing the FilenameRoot property, you can create as many plots as you wish without having the WinPLOT files overwritten.

Figure 3: Multiplot parameter dialog showing icons for an optical signal.

Properties


Outputs None



datafile enumerated NOTHING NOTHING

style enumerated Signal Signal, Spectrum, EyeDiagram, Chirp, Autocorrelator

FilenameRoot string

Domain enumerated Default Default, Baseband, Frequency, Wavelength

ShowNoise enumerated Yes No, Yes

Polarization enumerated sep_x_and_y x_only, y_only, sep_x_and_y, comb_x_and_y

PowerUnits enumerated Watt Watt, dBm

PowerThreshold double 1e-10 [ 0, 1e32 ] W seed integer 1 [ -1e8, 1 ] none

Toffset double 0 [ -1e15, 1e15 ] s


Sig_Representation enumerated Default Magnitude, Mag/Phase, Mag/Chirp, Mag/Noise, Real, Real/Imag, Default

Eye_Mode enumerated Default Default, Line, Contour, ContourBW

Eye_Representation enumerated Default Magnitude, Real, Default

Spec_Representation

enumerated Power Amp/Phase, Pow/Phase, Amp/Delay, Pow/Delay, Real/Imag, Power

Spec_ShowNoise enumerated Combined No, Separately, Combined, Both, Stochastic

IncludeDCoffset enumerated Yes No, Yes

Normalize enumerated No No, Yes

Resolution enumerated Limited Limited, Full ResolutionLimit integer 2049 [ 2, 1000000 ] points

filterType enumerated None None, Gaussian, Trapezoidal

filterSpecMode enumerated Frequency Frequency, Wavelength

filterBW double 10e9 [ 1e-32, 1e32 ] Hz or meters

filterBW0dB double 0.1e9 [ 0, 1e32 ] Hz or meters

filterThreshold double -100 [ -1000, 0 ] dB filterOrder integer 1 [ 1, 128 ] none

OSAsensitivityMode enumerated Unlimited Unlimited, Limited

OSAsensitivity double -65 [ -1000, 0 ] dBm

AC_Representation enumerated Power/Phase Power, Power/Phase, Power/WrappedPhas


e, Real/Imag


General Parameters datafile Stored datasets available for display FilenameRoot Filename roots for generated plots Domain Frequency representation for spectrum abscissa or chirp plot ordinate ShowNoise Include noise in display for Signal, EyeDiagram, and Autocorrelator plots Polarization Which polarization states should be included in plot PowerUnits Units for the magnitude power of optical signal: Watt or dBm PowerThreshold Optical signal power threshold above which the phase and frequency chirp may be

plotted Seed Random number generator seed for stochastically added noise. Follows OptSim seed

convention Toffset Time offset of the start of the plot Plot Whether to create plot data, display the plot, or skip it

Signal Analyzer Parameters Sig_Representation Format of the displayed signal: Magnitude/Phase, Magnitude/Chirp, Real/Imag etc

EyeDiagram Analyzer Parameters Eye_Mode Appearance of eye diagram: Line, Contour, Black/White Contour Eye_Representation Plot eye using magnitude or real part of the signal

Spectrum Analyzer Parameters Spec_Representation Which combinations of signal spectrum amplitude, power, phase, group delay, or

real/imaginary to plot Spec_ShowNoise Whether to plot the ASE noise of an optical signal in the same plot as the signal itself,

in a separate plot, both, stochastically combined, or neither. Valid only in wavelength or frequency domain

IncludeDCoffset Whether to include or ignore the DC offset of the signal when computing the spectrum

Normalize Whether to normalize the plot to dB or not Resolution Whether to use the Full simulated resolution in the plot, or a reduced resolution for a

smaller plot file ResolutionLimit The maximum number of data points to be included in plot data file if resolution is

reduced


filterType Whether a Gaussian, trapezoidal, or no OSA optical filter is used. Can be applied only when spectrum is in Power representation

filterSpecMode Whether OSA optical filter specs are specified in frequency units of Hz or wavelength units of m

filterBW Bandwidth of OSA optical filter in units determined by filterSpecMode filterBW0dB For trapezoidal OSA optical filter, the bandwidth of the flat top of the filter response

with 0dB loss filterOrder For Gaussian OSA optical filter, the order of the filter function FilterThreshold The filter loss at which all portions of filter response with a lower loss are set to

infinite loss OSAsensitivityMode Whether to display full OSA optical spectrum data or limit it to power which is above

the specified sensitivity. Can be applied when signal is in Power representation, in dBm units, and either wavelength or frequency domain.

OSAsensitivity The minimum optical power which the OSA registers; behaves as an optical noise floor

Autocorrelator Analyzer Parameters AC_Representation Format for plot: Power/Phase, Power/WrappedPhase, Real/Imag


Transfer Function Analysis Tool

This model provides the facility to measure a spectrum change or a frequency response between any two points in OptSim schematic. It can be applied to both optical and electrical signals.

The Tranfer Function Analysis Tool is an auxillary model that require use of other models to profuce the results. It does not have any input ports and uses outputs of other model for its calculations. If, for example, we need to measure frequency response or gain shape change of some device under test consisting of one or more components, first we have to add spectrum analyzers before and after the device under test. Figure 1 demonstrates the example of Transfer Function block usage. Here the device unbder test (DUT) consists of fiber and optical filter and we want to find out the frequency response of that device. In addition to Transfer Function block we have two Spectrum Analyzer blocks connected before and after the DUT. The parameter setting for both Spectrum Analyzers should be identical, specifically both have to use setting PowerUnits: dBm and Spec_Representation: Power. Then to calculate the frequency response the Transfer Funtion block will take a difference between two spectrums.

Figure 1: Topology demonstrating use of Transfer Function Analysis Tool.

In the properties dialog window for Tranfer Function Analysis Tool user has to specify names of input and output spectrum in string parameters InputSpecrtrum and OutputSpecrtrum, correspondingly. These names should correspond to parameter FilenameRoot of corresponding Spectrum Amalyzer, or in case if this field is blank, then to the name of Spectrum Analyzer block.

The frequency response can be displayed in wavelength, frequency, and baseband domain by setting parameter Domain in both Spectrum Analyzers. Also the spectrum can be normalized or not by setting parameter Normalize. Figure 2 shows the transfer function plots for the topology of Figure 1 in wavelength domain and baseband domain with normalized spectra.

Setting the spectra in baseband domain and normalized give us conventional presentation of transfer fuucntion (frequency response) and also allow determining graphically what is a 3-dB bandwidth for this transfer function. If we zoom in the baseband plot of Figure 2 to the - 3dB level – see Figure 3 - we can see that for this particular example 3-dB bandwidth is about 6.3 GHz.


Figure 2: Examples of frequency response plots in wavelength (left) and basedand (right) domains.

Figure 3: 3-dB bandwidth of tranfer function derived from a plot.


Properties

Inputs None

Outputs None

Parameter Values Name Type Default Range Units InputSpecrtrum string

OutputSpectrum string

FilenameRoot string Plot enumerated Save No, Save, Display

Parameter Descriptions InputSpecrtrum Name of the spectrum analyzer block for input data OutputSpectrum Name of the spectrum analyzer block for input data FilenameRoot Root of the filename used to store the plot data Plot Whether to create plot data, display the plot, or skip it

OptSim Models Reference: Block Mode Chapter 11: Data Storage and Meta Blocks •••• 407

Chapter 11: Data Storage and Meta Blocks

This chapter describes data storage and meta blocks.

• Save and Load Signal To/From File

save and load signal data to and from files

• Repeat Loop and Typed Repeat Loop

repeat segment of the link several times

• Delay Block

simulate iteration delay

• Fork

split input signal losslessly

• Typed Fork

split preset input signal type losslessly

• Hierarchical Input Signal Port Block

input port for thehierarchical block

• Hierarchical Output Signal Block

output port for the hierarchical block

• Write Once Read Many (WORM) Block

write signal once to cache and read many times

• Null Signal Block

feed null signal to the unused input ports of a compound component

408 •••• Chapter 11: Data Storage and Meta Blocks OptSim Models Reference: Block Mode

Save and Load Signal To/From File

and

The Save Signal to File and Load Signal from File blocks are used to save and load signal data to and from files. The save block saves the entire signal data structure so that future simulations can use the load block to use a previously saved signal as a driving signal in the current simulation. For example, this allows the simulations of the transmitter, channel, and receiver portions of an optical link to be simulated separately with results equivalent to performing a simulation of all portions of the link at once.

Any type of signal may be saved by the Save Signal block. The Load Signal block will then create the type of output signal which is specified in the data file.

The file is saved and loaded from the directory in which the simulation takes place, which also contains the topology data file. The data files are formatted in a straightforward fashion to allow the user to modify the signal data if desired or import it into other programs for further analysis.

File Formats There are three file formats corresponding to the three signal types: Binary, Electrical, and Optical. Below we summarize each format.

Binary Signal File Format BEGINSIG

SET <simulation ID #>

INNERSWEEP <inner-sweep iteration>

OUTERSWEEP <outer-sweep iteration>

STATRUN <statistical-sweep iteration>

LOOP <repetition-loop iteration>

BinSigType

bitRate <bitrate (bps)>

startTime <start time (s)>

noPoints <number of bits>

bitSequence

<bit-value #1>

<bit-value #2>

...

<bit-value #noPoints>

ENDSIG

Electrical Signal File Format BEGINSIG







ElecSigType

UnitType <CURRENT or VOLTAGE>

domain TIMEDOMAIN


patternLength <number of bits>

tStep <time step (s)>

startTime <start time (s)>

noPoints <number of samples>

elecSamples

<signal sample #1 (V)>

<signal sample #2 (V)>

...

<signal sample #noPoints (V)>

sigmaSamples

<rms noise sample #1 (V)>

<rms noise sample #2 (V)>

...

<rms noise sample #noPoints (V)>

ENDSIG

Optical Signal File Format Not all of the sections in the Optical Signal file format presented below will appear in a particular data file. In cases where no ASE spectrum is present, the OpSigASE … ENDASE section will be absent. When no Y polarization is present, no Ey_samples section will exist. Similarly, only for multimode signals will SpatialX (for the X polarization) and SpatialY (for the Y polarization) sections be present. Finally, when multiple optical signals are present, there will be multiple OpSigType … ENDOpSig sections.

BEGINSIG






OpSigASE

fstart <ASE start frequency (Hz)>

fstep <ASE frequency step (Hz)>

noPoints <number of ASE data points>

powerSamples

<ASE value #1 (W/Hz)>

<ASE value #2 (W/Hz)>


...

<ASE value #noPoints (W/Hz)>

ENDASE

OpSigType

wavelength <wavelength (m)>

domain TIMEDOMAIN


noPoints <number of samples>

patternLength <number of bits>

tStep <time step (s)>

startTime <time step (s)>

RIN <RIN (Hz-1)>

SpatialX <mode type> <mode-data file name>

<l mode index> <m mode index>

<internal scaling factor (complex)>

<x shift (µm)> <y shift (µm)> <rotation (rad)>

<xmin (µm)> <xmax (µm)> <ymin (µm)> <ymax (µm)>

<dx (µm)> <dy (µm)>

<effective-index wavelength (µm)> <effective index (complex)>

<group-velocity wavelength (µm)> <group velocity (complex, µm/s)>

<dispersion wavelength (µm)> <dispersion (complex, s/µm/µm)>

SpatialY <mode type> <mode-data file name>

<l mode index> <m mode index>

<internal scaling factor (complex)>

<x shift (µm)> <y shift (µm)> <rotation (rad)>

<xmin (µm)> <xmax (µm)> <ymin (µm)> <ymax (µm)>

<dx (µm)> <dy (µm)>

<effective-index wavelength (µm)> <effective index (complex)>

<group-velocity wavelength (µm)> <group velocity (complex, µm/s)>

<dispersion wavelength (µm)> <dispersion (complex, s/µm/µm)>

Ex_samples

<Ex sample #1 (complex, W1/2)>

<Ex sample #2 (complex, W1/2)>

...

<Ex sample #noPoints (complex, W1/2)>

Ey_samples

<Ey sample #1 (complex, W1/2)>

<Ey sample #2 (complex, W1/2)>

...


<Ey sample #noPoints (complex, W1/2)>

ENDOpSig

ENDSIG

Mode-Data File Formats As for the mode data files required by the SpatialX and SpatialY sections, there are different formats corresponding to each of the possible settings for <mode type> (gridded, griddedradial, lpfiber, lg, hg, donut, uniform, and mmfiber).

For gridded and griddedradial modes, the file formats are described in Section 9.3 of the OptSim User Guide.

For lpfiber modes, the file format is <l mode index>

<m mode index>

<cladding material index>

<core material index>

<core radius (µm)>

For lg modes, the file format is <l mode index>

<m mode index>

<spot size (µm)>

<inverse radius of curvature (µm-1)>

For hg modes, the file format is <l mode index>

<m mode index>

<x spot size (µm)>

<y spot size (µm)>

<x inverse radius of curvature (µm-1)>

<y inverse radius of curvature (µm-1)>

For donut modes, the file format is <l mode index>

<inner radius(µm)>

<outer radius (µm)>

For uniform modes, the file format is

<width (µm)>

For mmfiber modes, the file format is <l mode index>

<m mode index>


<core material index>

<delta>

<core radius (µm)>

Properties

Inputs #1: Electrical, Optical, or Binary signal

Outputs Same as Inputs

Parameter Values Name Type Default Range Units FileName string

Parameter Descriptions FileName Name of the file to be saved or loaded


Repeat Loop and Typed Repeat Loop

The repetition loops are used to indicate a section of the topology which is to be repeated during the simulation the specified number of times as if it were actually that many sets of component models connected together. The first input at the left hand of the icon above is used to indicate the start of a repetition loop, while the output at the right of the icon is used to indicate the end of a repetition loop. The number of times the loop is to be executed is defined by the NumReps parameter.

The Repeat Loop model will automatically take at the run time the type of signal being repeated depending upon its input port connection. On the other hand, a user needs to statically define the signal type in case of the Typed Repeat Loop model via its parameter _vergil_signalType.

The input and output ports at the bottom of the icon above connect to the segment actually to be repeated. The following schematic illustrates repetition loops:

The repetition loop icons may also be used to define network ring configurations. The loop begin icon would represent the initialization of the ring. The loop end icon would represent the point in the ring where the signal wraps around to the beginning, or the loop begin icon. Set the number of repetitions in the loop begin icon to represent the number of times you wish the signal to circulate in the ring before the simulation completes.

Nested repetition loops are supported.

Properties

Inputs #1-2: Electrical, Optical, or Binary signal

Outputs Same as Inputs


Parameter Values Name Type Default Range Units NumReps integer 1 [ 1, 100000 ] repetitions _vergil_signalType enumerated Unknown_Signal Logical_Signal

Electrical_Signal Optical_Signal Unknown_Signal Measurand_Signal LambdaSim_Signal

Parameter Descriptions NumReps Number of times the loop is to be repeated in the simulation _vergil_signalType Type of the signal undergoing repetition loop(s)


Delay Block

The Delay Block model has a single parameter, the number_of_iterations_delay. In OptSim, a Delay Block with this parameter equal to one simply puts out a single initial particle at its output port. This initial particle may enable a model, assuming that the destination model for the Delay Block requires one particle in order to fire. To avoid deadlock, all feedback loops must have a Delay Block. The simulation scheduler will flag an error if it finds a loop with no delays. The particle type put out by the delay block depends on the type of signal connected to its input. The model uses a FIFO queue to simulate delays greater than unity.

Properties

Inputs #1: Any Block-Mode signal type

Outputs #1: Same signal type as input signal


number_of_iterations_delay int 1 [ 1, 100000 ]

Parameter Descriptions number_of_iterations_delay the number of iterations to delay the signal at the input.


Fork

This block is used to represent the fork where input signal is losslessly split into many.

Properties

Inputs #1: signal of any type

Outputs #N: signals of input type


Typed Fork

This block is used to represent typed fork where input signal is losslessly split into many.

The type of the signal can be preset from the parameter box of this component.

Properties

Inputs #1: signal of specific type

Outputs #N: signals of input type


Hierarchical Input Signal Port Block

This block is used to represent the input ports to the superblock which the topology represents.

The input signal type is determined by the output signal type arriving at the input signal port block in the topology.

Properties

Inputs none

Outputs #1: signal of any type


Hierarchical Output Signal Port Block

This block is used to represent the output ports of the superblock which the topology represents.

The output signal type is determined by the input signal type to the output signal port block in the topology.

Properties


Outputs none


Write Once Read Many (WORM) Block

The Write-Once/Read-Many model is a special model that is used to satisfy an unusual scheduling situation in OptSim. In some cases the user will want to have a connection that branch into a reptition loop. The simulation scheduler would normally treat this as an ambiguous connection diagram and issue and error. The WO-RM model indicates a special situation to the scheduler. The scheduler ensures that the input signal is cached inside the model and is read as many times as requested inside the loop. This model must be used on all connections that branch into a repetition loop. The model has no parameters. The following schematic illustrates its usage and purpose:

Properties


Outputs #1: signal of the input type


Null Signal Block

When not all of the input ports of a compound component are used in the main project file, OptSim issues an error message. For example, if the compound component is a 4-channel transmitter, but the main project file needs to use only, say, three of these four channels, a null signal block should be connected to the fourth un-used input port of the compound component to fulfill the scheduling requirement. The following schematic illustrates its usage and purpose. A single Null Signal block can be used to feed many input ports at different points of the schematic. The model has no parameters.

Properties

Outputs #1: signal of any type

OptSim Models Reference: Block Mode Chapter 12: Transient Modules •••• 423

Chapter 12: Transient Modules

This chapter describes transient modules.

• Transient Pulse Generator

generate binary pulse

• Compact Transient EDFA

model dynamic behavior of EDFA via a “reservoir”-based model

• Transient EDFA

model dynamic behavior of EDFA via a set of well-established physical equations

• Static Optical Switch (2X2)

model a 2X2 static optical switch

• Transient Optical Switch (2X2)

model a 2X2 transient optical switch

• Dynamic Optical Switch (2X2)

model a 2X2 dynamic optical switch

• Transient Plotter

view generated live plots

424 •••• Chapter 12: Transient Modules OptSim Models Reference: Block Mode

Transient Pulse Generator

This model generates a binary pulse. The model should be used in the transient simulation regime. A single sample of the pulse is produced each iteration of a multiple iteration simulation. This model was created to drive the control input of the Dynamic Optical Switch model that will be described later in this section. The pulse can control the state of the optical switch. When the pulse is high the switch is in the cross state and when it is low it is in the bar state. The user can set the period and duty cycle of the pulse in effect controlling the regular interval in which the state of the optical switch will change. The user can also provide an array specifying an arbitrary sequence of bits that will be used at the beginning of the simulation.

Properties

Inputs None.

Outputs #1-N: Binary signal

Parameter Values Name Type Default Range Units pulse_period double 1.0e-4 [0,1] sec

pulse_duty_cycle double 50 [0,100] %

start_sequence enumerated “ONES” “ONES”, “ZEROS”

number_of_prebits int 0 [0, 10000 ]

prebits Integer array [0]

Parameter Descriptions pulse_period The period of the binary pulse. This should be set relative to the Iteration and Time Step Per

Iteration parameters that are available from the Multiple Iteration Simulation tab in the Run Window

pulse_duty_cycle The duty cycle of the pulse as a percent of the pulse period. start_sequence Determines if the pulse should occur at the beginning or at the end of the pulse period. number_of_prebits The number of special bits at the beginning of the simulation event. If this number N is

more than the number of bits M in the prebits array, the last bit in the array is repeated N-M times.

prebits An integer array of the form [1,0,1,0,1,1] specifying an optional binary sequence of bits that will be generated at the beginning of the simulation.


Compact Transient EDFA

This block models the dynamic operation of an erbium-doped fiber amplifier (EDFA) via a “reservoir”-based model adapted from the work of Bononi et al. [1],[2] and Sun et al. [3]. The model supports a variety of pump and signal configurations, as well as both fiber and waveguide amplifier geometries. (Strictly speaking, the latter geometry results in an EDWA, but henceforth we will refer to the amplifier as an EDFA.) Forward-propagating optical signals are launched into the EDFA via the first input node, while backward-propagating signals (e.g., counter-propagating pumps and signals) can enter via the second input node. OptSim’s multiplexer components can be used to combine signals and pumps at either input. The EDFA outputs are available at the output nodes, and include any signals, pumps, and amplified spontaneous emission (ASE) that are exiting the amplifier. Output signals will be produced at the backward output provided that there are input signals at the backward input.

Background The Compact Transient EDFA model is based on a set of equations developed by Bononi et al. [1],[2] and Sun et al. [3]. Instead of dealing with propagation along the length of the EDFA, a “reservoir” approach is used wherein the total EDFA upper-level population is used instead of the upper-level density as a function of length. This type of model can be derived from standard EDFA models such as those described in [4]-[6].

As in the Physical and Transient EDFA models, the erbium atomic manifolds of interest can be reduced to a three-level atomic system, as illustrated in Fig. 1(a), with population densities in each manifold assumed to be Boltzmann-distributed at thermal equilibrium. Optical signals with wavelengths near 980 nm (henceforth referred to as the 980-band) are used to pump from the first (4I15/2) to the third (4I11/2) level, while 1480-nm signals pump from the first to the second (4I13/2) level. In the former case, fast non-radiative decay from the third to the second level effectively eliminates any stimulated emission from the third to first level, allowing us to simplify the model to a two-level atomic system with zero stimulated emission in the 980-band [5]. This reduced arrangement is depicted in Fig. 2(b), where R13 is the stimulated absorption rate for 980-band transitions, R12 and R21 are stimulated absorption and emission rates between the 4I15/2 and 4I13/2 levels, respectively, and τ is the spontaneous emission lifetime of the 4I13/2 level. The optical signals being amplified by the EDFA usually have wavelengths ranging from 1530-1580-nm (the C-band) or 1580-1610-nm (the L-band), and therefore interact with the same atomic populations as 1480-nm pumps. Thus, R12 and R21 account for both signal and pump transitions at wavelengths typically ranging from 1450-1650-nm (henceforth referred to as the 1550-band). Amplified spontaneous emission (ASE) in the EDFA also occurs in this band of wavelengths.

4I11/2

4I13/2

4I15/2

1550 nm

980 nm

1480 nmpumps

level 2

level 1

R13

R21

R121/ τ

(a) (b)

Figure 1: Erbium atomic manifolds. (a) Three manifolds involved in predominant atomic transitions. (b) Simplified two-level model.

In order to describe the interaction between the erbium ions and local signal, pump, and noise powers, the model uses a single rate equation for the total level-2 erbium ion population. This population is the “reservoir” r [1]. As shown in [1]-


[3], once r is known, the full spectral gain of the EDFA can be calculated. After dividing the frequency range of interest into bins of width ∆ν, the basic form of the equation, neglecting background loss, ASE noise, and upconversion is [1]:

( )( ), ,

( ) ( ) ( ) ( ) 1 jG r tj in j in

j

dr t r t P t P t edt τ

+ − = − + + ⋅ − ∑

(1)

where j indexes the frequency bins at frequencies νj, , ( )in jP t+ and , ( )in jP t− are the forward and backward input signal

powers, τ is the Er metastable lifetime (metastable_lifetime), and Gj is the gain (as a function of r). As shown in [2], ASE noise generation can be added to the model if we assume a uniform inversion along the length of the EDFA. In this model, we extend this assumption in order to include background loss and uniform upconversion [7]. In many situations this approximation is a good one, giving results in good agreement with a full EDFA model. However, in cases where the inversion is not uniform, the model will not be as accurate. This is a natural consequence of the tradeoff between speed and accuracy, as this Compact Transient EDFA model is significantly faster than the full Transient EDFA model.

Model Implementation Below we present the model equations for both an overlap and Giles-based implementation. In these equations, 980-band powers are denoted by jQ± , 1550-band powers by kP± , and ASE powers by lA± . A number of assumptions are inherent to these equations. First, as noted above, uniform inversion along the length of the EDFA is assumed when including ASE generation, background loss, and uniform upconversion. Second, we have assumed that 980-band signals all have the same normalized mode profile ( )p rψ ; similarly, all 1550-band signals and ASE use a common profile ( )s rψ . Third, higher order effects such as excited-state absorption (ESA) [4]-[5] are neglected. Finally, effects such as dispersion and nonlinearities are also neglected, due to the relatively short lengths of most EDFA’s. It should also be noted that the model assumes no spectral overlap between separate optical signals. In cases where this is detected, the model will issue an appropriate warning.

Model Levels The different implementations of the Compact Transient EDFA model can be selected via the simulation_mode parameter, and can take on values of giles_params, constant_overlap, or calculated_overlap.

Level giles_params The Giles implementation should be used when measured gain and loss spectra (as opposed to emission and absorption cross sections) are available. This well known approach significantly reduces the number of model parameters that are necessary to model an EDFA. Such a model is generally valid for strongly confined erbium-doping profiles. The reservoir rate equation and output-power equations are:

( ) ( )( ) ( )( ) ( )

, ,2

, ,

, ,

1 1

1 1

1 1

1(2) 1

j j

k k

l l

l l

l l

j in j in G Ljup

j j j j

k in k in G Lk

k k k k

l in l in G Ll

l l l l

G LlG L

ll l

Q Q Ldr r c r edt h G L

P P L eh G L

A A L eh G L

L eD r e

G L

α

α

α

αα

ατ ν α

αν α

αν α

αν

α

+ −−

+ −−

+ −−

−−

+ = − − − ⋅ + ⋅ − −

+ − ⋅ + ⋅ − −

+ − ⋅ + ⋅ − −

⋅ −− ⋅ ∆ ⋅ ⋅ ⋅ ⋅ − +

−

∑

∑

∑

( )11l

l l l

LG L

αα

− − − ∑


(2)

, ,j jG L

j out j inQ Q e α−± ±= ⋅

(3)

, ,k kG L

k out k inP P e α−± ±= ⋅

(4)

, ,1(2)

l ll l

G LG L

l out l in l ll l

eA A e h D rG L

αα ν ν

α

−−± ± −= ⋅ + ⋅ ⋅ ∆ ⋅ ⋅ ⋅ −

(5)

where

2 2giles

up

cc

Lτ ζ=

(6)

, ,,

e j a jj a j

g gG r g L

ζτ+

= ⋅ −

(7)

,e ll

gD

ζτ=

(8) In the above equations, ge,j and ga,j are the measured Giles gain and loss spectra, respectively (where the gain is zero in the 980-band); ζ is the fiber saturation parameter (fiber_saturation_param); L is the device length (length); cgiles (giles_upconversion_coeff) is the Giles-model homogeneous upconversion coefficient; and h is Planck’s constant. Also, the factor of 2 in Equations (2) and (5) assumes that both polarizations are included in ASE generation. However, as noted below, this factor can be controlled via the parameters inject_ASE and ASE_polarization.

Levels constant_overlap, calculated_overlap In terms of computational complexity, the constant_overlap and calculated_overlap models are equivalent to the giles_params model, but support a more thorough parameterized description of the EDFA, including absorption/emission cross sections, doping profiles, and mode profiles. In these versions of the model, any explicit transverse spatial dependence is replaced by factors that account for the spatial overlap between the erbium population densities and optical modes [4]. In the calculated_overlap model, these overlap factors are determined from explicit doping and mode profiles, whereas in the constant_overlap model, the user provides these values directly. For 980-band signals, the overlap parameter is Γp (overlap_980), and for 1550-band signals, it is Γs (overlap_1550). Like the giles_params model, the overlap models are ideally suited for EDFA’s with strongly confined erbium-doping profiles. The core equations for these models are the same as Equations (2)-(5), but with the following definitions for cup, Gj, and Dl:

22

2 2 20

fup

eff eff

cU Ncc dAA L A L N

= = ⋅ ∫

(9)


, ,, 0

e j a jj j a j j

eff

G r LNA

σ σσ

+= ⋅Γ − Γ

(10)

,e l ll

eff

DA

σ Γ=

(11) where σe,j and σa,j are the emission and absorption spectra, respectively (where the emission is zero in the 980-band); Γj is the overlap factor, equal to Γs or Γp, depending on the frequency; N2 is the upper-level density; N0 is the average doping density; c (upconversion_coeff) is the homogeneous upconversion coefficient; and Aeff is the effective transverse doping area. Note that both Aeff and the transverse integral in Equation (9) are calculated internally by the model.

EDFA Configurations In specifying the configuration of the EDFA, the user must always specify a device length via the length parameter. They should also specify whether the device is a fiber or waveguide amplifier via the parameter amplifier_type. They may also provide coupling losses at both the input and output nodes via the parameters forward_input_loss, backward_input_loss, output_loss, and backward_output_loss.


Absorption/Emission Spectra For optical signals and ASE in the 1550-band, the user must specify the Giles gain/loss spectra (for the Giles model), or absorption/emission cross-section spectra (for the other models). By setting the spectra_1550_model parameter, the user may select either built-in default spectra or device-specific data that is to be read in from a file.



A loss or absorption spectrum must also be provided for signals in the 980-band. In this case, three choices are available via the parameter spectra_980_model. For either a rectangular profile (rectangular) or Gaussian profile (gaussian), the user must specify a center wavelength (spectra_980_center) and spectral width (spectra_980_width). For the Giles model, the center-wavelength loss is set via giles_loss_980. For the other models, the absorption cross-section center-wavelength value is set via absorption_980.


Depending on the provided data, the model can automatically determine the range of wavelengths that comprise the 980-band and 1550-band based on the range of the gain/loss or emission/absorption spectra. However, by setting


spectra_1550_auto_limit to no, the user may directly limit the 1550-band to a specific range of wavelengths via the parameters spectra_1550_low and spectra_1550_high. Similarly, by setting spectra_980_auto_limit to no, the 980-band limits can be set via the parameters spectra_980_low and spectra_980_high.

Thermal Dependence Thermal dependence of either the absorption/emission or loss/gain spectra is modeled based on the work of M. Bolshtyansky et al. in [8]. Given a set of reference spectra αref (absorption or loss) and gref (emission or gain) at a temperature Tref, spectra αnew and gnew at a new temperature Tnew are calculated as:

1 2

1 2

/ /

/ /

( )( ) ( )( )

new new

ref ref

T T T Tref

new ref T T T Tnew

T K e eT K e e

λα λ α λλ

− −

− −

+= ⋅ ⋅+

(12)

1 2

1 2

/ 1/ /

/ /

( )( )( ) ( )( )( )

ref newnew new

ref ref

T TT T T T


new ref

T gK e eg gT K e e

λλλ λα λλ

−− −

− −

+= ⋅ ⋅ ⋅ +

(13)


To activate this model, the user must set thermal_model equal to builtin or user, and specify the temperature (reference_temp) at which the reference spectra were measured, and the new temperature (actual_temp) at which the spectra should be modeled. The absorption/emission or loss/gain spectra specified earlier by the user are considered to be the reference spectra.





Doping Profile A number of choices are available for specifying the EDFA erbium doping profile. For fiber amplifiers, these options are available through the doping_model parameter. For both rectangular and gaussian doping profiles, a doping radius and peak density must be provided via the parameters doping_radius and doping_density, respectively. Alternatively, the user may choose to provide a data file containing the doping density profile by setting doping_model to file and specifying a file name through the parameter doping_file. The file format is described in the appendix.

For waveguide amplifiers, the parameter waveguide_doping_model is set to file, and the user must specify the doping-profile data file via the parameter waveguide_doping_file. The format of this file is described in the appendix.


Mode Profiles For the calculated_overlap and spatial models, 980-band and 1550-band mode profiles are required. For fiber amplifiers, different mode shapes may be selected using the parameter mode_model. If mode_model is set to lp, then LP01 mode solutions [5] are determined for a fiber with the specified numerical aperture (fiber_NA) and fiber core radius (fiber_core_radius). For all 1550-band signals, a wavelength of 1550 nm is used in the calculation, whereas a 980-nm wavelength is adopted for all 980-band signals.

If a rectangular or gaussian set of mode shapes is chosen, then relevant mode widths must be specified via the mode_1550_width and mode_980_width parameters. Because the calculated mode shapes are normalized, no additional information is required.


For waveguide amplifiers, mode shapes are input from a data file (waveguide_mode_model is set to file). The 1550-band file name must be provided through the waveguide_mode_1550_file parameter, while the 980-band file name must be entered through the waveguide_mode_980_file parameter. The file format is described in the appendix. Note that as before, mode shapes are automatically normalized by the model.

Background Loss In most fiber-amplifier simulations, fiber background loss can be neglected [5], in which case the parameter background_loss_model should be set to no_loss. However, other options are available if required.


Encrypted Data Files Note that with the exception of the waveguide doping and mode profiles, all of the model characteristics that may be specified via a data file support encrypted file formats. Data files from vendors may be provided in this manner. In this case, an accompanying password file would be included with the data file, and should be placed in the same directory as the link topology, or in the directory containing the RSoft software license files.

Numerical Settings A number of numerical parameters are available to optimize the numerical calculations of the model. Adjustment of these parameters may help overcome difficulties encountered during a simulation.

The nominal width for the discretized frequency intervals in the 980- and 1550-bands may be set using the spectral_step parameter. As this parameter is set in units of nanometers, an equivalent spectral step in Hertz is calculated internally by the model. The resolution of transverse calculations may also be adjusted by setting the radial_points parameter to the desired number of points in the radial direction.

The user can set the number of simulation iterations to calculate steady state solutions before entering the transient regime. This is done using the num_steady_state_iter parameter. This is often desirable, since it can take a number of simulation iterations before some system topologies reach a steady state. We typically want to perturb the system after it has reached a steady state operating point. A very important numerical parameter for the Transient EDFA model is the Time Step Per Iteration value that is set in the Multiple Iterations tab of the Run Window. This value is often referred to as the simulation time step. This is implemented as a global simulation parameter rather than being local to the model because many of the models that support transient simulations need to be synchronized to the same time step. The size of the simulation time step is often limited by some physical properties associated with the design of the EDFA. For instance, an all-optical fixed-gain


EDFA may impose limits on the maximum time step. The propagation delay around the lasing loop or lasing cavity must be taken into consideration. Otherwise, except for considerations related to numerical convergence, the user can adjust the simulation time step while trading accuracy for total simulation time.


Reference Plots In order to help study the performance of an EDFA within an optical link, a variety of reference plots can be generated by the model. The generate_plots parameter allows the user to turn on and off all plots. The units for the displayed data may be selected via the parameters spectral_units, power_units, and signal_gain_units (for gain and noise figure data). The plotting_interval parameter allows the user to set the simulation times at which the model will periodically generate internal plots. This level of control is very important when working with topologies that are simulated using OptSim’s multiple iteration capability. Generating a set of plots for each execution of the system under study could lead to an excessively large number of result files being saved to disk.

Below we summarize the different plots that may be generated, listing them by their root WinPlot file names.

• 1550_power_spectra_fwd.pcs:

Displays the forward input/output signal and ASE power spectra in the 1550-band.


Displays the backward input/output signal and ASE power spectra in the 1550-band.

• 980_power_spectra_fwd.pcs:

Displays the forward input and output 980-band power spectra.


Displays the backward input and output 980-band power spectra.

• gain_fwd.pcs:

Displays the overall forward 1550-band signal gain spectra (1480-nm pumps excluded).

• gain_bwd.pcs:

Displays the overall backward 1550-band signal gain spectra (1480-nm pumps excluded).

• absolute_gain_fwd.pcs:

Displays a spectrum of the absolute change in forward input-signal power.


Displays a spectrum of the absolute change in backward input-signal power.

• noise_figure_fwd.pcs:

Displays the overall forward 1550-band noise figure, calculated as a function of signal gain ( )G υ , output

ASE spectral density ( )ASEρ ν , and input ASE spectral density , ( )ASE inρ ν [4]:


,

( )1 1( )

( )1

ASE

ASE in

G hNF

h

ρ νν ν

ρ νν

⋅ + = +

Note that the input ASE is taken into account when performing this calculation.


Displays the overall backward 1550-band noise figure

Test Functions In selecting the input parameters for a particular EDFA, it may at times be necessary to visualize the various input spectra, doping profiles, and mode shapes. By setting the test_function parameter to the desired output and clicking the Test button in the component-parameter editing window, the user may display the EDFA characteristics summarized below. Furthermore, the default plot ranges for each characteristic may be overridden by setting test_default_settings to no, and specifying values for test_function_x_low, test_function_x_high, and test_function_points (the number of data points to plot). The appropriate units for each characteristics’ overrides are listed in parentheses below. Units for the displayed plots may be specified using the parameters spectral_units, radial_units, cross_section_units, density_3d_units, and loss_units.





• doping(um):

For fiber amplifiers, plots the erbium doping profile as a function of radius. For waveguide amplifiers, plots the doping profile as a function of X and Y using internally generated plot ranges.

• modes(um):

For fiber amplifiers, plots the 1550- and 980-band normalized mode profiles as functions of radius. For waveguide amplifiers, plots the 1550- and 980-band normalized profiles as functions of X and Y using internally generated plot ranges.



References [1] A. Bononi and L. A. Rusch, “Doped-fiber amplifier dynamics: A system perspective,” Journal of Lightwave Technology, 16, 945-956, (1998).

[2] A. Bononi, L. Barbieri, and L. A. Rusch, “Using Spice to simulate gain dynamics in doped-fiber amplifier chains,” OFC ’98 Workshop 204, February 1998.

[3] Y. Sun, J. L. Zyskind, and A. K. Srivastava, “Average inversion level, modeling, and physics of erbium-doped fiber amplifiers,” IEEE Journal of Selected Topics in Quantum Electronics, 3, 991-1007, (1997).

[4] P. C. Becker, N. A. Olsson, and J. R. Simpson, Erbium-Doped Fiber Amplifiers: Fundamentals and Technology. (San Diego, Academic Press, 1999).






Properties






constant_overlap, calculated_overlap








overlap _1550 double 0.35 0 ≤ x ≤ 1e32


upconversion_coeff double 0 0 ≤ x ≤ 1e32 m3/s

giles_upconversion_coeff double 0 0 ≤ x ≤ 1e32 m-1s-1




























T1_1550 double 90 -1e32 ≤ x ≤ 1e32 K

T2_1550 double 650 -1e32 ≤ x ≤ 1e32 K


T1_980 double 90 -1e32 ≤ x ≤ 1e32 K

T2_980 double 650 -1e32 ≤ x ≤ 1e32 K

amplifier_type enumerated fiber fiber, waveguide

doping_model enumerated rectangular rectangular, gaussian, file



doping_file string

waveguide_doping_model enumerated file file

waveguide_doping_file string

mode_model enumerated lp lp, gaussian, rectangular, file







waveguide_mode_model enumerated file file

waveguide_mode_1550_file string

waveguide_mode_980_file string






num_steady_state_iter integer 10 1 ≤ x ≤ 100000


generate_plots enumerated no yes, no

plotting_interval double 100e-6 0.0 ≤ x ≤ 100000.0








test_function enumerated 1550_spectra(nm) 1550_spectra(nm), 980_spectra(nm), doping(um), modes(um), background_loss(nm)






simulation_mode model-level options forward_input_loss forward-propagation input coupling backward_input_loss backward- propagation input coupling output_loss forward-propagation output coupling backward_output_loss backward- propagation output coupling length EDFA length metastable_lifetime metastable-level lifetime fiber_saturation_param Giles-model fiber-saturation parameter overlap_1550 constant 1550-band overlap factor overlap_980 constant 980-band overlap factor upconversion_coeff upconversion coefficient for non-Giles models giles_upconversion_coeff upconversion coefficient for Giles model spectra_1550_model 1550-band signal spectra options


spectra_1550_auto_limit option for automatically determining 1550-band spectral range spectra_1550_low lowest wavelength for 1550-band spectra spectra_1550_high highest wavelength for 1550-band spectra giles_loss_1550_file user-specified 1550-band Giles loss-spectrum data file giles_gain_1550_file user-specified 1550-band Giles gain-spectrum data file absorption_1550_file user-specified 1550-band absorption cross-section data file emission_1550_file user-specified 1550-band emission cross-section data file spectra_980_model 980-band signal spectrum options spectra_980_auto_limit option for automatically determining 980-band spectral range spectra_980_low lowest wavelength for 980-band spectra spectra_980_high highest wavelength for 980-band spectra giles_loss_980 center-value of 980-band Giles loss spectrum absorption_980 center-value of 980-band absorption cross-section spectra_980_center center-wavelength of 980-band spectrum spectra_980_width bandwidth of 980-band spectrum giles_loss_980_file user-specified 980-band Giles loss-spectrum data file absorption_980_file user-specified 980-band absorption cross-section data file background_loss_model background-loss options background_loss 1550-band background-loss value background_loss_980 980-band background-loss value background_loss_file user-specified background-loss data file thermal_model option for determining thermal dependency of gain/loss or

emission/absorption spectra reference_temp temperature at which gain/loss or emission/absorption spectra were measured actual_temp actual operating temperature of amplifier F1F2_ratio_1550_file empirical F1/F2 thermal data file for 1550-band spectra T1_1550 empirical T1 thermal parameter for 1550-band spectra T2_1550 empirical T2 thermal parameter for 1550-band spectra F1F2_ratio_980_file empirical F1/F2 thermal data file for 980-band spectra T1_980 empirical T1 thermal parameter for 980-band spectra T2_980 empirical T2 thermal parameter for 980-band spectra amplifier_type switch for selecting between fiber and waveguide geometries doping_model erbium-doping density options for fiber amplifier doping_radius doping radius for fiber amplifier doping_density center doping density (at r = 0) for fiber amplifier doping_file user-specified doping-profile data file for fiber amplifier waveguide_doping_model erbium-doping density options for waveguide amplifier waveguide_doping_file user-specified doping-profile data file for waveguide amplifier mode_model optical mode shape options for fiber amplifier fiber_NA fiber numerical aperture for fiber amplifier fiber_core_radius fiber core radius for fiber amplifier mode_1550_width 1550-band optical-mode width for fiber amplifier mode_980_width 980-band optical-mode width for fiber amplifier mode_1550_file user-specified 1550-band mode-profile data file for fiber amplifier mode_980_file user-specified 980-band mode-profile data file for fiber amplifier


waveguide_mode_model optical mode shape options for waveguide amplifier waveguide_mode_1550_file user-specified 1550-band mode-profile data file for waveguide amplifier waveguide_mode_980_file user-specified 980-band mode-profile data file for waveguide amplifier inject_ASE switch for injecting ASE within the EDFA ASE_polarization switch for injecting ASE into one or two polarizations spectral_step spacing between wavelengths in discretized power spectra radial_points number of radial points num_steady_state_iter number of steady-state iterations to calculate during transient simulation gain_application method of applying gain to each input signal generate_plots switch to override individual plot settings plotting_interval the regular time interval to generate internal plots spectral_units units for spectral data power_units units for power data radial_units units for radial data cross_section_units units for cross-section data density_3d_units units for density data loss_units units for loss data signal_gain_units units for gain and noise figure results test_function test-function output selection test_default_settings switch for plotting test-function output using default settings test_function_x_low user-specified lowest x-value for test-function output test_function_x_high user-specified highest x-value for test-function output test_function_points user-specified number of points for test-function output


Appendix: File formats For two-dimensional data files, the X-values should be monotonically increasing or decreasing. Furthermore, the field <num_pts> specifies the number of data lines in the file, while the choice of settings for the various unit fields are as follows:






Giles Gain/Loss Spectra Data files with gain and loss spectra for the Giles model are specified through the parameters giles_gain_1550_file, giles_loss_1550_file, and giles_loss_980_file. The X-values are in units of frequency, and the Y-values are in units of loss per distance.


<num_pts>





...


5

1450 0.3

1500 0.6

1530 2.0

1550 1.2

1600 0.4

Absorption/Emission Cross Sections Data files with absorption/emission cross sections are specified through the parameters absorption_1550_file, emission_1550_file, and absorption_980_file. The X-values are in units of frequency, and the Y-values are in units of area.


<num_pts>






...


5

1450 0.7e-25

1500 2.4e-25

1530 7.0e-25

1550 4.3e-25

1601 0.2e-25



<num_pts>





...


5

1500 0.40

1510 0.45

1520 0.50

1530 0.45

1540 0.40

Doping Profile For fiber amplifiers, a user-specified doping profile may be specified via the parameter doping_file. The X-values are in units of distance, and the Y-values are in units of cubic density.



<num_pts>





...


5

0 1.00e19

0.25 1.00e19

0.50 1.00e19

0.75 0.50e19

1.0 0.25e19

For waveguide amplifiers, a user-specified doping profile may be specified via the parameter waveguide_doping_file. The X- and Y-values should be in m, the doping density values in units of m-3. The file format is shown below, where <num_x_pts> (<num_y_pts>) is the number of points in the X- (Y-) direction, <x_min> (<y_min>) and <x_max> (<y_max>) are the minimum and maximum X- (Y-) values, respectively, and <data @ (x,y)> is the doping density at grid point (x,y).

Format: /rn,a,b/nx0/ls1

/r,qa,qb

<num_x_pts> <x_min> <x_max> 0 OUTPUT_REAL_3D

<num_y_pts> <y_min> <y_max>

<data @ (x_min,y_min)> … <data @ (x_min,y_max)>

…

<data @ (x_max,y_min)> … <data @ (x_max,y_max)>

Mode Profiles For fiber amplifiers, data files with mode shapes may be specified using the parameters mode_1550_file and mode_980_file. The X-values are in units of distance, and the Y-values are considered unitless (the mode profiles are automatically normalized by the model).


<num_pts>





...



5

0 1

1 0.7

2 0.4

3 0.25

5 0.15

For waveguide amplifiers, waveguide profiles may be specified via the parameters waveguide_mode_1550_file and waveguide_mode_980_file. The X- and Y-values should be in m, the mode values are unitless. The file format is shown below, where <num_x_pts> (<num_y_pts>) is the number of points in the X- (Y-) direction, <x_min> (<y_min>) and <x_max> (<y_max>) are the minimum and maximum X- (Y-) values, and <data @ (x,y)> is the mode value at grid point (x,y).

Format: /rn,a,b/nx0/ls1

/r,qa,qb

<num_x_pts> <x_min> <x_max> 0 OUTPUT_REAL_3D

<num_y_pts> <y_min> <y_max>

<data @ (x_min,y_min)> … <data @ (x_min,y_max)>

…

<data @ (x_max,y_min)> … <data @ (x_max,y_max)>



<num_pts>





...


5

1.40 0.25

1.45 0.21

1.50 0.19

1.55 0.17

1.60 0.21


Transient EDFA

This block models the dynamic operation of an erbium-doped fiber amplifier (EDFA) via a set of well-established physical equations. The component is based on the Giles model that was presented in the section that described the Physical EDFA model. The model supports a variety of pump and signal configurations. Figure 1 illustrates an OptSim schematic that utilizes the Transient EDFA model. Forward-propagating optical signals are launched into the EDFA via the first input node, while backward-propagating signals (e.g., counter-propagating pumps and signals) can enter via the second input node. OptSim’s multiplexer components can be used to combine signals and pumps at either input. The EDFA outputs are available at the output nodes, and include any signals, pumps, and amplified spontaneous emission (ASE) that are exiting the amplifier. When the model operates in bi-directional propagation mode output signals will also be produced at the backward output provided that there are input signals at the backward input.

Figure 1: OptSim topology depicting a Transient EDFA with forward propagating signals and a feedback loop.

Background Like the Physical EDFA model, the Transient EDFA model is based on a standard set of equations for an EDFA’s atomic manifold population densities and the evolution of optical powers along the length of the device [1]-[3]. Unlike the Physical EDFA model, the Transient EDFA model also models the transient response of the EDFA. Therefore, this model can be used to study the transitional interval between two steady-state operating points. Although the model can handle detailed sampled signals, it is more efficient to study transient effects using CW signals. The gain dynamics of the EDFA does not respond to the bit transitions in an optical signal so using CW signals does not affect the behavior of the amplifier.

Optical signals propagating along the EDFA interact with the local population densities, resulting in power gain or loss via stimulated emission and absorption. Spontaneous emission and its subsequent amplification also occur. The general background that was presented for the Physical EDFA model is repeated below for completeness.

Generally, the erbium atomic manifolds of interest can be reduced to a three-level atomic system, as illustrated in Fig.2(a), with population densities in each manifold assumed to be Boltzmann-distributed at thermal equilibrium. Optical signals with wavelengths near 980 nm (henceforth referred to as the 980-band) are used to pump from the first (4I15/2) to the third (4I11/2) level, while 1480-nm signals pump from the first to the second (4I13/2) level. In the former case, fast non-radiative decay from the third to the second level effectively eliminates any stimulated emission from the third to first level, allowing us to simplify the model to a two-level atomic system with zero stimulated emission in the 980-band [2]. This reduced arrangement is depicted in Fig.2(b), where R13 is the stimulated absorption rate for 980-band


transitions, R12 and R21 are stimulated absorption and emission rates between the 4I15/2 and 4I13/2 levels, respectively, and τ is the spontaneous emission lifetime of the 4I13/2 level. The optical signals being amplified by the EDFA usually have wavelengths ranging from 1530-1580-nm (the C-band) or 1580-1610-nm (the L-band), and therefore interact with the same atomic populations as 1480-nm pumps. Thus, R12 and R21 account for both signal and pump transitions at wavelengths typically ranging from 1450-1650-nm (henceforth referred to as the 1550-band). Amplified spontaneous emission (ASE) in the EDFA also occurs in this band of wavelengths.

4I11/2

4I13/2

4I15/2

1550 nm

980 nm

1480 nmpumps

level 2

level 1

R13

R21

R121/ τ

(a) (b)

Figure 2: Erbium atomic manifolds. (a) Three manifolds involved in predominant erbium atomic transitions. (b) Simplified two-level model.

In order to describe the interaction between the erbium ions and local signal, pump, and noise powers, the model uses a set of rate equations for erbium ion densities in each atomic level. However, because the sum of these two densities should equal the total erbium doping density N, we can in fact adopt a single rate equation for the level-2 population density N2. Furthermore, in most EDFA applications, the long lifetime τ of the metastable level-2 population acts to eliminate any significant transient changes in the level populations, thereby allowing us to set the time rate-of-change for N2 to zero. In other words, the EDFA’s operating characteristics depend on average optical powers. Thus [2],

12 212 2

13 1 12 1 21 2 1 2ASE ASEdN NR N R N R N R N R Ndt

= + − + − −τ

(1)

where N1 = N – N2 is the level-1 population density, RASE12 is the stimulated absorption rate for spontaneous emission, and RASE21 is the corresponding stimulated emission rate. Generally, because there is a continuum of transition frequencies between the erbium atomic manifolds, each of the transition rates R in Eq.(1) are of the form:

νν

ψνρνσ dh

rR ∫= )()()( r

(2)


is the normalized optical mode profile. By assuming homogeneous broadening of the atomic transitions, we have adopted a single frequency-dependent function for the transition cross sections [2]. Following the approach in Ref.[2], we can discretize the above integral over fixed frequency intervals ν∆ , in which case the transition rates take the form

( )i i i

i i

P PRh

σ ψν

+ −+=∑

(3)


where we have replaced the optical spectral density with forward ( iP+ ) and backward ( iP− ) signal powers in each frequency interval.

To complete the model, rate equations are necessary for describing the evolution of signal, pump, and noise powers along the EDFA. Separate equations for both forward and backward propagation are required for 980-band signals, 1550-band signals, and 1550-band ASE. Again following the approach taken in [2], we have adopted the following equations:

dQdz

N r r r dr N r r r dr Qja j p a j p

jj

±±= ± ⋅ ⋅ − ⋅ −

⋅∫∫2

22π σ ψ σ ψαπ, ,( ) ( ) ( ) ( )

(4)

dPdz

N r r r dr N r r r dr Pke k a k s a k s

kk

±±= ± ⋅ + ⋅ − ⋅ −

⋅∫∫222π σ σ ψ σ ψ απ

( ) ( ) ( ) ( ) ( ), , ,

(5)

( )

, , 2 ,

, 2

2 ( ) ( ) ( ) ( ) ( )2

2 ( ) ( ) 2

l le l a l s a l s l

e l s l


N r r r dr h


π σ ψ ν ν

±± = ± ⋅ + ⋅ − ⋅ − ⋅

± ⋅ ⋅ ⋅ ⋅ ⋅ ∆

∫ ∫

∫

(6)

where 980-band powers are denoted by jQ± , 1550-band powers by kP± , and ASE powers by lA± . Intrinsic background loss is accounted for by α , and locally generated spontaneous emission is included via the last term in Eq.(6).

A number of assumptions are inherent to Eqs.(4)-(6). First, we have assumed that the erbium-doping and mode profiles have no azimuthal dependence. Second, we have assumed that 980-band signals all have the same normalized mode profile ( )p rψ ; similarly, all 1550-band signals and ASE use a common profile ( )s rψ . Third, higher order effects such as excited-state absorption (ESA) [1]-[2] are neglected. However, higher order effects such as upconversion and pair-induced quenching, while not included in the above equations, are included in the model, and will be explained shortly. Finally, fiber effects such as dispersion and nonlinearities are also neglected, due to the relatively short lengths of most EDFA’s. It should also be noted that the model assumes no spectral overlap between separate optical signals. In cases where this is detected, the model will issue an appropriate warning.

The number of wavelength channels passing through an EDFA in a multiwavelength network will vary as a result of network reconfiguration, network growth to larger number of channels, or component failures that can cause one or more channel to drop out. Because these amplifiers are operating near saturation, and since the total output power of a saturated EDFA is very nearly constant, independent of the number of channels, the gain experienced by each channel will, therefore, depend on the number of channels present. This will induce time-varying perturbations, via transient cross-saturation in the amplifier, on other wavelengths. These perturbations, which in general accumulate along an amplifier chain, may grow large in systems that undergo reconfiguration. Cross-saturation in the network’s EDFAs will induce power transients in the surviving channels, the speed of which is proportional to the number of amplifiers in the network. The increased gain when channels are dropped can give rise to surviving channel errors since the power of the surviving channels may surpass the thresholds for nonlinear effects such as Brillouin scattering. Rapidly changing gain, due to channel drop or addition, can lead to errors as the receiver’s ability to adapt to changing power levels may be exceeded. An important design goal in multiwavelength optical networks is to ensure that information in one channel is not degraded by any changes in the other channels. Therefore, both steady state gain and fast transients must be controlled. The transient gain changes that occur at the time of channel dropping present a particular challenge in amplifier cascades. While typical time scales for gain changes in a single amplifier are tens of microseconds, the time constant for a chain of N amplifiers is 1/N times shorter than that of a single amplifier [6]. Thus long chains of amplifiers will require faster control to limit the undesirable power excursions, presenting a greater gain stabilization challenge.


Model Implementation The Transient EDFA model is based on the spectrally resolved numerical model of Giles [3] which assumes a homogeneous broadened gain medium.. As shown in [3], the model equations presented above can be significantly simplified by dealing with the average level-population densities and eliminating their explicit spatial dependence via measured gain and loss spectra (as opposed to emission and absorption cross sections) that are weighted by the local doping and mode profiles. The Giles model uses these simplifications. Such a model is generally valid for strongly confined erbium-doping profiles.The fiber is completely characterized by only four parameters; the Er3+ absorption coefficient a(λ), the gain coefficient g(λ), the fiber intrinsic background loss α, and fiber saturation parameter ζ ( fiber_saturation_param) = Ant/τ . Here A is the erbium core area, nt is the ion density, and τ is the metastable lifetime. The model uses both the time and space dependent rate (Eqn. 20) and propagation (Eqn. 18) equations of [3] which are repeated below as equations (7) and (8) respectively:

.)()(

22

22,2

11,2

τππσσ nnPnPnd

k effk

kekk

k effk

kakk

bhv

z

bhv

zdt

−−= ∑ Γ∑ Γ (7)

dPdz

u g nn

P z u g nn

mhv u l Pkk k k

tk k k

tk k k k k= + + − +( ) ( ) ( )* *

.α α2 2 (8)

α(λ) = σa(λ)Γ(λ)nt (9)

g∗(λ) = σe(λ)Γ(λ)nt (10)

tn n nr z r z r z( , , ) ( , , ) ( , , ).φ φ φ= +1 2 (11)

The first term of equation (7) represents the metastable population through absorption of light, the second term represents the stimulated emission which causes depletion of the metastable level and the last term represents spontaneous emission from the metastable level. Using equations (9) and (10), defining a new fiber parameter,

ζ π τ= b neff

t2 , the ratio of the linear density of ions to the metastable lifetime, assuming that Γk,1 and Γk,2 are nearly

equal (i.e. the erbium ions are well confined to the center of the optical modes), equation (7) becomes:

2

1

2

2

22dn

nn

nn n

dt hv bP z

g

hv bP z

kt

k effk

k

kt

k effkk= − −∑ ∑

απ π τ( ) ( )

*

(12)

simplifying further we have:

2 1 2 2dn n n ndt hv

P zghv

P zk

kk

k

k

kkk= − −∑ ∑α

ζτ ζτ τ( ) ( )

*

(13)

Letting n nnt

= 2 and using equation (11), equation (13) becomes

dndt

P zhv

n g P zhv

n nk k

kk

k k

kk

= − − −∑ ∑αζτ ζτ τ( )

( )( )*

1 (14)

The parameter ζ can be determined from measurements of the fiber saturation power Pksat as,

ζ α= +P g hvksat

k k k( ) /* . Solving equations (8) and (14) iteratively is the foundation of the numerical Transient EDFA model. The boundary conditions at z = 0 and L for the k beams must be specified in order to solve the equations


numerically. Both equations are integrated over space (z), optical frequency (k), and time (t). The fourth-order Runge-Kutta method is used to solve the equations. The space and time are decomposed into a grid of M x N discrete bins ∆z and ∆t, respectively. The space equations are integrated iteratively for each time t = m∆t (m = 1, 2,…,N). The initial conditions of the system for the metastable state population [i.e., n2(z,t = 0)], are given by solving the steady state inversion equation (Eqn. 19) of [3], repeated below as equation (15). In the steady state regime where the populations are

time invariant, i.e. dndt

i = 0 , equation (7) provides the static solution:

2

1

nnt

k k

kk

k k k

kk

P zhv

P z ghv

=+

+

∑

∑

( )

( )( )*

αζ

αζ

(15)

This condition is applicable for CW beams and those modulated at frequencies in excess of 10 kHz [3]. The initial set of power levels at each discrete point of the fiber is obtained by substituting equation (15) into equation (8). The algorithm uses the initial conditions to evaluate the inversion for the first time step ∆t. The calculated value of the inversion is then used to calculate the signal, ASE noise spectrum, and power in the pump as a function of position. These powers are then used to evaluate the inversion at the next time step 2∆t. By this computationally intense iterative process, we are able to capture and accurately characterize the dynamic behavior of the EDFA.

Higher-Order Effects The above equations ignore higher-order effects such as homogeneous upconversion [4] and pair-induced quenching [5]. Based on the treatments in [1], [4], and [5], we have incorporated these effects into the Physical EDFA model. Homogeneous upconversion is accounted for via the parameter giles_upconversion_coeff . Pair-induced quenching is incorporated via the parameter pair_fraction, which should be set equal to the fraction of Er ions that appear in pairs within the fiber. This parameter is equal to twice the parameter k from [1] and [5].

EDFA Configurations In specifying the configuration of the EDFA, the user must always specify a fiber length via the length parameter. They may also provide coupling losses at both the input and output nodes via the parameters forward_input_loss, backward_input_loss, and output_loss. Furthermore, they may select to have any pumps (i.e., optical signals with wavelengths near 980 or 1480 nm) excluded from the model output via the output_pumps parameter. Setting this option to no is useful in cases where the pump signal no longer impacts system performance in components that follow the EDFA.

The EDFA model may also be used to simulate bidirectional signal propagation. In this case, the parameter bidirectional should be set to yes. The user may then provide input signals at both the forward and backward input nodes. The backward output appears at the backward output node of the model. The user may specify a backward output coupling loss via the parameter backward_output_loss. Note that the parameter bidirectional must be set to yes if the user is only providing input signals at the backward input.

In addition to these basic configuration settings, the Transient EDFA model also supports various EDFA pump/signal recycling schemes [2]. By placing mirrors at either end of an EDFA, pumps and/or signals may be recycled, thereby providing the opportunity for enhanced amplification. Such configurations may be selected through the mirror_configuration parameter. The most basic option is no_mirror, in which no pump/signal reflectors are included in the EDFA (This configuration is always activated if bidirectional is set to yes.). In cases where forward-propagating optical inputs are to be recycled (typically pumps), the forward_mirror option should be selected. This arrangement is illustrated in Fig.3(a). Alternatively, backward-propagating inputs may be recycled via the backward_mirror option, shown in Fig.3(b). In this case, the user may select to have the reflected signal included in the EDFA output via the output_reflected option. Finally, the user may choose to adopt a signal recycling scheme, such as that depicted in Fig.3(c), wherein the EDFA input and output are actually at the same end of the device, with an optical circulator providing separation between the two. Both signal and pump may be recycled in this manner. This option may be chosen by setting mirror_configuration to signal_mirror.


input

EDFA

output

EDFA

inputoutput

O

reflector

reflector

input

EDFA

output

reflector

input (bwd)

circulator

(a) (b)

(c)

Figure 3: EDFA pump/signal recycling configurations. (a) Co-propagating pump reflector. (b) Counter-propagating pump reflector. (c) Signal/pump recycler.

The spectral characteristics of any mirrors included in the EDFA are set via the mirror_model, mirror_reflectivity, mirror_center, mirror_bandwidth, and mirror_file parameters. Setting mirror_model to rectangular implements a rectangular spectra with the in-band reflectivity specified by mirror_reflectivity, a center wavelength specified by mirror_center, and a bandwidth specified by mirror_bandwidth. The Gaussian option for mirror_model uses the same parameters, but of course implements a Gaussian wavelength dependence. Alternatively, a reflectivity spectrum may be read in directly from a file by setting mirror_model to file, and providing a file name in mirror_file. The file format is described in the appendix.


Absorption/Emission Spectra For optical signals and ASE in the 1550-band, the user must specify the Giles gain/loss spectra . By setting the spectra_1550_model parameter, the user may select either built-in default spectra or device-specific data that is to be read in from a file.


If spectra_1550_model is set to user_specified, then spectra must be provided through input files. The appropriate Giles gain/loss files are identified with the parameters giles_loss_1550_file and giles_gain_1550_file. The file format is described in the appendix.

A loss or absorption spectrum must also be provided for signals in the 980-band. In this case, three choices are available via the parameter spectra_980_model. For either a rectangular profile (rectangular) or Gaussian profile (Gaussian), the


user must specify a center wavelength (spectra_980_center) and spectral width (spectra_980_width). The center-wavelength loss is set via giles_loss_980. If spectra_980_model is set to file, then the 980-band spectrum must be provided through an input file. The Giles loss file is identified with the parameter giles_loss_980_file. The file format is described in the appendix.


Thermal Dependence Thermal dependence of the loss/gain spectra is modeled based on the work of M. Bolshtyansky et al. in [7]. Given a set of reference spectra αref (loss) and gref (gain) at a temperature Tref, spectra αnew and gnew at a new temperature Tnew are calculated as:

1 2

1 2

/ /

/ /

( )( ) ( )( )

new new

ref ref

T T T Tref

new ref T T T Tnew

T K e eT K e e

λα λ α λλ

− −

− −

+= ⋅ ⋅+

(16)

1 2

1 2

/ 1/ /

/ /

( )( )( ) ( )( )( )

ref newnew new

ref ref

T TT T T T


new ref

T gK e eg gT K e e

λλλ λα λλ

−− −

− −

+= ⋅ ⋅ ⋅ +

(17)


To activate this model, the user must set thermal_model equal to builtin or user, and specify the temperature (reference_temp) at which the reference spectra were measured, and the new temperature (actual_temp) at which the spectra should be modeled. The loss/gain spectra specified earlier by the user are considered to be the reference spectra.






If background_loss_model is set to constant, then uniform background loss is assumed, the value of which may be specified with the parameter background_loss. Alternatively, different loss values can be provided for signals in the


1550- and 980-bands by setting background_loss_model to two_constant and specifying an additional 980-band loss value through the background_loss_980 parameter. Finally, a background loss spectrum may be read in from a file by setting background_loss_model to file and specifying a file name with the background_loss_file parameter. The file format is described in the appendix.


Numerical Settings A number of numerical parameters are available to optimize the numerical solution of the bidirectional rate equations for the pump/signal/ASE power evolution. Adjustment of these parameters may help overcome any convergence difficulties encountered during a simulation.

The numerical integration of the power evolution rate equations is handled via a constant-step-size fourth-order Runge-Kutta algorithm. The integration step along the length of the EDFA may be adjusted using the z_step parameter. Similarly, the nominal width for the discretized frequency intervals in the 980- and 1550-bands may be set using the spectral_step parameter. As this parameter is set in units of nanometers, an equivalent spectral step in Hertz is calculated internally by the model.

As mentioned before, the initial conditions of the model is calculated by solving the steady-state equation shown in equation (15). The bidirectional nature of the power-evolution rate equations requires an iterative solution scheme in this case, the control of which may be achieved through the iterative_damping and convergence_tolerance parameters. The iterative_damping controls the rate at which the solution is allowed to progress to a final answer. Larger values slow down this process, and may be required when convergence is difficult to achieve. Parameter convergence_tolerance is the convergence criterion. Convergence is achieved when the largest change in the power solution between successive iterations is less than convergence_tolerance. To limit the time spent on a poorly converging solution, the maximum number of iterations may be set with the parameter max_iterations. The user can set the number of simulation iterations to calculate steady state solutions before entering the transient regime. This is done using the num_steady_state_iter parameter. This is often desirable, since it can take a number of simulation iterations before some system topologies reach a steady state. We typically want to perturb the system after it has reached a steady state operating point. A very important numerical parameter for the Transient EDFA model is the Time Step Per Iteration value that is set in the Multiple Iterations tab of the Run Window. This value is often referred to as the simulation time step. This is implemented as a global simulation parameter rather than being local to the model because many of the models that support transient simulations need to be synchronized to the same time step. The size of the simulation time step is often limited by some physical properties associated with the design of the EDFA. For instance, an all-optical fixed-gain EDFA may impose limits on the maximum time step. The propagation delay around the lasing loop or lasing cavity must be taken into consideration. Otherwise, except for considerations related to numerical convergence, the user can adjust the simulation time step while trading accuracy for total simulation time.


Reference Plots In order to help study the performance of an EDFA within an optical link, a variety of reference plots can be generated by the model in order to study internal power evolution, signal gain, noise figure, and the atomic-manifold population densities. What plots are generated is determined via the parameters power_plots, spectra_plots, density_plots,


gain_nf_plot (by setting them to either yes or no); they may be displayed at the conclusion of a simulation by double-clicking on the EDFA’s icon. The generate_plots parameter allows the user to turn on and off all plots using a single control. The units for the displayed data may be selected via the parameters spectral_units, power_units, length_units, radial_units, density_3d_units, and signal_gain_units (for gain and noise figure data). . Below we summarize the different plots that may be generated, listing them by their root WinPlot file names. The plotting_interval parameter allows the user to set the simulation times at which the model will periodically generate internal plots. This level of control is very important when working with topologies that are simulated using OptSim’s multiple iteration capability. Generating a set of plots for each execution of the system under study could lead to an excessively large number of result files being saved to disk. power_plots


Displays the evolution of total power in the 1550-band signals (1480-nm pumps excluded) and ASE along the length of the EDFA. Both forward and backward directions of propagation are included.




Displays the evolution of signal gain along the EDFA.


Displays a contour plot of the complete forward-propagating 1550-band power spectra solution along the length of the EDFA.



.

spectra_plots










Displays the internal ASE power spectra at both ends of the EDFA.

• gain.pcs:


• gain_bwd.pcs:








Displays the overall 1550-band noise figure, calculated as a function of signal gain ( )G υ and output ASE

spectral density ( )ASEρ ν [1]:

( )1 1

( )ASENF

G hρ ν

ν ν = ⋅ +


Bidirectional mode only. Displays the overall backward 1550-band noise figure

density_plots


Displays the average population densities along the length of the EDFA in both the upper and lower atomic levels.

• n2(r)_vs_z.pcs:

(spatial model only) Displays a contour plot of the transverse population density along the EDFA for the upper atomic level.

gain_nf_plot

• gain_scan.pcs:

Displays the gain of the EDFA at the wavelength specified via the parameter target_wavelength. If the topology is simulated via a parameter scan, then this plot shows the gain as a function of the scanned parameters. This plot can be useful for gain optimization of the EDFA.


Bidirectional mode only. Displays the backward gain of the EDFA at the wavelength specified via the parameter target_wavelength.


Displays the noise figure of the EDFA at the wavelength specified via the parameter target_wavelength. If the topology is simulated via a parameter scan, then this plot shows the noise figure as a function of the scanned parameters.


Bidirectional mode only. Displays the backward noise figure of the EDFA at the wavelength specified via the parameter target_wavelength.

Test Functions In selecting the input parameters for a particular EDFA, it may at times be necessary to visualize the various input spectra, doping profiles, and mode shapes. By setting the test_function parameter to the desired output and clicking the Test button in the component-parameter editing window, the user may display the EDFA characteristics summarized below. Furthermore, the default plot ranges for each characteristic may be overridden by setting test_default_settings


to no, and specifying values for test_function_x_low, test_function_x_high, and test_function_points (the number of data points to plot). The appropriate units for each characteristics’ overrides are listed in parentheses below. Units for the displayed plots may be specified using the parameters spectral_units, density_3d_units, loss_units, and reflectivity_units.







• mirror(nm):






[5] E. Delevaque, T. Georges, M. Monerie, P. Lamouler, and J.-F. Bayon, “Modeling of pair-induced quenching in erbium-doped silicate fibers,” IEEE Photonics Technology Letters, 5, 73-75 (1993).

[6] J. L. Zyskind, Y. Sun, A. K. Srivastava, J. W. Sulhoff, A. J. Lucero, C. Wolf, and R. W. Tkach, “Fast Power Transients in Optically Amplified Optical Networks,” Optical Fiber Communication Conference (Optical Society of America) postdeadline paper PD31.


Properties



Outputs #1: Optical signals

Parameters Values Name Type Default Range Unit


simulation_mode Noneditable string

“giles_params”


mirror_configuration enumerated “no_mirror” no_mirror, forward_mirror, backward_mirror, signal_mirror











pair_fraction double 0.0 0 ≤ x ≤ 1








spectra_980_model enumerated rectangular rectangular, Gaussian, file












T1_1550 double 90 -1e32 ≤ x ≤ 1e32 K

T2_1550 double 650 -1e32 ≤ x ≤ 1e32 K


T1_980 double 90 -1e32 ≤ x ≤ 1e32 K

T2_980 double 650 -1e32 ≤ x ≤ 1e32 K

background_loss_model enumerated no_loss no_loss, constant,


two_constant, file




mirror_model enumerated rectangular rectangular, Gaussian, file




mirror_file string






num_steady_state_iter integer 10 0 ≤ x ≤ 100000







generate_plots Enumerated “no” “yes”,”no”

plotting_interval double 100e-6 0 ≤ x ≤ 100 sec








test_function enumerated 1550_spectra(nm) 1550_spectra(nm), 980_spectra(nm), doping(um), modes(um), background_loss(nm), mirror(nm)







simulation_mode fixed to “giles_param” for now. bidirectional switch for simulating bidirectional signal propagation mirror_configuration pump/signal recycling options output_reflected switch for including any reflected backward inputs in the output output_pumps switch for including pump signals in the output inject_ASE switch for injecting ASE within the EDFA ASE_polarization switch for injecting ASE into one or two polarizations forward_input_loss forward-propagating input coupling backward_input_loss backward-propagating input coupling output_loss output coupling length EDFA length metastable_lifetime metastable-level lifetime fiber_saturation_param Giles-model fiber-saturation parameter pair_fraction fraction of dopant ions that are paired giles_upconversion_coeff upconversion coefficient for Giles model spectra_1550_model 1550-band signal spectra options spectra_1550_auto_limits option for automatically determining 1550-band spectral range spectra_1550_low lowest wavelength for 1550-band spectra spectra_1550_high highest wavelength for 1550-band spectra giles_loss_1550_file user-specified 1550-band Giles loss-spectrum data file giles_gain_1550_file user-specified 1550-band Giles gain-spectrum data file spectra_980_model 980-band signal spectrum options spectra_980_auto_limits option for automatically determining 980-band spectral range spectra_980_low lowest wavelength for 980-band spectra spectra_980_high highest wavelength for 980-band spectra giles_loss_980 center-value of 980-band Giles loss spectrum spectra__980_center center-wavelength of 980-band spectrum spectra_980_width bandwidth of 980-band spectrum giles_loss_980_file user-specified 980-band Giles loss-spectrum data file thermal_model option for determining thermal dependency of gain/loss or

emission/absorption spectra reference_temp temperature at which gain/loss or emission/absorption spectra were measured actual_temp actual operating temperature of amplifier F1F2_ratio_1550_file empirical F1/F2 thermal data file for 1550-band spectra T1_1550 empirical T1 thermal parameter for 1550-band spectra T2_1550 empirical T2 thermal parameter for 1550-band spectra F1F2_ratio_980_file empirical F1/F2 thermal data file for 980-band spectra T1_980 empirical T1 thermal parameter for 980-band spectra T2_980 empirical T2 thermal parameter for 980-band spectra background_loss_model background-loss options background_loss 1550-band background-loss value


background_loss_980 980-band background-loss value background_loss_file user-specified background-loss data file mirror_model mirror power-reflectivity spectrum options mirror_reflectivity mirror power reflectivity at center wavelength mirror_center mirror-spectrum center wavelength mirror_bandwidth mirror-spectrum bandwidth mirror_file user-specified mirror-spectrum data file z_step integration step along EDFA length spectral_step spacing between wavelengths in discretized power spectra iterative_damping iterative damping factor for numerical solution convergence_tolerance convergence tolerance for numerical solution max_iterations maximum number of iterations during bidirectional solution generate_plots switch to override individual plot settings plotting_interval the regular time interval to generate internal plots gain_application method of applying gain to each input signal power_plots switch for power-solution reference plots spectra_plots switch for power-spectra reference plots density_plots switch for level-density reference plots gain_nf_plot switch for plots of gain and noise-figure scans target_wavelength target wavelength for plots of gain and noise-figure scans spectral_units units for spectral data power_units units for power data length_units units for positional data density_3d_units units for density data loss_units units for loss data signal_gain_units units for gain and noise figure results reflectivity_units units for mirror data test_function test-function output selection test_default_settings switch for plotting test-function output using default settings test_function_x_low user-specified lowest x-value for test-function output test_function_x_high user-specified highest x-value for test-function output test_function_points user-specified number of points for test-function output


Appendix: File formats For all data files, the x values should be monotonically increasing or decreasing. Furthermore, the field <num_pts> specifies the number of data lines in the file, while the choice of settings for the various unit fields are as follows:







Giles Gain/Loss Spectra Data files with gain and loss spectra for the Giles model are specified through the parameters giles_gain_1550_file, giles_loss_1550_file, and giles_loss_980_file. The x-values are in units of frequency, and the y-values are in units of loss per distance.


<num_pts>





...


5

1450 0.3

1500 0.6

1530 2.0

1550 1.2

1600 0.4



<num_pts>






...


5

1500 0.40

1510 0.45

1520 0.50

1530 0.45

1540 0.40



<num_pts>





...


5

1.40 0.25

1.45 0.21

1.50 0.19

1.55 0.17

1.60 0.21

Mirror Reflectivity Spectrum A data file with the mirror reflectivity spectrum may be specified using the parameter mirror_file. The x-values are in units of frequency, and the y-values are in units of power-reflectivity.


<num_pts>






...


5

970 10

975 100

980 100

985 100

990 10


Static Optical Switch (2x2)

This model represents a 2X2 static optical switch. It takes an optical input signal on each port and switches it to one of the two output ports depending on the state of the switch. The model allows the user to set independent loss values for both the bar and the cross states. The crosstalk levels for both states can also be set independent of each other. The switch state can be set to “bar” or “cross”. In “bar” state the signal at input #1 is connected to output #1. In “cross” state the signal at input #1 is connected to output #2. Input #2 will have the opposite connectivity of the signal at input #1. A third option allows the user to set the switch state using an indexed value of an array. In this case, the array is specified by the switch_array parameter and the particular index is set using the array_index parameter. An index of “0” refers to the first element of the array. The array should contain binary values where a “0” indicates bar state and a “1” indicates cross state. The array option is very useful when this simple 2X2 component is used to create larger switch architectures in the context of a Compound Component (CC). All the switch states can be controlled from a single array with the individual switch blocks indexing different elements of the array to control their individual states.

Crosstalk in switches can be categorized by its order. When a 2X2 elementary switch is in the “bar” or “cross” states, a fraction ε of the iput optical power to the switch will leak into the wrong output port. The leakage fraction ε, expressed in dB, is called first order crosstalk. Another terminology that is often used is extinction ration, which is defined as:

Power at inactive port Power at active port 1

Extinction Ratio ∈= =−∈

A crosstalk signal through a switching fabric that passes through K of these wrong-way hops is referred to as a k-th order crosstalk term. The parameter single_band_mode determines if the signal and crosstalk terms are combined in single-band or multi-band representation at the output ports. Single-band mode should only be used for time-sequenced signals.

Properties


#2: Optical signal


#2: Optical signal

Parameter Values Name Type Default Range Units bar_loss double 0.0 [ 0, 1e32 ] dB

cross_loss double 0.0 [ 0, 1e32 ] dB

bar_state_xtalk_level double -30 [ -1e32, 0 ] dB cross_state_xtalk_level double -30 [ -1e32, 0 ] dB

switch_state enumerated “bar” “bar”, “cross”, “array_index”

none

switch_array Integer array [0,0,0,0,0,0,0,0] none


array_index integer 0 [ 0,64] none

single_band_mode enumerated “No” “No”, “Yes” none drop enumerated “No” “No”, “Yes” none

drop_threshold double -1e-32 [ -1e32, 1e32 ] dBm

Parameter Descriptions bar_loss the loss experienced by signals when the switch is in bar state cross_loss the loss experienced bys signals when the switch is in the cross state bar_state_xtalk_level the bar state crosstalk level of the switch cross_state_xtalk_level the cross state crosstalk level of the switch switch_state the state of the switch. If it is set to “array_index” the array_index parameter is used

to index into the switch_array integer array to read a binary value that is used to set the state. A “0” means bar state and a “1” means cross statet

swtch_array An array containing binary values that can be used to set the switch state. aray_index the index of the switch_array array that is used to set the switch state. single_band_mode turns on and off single-band signal combination mode. drop turns on and off the option to drop signals with power levels below a threshold value drop_threshold The threshold level below which signals are dropped if the drop parameter is set to

“Yes”


Transient Optical Switch (2x2)

This model represents a transient 2X2 switch that is capable of a single state change during the course of a simulation event. The initial state can be controlled by the initial_state parameter. The initial state can also be set from using the switch_array parameter. In this case, the state will be taken from the array using the array_index parameter, where an index value of “0” represents the first member of the array. As was the case in the Static Optical Switch model, a “0” represents bar state and a “1” is used to represent cross state. The model has a number of parameters that control the switching time of the model. Figure 4 shows the optical power at an output of the switch for two different wavelengths at the input. The other output port will experience a similar state transition, but the nominal wavelengths will be interchanged. The parameters start_first_trans, end_first_trans, start_second_trans, and end_second_trans occur at 100 µs, 150 µs, 200 µs, and 250 µs respectively, in Figure 4 below. The parameter single_band_mode determines if the signal and crosstalk terms are combined in single-band or multi-band representation at the output ports. Single-band mode should not be used for CW input signals.

Figure 4: Switch Transition Times

Properties


#2: Optical signal


#2: Optical signal

Parameter Values Name Type Default Range Units bar_loss double 0.0 [ 0, 1e32 ] dB


cross_loss double 0.0 [ 0, 1e32 ] dB

bar_state_xtalk_level double -30 [ -1e32, 0 ] dB cross_state_xtalk_level double -30 [ -1e32, 0 ] dB

initial_state enumerated “bar” “bar”, “cross”, “array_index”

none

switch_array Integer array [0,0,0,0,0,0,0,0] none array_index integer 0 [ 0,64] none

single_band_mode enumerated “No” “No”, “Yes” none

drop enumerated “No” “No”, “Yes” none

drop_threshold double -1e-32 [ -1e32, 1e32 ] dBm start_first_trans double 10e-6 [0,1] sec

end_first_trans double 20e-6 [0,1] sec

start_second_trans double 40e-6 [0,1] sec

end_second_trans double 50e-6 [0,1] sec

Parameter Descriptions bar_loss the loss experienced by signals when the switch is in bar state cross_loss the loss experienced bys signals when the switch is in the cross state bar_state_xtalk_level the bar state crosstalk level of the switch cross_state_xtalk_level the cross state crosstalk level of the switch switch_state the state of the switch. If it is set to “array_index” the array_index parameter is used

to index into the switch_array integer array to read a binary value that is used to set the state. A “0” means bar state and a “1” means cross statet

swtch_array An array containing binary values that can be used to set the switch state. aray_index the index of the array that is used to set the switch state. single_band_mode turns on and off single-band signal combination mode. drop turns on and off the option to drop signals with power levels below a threshold value drop_threshold the threshold level below which signals are dropped if the drop parameter is set to

“Yes” start_first_trans time at which the switch starts changing state end_first_trans time at which the switch puts out no light start_second_trans time at which the switch starts putting out a minimum amout of light end_second_trans time at which the state change is complete.


Dynamic Optical Switch (2x2)

This model represents a dynamic optical switch. It takes two optical input signals and a single binary input signal on the input ports. There are two optical output ports. The binary signal acts as the control signal of the switch. At “1” buts it in cross state and a “0” puts it in bar state. The switch itself has a set of parameters that control the time it takes for it to change state. The time_to_turn_off is the time that it takes the light out at the output ports to go from maximum output to zero output. The time_off is the time that the light output from either output port is exactly zero. Practically, this can occur if the switch is based on moving mechanical parts. The time_to_turn_on parameter determines how much time it takes for the light output to go from zero to maximum optical power at the output of the switch. The total of these three times is the total switching time of the switch. The other parameters have the same meaning as the other switches that were described in the previous two sections. The parameter single_band_mode determines if the signal and crosstalk terms are combined in single-band or multi-band representation at the output ports. Single-band mode should not be used for CW input signals.

Properties


#2: Optical signal

#3: Binary signal


#2: Optical signal

Parameter Values

Name Type Default Range Units bar_loss double 0.0 [ 0, 1e32 ] dB cross_loss double 0.0 [ 0, 1e32 ] dB

bar_state_xtalk_level double -30 [ -1e32, 0 ] dB

cross_state_xtalk_level double -30 [ -1e32, 0 ] dB

single_band_mode enumerated “No” “No”, “Yes” none drop enumerated “No” “No”, “Yes” none

drop_threshold double -1e-32 [ -1e32, 1e32 ] dBm

time_to_turn_off double 5e-6 [0,1] sec

Time_off double 1e-6 [0,1] sec Time_to_turn_on double 5e-6 [0,1] sec

Parameter Descriptions bar_loss the loss experienced by signals when the switch is in bar state


cross_loss the loss experienced bys signals when the switch is in the cross state bar_state_xtalk_level the bar state crosstalk level of the switch cross_state_xtalk_level the cross state crosstalk level of the switch Single_band_mode turns on and off single-band signal combination mode. drop turns on and off the option to drop signals with power levels below a threshold value drop_threshold the threshold level below which signals are dropped if the drop parameter is set to

“Yes” time_to_turn_off time at which the switch starts changing state time_off time at which the switch puts out no light time_to_turn_on time at which the switch starts putting out a minimum amout of light


Transient Plotter

This model plots the signals at its input using a Java plotting package that is capable of generating live plots (i.e. the user can start the display immediately after starting the simulation and observe the progress of the simulation). The graph is automatically rescaled every fillInterval iterations to enhance the viewing for the user. The horizontal scale is in units of seconds and is automatically calibrated by the iterations and the Time Step Per Iteration global simulation parameters. The model also allows for the monitoring of a single signal generated from the ASE noise of optical amplifiers. The aseCenterWavelength can be used to set the center wavelength of the region of the ASE spectrum that should be plotted. The plotAseSignal parameter is used to turn on and off the plotting of this signal. The power values of the signal can be generated either in units of dBm or mW. If the unit is specified in dBm, the zeroPowerFloor parameter becomes relevant. It is used to truncate the display of power values that are nominally at 0 Watts. The model has a multiporthole input. This allows several input signals to be simultaneously connected to the component and displayed for comparison. The order of the plots as shown in the legend on the plot window depends on the order in which the connections are made. This model can also be used inside an iteration loop. However, in this case, the number of connections is restricted to one. The plotter will output signals generated at each loop iteration. The legend in the plot window indicates the number of the loop iteration associated with a particular plot.

Properties


Parameter Values Name Type Default Range Units plotTitle string Transient Plots

powerUnits enumerated “mW” “dBm”, “mW”

displayOptions string "-tk =500x300 -x seconds -y dBm "

fillInterval integer 1000

plotAseSignal enumerated “No” “No”, “Yes” aseCenterWavelength double 1531 nm

zeroPowerFloor double -90 dBm

Parameter Descriptions plotTitle The title of the plot powerUnits the units to use for the vertical scale. This overrides the vertical label that is set in

transientOptions displayOptions options that set the size of the frame and optionally the labels for the vertical and

horizontal scales. fillInterval Regular interval at which the live plot is automatically rescaled to allow better

viewing plotAseSignal Indicates if an ASE signal should be plotted or not. Useful for a ring laser

configuration. aseCenterWavelength optional center wavelength to plot an amplified spontaneous emission (ASE) noise

i l hi i l d l i l l i i l h i d f h


signal. This is mostly used to plot a particular lasing signal that is generated from the ASE noise.

zeroPowerFloor the zero power level that is used when powerUnits is set to “dBm”

OptSim Models Reference: Block Mode Chapter 13: Multimode Modules •••• 469

Chapter 13: Multimode Modules

This chapter describes the spatial multimode models.

• Spatial Adder attach a spatial field to an optical signal

• Spatial Direct Modulated Laser the spatially enabled version of the direct-modulated laser

• Spatial Mode-Locked Laser the spatially enabled version of the mode-locked laser

• Spatial CW Laser the spatially enabled version of the continuous-wave laser

• Spatial VCSEL the spatially enabled version of the vertical-cavity surface emitting laser

• Spatial LED the spatially enabled version of the light-emitting diode

• Thin Lens focus a spatial field

• Vortex Lens focus a spatial field with vortex phase transformation

• Spatial Coupler perform rotational and translational offsets as well as free-space propagation

• Spatial BeamPROP Interface provide interface to RSoft’s BPM simulation tool BeamPROP

• Multimode Fiber model for the multimode fiber

• Spatial Aperture pass spatial fields through an aperture, cropping when necessary

• Spatial Photodetector the spatially enabled version of the photodetector

• Spatial Receiver the spatially enabled version of the photoreceiver

470 •••• Chapter 13: Multimode Modules OptSim Models Reference: Block Mode

• Spatial Analyzer perform various calculations on spatial fields and produce spatial field plots

• Encircled Flux Analysis Tool calculate the encircled flux of a laser/fiber pair

• Differential Mode Delay Analysis Tool measure the differential mode delay of a given fiber

• Signal Band Converter collapse a linked list of spatially orthogonal fields into a single signal

• Gridded Field Converter convert an optical signal’s spatial mode profiles into two-dimensional gridded representations


Spatial Adder

The spatial adder takes multiple optical inputs, and attaches transverse mode profiles to their X and Y polarizations. Different profiles can be attached to each polarization, and a single-mode input can be converted into a set of multimode optical signals with identical time-domain shapes (scaled by a specified power distribution) but different mode shapes. The available mode profiles include those presented in Chapter 6 of the OptSim User Guide.

While the spatial adder supports multiple input nodes, with potentially multiple optical signals at each node, the entire set of input signals is treated as if it had appeared on a single node. The output signals at each node, however, do correspond to the correct input.

General Options The spatial adder can be deactivated by setting spatial_effects to off. With this parameter on, however, the full range of options become available. The first of these is the mode_polarization parameter, and it determines how modes are attached to an input field’s different polarizations. If mode_polarization=x_only, then a transverse mode profile is only attached to an input signal’s X polarization. If mode_polarization=y_only, then a profile is only attached to the Y polarization. If mode_polarization=xy_same, then the same profile that is attached to the X polarization is attached to the Y polarization. Finally, if mode_polarization=xy_unique, then different mode profiles must be specified for each polarization.

Each transverse mode profile in OptSim has a default grid spacing and domain, in order to facilitate numerical calculations. The grid spacing is specified on a rectangular XY grid. The user may override the default grid spacings by setting mode_default_grid_spacing=no, and then adjusting parameters mode_dx and mode_dy. The domain, meanwhile, determines the default rectangular space over which the field is considered to have non-negligible magnitude. Again, the user may override the default by setting mode_default_domain=no, and adjusting mode_width_x and mode_width_y. The domain is automatically centered about the origin.

Mode-Attachment Styles The user has a variety of options for the style in which mode profiles are attached to the Spatial Adder’s optical inputs, and these are selected via the mode_mapping parameter. The three options are: single, multi, and multi_from_single. In the descriptions that follow, parameters for the X polarization are prefixed with x_, while parameters for the Y polarization are prefixed with y_.

• single

With this option, the user identifies specific mode profiles for either or both polarizations, and these profiles are attached to every optical input. For each polarization, the user selects a mode shape via the parameter x_single_mode_type (y_single_mode_type). The available mode types are: file, radial_file, LG, HG, LP, spot, and donut. These will be described shortly. The user must also identify mode indices via the parameters x_l (y_l) and x_m (y_m).

• multi

With this option, the user specifies a family of m mode profiles for either or both polarizations. These profiles are attached in sequence to the Spatial Adder’s n optical input signals. If m > n, then only the first n profiles are used. If n > m, then the profiles are reused in sequence until each input has been handled. For each polarization, the user selects a


mode shape via the parameter x_multi_mode_type (y_multi_mode_type). The available types are: file, LG, HG, and LP. For built-in mode shapes, the user must then specify a range of values for the mode indices. This is accomplished via the parameters x_l_lo (y_l_lo), x_l_hi (y_l_hi), x_m_lo (y_m_lo) and x_m_hi (y_m_hi) In cycling through these mode indices during processing of the input signals, the Spatial Adder first loops through the l index, and for each value, loops through all of the m indices.

If x_multi_mode_type (y_multi_mode_type) is set to file, then the user must provide a file which specifies a list of different mode shapes. The advantage of this approach is that different profile types can be mixed, such as LP modes with Laguerre-Gaussian modes. The name of this file is specified through the parameter x_multimode_file (y_multimode_file). The format for this file is described later.

• multi_from_single

The Spatial Adder can also convert individual optical inputs into multiple output signals, each with different mode shapes, but with time-domain behaviors identical except for a power distribution scaling factor. The user specifies the family of modes for each polarization in the same manner as was used for the multi setting. However, in this case each input is copied into multiple signals, and the modes are then attached to these copies. This technique is useful for converting a single-mode optical signal into a multimode one.

The number of copies of each signal is determined by the multi_mode_powers parameter. This parameter should be set to a list of power values which describe the power distribution between the multiple modes. These values are normalized so that they add to 1.0, and are then used to scale each copy. For example, if multi_mode_powers is set to:

1.0 2.0 3.0 4.0

then each optical input is converted into four copies, each with 10, 20, 30, and 40% of the input power, respectively. The user may also use the multi_mode_powers parameter to specify a file in which these values are stored.

Mode Types The different mode types available in the Spatial Adder (as specified via the various mode_type parameters) are as follows. (Unless otherwise noted, detailed mode descriptions can be found in Chapter 6 of the OptSim User Guide.) As before, parameters for the X polarization are prefixed with x_, while parameters for the Y polarization are prefixed with y_.

• file

Using the x_mode_file (y_mode_file) parameter, the user must specify a data file name for a two-dimensional gridded mode.

• radial_file

In this case, the user must specify a data file name for a one-dimensional gridded mode.

• LG

Laguerre-Gaussian modes will be used. The mode spot size and inverse radius of curvature are specified via the parameters x_wo (y_wo) and x_iRo (y_iRo), respectively.

• HG

Hermite-Gaussian modes will be used. The X-axis spot size and inverse radius of curvature are specified via the parameters x_wox (y_wox) and x_iRox (y_iRox), respectively; the Y-axis parameters are specified via x_woy (y_woy) and x_iRoy (y_iRoy), respectively.

• LP


Linearly polarized modes will be used. The cladding index is specified via x_Nclad (y_Nclad); the core index via x_Ncore (y_Ncore); and the core radius via x_core_radius (y_core_radius).

• spot

Spot modes will be used, with radius specified via x_outer_radius (y_outer_radius).

• donut

Donut modes will be used, with inner radius specified via x_inner_radius (y_inner_radius); and outer radius specified via x_outer_radius (y_outer_radius).

Multimode File Format The multimode settings of the Spatial Adder support the use of a data file for describing the modes. Each line of this data file contains a brief description of the modes. The format is: <# of modes>

<mode description 1>


...

Line 1 always contains the number of modes in the file. Each subsequent line contains a description for one of these modes. The supported mode types are gridded modes (one- and two-dimensional), LG modes, HG modes, and LP modes. The line format for each mode is as follows:

• Two-dimensional gridded mode: gridded <l> <m> <filename>

where l and m are mode indices, and filename is the name of the file containing the field data.

• One-dimensional gridded mode: griddedradial <l> <m> <filename>


• LG mode: lg <l> <m> <wo> <iRo>

where l and m are mode indices, wo is the spot size in microns, and iRo is the inverse radius of curvature in inverse microns.

• HG mode: hg <l> <m> <wox> <woy> <iRox> <iRoy>

where l and m are mode indices; wox and woy are the X- and Y-axis spots sizes, respectively, in microns; and iRox and iRoy are the X- and Y-axis radii of curvature, respectively, in inverse microns.

• LP mode lpfiber <l> <m> <Nclad> <Ncore> <rcore>

where l and m are mode indices, Nclad is the cladding index, Ncore is the core index, and rcore is the core radius in microns.


Test Parameters A number of parameters are available for testing the Spatial Adder settings. First, test_mode_polarization is used to select which polarization to use when plotting the mode profiles. test_mode_number is used when the Spatial Adder is in multi or multi_from_single mode. The available modes are numbered from zero, and in the order by which they are attached to optical signals. test_mode_wavelength determines the field wavelength at which to plot the test mode. Finally, test_mode_plot determines the type of plot to generate: magnitude plots the profile’s transverse magnitude, mag_phase plots its magnitude and phase, and real_imag plots the real and imaginary parts of the mode.

Properties

Inputs #1-#512: Optical Signal

Outputs #1-#512: Optical Signal

Parameter Values Name Type Default Range Units spatial_effects enumerated on on, off mode_mapping enumerated single single,

multi_from_single, multi

multi_mode_powers string 1.0 mode_polarization enumerated x_only x_only, y_only,

xy_same, xy_unique

mode_default_domain enumerated yes yes, no

mode_width_x double 1 [ 0, 1e32 ] um

mode_width_y double 1 [ 0, 1e32 ] um

mode_default_grid_spacing enumerated yes yes, no mode_dx double 0.1 [ 0, 1e32 ] um

mode_dy double 0.1 [ 0, 1e32 ] um

x_single_mode_type enumerated LG file, radial_file, LG, HG, LP, spot, donut

x_multi_mode_type enumerated LG file, LG, HG, LP x_mode_file string

x_multimode_file string

x_l integer 0 [ -75, 75 ] none

x_m integer 0 [ 0, 75 ] none x_l_lo integer 0 [ -75, 75 ] none

x_l_hi integer 0 [ -75, 75 ] none

x_m_lo integer 0 [ 0, 75 ] none

x_m_hi integer 0 [ 0, 75 ] none x_wo double 1 [ 0, 1e32 ] um

x_iRo double 0 [ -1e32, 1e32 ] um^-1

x_wox double 1 [ 0, 1e32 ] um


x_iRox double 0 [ -1e32, 1e32 ] um^-1

x_woy double 1 [ 0, 1e32 ] um x_iRoy double 0 [ -1e32, 1e32 ] um^-1

x_Nclad double 1 [ 0, 1e32 ] none

x_Ncore double 1 [ 0, 1e32 ] none

x_core_radius double 1 [ 0, 1e32 ] um x_outer_radius double 1 [ 0, 1e32 ] um

x_inner_radius double 0 [ 0, 1e32 ] um

y_single_mode_type enumerated LG file, radial_file, LG, HG, LP, spot, donut

y_multi_mode_type enumerated LG file, LG, HG, LP

y_mode_file string y_multimode_file string

y_l integer 0 [ -75, 75 ] none

y_m integer 0 [ 0, 75 ] none

y_l_lo integer 0 [ -75, 75 ] none y_l_hi integer 0 [ -75, 75 ] none

y_m_lo integer 0 [ 0, 75 ] none

y_m_hi integer 0 [ 0, 75 ] none

y_wo double 1 [ 0, 1e32 ] um y_iRo double 0 [ -1e32, 1e32 ] um^-1

y_wox double 1 [ 0, 1e32 ] um

y_iRox double 0 [ -1e32, 1e32 ] um^-1

y_woy double 1 [ 0, 1e32 ] um y_iRoy double 0 [ -1e32, 1e32 ] um^-1

y_Nclad double 1 [ 0, 1e32 ] none

y_Ncore double 1 [ 0, 1e32 ] none

y_core_radius double 1 [ 0, 1e32 ] um y_outer_radius double 1 [ 0, 1e32 ] um

y_inner_radius double 0 [ 0, 1e32 ] um

test_mode_polarization enumerated x x, y

test_mode_number integer 0 [ 0, 100000 ] none test_mode_wavelength double 1 [ 0, 1e32 ] um

test_mode_plot enumerated magnitude magnitude, mag_phase, real_imag

Parameter Descriptions spatial_effects Switch to turn spatial-addition by model block on or off.

mode_mapping Transverse mode profile style selection (single- vs. multimode).

multi_mode_powers List of mode powers, or file-name with this data. mode_polarization Polarization mode-attachment selection.

mode_default_domain Default mode domain override.

mode_width_x User-specified mode domain width along the X axis.


mode_width_y User-specified mode domain width along the Y axis.

mode_default_grid_spacing Default grid spacing override.

mode_dx User-specified X-axis grid spacing. mode_dy User-specified Y-axis grid spacing.

x_single_mode_type Single-mode profile selection for X polarization.

x_multi_mode_type Multimode profile selection for X polarization.

x_mode_file Single-mode gridded mode data file for X polarization. x_multimode_file Multimode data file for X polarization.

x_l Mode index l for X polarization.

x_m Mode index m for X polarization.

x_l_lo Lower bound for X polarization sweep of mode index l. x_l_hi Upper bound for X polarization sweep of mode index l. x_m_lo Lower bound for X polarization sweep of mode index m.

x_m_hi Upper bound for X polarization sweep of mode index m.

x_wo Laguerre-Gaussian spot size for X polarization. x_iRo Inverse of Laguerre-Gaussian radius of curvature for X polarization.

x_wox Hermite-Gaussian X-axis spot size for X polarization.

x_iRox Inverse of Hermite-Gaussian X-axis radius of curvature for X polarization.

x_woy Hermite-Gaussian Y-axis spot size for X polarization. x_iRoy Inverse of Hermite-Gaussian Y-axis radius of curvature for X-polarization.

x_Nclad Cladding index for X-polarized LP mode.

x_Ncore Core index for X-polarized LP mode.

x_core_radius Core radius for X-polarized LP mode. x_outer_radius Outer radius for X-polarized donut and spot modes.

x_inner_radius Inner radius for X-polarized donut mode.

y_single_mode_type Single-mode profile selection for Y polarization.

y_multi_mode_type Multimode profile selection for Y polarization. y_mode_file Single-mode gridded mode data file for Y polarization.

y_multimode_file Multimode data file for Y polarization.

y_l Mode index l for Y polarization.

y_m Mode index m for Y polarization. y_l_lo Lower bound for Y polarization sweep of mode index l. y_l_hi Upper bound for Ypolarization sweep of mode index l. y_m_lo Lower bound for Y polarization sweep of mode index m.

y_m_hi Upper bound for Y polarization sweep of mode index m. y_wo Laguerre-Gaussian spot size for Y polarization.

y_iRo Inverse of Laguerre-Gaussian radius of curvature for Y polarization.

y_wox Hermite-Gaussian X-axis spot size for Y polarization.

y_iRox Inverse of Hermite-Gaussian X-axis radius of curvature for Y polarization. y_woy Hermite-Gaussian Y-axis spot size for Y polarization.

y_iRoy Inverse of Hermite-Gaussian Y-axis radius of curvature for Y-polarization.

y_Nclad Cladding index for Y-polarized LP mode.

y_Ncore Core index for Y-polarized LP mode.


y_core_radius Core radius for Y-polarized LP mode.

y_outer_radius Outer radius for Y-polarized donut and spot modes.

y_inner_radius Inner radius for Y-polarized donut mode. test_mode_polarization Test plot field polarization selection.

test_mode_number Test plot mode number.

test_mode_wavelength Test plot field wavelength.

test_mode_plot Test plot style selection.


Spatial Direct Modulated Laser

This block models a semiconductor laser directly modulated with an electrical signal. It is identical to the standard Direct Modulated Laser model, except for the ability to include transverse mode profiles in the optical output.

This model computes the electrical current injected into the laser’s optical cavity and solves the laser rate equations for the optical output. The behavior of the model can be partitioned into three blocks, as shown in Fig. 1.

The driving source consists of the electrical signal input into the model. The parasitics consist of a bond inductance and shunting capacitance. Finally, the laser cavity is modeled via a simplified current-voltage (IV) relationship and the laser rate equations.

Figure 1: Main components of the Direct Modulated Laser model

Driving Source The Direct Modulated Laser is driven by a combination of the electrical signal at its input and, when applicable, a dc bias current specified by Io (Io). Io can also be specified via a dc bias output power Po (Po). The user determines which will be used via the parameter Bias_Value. The other parameter is then calculated automatically.

The input electrical signal can be interpreted in a variety of ways, based on the settings of the model parameter Drive_Scheme. This parameter can take on values of direct_drive, bias_tee, or bias_tee_old.

• Direct_drive The input signal is assumed to come from either an ideal current source or zero-impedance voltage source. In other words, the laser is assumed to be undergoing direct-drive modulation. If the input signal is a current, then it is combined with the bias current Io to form the total input current. Figure 2(a) illustrates this scenario. If the input signal is a voltage, then the bias current Io is ignored. Note that the input voltage should be larger than the laser’s turn-on voltage Von; otherwise, the resulting current is zero. This arrangement is illustrated in Fig. 2(b).



• bias_tee The input signal is assumed to be generated by a source with output impedance Rs (Rs), connected to the laser via an ideal bias tee. The bias current Io is similarly connected. The input signal can be either a current or a voltage. Figure 3 depicts a bias tee setup for both the voltage- and current-source cases.


• bias_tee_old This choice implements the driving scheme from OptSim versions prior to 3.0. Note that these previous implementations intended to model the bias tee as described above, but did not remove any dc component which may have been present in the electrical input signal that would naturally occur via the tee’s capacitive leg. While future releases of OptSim may no longer support this option, it is included here for compatibility.

Parasitics The parasitics consist of a bond inductance Lb (Lb) and shunting capacitance Cp (Cp), as shown in Fig. 1. These can be turned on or off via the parameter Parasitics.

Laser Cavity Both electrical and optical effects are modeled within the laser cavity.

Electrical The electrical model of the laser cavity is that of a simplified diode IV relationship, consisting of a series resistance Rd (Rd) and turn-on voltage Von (Von). During solution of the cavity current I, the model ensures that negative currents are effectively limited to zero.

Rate Equations The core of the Direct Modulated Laser block are the semiconductor laser rate equations, which determine the optical output in response to the cavity current I. Relative intensity noise is modeled via a constant value RIN (RIN). The rate equations, based largely on those discussed in [1], are:


dNdt

IqV

R N R N G N N Nisp nr p p= − − − ⋅

η( ) ( ) ( , )

(1)

dN

dtN

R N G N N Np p

psp sp p p= − + + ⋅

τβΓ Γ( ) ( , )

(2)

ddt

G N NN

pp

p

p

φ ατ

ετ

= ⋅ − +′

2

1Γ ( , )

(3)

P c N V hc v AFout fact p

g mir= ⋅ = ⋅ ⋅ ⋅λ Γ

(4)

where I is the injection current, N is the carrier density, Np is the photon density, φ is the optical phase, ηI (effint) is the current injection efficiency, q is the electron charge, V is the cavity volume, τp is the photon lifetime, Γ (K) is the confinement factor, βsp (b) is the spontaneous emission coupling coefficient, α (a) is the linewidth enhancement factor, ε ′ is a modified gain saturation factor (see Gain section below), h is Planck’s constant, c is the speed of light in a vacuum, λ (wavelength) is the lasing wavelength, vg is the group velocity (c/n, where n (indx) is the material index), Amir is the mirror, or facet, loss, F is the fraction of light that escapes the output facet, Rsp(N) is the radiative recombination, Rnr(N) is the nonradiative recombination, and G(N,Np) is the laser gain. The recombination terms are modeled using:

R N BNsp ( ) = 2

(5)

R N AN CNnr ( ) = + 3

(6)

where A (A) is the unimolecular recombination coefficient, B (B) is the radiative recombination coefficient, and C (C) is the Auger recombination coefficient.

Device Geometry The device geometry can be modeled in a number of ways via the parameter geometry, which can take on values of rectangular, cylindrical, or volumetric.

• Rectangular The laser is assumed to be an edge emitting laser with cavity width Lstp (Lstp), cavity thickness Lact (Lact), and cavity length Lcav (Lcav). The volume V is then computed as:

V L L Lstp act cav= ⋅ ⋅

(7)

• cylindrical The laser is assumed to be a vertical-cavity device with cavity diameter W (W), cavity thickness Lact, and total cavity length Lcav. The volume V is then computed as:

V L Wact= ⋅ π 2

4


(8)

• volumetric The volume is directly specified via the parameter V (V).

Mirror Effects When the parameter mirror_effects is set to defined, the mirror parameters Amir and F are directly specified by the parameters Amir and F, respectively. When mirror_effects is set to calculated, the mirror parameters are calculated using Lcav and the mirror reflectivities R1 (R1) and R2 (R2), where the optical output is assumed to exit from mirror #1 [2]:

AR RLmir

cav= −

ln( )1 22

(9)

FR

R R R R=

−− + − ⋅

11 1

1

1 2 1 2( ) /

(10)

Intrinsic Loss If the parameter intrinsic_loss is set to defined, then the intrinsic loss of the cavity is directly specified as Aint (Aint). When intrinsic_loss is set to calculated, then Aint is defined in relation to Amir via the scaling factor loss_ratio (loss_ratio):

A loss ratio Amirint = ⋅_

(11)

Photon Lifetime When parameter photon_lifetime is set to defined, the photon lifetime is directly specified via τp (tp). When photon_lifetime is set to calculated, the photon lifetime is calculated using [2]:

τ pgr mirv A A

=⋅ +

1( )int

(12)

Gain The laser gain G(N,Np) consists of the material gain g(N) and the optical gain saturation Φ(Np), with both terms able to take on a number of forms via the setting of the gain and saturation parameters:

G N N v g N Np gr p( , ) ( ) ( )= ⋅ ⋅ Φ

(13)

The gain parameter can take on values of logarithmic:N, logarithmic:R(N), or linear:

• Logarithmic:N The material gain is modeled as a logarithmic function of N [1]:

g N GN N

N Nos

tr s( ) ln=

++

(14)


where Go (Go) is the gain coefficient, Ntr (Ntr) is the transparency density, and Ns (Ns) is an adjustable correction parameter [1].

• Logarithmic:R(N)

The material gain is modeled as a logarithmic function of the carrier recombination [3],[4]. This approach is a more general version of the simple logarithmic gain.

g N GR N R N N

R N R N Nosp nr s

sp tr nr tr s( ) ln

( ) ( )( ) ( )

=+ ++ +

(15)

• linear

The material gain is assumed to be linearized about the transparency density Ntr, and is described by [2]:

g N G V N No tr( ) ( )= ⋅ ⋅ −

(16)

Gain Saturation The saturation parameter can also take on three values: Channin, Agrawal, or linear.

• Channin

The gain saturation is modeled following the approach in [1] and [5]:

( )p

p NN

ε+=Φ

11

(17)

where ε (e) is the gain saturation factor. In this case, the modified gain saturation factor is simply ′ =ε ε .

• Agrawal

The gain saturation is modeled following the approach in [4] and [6], and is applicable when intraband effects are important:

Φ( )NNp

p=

+1

1 ε

(18)

In this case, the modified gain saturation factor is ′ =ε ε / 2 .

• Linear

The gain saturation is assumed to be linear, though the expression is strictly only valid when Np < 1/ε. The modified gain saturation factor is ′ =ε ε .

Φ( )N Np p= −1 ε

(19)






Spatial Effects This model allows different transverse mode profiles to be attached to the output signal’s X and Y polarizations. Different profiles can be attached to each polarization, and a single-mode input can be converted into a set of multimode optical signals with identical time-domain shapes (scaled by a specified power distribution) but different mode shapes. The available mode profiles include those presented in Chapter 6 of the OptSim User Guide.

General Options The spatial effects can be deactivated by setting spatial_effects to off. With this parameter on, however, the full range of options become available. The first of these is the mode_polarization parameter, and it determines how modes are attached to an output field’s different polarizations. If mode_polarization=xy_same, then the same profile that is attached to the X polarization is attached to the Y polarization. If mode_polarization=xy_unique, then different mode profiles must be specified for each polarization.


Mode-Attachment Styles The user has a variety of options for the style in which mode profiles are attached to the laser’s optical outputs, and these are selected via the mode_mapping parameter. The two options are: single and multi_from_single. In the descriptions that follow, parameters for the X polarization are prefixed with x_, while parameters for the Y polarization are prefixed with y_.

• single

With this option, the user identifies specific mode profiles for either or both polarizations. For each polarization, the user selects a mode shape via the parameter x_single_mode_type (y_single_mode_type). The available mode types are: file, radial_file, LG, HG, LP, spot, and donut. These will be described shortly. The user must also identify mode indices via the parameters x_l (y_l) and x_m (y_m).


The model can also convert an individual optical output into multiple output signals, each with different mode shapes, but with time-domain behaviors identical except for a power distribution scaling factor. The optical output is copied into multiple signals, and different


modes are attached to each copy. This technique is useful for converting a single-mode optical signal into a multimode one.


1.0 2.0 3.0 4.0

then the output signal is converted into four copies, each with 10, 20, 30, and 40% of the original power, respectively. The user may also use the multi_mode_powers parameter to specify a file in which these values are stored.

The user must also specify a family of m mode profiles for either or both polarizations. These profiles are attached in sequence to the multiple output signals. Let n denote the number of signals into which the single output has been split. If m > n, then only the first n profiles are used. If n > m, then the profiles are reused in sequence until each input has been handled. For each polarization, the user selects a mode shape via the parameter x_multi_mode_type (y_multi_mode_type). The available types are: file, LG, HG, and LP. For built-in mode shapes, the user must then specify a range of values for the mode indices. This is accomplished via the parameters x_l_lo (y_l_lo), x_l_hi (y_l_hi), x_m_lo (y_m_lo) and x_m_hi (y_m_hi) In cycling through these mode indices during processing of the signals, the model first loops through the l index, and for each value, loops through all of the m indices.


Mode Types The different mode types available in the model (as specified via the various mode_type parameters) are as follows. (Unless otherwise noted, detailed mode descriptions can be found in Chapter 6 of the OptSim User Guide.) As before, parameters for the X polarization are prefixed with x_, while parameters for the Y polarization are prefixed with y_.

• file


• radial_file


• LG


• HG

Hermite-Gaussian modes will be used. The X-axis spot size and inverse radius of curvature are specified via the parameters x_wox (y_wox) and x_iRox(y_iRox), respectively; the Y-axis parameters are specified via x_woy (y_woy) and x_iRoy (y_iRoy), respectively.

• LP



• spot


• donut


Multimode File Format The multimode setting of the model supports the use of a data file for describing the modes. Each line of this data file contains a brief description of the modes. The format is: <# of modes>



...










• LP mode: lpfiber <l> <m> <Nclad> <Ncore> <rcore>


Multi-Line Output It is frequently necessary to produce several signals with similar properties but different wavelengths. In DWDM simulations especially, a series of regularly spaced optical sources is common. The Direct Modulated Laser model provides a number of convenient facilities so that many or all required lines can be generated from a single icon. To produce a series of lines spaced equally in wavelength or frequency, set


the parameter mode=LambdaGrid or mode=FreqGrid, respectively. The number of sources is controlled by the number of electrical inputs to the laser, and in both modes the parameter wavelength specifies the first line in a series of ascending wavelengths or ascending frequencies, respectively. The source separation in wavelength or frequency is specified with deltaFreq. To set the number of electrical inputs to the laser, select the direct modulated laser icon and open the menu item Properties. In the Ports tab, number_input_ports field, enter the number of inputs.

Multi-Line Multi-Node Output The comb of sources may be emitted either through a single node (the default) or with one channel per output node. To achieve the latter behavior, select the direct modulated laser icon and open the menu item Properties. Set the value of the number_output_ports field in Ports tab to that of the number_input_ports field.


Test Parameters In order to ascertain whether the parameter settings for the Direct Modulated Laser block provide the component performance desired, the user may test them from the component parameter editing window. Depending on the setting of the parameter test_function, this test produces a light-current (LI) curve (test_function set to LI), small-signal frequency response curves (ac), performance report (report), or mode profile (mode_profile). These are typically used to fit rate-equation laser model parameters to the performance of actual semiconductor lasers. The test can be controlled via model parameters carrying a prefix Test_. These parameters allow the user to set sweep limits, bias conditions, etc.

LI Curve The LI curve is controlled via the parameters Test_LItype, Test_Imin, Test_Imax, and Test_Lipoints. When Test_Litype is set to auto-scaled, an LI curve is generated over currents ranging from 0 to Imax, where Imax is the larger of 3Ith or 2Io, Ith being the calculated laser threshold current. With Test_Litype set to user-specified, the LI curve is generated over currents ranging from Test_Imin to Test_Imax. The total number of points in the LI curve is determined by Test_Lipoints. A sample LI curve is shown in Fig. 4.

Figure 4: Sample test LI curve, generated using automatic scaling


Small-Signal Frequency Responses Two sets of frequency response curves are generated. The first set depicts the device’s transfer function Tf, i.e. the small-signal output power vs. small-signal input current. This calculation includes parasitics if any are present. The second set of curves depicts the device’s normalized S21 response using the relationship S21 = 2Tf/(Zin+50), where Zin is the device input impedance and a 50-Ω test setup is assumed. Both sets of curves are generated at five different bias currents. With the parameter Test_actype set to Ith-based, these currents are 1.1Ith, 1.5Ith, 2.0Ith, 3.0Ith, and Io. When Test_actype is set to user-specified, these currents are defined by the parameters Test_I1ac, Test_I2ac, Test_I3ac, Test_I4ac, and Test_I5ac. Parameters Test_acxscale and Test_acyscale control the type of scales on the x- and y-axes, respectively. Log results in logarithmic scaling, and linear in linear scaling. Test_freqlow and Test_freqhigh specify the range of frequencies over which to generate the response curves, and Test_acpoints defines the number of points in each curve. Sample frequency-response curves are shown in Fig. 5.

Figure 5: Sample transfer-function and S21 frequency-response curves at threshold-based bias values

Performance Report In addition to the above plots, a report is generated highlighting some of the important performance metrics of the laser. These include the threshold current, differential quantum efficiency, power, or slope, efficiency, bias values Io and Po, as well as bandwidths and relaxation-peak locations for the transfer-function, S21, and intrinsic (i.e., no parasitics) frequency responses at the five bias currents of interest.

Mode Profile A number of parameters are available for testing the spatial settings. First, test_mode_polarization is used to select which polarization to use when plotting the mode profiles. test_mode_number is used when the model is in multi_from_single mode. The available modes are numbered from zero, and in the order by which they are attached to optical signals. test_mode_wavelength determines the field wavelength at which to plot the test mode. Finally, test_mode_plot determines the type of plot to generate: magnitude plots the profile’s transverse magnitude, mag_phase plots its magnitude and phase, and real_imag plots the real and imaginary parts of the mode.

Compatibility with OptSim version 2.1 VCSEL Model With this release of OptSim, the OptSim 2.1 VCSEL model can be invoked through the use of the Direct Modulated Laser block. In most cases, the old VCSEL parameters can be mapped into the present ones


with little or no change. However, the old parameters τn, τs, a, and ε can be converted to new parameters by setting ( ) ( ) ( ) ( )( ) /( )n old s old n old s oldA τ τ τ τ= + , ( ) /o oldG a V= , and ( ) /oldVε ε= Γ .

References [1] L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (John Wiley & Sons, New York, 1995).


[3] T. A. DeTemple and C. M. Herzinger, “On the semiconductor laser logarithmic gain-current density relation,” IEEE Journal of Quantum Electronics, 29, 1246 (1993).

[4] P. V. Mena, S.-M. Kang, and T. A. DeTemple, “Rate-equation-based laser models with a single solution regime,” Journal of Lightwave Technology, 15, 717 (1997).

[5] D. J. Channin, “Effect of gain saturation on injection laser switching,” Journal of Applied Physics, 50, 3858 (1979).

[6] G. P. Agrawal, “Effect of gain and index nonlinearities on single-mode dynamics in semiconductor lasers,” IEEE Journal of Quantum Electronics, 26, 1901 (1990).


Properties



Parameter Values Name Type Default Range Units wavelength double 1550e-9 [ 0, 1 ] m





effint double 0.9 [ 0, 1 ] none

K double 0.1 [ 0, 1 ] none

indx double 4.1 [ 0, 1e32 ] none geometry enumerated rectangular rectangular,

cylindrical, volumetric

Lact double 500e-8 [ 0, 1e32 ] cm

Lstp double 2.5e-4 [ 0, 1e32 ] cm


Lcav double 345e-4 [ 0, 1e32 ] cm

W double 1.5e-3 [ 0, 1e32 ] cm V double 4.3125e-11 [ 0, 1e32 ] cm^3

mirror_effects enumerated defined defined, calculated

Amir double 33 [ 0, 1e32 ] 1/cm

F double 1 [ 0, 1 ] none R1 double 0.9966 [ 0, 1e32 ] none

R2 double 0.9989 [ 0, 1e32 ] none

intrinsic_loss enumerated defined defined, calculated

Aint double 40 [ 0, 1e32 ] 1/cm loss_ratio double 2.0 [ 0, 1e32 ] none

b double 1e-4 [ 0, 1 ] none

photon_lifetime enumerated calculated calculated, defined

tp double 1e-12 [ 0, 1e32 ] s A double 0 [ 0, 1e32 ] 1/s

B double 1e-10 [ 0, 1e32 ] cm^3/s

C double 30e-30 [ 0, 1e32 ] cm^6/s

gain enumerated logarithmic_N logarithmic_N, logarithmic_RN, linear

Go double 1537 [ 0, 1e32 ] 1/cm

Ntr double 1.5e18 [ 0, 1e32 ] 1/cm^3 Ns double 0 [ 0, 1e32 ] none

saturation enumerated Channin Channin, Agrawal, linear

e double 10e-17 [ 0, 1e32 ] cm^3

a double 2 [ 0, 1e32 ] none

RIN double -150 [ -1e32, 1e32 ] dB/Hz Rd double 5 [ 0, 1e32 ] ohm

Von double 2.0 [ 0, 1e32 ] V

Drive_Scheme enumerated bias_tee_old direct_drive, bias_tee, bias_tee_old


Bias_Value enumerated Io Io, Po Io double 30e-3 [ 0, 1e32 ] A

Po double 0 [ 0, 1e32 ] W

Parasitics enumerated on on, off

Lb double 0.3e-9 [ 0, 1e32 ] H Cp double 2e-12 [ 0, 1e32 ] F

eps double 1e-6 [ 0, 1e32 ] none

initial_tstep double 1e-13 [ 0, 1e32 ] s

min_tstep double 0 [ 0, 1e32 ] s spatial_effects enumerated on on, off

mode_mapping enumerated single single,


multi_from_single multi_mode_powers string 1.0

mode_polarization enumerated xy_same xy_same, xy_unique

mode_default_domain enumerated yes yes, no mode_width_x double 1 [ 0, 1e32 ] um


mode_default_grid_spacing enumerated yes yes, no

mode_dx double 0.1 [ 0, 1e32 ] um mode_dy double 0.1 [ 0, 1e32 ] um


x_multi_mode_type enumerated LG file, LG, HG, LP

x_mode_file string x_multimode_file string


x_m integer 0 [ 0, 75 ] none

x_l_lo integer 0 [ -75, 75 ] none x_l_hi integer 0 [ -75, 75 ] none


x_m_hi integer 0 [ 0, 75 ] none

x_wo double 1 [ 0, 1e32 ] um x_iRo double 0 [ -1e32, 1e32 ] um^-1



















y_iRox double 0 [ -1e32, 1e32 ] um^-1 y_woy double 1 [ 0, 1e32 ] um

y_iRoy double 0 [ -1e32, 1e32 ] um^-1


y_Ncore double 1 [ 0, 1e32 ] none y_core_radius double 1 [ 0, 1e32 ] um

y_outer_radius double 1 [ 0, 1e32 ] um


test_function enumerated LI LI, ac, report, mode_profile

Test_LItype enumerated auto-scaled auto-scaled, user-specified



Test_actype enumerated Ith-based Ith-based, user-specified


Test_I2ac double 2.5e-3 [ 0, 1e32 ] A Test_I3ac double 5e-3 [ 0, 1e32 ] A



Test_acxscale enumerated log log, linear Test_acyscale enumerated linear log, linear

Test_freqlow double 1e7 [ 0, 1e32 ] Hz

Test_freqhigh double 1e12 [ 0, 1e32 ] Hz

Test_acpoints integer 201 [ 0, 100000 ] none test_mode_polarization enumerated x x, y

test_mode_number integer 0 [ 0, 100000 ] none

test_mode_wavelength double 1 [ 0, 1e32 ] um


Parameter Descriptions wavelength Wavelength of the laser, λ mode Type of wavelength grid deltaFreq Frequency or wavelength grid spacing for multi-line output effint Current injection efficiency, ηI K Optical confinement factor, Γ indx Optical mode index, n geometry Device geometry: rectangular, cylindrical, volumetric Lact Laser active region thickness, Lact Lstp Laser active region width, Lstp


Lcav Laser cavity length, Lcav W Laser diameter (for cylindrical geometries), W V Laser cavity volume, V mirror_effects Mirror effects: defined, calculated Amir Laser mirror loss, Amir F Fraction of power that escapes from the output mirror, F R1 Reflectivity of mirror #1, R1 R2 Reflectivity of mirror #2, R2 intrinsic_loss Intrinsic loss: defined, calculated Aint Laser internal loss, Aint loss_ratio Ratio of intrinsic loss to mirror loss, loss_ratio b Spontaneous emission coupling coefficient, βsp photon_lifetime Photon lifetime definition: calculated, defined tp Photon lifetime, τp A Unimolecular recombination coefficient, A B Radiative recombination coefficient, B C Auger recombination coefficient, C gain Gain definition: logarithmic:N, logarithmic:R(N), linear Go Gain coefficient, Go Ntr Carrier transparency density, Ntr Ns Logarithmic gain correction factor, Ns saturation Gain saturation definition: Channin, Agrawal, linear e Gain saturation factor, ε a Linewidth enhancement factor, α RIN Laser relative intensity noise, RIN Rd Laser cavity resistance, Rd Von Laser turn-on voltage, Von Drive_Scheme Drive signal definition: direct_drive, bias_tee, bias_tee_old Rs Source impedance, Rs Bias_Value Bias definition: Io, Po Io Laser bias current, Io Po Laser dc power level, Po Parasitics Parasitics flag: on, off Lb Laser bond inductance, Lb Cp Laser parasitic capacitance, Cp eps Accuracy of ODE solver initial_tstep Initial time step taken by ODE solver min_tstep Minimum time step taken by ODE solver spatial_effects Switch to turn spatial-addition by model block on or off. mode_mapping Transverse mode profile style selection (single- vs. multimode). multi_mode_powers List of mode powers, or file-name with this data. mode_polarization Polarization mode-attachment selection. mode_default_domain Default mode domain override. mode_width_x User-specified mode domain width along the X axis.


mode_width_y User-specified mode domain width along the Y axis. mode_default_grid_spacing Default grid spacing override. mode_dx User-specified X-axis grid spacing. mode_dy User-specified Y-axis grid spacing. x_single_mode_type Single-mode profile selection for X polarization. x_multi_mode_type Multimode profile selection for X polarization. x_mode_file Single-mode gridded mode data file for X polarization. x_multimode_file Multimode data file for X polarization. x_l Mode index l for X polarization. x_m Mode index m for X polarization. x_l_lo Lower bound for X polarization sweep of mode index l. x_l_hi Upper bound for X polarization sweep of mode index l. x_m_lo Lower bound for X polarization sweep of mode index m. x_m_hi Upper bound for X polarization sweep of mode index m. x_wo Laguerre-Gaussian spot size for X polarization. x_iRo Inverse of Laguerre-Gaussian radius of curvature for X

polarization. x_wox Hermite-Gaussian X-axis spot size for X polarization. x_iRox Inverse of Hermite-Gaussian X-axis radius of curvature for X

polarization. x_woy Hermite-Gaussian Y-axis spot size for X polarization. x_iRoy Inverse of Hermite-Gaussian Y-axis radius of curvature for X-

polarization. x_Nclad Cladding index for X-polarized LP mode. x_Ncore Core index for X-polarized LP mode. x_core_radius Core radius for X-polarized LP mode. x_outer_radius Outer radius for X-polarized donut and spot modes. x_inner_radius Inner radius for X-polarized donut mode. y_single_mode_type Single-mode profile selection for Y polarization. y_multi_mode_type Multimode profile selection for Y polarization. y_mode_file Single-mode gridded mode data file for Y polarization. y_multimode_file Multimode data file for Y polarization. y_l Mode index l for Y polarization. y_m Mode index m for Y polarization. y_l_lo Lower bound for Y polarization sweep of mode index l. y_l_hi Upper bound for Ypolarization sweep of mode index l. y_m_lo Lower bound for Y polarization sweep of mode index m. y_m_hi Upper bound for Y polarization sweep of mode index m. y_wo Laguerre-Gaussian spot size for Y polarization. y_iRo Inverse of Laguerre-Gaussian radius of curvature for Y

polarization. y_wox Hermite-Gaussian X-axis spot size for Y polarization. y_iRox Inverse of Hermite-Gaussian X-axis radius of curvature for Y

polarization. y_woy Hermite-Gaussian Y-axis spot size for Y polarization. y_iRoy Inverse of Hermite-Gaussian Y-axis radius of curvature for Y-

l i i


polarization. y_Nclad Cladding index for Y-polarized LP mode. y_Ncore Core index for Y-polarized LP mode. y_core_radius Core radius for Y-polarized LP mode. y_outer_radius Outer radius for Y-polarized donut and spot modes. y_inner_radius Inner radius for Y-polarized donut mode. test_function Test function selection: LI curve, ac curves, or performance

report Test_LItype LI curve definition: auto-scaled, user-specified Test_Imin Minimum current for LI curve Test_Imax Maximum current for LI curve Test_Lipoints Number of points in LI curve Test_actype Frequency-response curve definition: Ith-based, user-specified Test_I1ac Bias current for first ac curve Test_I2ac Bias current for second ac curve Test_I3ac Bias current for third ac curve Test_I4ac Bias current for fourth ac curve Test_I5ac Bias current for fifth ac curve Test_acxscale Scale for x axis: log, linear Test_acyscale Scale for y axis: log, linear Test_freqlow Minimum frequency for ac curves Test_freqhigh Maximum frequency for ac curves Test_acpoints Number of points per ac curve test_mode_polarization Test plot field polarization selection. test_mode_number Test plot mode number. test_mode_wavelength Test plot field wavelength. test_mode_plot Test plot style selection.


Spatial Mode-Locked Laser

This models a mode-locked laser. It is identical to the standard Mode-Locked Laser model, except for the ability to include transverse mode profiles in the optical output.

The following pulse types are presently supported for the output optical signal (u(t) is the field amplitude):

• gaussian

( )[ ]205.0exp)( Tttu −=

• sech

( )0sec)( Tthtu =

• on_off

|u(t)|2 = linear ramp to 1 for tr; flat at 1 for tFWHM-tr/2-tf/2; linear ramp to 0 for tf.

• raisedCosAmp

( )02cos5.05.0)( Tttu π+=

• raisedCosPow

( ) ( )02 2cos5.05.0 Tttu π+=

• supGaussian

( )[ ]mTttu 205.0exp)( −=

supGaussian (Super Gaussian) is an approximation to square pulses but with rounded edges. The m parameter is determined from the rise time of the pulse. Although tr and tf may be different for on_off pulse type, they must be the same for supGaussian type. The 10%-90% rise time rule is used for the supGaussian while for the on_off it is the total rise or fall time.

The FWHM of these pulse types are:

• gaussian

02ln2 TTFWHM ⋅=

• sech

)21ln(2 0 +⋅= TTFWHM

• raisedCosAmp

)12cos(10 −⋅

π= aTTFWHM

• raisedCosPow

2/0TTFWHM =

• supGaussian


mFWHM TT 21

0 )2(ln2 ⋅=

Arbitrary amounts of linear chirp may be added to each of these pulse forms. The chirp is included via a chirp factor C. The expression for the field amplitude of a chirped pulse uchirp(t) is given by (this is a general result irrespective of the pulse shape, see G. P. Agrawal, Nonlinear Fiber Optics, Academic Press),

−= 2

202

exp)()( TTCjtutuchirp

The expression for the additional spectral width due to pulse chirp is given by,

λ−

π

−λ

=λ∆−

0

1

00 412

cTC

In the case of mode-locked sources, phase shift between adjacent bits is allowed so that the phase shifted soliton transmission technique can be studied.

Note that this model’s output can be disabled by setting its peak-power value to 0.







Each transverse mode profile in OptSim has a default grid spacing and domain, in order to facilitate numerical calculations. The grid spacing is specified on a rectangular XY grid. The user may override the default grid spacings by setting mode_default_grid_spacing=no, and then adjusting parameters mode_dx and mode_dy. The domain, meanwhile, determines the default rectangular space over which the field is


considered to have non-negligible magnitude. Again, the user may override the default by setting mode_default_domain=no, and adjusting mode_width_x and mode_width_y. The domain is automatically centered about the origin.


• single



The model can also convert an individual optical output into multiple output signals, each with different mode shapes, but with time-domain behaviors identical except for a power distribution scaling factor. The optical output is copied into multiple signals, and different modes are attached to each copy. This technique is useful for converting a single-mode optical signal into a multimode one.


1.0 2.0 3.0 4.0






• file


• radial_file


• LG


• HG


• LP


• spot


• donut





...













Multi-Line Output It is frequently necessary to produce several signals with similar properties but different wavelengths. In DWDM simulations especially, a series of regularly spaced optical sources is common. The Mode-Locked Laser model provides a number of convenient facilities so that many or all required lines can be generated from a single icon. To produce a series of lines spaced equally in wavelength or frequency, set the parameter mode=LambdaGrid or mode=FreqGrid, respectively. The number of sources is controlled by noSources, and in both modes the parameter wavelength specifies the first line in a series of ascending wavelengths or ascending frequencies, respectively. The source separation in wavelength or frequency is specified with deltaFreq.


MultiModeLockedLaserFormat1 <freq_format> <power_format>

<freq_1> <power_1>

<freq_2> <power_2>

etc..


Multi-Line Multi-Node Output The comb of sources may be emitted either through a single node (the default) or with one channel per output node. To achieve the latter behavior, first set multiNodeOutput=YES. Then select the mode-locked laser icon and open the menu item Properties. In the Ports tab, number_output_ports field, enter the number of lines that the model will generate (i.e. either noSources or the number of lines in the user data file if mode=File).

Test Parameters A number of parameters are available for testing the model’s spatial settings. First, test_mode_polarization is used to select which polarization to use when plotting the mode profiles. test_mode_number is used when the model is in multi_from_single mode. The available modes are numbered from zero, and in the order by which they are attached to optical signals. test_mode_wavelength determines the field wavelength at which to plot the test mode. Finally,


test_mode_plot determines the type of plot to generate: magnitude plots the profile’s transverse magnitude, mag_phase plots its magnitude and phase, and real_imag plots the real and imaginary parts of the mode.

References [1] M. Born and E. Wolf, Principles of Optics, 7th. Ed. (Cambridge University Press, Cambridge, 1999).

Properties

Inputs None


Parameter Values Name Type Default Range Units pattern enumerated Multiple Single, Multiple type enumerated gaussian sech, gaussian,

supGaussian, on_off, raisedCosPow, raisedCosAmp

peakPower double 1e-3 [ 0, 1e32 ] Watts



multiNodeOutput enumerated NO NO, YES

noSources integer 10 [ 1, 1000 ] none deltaFreq double 100e9 [ 0, 1e18 ] meters or Hz

filename string



width double 10e-12 [ 0, 1e32 ] sec

patternLength integer 7 [ 0, 27 ] 2^x_bits pointsPerBit integer 5 [ 1, 27 ] 2^x_bits

repRate double 10e9 [ 1, 1e32 ] Bit/s


ChirpFactor double 0 [ -1e32, 1e32 ] none phaseShift double 0 [ -6.284, 6.284 ] rad

riseTime double 10e-12 [ 0, 1e32 ] sec

fallTime double 10e-12 [ 0, 1e32 ] sec

spatial_effects enumerated on on, off


mode_mapping enumerated single single, multi_from_single

multi_mode_powers string 1.0

mode_polarization enumerated xy_same xy_same, xy_unique mode_default_domain enumerated yes yes, no






















y_multi_mode_type enumerated LG file, LG, HG, LP y_mode_file string

y_multimode_file string


y_m integer 0 [ 0, 75 ] none y_l_lo integer 0 [ -75, 75 ] none

y_l_hi integer 0 [ -75, 75 ] none


y_m_hi integer 0 [ 0, 75 ] none y_wo double 1 [ 0, 1e32 ] um


y_iRo double 0 [ -1e32, 1e32 ] um^-1

y_wox double 1 [ 0, 1e32 ] um y_iRox double 0 [ -1e32, 1e32 ] um^-1

y_woy double 1 [ 0, 1e32 ] um


y_Nclad double 1 [ 0, 1e32 ] none y_Ncore double 1 [ 0, 1e32 ] none

y_core_radius double 1 [ 0, 1e32 ] um

y_outer_radius double 1 [ 0, 1e32 ] um

y_inner_radius double 0 [ 0, 1e32 ] um test_mode_polarization enumerated x x, y

test_mode_number integer 0 [ 0, 100000 ] none

test_mode_wavelength double 1 [ 0, 1e32 ] um


Parameter Descriptions mode Type of wavelength grid multiNodeOutput Select multi-line output on a single or multiple output nodes noSources Number of lines in multi-line output deltaFreq Frequency or wavelength grid spacing for multi-line output filename Filename for user-specified source grid type Mode-locked source type: gaussian, supGaussian, sech, on_off,

raisedCosAmp, raisedCosPow repRate Repetition rate of the source patternLength Bit pattern length = 2x, x is input pointsPerBit Number of sampling points per bit period in output optical signal peakPower Peak power wavelength Wavelength of the laser RIN Relative intensity noise of the laser width Pulse FWHM Parameter ChirpFactor Chirp factor C phaseShift Phase Shift Between Adjacent Pulses pattern Single or multiple pulses in bit stream riseTime Pulse Rise Time for on-off and supGaussian types fallTime Pulse Fall Time for on-off type spatial_effects Switch to turn spatial-addition by model block on or off. mode_mapping Transverse mode profile style selection (single- vs. multimode). multi_mode_powers List of mode powers, or file-name with this data. mode_polarization Polarization mode-attachment selection. mode_default_domain Default mode domain override. mode_width_x User-specified mode domain width along the X axis. mode_width_y User-specified mode domain width along the Y axis. mode_default_grid_spacing Default grid spacing override.


mode_dx User-specified X-axis grid spacing. mode_dy User-specified Y-axis grid spacing. x_single_mode_type Single-mode profile selection for X polarization. x_multi_mode_type Multimode profile selection for X polarization. x_mode_file Single-mode gridded mode data file for X polarization. x_multimode_file Multimode data file for X polarization. x_l Mode index l for X polarization. x_m Mode index m for X polarization. x_l_lo Lower bound for X polarization sweep of mode index l. x_l_hi Upper bound for X polarization sweep of mode index l. x_m_lo Lower bound for X polarization sweep of mode index m. x_m_hi Upper bound for X polarization sweep of mode index m. x_wo Laguerre-Gaussian spot size for X polarization. x_iRo Inverse of Laguerre-Gaussian radius of curvature for X polarization. x_wox Hermite-Gaussian X-axis spot size for X polarization. x_iRox Inverse of Hermite-Gaussian X-axis radius of curvature for X


polarization. x_Nclad Cladding index for X-polarized LP mode. x_Ncore Core index for X-polarized LP mode. x_core_radius Core radius for X-polarized LP mode. x_outer_radius Outer radius for X-polarized donut and spot modes. x_inner_radius Inner radius for X-polarized donut mode. y_single_mode_type Single-mode profile selection for Y polarization. y_multi_mode_type Multimode profile selection for Y polarization. y_mode_file Single-mode gridded mode data file for Y polarization. y_multimode_file Multimode data file for Y polarization. y_l Mode index l for Y polarization. y_m Mode index m for Y polarization. y_l_lo Lower bound for Y polarization sweep of mode index l. y_l_hi Upper bound for Ypolarization sweep of mode index l. y_m_lo Lower bound for Y polarization sweep of mode index m. y_m_hi Upper bound for Y polarization sweep of mode index m. y_wo Laguerre-Gaussian spot size for Y polarization. y_iRo Inverse of Laguerre-Gaussian radius of curvature for Y polarization. y_wox Hermite-Gaussian X-axis spot size for Y polarization. y_iRox Inverse of Hermite-Gaussian X-axis radius of curvature for Y


polarization. y_Nclad Cladding index for Y-polarized LP mode. y_Ncore Core index for Y-polarized LP mode. y_core_radius Core radius for Y-polarized LP mode.


y_outer_radius Outer radius for Y-polarized donut and spot modes. y_inner_radius Inner radius for Y-polarized donut mode. test_mode_polarization Test plot field polarization selection. test_mode_number Test plot mode number. test_mode_wavelength Test plot field wavelength. test_mode_plot Test plot style selection.


Spatial CW Laser

This model produces the optical signal output of one or more CW lasers. It is most commonly used in conjunction with the external modulator model to encode a binary signal upon the CW source. It is identical to the standard CW Laser model, except for the ability to include transverse mode profiles in the optical output.

In this model, the CW source is characterized completely by its power, wavelength, linewidth, relative intensity noise (RIN) and phase. These are controlled directly through the parameters peakPower, wavelength, linewidth, RIN and phase. The laser can also be assigned a random phase by setting randomPhase=YES. Note that by setting peakPower to 0, you can disable this model’s output.

There are two options for the temporal representation of the laser output selected by the parameter signalType. For topologies in which a CW laser model provides direct input to a modulator model or the pumps of a Raman fiber amplifier, the PowerValue signal representation is most convenient. In this representation, the optical signal holds a single complex value describing the field amplitude and phase. For topologies in which a CW laser model’s output is used as input to other component models or is to be multiplexed with different optical signals from other types of sources, the TimeSequence signal representation should be used. This is the standard time-sampled representation of optical signals. For this representation, the timeStep and noSamples parameters must be set appropriately to match the sampling rate and number of data samples of any other signals with which it will interact in the simulation. The nominalBitRate parameter should also be set to an appropriate data rate.

Linewidth Inclusion of a source linewidth in the laser output is controlled via the parameter linewidth_model, which by default is set to none (for a linewidth of zero). If linewidth_model=phase_noise, and the laser output uses the TimeSequence signal representation, then linewidth is added to the output via phase noise. These random phase variations (seeded via the same phaseSeed parameter that controls random initial phase values) result in a Lorentzian output power spectrum [1].











• single





1.0 2.0 3.0 4.0


The user must also specify a family of m mode profiles for either or both polarizations. These profiles are attached in sequence to the multiple output signals. Let n denote the number of


signals into which the single output has been split. If m > n, then only the first n profiles are used. If n > m, then the profiles are reused in sequence until each input has been handled. For each polarization, the user selects a mode shape via the parameter x_multi_mode_type (y_multi_mode_type). The available types are: file, LG, HG, and LP. For built-in mode shapes, the user must then specify a range of values for the mode indices. This is accomplished via the parameters x_l_lo (y_l_lo), x_l_hi (y_l_hi), x_m_lo (y_m_lo) and x_m_hi (y_m_hi) In cycling through these mode indices during processing of the signals, the model first loops through the l index, and for each value, loops through all of the m indices.



• file


• radial_file


• LG


• HG


• LP


• spot


• donut






...












Multi-Line Output It is frequently necessary to produce several CW signals with similar properties but different wavelengths. In DWDM simulations especially, a series of regularly spaced optical sources is common. The CW Laser model provides a number of convenient facilities so that many or all required lines can be generated from a single icon. To produce a series of lines spaced equally in wavelength or frequency, set the parameter mode=LambdaGrid or mode=FreqGrid, respectively. The number of sources is controlled by noSources, and in both modes the parameter wavelength specifies the first line in a series of ascending wavelengths or ascending frequencies, respectively. The source separation in wavelength or frequency is specified with deltaFreq.


MultiCWLaserFormat1 <freq_format> <power_format>



etc..



Multi-Line Multi-Node Output The comb of sources may be emitted either through a single node (the default) or with one channel per output node. To achieve the latter behavior, first set multiNodeOutput=YES. Then select the cw laser icon and open the menu item Properties. In the Ports tab, number_output_ports field, enter the number of lines that the model will generate (i.e. either noSources or the number of lines in the user data file if mode=File). Figure 1 indicates the use of multi-node output mode to produce a series of WDM sources.

Figure 1: CW laser in multi-node output mode for WDM sources.

Test Parameters A number of parameters are available for testing the model’s spatial settings. First, test_mode_polarization is used to select which polarization to use when plotting the mode profiles. test_mode_number is used when the model is in multi_from_single mode. The available modes are numbered from zero, and in the order by which they are attached to optical signals. test_mode_wavelength determines the field wavelength at which to plot the test mode. Finally, test_mode_plot determines the type of plot to generate: magnitude plots the profile’s transverse magnitude, mag_phase plots its magnitude and phase, and real_imag plots the real and imaginary parts of the mode.




Properties

Inputs None


Parameter Values Name Type Default Range Units peakPower double 1e-3 [ 0, 1e18 ] Watts



multiNodeOutput enumerated NO NO, YES

noSources integer 10 [ 1, 1000 ] none deltaFreq double 100e9 [ 0, 1e18 ] meters or Hz

filename string








timeStep double 0.0 [ 0, 1 ] seconds nominalBitRate double 10e9 [ 0, 1e32 ] Hz

noSamples integer 0 [ 0, 27 ] 2^x_samples

randomPhase enumerated NO NO, YES

phase double 0 [ -180, 180 ] degrees phaseSeed integer 0 [ -1e8, 1 ] none



multi_mode_powers string 1.0 mode_polarization enumerated xy_same xy_same, xy_unique






























y_m_hi integer 0 [ 0, 75 ] none y_wo double 1 [ 0, 1e32 ] um

y_iRo double 0 [ -1e32, 1e32 ] um^-1


y_iRox double 0 [ -1e32, 1e32 ] um^-1 y_woy double 1 [ 0, 1e32 ] um










Parameter Descriptions mode Type of wavelength grid multiNodeOutput Select multi-line output on a single or multiple output nodes peakPower Peak power (also average power for CW laser) wavelength Laser wavelength noSources Number of lines in multi-line output deltaFreq Frequency or wavelength grid spacing for multi-line output filename Filename for user-specified source grid linewidth_model Select linewidth representation linewidth_units Select linewidth units linewidth Linewidth value RIN Relative intensity noise of the laser signalType Select representation of a power or time sequence timeStep For TimeSequence representation, the signal sampling time nominalBitRate For TimeSequence representation, the signal bit rate noSamples For TimeSequence representation, the number of signal samples randomPhase Select randomization of laser phase phase Select amplitude of phase randomization phaseSeed Random number seed for phase randomization. (Standard OptSim seed convention). spatial_effects Switch to turn spatial-addition by model block on or off. mode_mapping Transverse mode profile style selection (single- vs. multimode). multi_mode_powers List of mode powers, or file-name with this data. mode_polarization Polarization mode-attachment selection. mode_default_domain Default mode domain override. mode_width_x User-specified mode domain width along the X axis. mode_width_y User-specified mode domain width along the Y axis. mode_default_grid_spacing

Default grid spacing override.

mode_dx User-specified X-axis grid spacing. mode_dy User-specified Y-axis grid spacing. x_single_mode_type Single-mode profile selection for X polarization. x_multi_mode_type Multimode profile selection for X polarization. x_mode_file Single-mode gridded mode data file for X polarization. x_multimode_file Multimode data file for X polarization.


x_l Mode index l for X polarization. x_m Mode index m for X polarization. x_l_lo Lower bound for X polarization sweep of mode index l. x_l_hi Upper bound for X polarization sweep of mode index l. x_m_lo Lower bound for X polarization sweep of mode index m. x_m_hi Upper bound for X polarization sweep of mode index m. x_wo Laguerre-Gaussian spot size for X polarization. x_iRo Inverse of Laguerre-Gaussian radius of curvature for X polarization. x_wox Hermite-Gaussian X-axis spot size for X polarization. x_iRox Inverse of Hermite-Gaussian X-axis radius of curvature for X polarization. x_woy Hermite-Gaussian Y-axis spot size for X polarization. x_iRoy Inverse of Hermite-Gaussian Y-axis radius of curvature for X-polarization. x_Nclad Cladding index for X-polarized LP mode. x_Ncore Core index for X-polarized LP mode. x_core_radius Core radius for X-polarized LP mode. x_outer_radius Outer radius for X-polarized donut and spot modes. x_inner_radius Inner radius for X-polarized donut mode. y_single_mode_type Single-mode profile selection for Y polarization. y_multi_mode_type Multimode profile selection for Y polarization. y_mode_file Single-mode gridded mode data file for Y polarization. y_multimode_file Multimode data file for Y polarization. y_l Mode index l for Y polarization. y_m Mode index m for Y polarization. y_l_lo Lower bound for Y polarization sweep of mode index l. y_l_hi Upper bound for Ypolarization sweep of mode index l. y_m_lo Lower bound for Y polarization sweep of mode index m. y_m_hi Upper bound for Y polarization sweep of mode index m. y_wo Laguerre-Gaussian spot size for Y polarization. y_iRo Inverse of Laguerre-Gaussian radius of curvature for Y polarization. y_wox Hermite-Gaussian X-axis spot size for Y polarization. y_iRox Inverse of Hermite-Gaussian X-axis radius of curvature for Y polarization. y_woy Hermite-Gaussian Y-axis spot size for Y polarization. y_iRoy Inverse of Hermite-Gaussian Y-axis radius of curvature for Y-polarization. y_Nclad Cladding index for Y-polarized LP mode. y_Ncore Core index for Y-polarized LP mode. y_core_radius Core radius for Y-polarized LP mode. y_outer_radius Outer radius for Y-polarized donut and spot modes. y_inner_radius Inner radius for Y-polarized donut mode. test_mode_polarization Test plot field polarization selection. test_mode_number Test plot mode number. test_mode_wavelength Test plot field wavelength. test_mode_plot Test plot style selection.


Spatial VCSEL

This block models a vertical-cavity surface-emitting laser (VCSEL) directly modulated with an electrical signal. It is identical to the standard VCSEL model, except for the ability to include transverse mode profiles in the optical output.

The model computes the electrical current injected into the laser’s optical cavity and solves the laser rate equations for the optical output. Important VCSEL behaviors such as spatial hole burning, lateral carrier diffusion, thermally dependent gain, and thermal carrier leakage are all accounted for. The behavior of the model can be partitioned into three blocks, as shown in Fig. 1.

Figure 1: Main components of the VCSEL model

The driving source consists of the electrical signal input into the model. The parasitics consist of a bond inductance and shunting capacitance. Finally, the VCSEL cavity is modeled via a simplified current-voltage (IV) relationship and spatially independent VCSEL rate equations.

Driving Source The VCSEL is driven by a combination of the electrical signal at its input and, when applicable, a dc bias current specified by Io (Io). The input electrical signal can be interpreted in a variety of ways, based on the settings of the model parameter Drive_Scheme. This parameter can take on values of direct_drive, direct_user_iv, or bias_tee.

• direct_drive

The input signal is assumed to come from either an ideal current source or zero-impedance voltage source. In other words, the laser is assumed to be undergoing direct-drive modulation. If the input signal is a current, then it is combined with the bias current Io to form the total input current. Figure 2(a) illustrates this scenario. If the input signal is a voltage, then the bias current Io is ignored. Note that the input voltage should be larger than the laser’s turn-on voltage Von; otherwise, the resulting current is zero. This arrangement is illustrated in Fig. 2(b).

OptSim 4.0 Models Reference: Block Mode Chapter 13: Multimode Modules •••• 515


• direct_user_iv

Equivalent to direct_drive, but with support for a user-specified equation for the cavity voltage (see below).

• bias_tee

The input signal is assumed to be generated by a source with output impedance Rs (Rs), connected to the laser via an ideal bias tee. The bias current Io is similarly connected. The input signal can be either a current or a voltage. Figure 3 depicts a bias tee setup for both the voltage- and current-source cases.


Parasitics The parasitics consist of a bond inductance Lb (Lb) and shunting capacitance Cp (Cp). These can be turned on or off via the parameter Parasitics.

VCSEL Cavity Both electrical and optical effects are modeled within the VCSEL cavity.

Electrical In both the direct_drive and bias_tee modes of operation, the electrical model of the VCSEL cavity is that of a simplified diode IV relationship, consisting of a series resistance Rd (Rd) and turn-on voltage Von (Von). During solution of the cavity current I, the model ensures that negative currents are effectively limited to zero.

For the direct_user_iv mode of operation, the user may specify a nonlinear equation for the cavity voltage via the parameter voltage_equation. This equation should be a function of cavity current I and device temperature T. For example, to implement a simple diode, the user might set voltage_equation equal to “I*100+17.2e-5*T*log(I/1e-18+1.0)”, where we have assumed a 100-Ω series resistance, a saturation current of 10-18 A, and a diode ideality factor of 2.


Rate Equations At the core of the VCSEL block are spatially independent semiconductor laser rate equations, which determine the optical output in response to the cavity current I [1]. Relative intensity noise is modeled via a constant value RIN (RIN), and the optical emission frequency is set by λ (wavelength). The model rate equations are based on the following single-mode spatially dependent equations [1]:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2

2, , , ,, ,effi l

n n

LN r t I r t N r t I N TG r t S t r N r t

t q qη

ψτ τ

∂= − − + ∇ −

∂

r r rr r r

(1)

∂∂ τ

βτ

ψS tt

S tV

N r t dvV

G r t S t r dvp n V V

( ) ( ) ( , ) ( , ) ( ) ( )= − + ⋅ ⋅ + ⋅∫ ∫1 1v v v

(2)

In (1), I is the spatially dependent injection current, N is the carrier density scaled by the effective active-layer volume V, S and ψ are the total photon number and normalized transverse mode profile, T is the device temperature, G is the gain, Il is the thermal leakage current, ηI (effint) is the current-injection efficiency, τn (tn) is the carrier lifetime, Leff is the effective carrier diffusion length, and q is the electron charge. In (2), β (b) is the spontaneous-emission coupling coefficient, and τp (tp) is the photon lifetime. Following the approach taken in [1], we can eliminate the explicit spatial dependence from (1) and (2) by assuming a cylindrical geometry, neglecting azimuthal variations, and adopting a two-term Bessel-series expansion for the carrier profile:

N N J r R0 1 0 1− ( / )σ

(3)

where σ1 is the first nonzero root of J1(x), and R is the active-layer effective radius. If we further assume a uniform current distribution and linear gain, we can eliminate the explicit spatial dependence from (1) and (2), thereby obtaining the following spatially independent VCSEL rate equations:

dNdt

Iq

N G T N N T NS

SI N T

qi

n

t l0 0 00 0 01 1 01

= − −⋅ − −

+⋅ −

ητ

γ γε

( ) [ ( ( )) ] ( , )

(4)

dNdt

N h G T N N T NS

Sn

difft1 1 100 0 101 11

1= − ⋅ + + ⋅ − −

+⋅

τφ φ

ε( ) ( ) [ ( ( )) ]

(5)

dSdt

S N G T N N T NS

Sp n

t= − + + ⋅ − −+

⋅τ

βτ

γ γε

0 00 0 01 11

( ) [ ( ( )) ]

(6)

In (4)-(6), the spatial dependence of the gain is now accounted for via the overlap coefficients γ00 (gam00), γ01 (gam01), φ100 (phi100), and φ101 (phi101), while diffusive effects are taken care of via hdiff (hdiff), which is equal to 2

1( / )effL Rσ . The thermal dependence of the gain is taken into account via the thermally

dependent gain constant G(T) and transparency number Nt(T), while gain saturation is modeled via ε (e), the gain saturation factor. Furthermore, the leakage Il is now a function of N0 (the average carrier number), as opposed to N. Finally, the photon number S is converted to an output power via the output-power coupling coefficient kf (kf) using the expression Pout = kfS.


Phase Rate Equation A rate equation for the optical phase is also included in the model, and is based on work presented in [7] and [8]:

00 0 01 1( ) [ ( ) ]2 1

thG T N N Nddt S

γ γφ αε

⋅ − −= ⋅

+

(7)

φ is the optical phase, α (alpha) is the linewidth enhancement factor, and Nth is the room-temperature threshold carrier number (the model assumes that the specified laser wavelength is defined at room temperature).

Mode-Carrier Overlap If the transverse mode profile ψI is normalized such that

2 12

0Rr r dr⋅ ⋅ =

∞

∫ψ ( )

(8)

then the mode overlap coefficients can be calculated as [1]:

γ σ ψ0 2 00

2i i

R

RJ r R r r dr= ⋅ ⋅ ⋅∫ ( / ) ( )

(9)

φσ

σ σ ψ10 202

10 0 1

0

2i i

R

R JJ r R J r R r r dr= ⋅ ⋅ ⋅ ⋅∫( )

( / ) ( / ) ( )

(10)

where I = 0 or 1, and σ0 = 0. By setting the parameter overlap_calculation to off, the user is free to calculate specific values for the overlap coefficients depending on their particular choice of mode profile.

In many situations, one can model the mode profile of a single mode device as a Gaussian with characteristic radius Rm. In this case, it can be shown that the overlap coefficients reduce to functions of ρ (overlap), where ρ = Rm/R [2]. By setting overlap_calculation to on, the overlap coefficients are calculated as functions of ρ, overriding the specified values.

Thermally Dependent Gain In order to account for a VCSEL gain’s unique thermal dependence, the gain is modeled as a linear function of carrier number N, with thermally dependent gain constant and transparency number. The gain constant G(T) and transparency number Nt(T) are described via the following empirical relationships [1]:

G T Ga a T a T

b b T b To

g g g

g g g( ) = ⋅

+ +

+ +0 1 2

2

0 1 22

(11)

N T N c c T c Tt tr n n n( ) ( )= ⋅ + +0 1 22


(12)

where Go (Go) is a gain constant, Ntr (Ntr) is a transparency number, and ag0-ag2 (ag0-ag2), bg0-bg2 (bg0-bg2), and cn0-cn2 (cn0-cn2) are fitting parameters. Generally, the gain constant will be peaked about some optimal temperature value, as a result of the temperature-dependent mismatch between lasing wavelength and gain peak. An example of the gain constant based on the model’s default values is illustrated in Fig. 4. The transparency number generally increases with temperature.

Figure 4: Sample thermally dependent gain constant based on default model values

Thermal Carrier Leakage Thermally dependent carrier leakage is modeled using the following empirical relationship [1], based on the analysis of [3]:

I N T Ia a N a N T a N

Tl lo( , ) exp/

00 1 0 2 0 3 0= ⋅

− + + −

(13)

where Ilo (Ilo) is the leakage current factor, and a0-a3 (a0-a3) are fitting parameters. This expression accounts for the interdependence of the carrier number and temperature in determining the total leakage current. An example of (13) for various temperatures is illustrated in Fig. 5.

Figure 5. Sample thermal leakage current based on default model values


Temperature Rate Equation To complete the thermal aspect of the model, a rate equation for the device temperature as a function of dissipated heat is included. Heat generation is assumed to arise from any power not dissipated as part of the optical output. The resulting equation is [1],[4]:

T T I V P R dTdto tot out th th= + − ⋅ − ⋅( ) τ

(14)

where To (To) is the ambient temperature, Itot is the total current flowing through the VCSEL (including current through the intrinsic parasitic shunting capacitance Cp), Rth (Rth) is the device thermal impedance, and τth (tth) is the thermal time constant.








Mode-Attachment Styles The user has a variety of options for the style in which mode profiles are attached to the laser’s optical outputs, and these are selected via the mode_mapping parameter. The two options are: single and


multi_from_single. In the descriptions that follow, parameters for the X polarization are prefixed with x_, while parameters for the Y polarization are prefixed with y_.

• single





1.0 2.0 3.0 4.0





• file


• radial_file



• LG


• HG


• LP


• spot


• donut





...













Multi-Line Output It is frequently necessary to produce several signals with similar properties but different wavelengths. In DWDM simulations especially, a series of regularly spaced optical sources is common. The VCSEL model provides a number of convenient facilities so that many or all required lines can be generated from a single icon. To produce a series of lines spaced equally in wavelength or frequency, set the parameter mode=LambdaGrid or mode=FreqGrid, respectively. The number of sources is controlled by the number of electrical inputs to the laser, and in both modes the parameter wavelength specifies the first line in a series of ascending wavelengths or ascending frequencies, respectively. The source separation in wavelength or frequency is specified with deltaFreq. To set the number of electrical inputs to the laser, select the VCSEL icon and open the menu item Properties. In the Ports tab, number_input_ports field, enter the number of inputs.

Multi-Line Multi-Node Output The comb of sources may be emitted either through a single node (the default) or with one channel per output node. To achieve the latter behavior, select the VCSEL icon and open the menu item Properties. Set the value of the number_output_ports field in the Ports tab to that of the number_input_ports field.


Test Parameters In order to ascertain whether the parameter settings for the VCSEL block provide the component performance desired, the user may test them from the component parameter editing window. Spatial settings may also be tested. The test can be controlled via model parameters carrying a prefix Test_. These parameters allow the user to set sweep limits, etc.

LI Curve If test_function=LI, then a family of LI curves will be generated. The LI curve is controlled via the parameters Test_Imin, Test_Imax, Test_Lipoints, Test_Tomin, Test_Tomax, and Test_Topoints. Based on these parameters, a family of LI curves is generated over a range of ambient temperatures from Test_Tomin to Test_Tomax. The number of curves is determined by Test_Topoints. Each LI curve is generated over currents ranging from Test_Imin to Test_Imax, with the total number of points per curve specified by Test_LIpoints. A sample family of LI curves is shown in Fig. 6. Note that if a user-specified cavity voltage equation is being used, a plot of cavity voltage versus current will also be displayed.


Figure 6: Sample test family of LI curves at ambient temperatures of 25-75 C

Mode Profile If test_function=mode_profile, then a number of parameters are available for testing the spatial settings. First, test_mode_polarization is used to select which polarization to use when plotting the mode profiles. test_mode_number is used when the model is in multi_from_single mode. The available modes are numbered from zero, and in the order by which they are attached to optical signals. test_mode_wavelength determines the field wavelength at which to plot the test mode. Finally, test_mode_plot determines the type of plot to generate: magnitude plots the profile’s transverse magnitude, mag_phase plots its magnitude and phase, and real_imag plots the real and imaginary parts of the mode.

References [1] P. V. Mena, J. J. Morikuni, S.-M. Kang, A. V. Harton, and K. W. Wyatt, “A comprehensive circuit-level model of vertical-cavity surface-emitting lasers,” Journal of Lightwave Technology, 17, 2612 (1999).

[2] P. V. Mena, J. J. Morikuni, and K. W. Wyatt, “Compact representations of mode overlap for circuit-level VCSEL models,” IEEE/LEOS Annual Meeting Conference Proceedings, 234 (2000).

[3] J. W. Scott, R. S. Geels, S. W. Corzine, and L. A. Coldren, “Modeling temperature effects and spatial hole burning to optimize vertical-cavity surface-emitting laser performance,” IEEE Journal of Quantum Electronics, 29, 1295 (1993).

[4] N. Bewtra, D. A. Suda, G. L. Tan, F. Chatenoud, and J. M. Xu, “Modeling of quantum-well lasers with electro-opto-thermal interaction,” IEEE Journal of Selected Topics in Quantum Electronics, 1, 331 (1995).


[6] G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd. Ed. (Van Nostrand Reinhold, New York, 1993).

[7] M. X. Jungo, D. Erni, and W. Bächtold, “VISTAS: A comprehensive system-oriented spatiotemporal VCSEL model,” IEEE Journal of Selected Topics in Quantum Electronics, 9, 939 (2003).

[8] L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (John Wiley and Sons, Inc., New York, 1995).


Properties








effint double 1.0 [ 0, 1e32 ] none

kf double 1.5e-8 [ 0, 1e32 ] W b double 1e-3 [ 0, 1e32 ] none

tp double 2.5e-12 [ 0, 1e32 ] s

tn double 2.5e-9 [ 0, 1e32 ] s

Go double 3e4 [ 0, 1e32 ] 1/s ag0 double -0.4 [ -1e32, 1e32 ] none

ag1 double 0.00147 [ -1e32, 1e32 ] 1/K

ag2 double 7.65e-7 [ -1e32, 1e32 ] 1/K^2 bg0 double 1.3608 [ -1e32, 1e32 ] none

bg1 double -0.00974 [ -1e32, 1e32 ] 1/K

bg2 double 1.8e-5 [ -1e32, 1e32 ] 1/K^2

Ntr double 1e7 [ 0, 1e32 ] none cn0 double -1 [ -1e32, 1e32 ] none

cn1 double 0.008 [ -1e32, 1e32 ] 1/K

cn2 double 6e-6 [ -1e32, 1e32 ] 1/K^2

e double 5e-7 [ 0, 1e32 ] none alpha double 0 [ 0, 1e32 ] none

Ilo double 9.61 [ 0, 1e32 ] A

a0 double 4588.24 [ -1e32, 1e32 ] K

a1 double 2.12e-5 [ -1e32, 1e32 ] K a2 double 8e-8 [ -1e32, 1e32 ] none

a3 double 9.01e9 [ -1e32, 1e32 ] K

overlap_calculation enumerated off on, off

overlap double 1 [ 0.01, 10.00 ] none gam00 double 1 [ -1e32, 1e32 ] none

gam01 double 0.37978 [ -1e32, 1e32 ] none


phi100 double 2.3412 [ -1e32, 1e32 ] none

phi101 double 1.8193 [ -1e32, 1e32 ] none hdiff double 15 [ 0, 1e32 ] none


To double 25 [ -1e32, 1e32 ] C

Rth double 900 [ 0, 1e32 ] K/W tth double 1e-6 [ 0, 1e32 ] s

Rd double 105 [ 0, 1e32 ] ohm

Von double 1.75 [ 0, 1e32 ] V

Drive_Scheme enumerated direct_drive direct_drive, bias_tee

Rs double 50 [ 0, 1e32 ] ohm Io double 1.5e-3 [ 0, 1e32 ] A

Parasitics enumerated on on, off

Lb double 0.3e-9 [ 0, 1e32 ] H

Cp double 2e-12 [ 0, 1e32 ] F eps double 1e-6 [ 0, 1e32 ] none

initial_tstep double 1e-13 [ 0, 1e32 ] s

min_tstep double 0 [ 0, 1e32 ] s

spatial_effects enumerated on on, off mode_mapping enumerated single single,

multi_from_single


mode_polarization enumerated xy_same xy_same, xy_unique


mode_width_x double 1 [ 0, 1e32 ] um mode_width_y double 1 [ 0, 1e32 ] um

mode_default_grid_spacing enumerated yes yes, no

mode_dx double 0.1 [ 0, 1e32 ] um

mode_dy double 0.1 [ 0, 1e32 ] um x_single_mode_type enumerated LG file, radial_file, LG,

HG, LP, spot, donut


x_mode_file string

x_multimode_file string x_l integer 0 [ -75, 75 ] none


x_l_lo integer 0 [ -75, 75 ] none

x_l_hi integer 0 [ -75, 75 ] none x_m_lo integer 0 [ 0, 75 ] none


x_wo double 1 [ 0, 1e32 ] um

x_iRo double 0 [ -1e32, 1e32 ] um^-1 x_wox double 1 [ 0, 1e32 ] um



















y_woy double 1 [ 0, 1e32 ] um y_iRoy double 0 [ -1e32, 1e32 ] um^-1





test_function enumerated LI LI, mode_profile



Test_Tomin double 25 [ 0, 1e32 ] C Test_Tomax double 75 [ 0, 1e32 ] C

Test_Topoints integer 3 [ 0, 20 ] none





Parameter Descriptions wavelength Wavelength of the laser, λ mode Type of wavelength grid deltaFreq Frequency or wavelength grid spacing for multi-line output effint Current injection efficiency, ηI kf Output power coupling coefficient, kf b Spontaneous emission coupling coefficient, β tp Photon lifetime, τp tn Carrier lifetime, τn Go Gain coefficient, Go ag0-ag2, bg0-bg2 Gain coefficient empirical parameters, ag0-ag2, bg0-bg2 Ntr Carrier transparency number, Ntr cn0-cn2 Transparency number empirical parameters, cn0-cn2 e Gain saturation factor, ε alpha Linewidth enhancement factor, α Ilo Leakage current factor, A a0-a3 Leakage current empirical parameters, a0-a3 overlap_calculation Overlap calculation flag: on, off overlap Overlap parameter, ρ gam00, gam01 Overlap coefficients for N0 and S rate equations, γ00 and γ01 phi100, phi101 Overlap coefficients for N1 rate equation, φ100 and φ101 hdiff Diffusion parameter, hdiff

RIN VCSEL relative intensity noise, RIN To Ambient temperature, To

Rth VCSEL thermal impedance, Rth tth Thermal time constant, τth Rd VCSEL cavity resistance, Rd Von VCSEL turn-on voltage, Von Drive_Scheme Drive signal definition: direct_drive, bias_tee Rs Source impedance, Rs Io Laser bias current, Io Parasitics Parasitics flag: on, off Lb VCSEL bond inductance, Lb Cp VCSEL parasitic capacitance, Cp eps Accuracy of ODE solver initial_tstep Initial time step taken by ODE solver min_tstep Minimum time step taken by ODE solver spatial_effects Switch to turn spatial-addition by model block on or off. mode_mapping Transverse mode profile style selection (single- vs. multimode). multi_mode_powers List of mode powers, or file-name with this data. mode_polarization Polarization mode-attachment selection. mode_default_domain Default mode domain override. mode_width_x User-specified mode domain width along the X axis. mode_width_y User-specified mode domain width along the Y axis.


mode_default_grid_spacing Default grid spacing override. mode_dx User-specified X-axis grid spacing. mode_dy User-specified Y-axis grid spacing. x_single_mode_type Single-mode profile selection for X polarization. x_multi_mode_type Multimode profile selection for X polarization. x_mode_file Single-mode gridded mode data file for X polarization. x_multimode_file Multimode data file for X polarization. x_l Mode index l for X polarization. x_m Mode index m for X polarization. x_l_lo Lower bound for X polarization sweep of mode index l. x_l_hi Upper bound for X polarization sweep of mode index l. x_m_lo Lower bound for X polarization sweep of mode index m. x_m_hi Upper bound for X polarization sweep of mode index m. x_wo Laguerre-Gaussian spot size for X polarization. x_iRo Inverse of Laguerre-Gaussian radius of curvature for X

polarization. x_wox Hermite-Gaussian X-axis spot size for X polarization. x_iRox Inverse of Hermite-Gaussian X-axis radius of curvature for X


polarization. x_Nclad Cladding index for X-polarized LP mode. x_Ncore Core index for X-polarized LP mode. x_core_radius Core radius for X-polarized LP mode. x_outer_radius Outer radius for X-polarized donut and spot modes. x_inner_radius Inner radius for X-polarized donut mode. y_single_mode_type Single-mode profile selection for Y polarization. y_multi_mode_type Multimode profile selection for Y polarization. y_mode_file Single-mode gridded mode data file for Y polarization. y_multimode_file Multimode data file for Y polarization. y_l Mode index l for Y polarization. y_m Mode index m for Y polarization. y_l_lo Lower bound for Y polarization sweep of mode index l. y_l_hi Upper bound for Ypolarization sweep of mode index l. y_m_lo Lower bound for Y polarization sweep of mode index m. y_m_hi Upper bound for Y polarization sweep of mode index m. y_wo Laguerre-Gaussian spot size for Y polarization. y_iRo Inverse of Laguerre-Gaussian radius of curvature for Y

polarization. y_wox Hermite-Gaussian X-axis spot size for Y polarization. y_iRox Inverse of Hermite-Gaussian X-axis radius of curvature for Y


polarization.


y_Nclad Cladding index for Y-polarized LP mode. y_Ncore Core index for Y-polarized LP mode. y_core_radius Core radius for Y-polarized LP mode. y_outer_radius Outer radius for Y-polarized donut and spot modes. y_inner_radius Inner radius for Y-polarized donut mode. test_function Test-function selection. Test_Imin Minimum current for LI curves Test_Imax Maximum current for LI curves Test_LIpoints Number of points per LI curve Test_Tomin Minimum ambient temperature for LI curve generation Test_Tomax Maximum ambient temperature for LI curve generation Test_Topoints Number of LI curves (one per ambient temperature setting) test_mode_polarization Test plot field polarization selection. test_mode_number Test plot mode number. test_mode_wavelength Test plot field wavelength. test_mode_plot Test plot style selection.


Spatial Light Emitting Diode (LED)

This block models a Light Emitting Diode (LED) directly modulated with an electrical signal. It is identical to the standard LED model, except for the ability to include transverse mode profiles in the optical output.

The model computes the electrical current injected into the LED’s optical cavity and generates the optical response, including linewidth and relative intensity noise. The behavior of the model can be partitioned into three blocks, as shown in Fig. 1.

The driving source consists of the electrical signal input into the model. The parasitics consist of a bond inductance and shunting capacitance. Finally, the LED optical cavity is modeled via a simplified current-voltage (IV) relationship and linear carrier rate equation.

Figure 1: Main components of the Light Emitting Diode model.

LED Optical Response The LED cavity’s optical response is modeled via a linear rate equation for the cavity carrier density. Relative intensity noise is modeled via a constant value RIN (RIN). The carrier rate equation, following the treatment in [1], is:

n

dN I Ndt qV τ

= −

(1)

where I is the injection current, N is the carrier density, q is the electron charge, V is the cavity volume, and nτ is the carrier lifetime. The optical output power at wavelength λ (wavelength) is proportional to

N/ nτ . When the parasitic response time is much shorter than nτ , the carrier lifetime is proportional to the intrinsic LED rise/fall time rfτ (rise_fall_time), where ln(9)rf nτ τ= ⋅ . Because (1) is linear and the optical output is proportional to N, we can readily convert (1) into a frequency-domain transfer function for the output power versus input current, whose dc value is simply the LED responsivity ℜ (responsivity), measured in W/A. Thus, the intrinsic optical response is (in the frequency domain):

( ) ( )1out

n

P Ij

ω ωωτ

ℜ= ⋅+

(2)

where Pout is the optical output power and I is the cavity current. Note that when the parasitic response time contributes significantly to the rise/fall time, the value of the parameter rise_fall_time should be


adjusted accordingly by removing any parasitic contributions which are accounted for via the model’s parasitic element parameters.

In order to add a constant optical phase to this output, the user may specify either a fixed value (initial_phase), or request a randomly generated value (by setting randomInitialPhase to yes). In the latter case, the user must provide an integer seed (randomSeed) in the range –1e8 to 1. If randomSeed < 0, then randomSeed itself acts as the seed value for random number generation. If randomSeed = 0, then the LED component name acts as the seed. In both cases, the same random number is generated in successive simulations. Alternatively, if randomSeed = 1, then the system clock setting serves as the seed value, in which case different random numbers will be generated in successive simulations.

Linewidth Inclusion of a source linewidth in the LED output is controlled via the parameter linewidth_model, which by default is set to none (for a linewidth of zero). If linewidth_model=phase_noise, then linewidth is added to the output via phase noise. These random phase variations (seeded via the same randomSeed parameter that controls the random initial phase value) result in a Lorentzian output power spectrum [2].


LED Electrical Model The electrical model of the LED’s optical cavity is that of a simplified diode IV relationship, consisting of a series resistance Rd (Rd) and turn-on voltage Von (Von). During solution of the cavity current I, the model ensures that negative currents are effectively limited to zero.

Device parasitics may consist of a bond inductance Lb (Lb) and shunting capacitance Cp (Cp), as shown in Fig. 1. These can be turned on or off via the parameter parasitics.

Driving Source The Light Emitting Diode is driven by a combination of the electrical signal at its input and, when applicable, a dc bias current specified by Ibias (Ibias).

The input electrical signal can be interpreted in a variety of ways, based on the settings of the model parameter drive_scheme. This parameter can take on values of direct_drive or bias_tee.

• direct_drive

The input signal is assumed to come from either an ideal current source or zero-impedance voltage source. In other words, the LED is assumed to be undergoing direct-drive modulation. If the input signal is a current, then it is combined with the bias current Ibias to form the total input current. Figure 2(a) illustrates this scenario. If the input signal is a voltage, then Ibias is ignored. Note that the input voltage should be larger than the LED’s turn-on voltage Von; otherwise, the resulting current is zero. This arrangement is illustrated in Fig. 2(b).

• bias_tee

The input signal is assumed to be generated by a source with output impedance Rs (Rs), connected to the LED via an ideal bias tee. The bias current Ibias is similarly connected. The input signal can be either a current or a voltage. Figure 3 depicts a bias tee setup for both the voltage- and current-source cases.


Figure 2: Direct-drive modulation schemes: (a) current and (b) voltage.

Figure 3: Bias-tee modulation schemes: (a) voltage source and (b) current source.










• single





1.0 2.0 3.0 4.0






• file


• radial_file


• LG


• HG


• LP


• spot


• donut





...


• Two-dimensional gridded mode:


gridded <l> <m> <filename>










Multi-Line Output It is frequently necessary to produce several signals with similar properties but different wavelengths. In DWDM simulations especially, a series of regularly spaced optical sources is common. The LED model provides a number of convenient facilities so that many or all required lines can be generated from a single icon. To produce a series of lines spaced equally in wavelength or frequency, set the parameter mode=LambdaGrid or mode=FreqGrid, respectively. The number of sources is controlled by the number of electrical inputs to the laser, and in both modes the parameter wavelength specifies the first line in a series of ascending wavelengths or ascending frequencies, respectively. The source separation in wavelength or frequency is specified with deltaFreq. To set the number of electrical inputs to the laser, select the LED icon and open the menu item Properties. In the Ports tab, number_input_ports field, enter the number of inputs.

Multi-Line Multi-Node Output The comb of sources may be emitted either through a single node (the default) or with one channel per output node. To achieve the latter behavior, select the LED icon and open the menu item Properties. Set the value of the number_output_ports field in the Ports tab to that of the number_input_ports field.

Test Functions In order to ascertain whether the parameter settings for the Light Emitting Diode block provide the component performance desired, the user may test them from the component parameter editing window. This test produces either a light-current curve, small-signal frequency response curves, or a mode profile. By setting the test_function parameter to the desired output and clicking the Test button in the component-parameter editing window, the user may display the LED characteristics summarized below. Furthermore, the default plot ranges for most of the characteristics may be overridden by setting test_default_settings to no, and specifying values for test_function_x_low, test_function_x_high, and test_function_points


(the number of data points to plot). For frequency response curves, the user may also select whether or not to display the x-axis in log scale (test_log_x).

• LI: Plots the LED’s light-current (LI) characteristic. The plot range should be specified in amperes. The plot units can be set via the parameters power_units and current_units.

• transfer_function: Displays the LED’s small-signal optical transfer function (optical power versus current), including parasitic effects. The plot range should be specified in Hertz. The plot units can be set via the parameters response_units and frequency_units.

• S21: Displays the LED’s small-signal S21 response. The plot range should be specified in Hertz. Plot units can be set via the parameters response_units and frequency_units.

• mode_profile: A number of parameters are available for testing the spatial settings. First, test_mode_polarization is used to select which polarization to use when plotting the mode profiles. test_mode_number is used when the model is in multi_from_single mode. The available modes are numbered from zero, and in the order by which they are attached to optical signals. test_mode_wavelength determines the field wavelength at which to plot the test mode. Finally, test_mode_plot determines the type of plot to generate: magnitude plots the profile’s transverse magnitude, mag_phase plots its magnitude and phase, and real_imag plots the real and imaginary parts of the mode.

References [1] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, New York, 1991).



Properties








responsivity double 1.0 [ 0, 1e32 ] W/A rise_fall_time double 1e-9 [ 0, 1e32 ] s

linewidth_model enumerated phase_noise none, phase_noise, value



linewidth double 10e12 [ 0, 1e32 ] Hz or m randomInitialPhase enumerated no no, yes

initial_phase double 0 [ -180, 180 ] degrees


RIN double -150.0 [ -1e32, 1e32 ] dB/Hz Rd double 5.0 [ 0, 1e32 ] ohm

Von double 2.0 [ 0, 1e32 ] V

drive_scheme enumerated direct_drive direct_drive, bias_tee

Rs double 50.0 [ 0, 1e32 ] ohm Ibias double 25e-3 [ 0, 1e32 ] A

parasitics enumerated no no, yes

Cp double 2e-12 [ 0, 1e32 ] F

Lb double 0.3e-9 [ 0, 1e32 ] H spatial_effects enumerated on on, off



mode_polarization enumerated xy_same xy_same, xy_unique mode_default_domain enumerated yes yes, no












x_m_hi integer 0 [ 0, 75 ] none x_wo double 1 [ 0, 1e32 ] um

x_iRo double 0 [ -1e32, 1e32 ] um^-1


x_iRox double 0 [ -1e32, 1e32 ] um^-1 x_woy double 1 [ 0, 1e32 ] um

x_iRoy double 0 [ -1e32, 1e32 ] um^-1




x_core_radius double 1 [ 0, 1e32 ] um

x_outer_radius double 1 [ 0, 1e32 ] um x_inner_radius double 0 [ 0, 1e32 ] um



y_mode_file string

y_multimode_file string y_l integer 0 [ -75, 75 ] none


y_l_lo integer 0 [ -75, 75 ] none

y_l_hi integer 0 [ -75, 75 ] none y_m_lo integer 0 [ 0, 75 ] none


y_wo double 1 [ 0, 1e32 ] um

y_iRo double 0 [ -1e32, 1e32 ] um^-1 y_wox double 1 [ 0, 1e32 ] um


y_woy double 1 [ 0, 1e32 ] um

y_iRoy double 0 [ -1e32, 1e32 ] um^-1 y_Nclad double 1 [ 0, 1e32 ] none


y_core_radius double 1 [ 0, 1e32 ] um

y_outer_radius double 1 [ 0, 1e32 ] um y_inner_radius double 0 [ 0, 1e32 ] um

test_function enumerated LI LI, transfer_function, S21, mode_profile


test_function_x_low double 0 [ 0, 1e32 ] none

test_function_x_high double 100e-3 [ 0, 1e32 ] none test_function_points integer 201 [ 2, 100000 ] none

test_log_x enumerated yes yes, no





current_units enumerated mA nA, uA, mA, A frequency_units enumerated Hz Hz, kHz, MHz, GHz, THz

response_units enumerated dB linear, dB

Parameter Descriptions wavelength Wavelength of the LED, λ


mode Type of wavelength grid deltaFreq Frequency or wavelength grid spacing for multi-line output responsivity Optical response coefficient ℜ

rise_fall_time Intrinsic rise/fall time ln(9)rf nτ τ= ⋅

linewidth_model Linewidth model selection linewidth_units Linewidth units selection linewidth Linewidth value randomInitialPhase Random phase setting initial_phase Constant phase value, when randomInitialPhase = no randomSeed Random-number-generation seed value RIN Relative intensity noise, RIN Rd Cavity resistance Rd Von Turn-on voltage Von drive_scheme Drive signal setting Rs Source impedance Rs Ibias Bias current Ibias parasitics Parasitics setting Cp Parasitic capacitance Cp Lb Bond inductance Lb

power_units Units for optical power current_units Units for electrical current frequency_units Units for modulation frequency response_units Units for small-signal response data spatial_effects Switch to turn spatial-addition by model block on or off. mode_mapping Transverse mode profile style selection (single- vs. multimode). multi_mode_powers List of mode powers, or file-name with this data. mode_polarization Polarization mode-attachment selection. mode_default_domain Default mode domain override. mode_width_x User-specified mode domain width along the X axis. mode_width_y User-specified mode domain width along the Y axis. mode_default_grid_spacing Default grid spacing override. mode_dx User-specified X-axis grid spacing. mode_dy User-specified Y-axis grid spacing. x_single_mode_type Single-mode profile selection for X polarization. x_multi_mode_type Multimode profile selection for X polarization. x_mode_file Single-mode gridded mode data file for X polarization. x_multimode_file Multimode data file for X polarization. x_l Mode index l for X polarization. x_m Mode index m for X polarization. x_l_lo Lower bound for X polarization sweep of mode index l. x_l_hi Upper bound for X polarization sweep of mode index l. x_m_lo Lower bound for X polarization sweep of mode index m. x_m_hi Upper bound for X polarization sweep of mode index m. x_wo Laguerre-Gaussian spot size for X polarization.


x_iRo Inverse of Laguerre-Gaussian radius of curvature for X polarization. x_wox Hermite-Gaussian X-axis spot size for X polarization. x_iRox Inverse of Hermite-Gaussian X-axis radius of curvature for X polarization. x_woy Hermite-Gaussian Y-axis spot size for X polarization. x_iRoy Inverse of Hermite-Gaussian Y-axis radius of curvature for X-polarization. x_Nclad Cladding index for X-polarized LP mode. x_Ncore Core index for X-polarized LP mode. x_core_radius Core radius for X-polarized LP mode. x_outer_radius Outer radius for X-polarized donut and spot modes. x_inner_radius Inner radius for X-polarized donut mode. y_single_mode_type Single-mode profile selection for Y polarization. y_multi_mode_type Multimode profile selection for Y polarization. y_mode_file Single-mode gridded mode data file for Y polarization. y_multimode_file Multimode data file for Y polarization. y_l Mode index l for Y polarization. y_m Mode index m for Y polarization. y_l_lo Lower bound for Y polarization sweep of mode index l. y_l_hi Upper bound for Ypolarization sweep of mode index l. y_m_lo Lower bound for Y polarization sweep of mode index m. y_m_hi Upper bound for Y polarization sweep of mode index m. y_wo Laguerre-Gaussian spot size for Y polarization. y_iRo Inverse of Laguerre-Gaussian radius of curvature for Y polarization. y_wox Hermite-Gaussian X-axis spot size for Y polarization. y_iRox Inverse of Hermite-Gaussian X-axis radius of curvature for Y polarization. y_woy Hermite-Gaussian Y-axis spot size for Y polarization. y_iRoy Inverse of Hermite-Gaussian Y-axis radius of curvature for Y-polarization. y_Nclad Cladding index for Y-polarized LP mode. y_Ncore Core index for Y-polarized LP mode. y_core_radius Core radius for Y-polarized LP mode. y_outer_radius Outer radius for Y-polarized donut and spot modes. y_inner_radius Inner radius for Y-polarized donut mode. test_function Test function output selection. test_default_settings Selection for default settings in test function output test_function_x_low Lowest x-value for test function output test_function_x_high Highest x-value for test function output test_function_points Number of points in test function output test_log_x Switch for using log scale for test function x-values test_mode_polarization Test plot field polarization selection. test_mode_number Test plot mode number. test_mode_wavelength Test plot field wavelength. test_mode_plot Test plot style selection.


Thin Lens

This block models an optical lens using the thin lens approximation [1]. For a set of optical input signals, a phase transformation is applied to each signal’s transverse mode profile. This transformation changes an incident signal’s phase front, thereby affecting focusing (positive focal length) or defocusing (negative focal length) of the beam. The applied phase transformation is:

( ) ( )

λ+π

−=f

yxjyxt22

exp,

(1)

where x and y are transverse coordinates, λ is the signal wavelength, and f (focal_length) is the lens focal length. It should be noted that for spherical lenses with surface radii of curvature R1 and R2, and material index n, the focal length can be expressed as:

( )

−−=

21

1111RR

nf

(2)

Typical lenses will not transmit all of the incident power. To account for this, the model provides a power-reflectivity parameter, lens_reflectance. Aperturing due to finite lens dimensions can also be activated via the parameter aperturing. In this case, the lens is assumed to be circular, with diameter lens_diameter. Outside of this diameter, no phase transformation occurs, but a secondary power reflectivity may be specified via outer_reflectance. By setting outer_reflectance to 100%, the aperture completely blocks the field outside of the lens boundaries.

Numerical Effects If aperturing is turned off, then for incident optical signals with Gaussian (e.g., Laguerre- or Hermite-Gaussian) transverse mode profiles that have not been translationally or rotationally modified (e.g., by the Spatial Coupler), the Thin Lens block directly modifies the mode profile’s radius of curvature. This technique preserves the mode’s exact analytical description. In all other cases, the output signal is generated as a gridded mode, with data stored over a rectangular grid of points. The grid spacing is determined by the default settings of the input signal. However, to ensure that this grid spacing is sufficiently small to capture the fine detail of the lens’s phase transformation, the model parameter grid_correction should be set to yes.

References [1] J. W. Goodman, Introduction to Fourier Optics, 2nd. ed. (McGraw-Hill, New York, 1996).

Properties




Parameter Values Name Type Default Range Units focal_length double 10 [ -1e32, 1e32 ] mm

aperturing enumerated no yes, no

lens_diameter double 5 [ 0, 1e32 ] mm

lens_reflectance double 0 [ 0, 100 ] % outer_reflectance double 0 [ 0, 100 ] %

grid_correction enumerated yes yes, no

Parameter Descriptions focal_length Lens focal length.

aperturing Lens aperturing selection. lens_diameter Diameter of circular lens. lens_reflectance Power reflectance within lens boundaries.

outer_reflectance Power reflectance outside of lens boundaries. grid_correction Output-field grid auto-correction setting.


Vortex Lens

This block models a diffractive optical element that combines incident signal’s phase front transformation like in thin parabolic lens with phase vortex [1]. In general, solutions of propagation equation in graded index multimode fiber yields a complex set of propagating modes corresponding to both axial/meridional and skew rays. Vortex lens can be used to improve the coupling into the skew rays of graded index fiber. This approach offers an alternative to existing methods based on tilting and angular offsets for the conditioned launch problem.

For a set of optical input signals, a phase transformation is applied to each signal’s transverse mode profile. This transformation changes an incident signal’s phase front, thereby affecting focusing (positive focal length) or defocusing (negative focal length) of the beam. The applied phase transformation is:

+−= θ

λπ m

frnjyxt

2exp),(

2

,

r2 = x2 + y2 and θ = atan (y/x),

(1)

where x and y are transverse coordinates, λ is the signal wavelength, n (lens_index), is material index and f (focal_length) is the lens focal length. The parameter m is a vortex order. The skew rays correspond to all modes with m > 0, which correlates to rays spiraling both clockwise and counterclockwise. Therefore, the skew rays can be selectively excited by properly choosing the phase profile of the incident beam with the desired dependence.

Typical lenses will not transmit all of the incident power. To account for this, the model provides a power-reflectivity parameter, lens_reflectance. Aperturing due to finite lens dimensions can also be activated via the parameter aperturing. In this case, the lens is assumed to be circular, with diameter lens_diameter. Outside of this diameter, no phase transformation occurs, but a secondary power reflectivity may be specified via outer_reflectance. By setting outer_reflectance to 100%, the aperture completely blocks the field outside of the lens boundaries.

Numerical Effects If aperturing is turned off, then for incident optical signals with Gaussian (e.g., Laguerre- or Hermite-Gaussian) transverse mode profiles that have not been translationally or rotationally modified (e.g., by the Spatial Coupler), the Vortex Lens block directly modifies the mode profile’s radius of curvature. This technique preserves the mode’s exact analytical description. In all other cases, the output signal is generated as a gridded mode, with data stored over a rectangular grid of points. The grid spacing is determined by the default settings of the input signal. However, to ensure that this grid spacing is sufficiently small to capture the fine detail of the lens’s phase transformation, the model parameter grid_correction should be set to yes.

References [1] E. G. Johnson, J. Stack, and C. Koehler, “Light Coupling by a Vortex Lens into Graded Index Fiber”, J. of Lightwave Technology, vol.19, no.5, p.753 (2001)


Properties



Parameter Values Name Type Default Range Units vortex_order integer 0 [ 0, 1000 ]

focal_length double 1 [ -1e32, 1e32 ] mm

aperturing enumerated no yes, no lens_diameter double 5 [ 0, 1e32 ] mm

lens_index double 1.4 [ 1, 10 ]

lens_reflectance double 0 [ 0, 100 ] %

outer_reflectance double 0 [ 0, 100 ] % grid_correction enumerated yes yes, no

Parameter Descriptions vortex_order Vortex order.

focal_length Lens focal length. aperturing Lens aperturing selection. lens_diameter Diameter of circular lens..

lens_index Power reflectance outside of lens boundaries. lens_reflectance Power reflectance within lens boundaries.. outer_reflectance Power reflectance outside of lens boundaries grid_correction Output-field grid auto-correction setting


Spatial Coupler

This model is used to connect the output of one spatial model to the input of another. It allows the spatial models to be translationally and rotationally offset, and it performs a free-space propagation between the two components. Consider the topology of Figure 1 as an example:

Figure 1: Spatial coupler topology.

In this figure, the output of a spatial VCSEL model is to be connected to the input of the multimode fiber model. The coupler has six position-dependent parameters; the first three ),,( zyx ∆∆∆ allow the user to specify a translational offset between the two components. The parameters x∆ (xoffset) and y∆ (yoffset) enable the user to attribute a transverse offset between the two models while the parameter z∆ (distance) specifies the free-space distance between them (Figure 2).

Figure 2: Translational offsets in the spatial coupler model.

The remaining three positional parameters ),,( zyx φφφ specify rotation about the x, y, and z axes, respectively. These parameters are labeled phi_x, phi_y, and phi_z. Because the commutative property does not apply to rotations, it is necessary to specify the order in which the rotations occur. The convention adopted in OptSim is to first apply the rotation about the z-axis zφ , then the rotation about the x-axis

xφ and finally yφ . The rotations are defined in Figure 3. Note that the coordinate system is defined to be right handed with propagation in the +z direction.


Figure 3: Rotational definitions.

The spatial coupler functions by first applying five of the six transformations ),,,,( zyxyx φφφ∆∆ to the spatial field that emerges from the first model (e.g., the VCSEL in Figure 1). Once these transformations have been completed, the field is expressed in a new coordinate system ),,( zyx ′′′ [1]. The spatial coupler performs a free-space propagation over the specified distance in this new coordinate system then transforms the field back into the ),,( zyx coordinate system that is expected by the second model (e.g., the multimode fiber in Figure 1).

Modification of the spatial field as a result of translation, rotation, and propagation is only one function of the coupler model. If the spatial field is rotated, this effect must also be accounted for in the time-varying fields. If the incoming signal is expressed as ),,,( tzyxE , once it is rotated and translated, it is expressed as ),,,( tzyxE ′′′ . It is then propagated in the ),,( zyx ′′′ coordinate system after which it is mapped back into the ),,( zyx domain. Here the primed quantities denote values in the transformed coordinate system. The coordinate transformations can be expressed as:

zcybxax ˆˆˆˆ ++=′

zfyexdy ˆˆˆˆ ++=′

ziyhxgz ˆˆˆˆ ++=′

where a-i are the direction cosines of the transformation [1]. If the spatial fields are denoted by ψ , the input field can be expressed as:

yzyxtExzyxtEtzyxE yyxx ˆ),,()(ˆ),,()(),,,( ψ+ψ=

The time-varying and spatial fields are clearly related here through multiplication. Note also that the spatial fields have only x and y components; the OptSim convention is that spatial components in the z direction are either nonexistent or are negligible. After transformation, the field is expressed as:

zyzyxtExzyxtEtzyxE yyxx ′+′′′′ψ′+′′′′ψ′=′′′ ˆ0ˆ),,()(ˆ),,()(),,,(

Note that the coordinate transform is defined so that there is no field component in the z′ˆ direction. However, as evidenced by the )ˆ,ˆ,ˆ()ˆ,ˆ,ˆ( zyxzyx →′′′ coordinate system mappings described above, the transformation back to ),,( zyx will result in a nonzero field component in the z direction.

[ ][ ][ ]zzyxtfEzyxtcE

yzyxteEzyxtbE

xzyxtdEzyxtaEtzyxE

yyxx

yyxx

yyxx

ˆ),,()(),,()(

ˆ),,()(),,()(

ˆ),,()(),,()(),,,(

ψ+ψ+

ψ+ψ+

ψ+ψ=

As mentioned previously, the OptSim convention is to assume that the majority of power is contained in the transverse spatial fields; thus, the z component above is ignored. This is typically an excellent assumption as long as the amount of tilt is not too large and the propagation distance is not too great. To ensure this, the tilt is limited to o10± and the propagation distance to 100 mµ . However, in the event that 10% or more of the power is lost as a result of ignoring fields in the z direction, a warning will be issued.

The equation above has one major consequence: Each polarization of the output spatial field contains components of both polarizations of the input field. Because the temporal and spatial fields are stored as separate entities in OptSim, there is no way to internally store the sum

),,()(),,()( zyxtEzyxtE yyxx ψ+ψ


as a single entity. Thus, these two components have to be separated: The end result is that a single incident optical signal will, in general, produce two output optical signals (OpSigs) from the coupler. Rewriting the previous expression:

[ ] [ ]zfyexdzyxtEzcybxazyxtEtzyxE yyxx ˆˆˆ),,()(ˆˆˆ),,()(),,,( ++ψ+++ψ=

As before, all z-component data is neglected so:

[ ] [ ]

2#1#

ˆˆ),,()(ˆˆ),,()(),,,(

OpSigOpSig

yexdzyxytyEybxazyxxtxEtzyxE

+=

+ψ++ψ=

There are a few special cases where a single input OpSig will produce only a single output OpSig; e.g., ),,( zyx φφφ = (0, 0, 0), )0,,0( yφ , )0,0,( xφ . Additionally, a single output OpSig is produced if either the

Ex(t) or Ey(t) fields of the input signal is 0.

The entire preceding discussion assumes that it is important to determine how the power in the incident spatial field is decomposed into linear x and y components as a result of rotation. In one special situation it is unnecessary to maintain this information. When all subsequent spatial models in the topology are circularly symmetric, and when the only rotation occurs about the z-axis ),0,0( zφ , this step is not needed. In this case, the user can choose the parameter setting maintain_polarization=No. This will circumvent the storage of the second OpSig as described above and should result in a faster, more computationally efficient simulation.

Note that in the course of the coupling calculation, the model typically enforces that the grid spacing is not larger than a quarter-wavelength. If the user does not want this rule enforced, the flag grid_correction should be set to no.

Furthermore, the flag reduce_field allows the user to direct the coupler to make the output field as compact as possible. Figure 4 shows a representative transverse mode. Notice that the spatial representation used here pads the field with zeros on all four sides. When reduce_field=yes, OptSim will strip off these zero values which creates a smaller object and makes the simulation of subsequent spatial components more efficient.

Figure 4: Original field profile (left), reduced profile (right).

Beware that this operation can be quite time consuming when the object is surrounded by a lot of zeros; it is for this reason that the user is given the option to deactivate this feature.

Finally, the user may attenuate the signal passing through the coupler via the insertion_loss parameter.


References [1] M.R. Spiegel, Mathematical Handbook of Formulas and Tables. New York, NY: McGraw-Hill, 1968.

Properties

Inputs #1: Optical signal with spatial object

Outputs #1: Optical signal with spatial object

Parameter Values Name Type Default Range Units xoffset double 0 [ -100,100 ] Microns yoffset double 0 [ -100,100 ] Microns

distance double 0 [0,100 ] Microns

phi_x double 0 [ -10,10 ] Degrees

phi_y double 0 [ -10,10 ] Degrees phi_z double 0 [ -10,10 ] Degrees

insertion_loss double 0 [0,100] dB

grid_correction enumerated yes yes, no

maintain_polarization enumerated yes yes, no reduce_field enumerated yes yes, no

Parameter Descriptions xoffset Amount of translation of the input spatial field in the x-direction yoffset Amount of translation of the input spatial field in the y-direction distance Free-space propagation distance phi_x Amount of rotation of the input spatial field about the x-axis phi_y Amount of rotation of the input spatial field about the y-axis phi_z Amount of rotation of the input spatial field about the z-axis insertion_loss Amount of attenuation suffered as signal passes through coupler grid_correction Flag to control enforcement of quarter-wavelength grid spacing maintain_polarization Flag to either maintain or ignore the exact x or y polarization of the coupler output signal reduce_field Flag that determines whether extraneous zero values should be stripped from field


Spatial BeamPROP Interface

The Spatial BeamPROP Interface provides the user with an interface to BeamPROP, RSoft Design Group’s BPM simulation tool, whereby the spatial fields associated with an optical signal are propagated through an index structure specified via a BeamPROP index file. Much like the Spatial Coupler, which propagates the spatial fields through free space, this model propagates the fields through an arbitrary 3D index structure (2D index structures are incompatible with the spatial fields generated by OptSim). For example, the user may design a lens assembly in BeamPROP, and then simulate it in the context of a multimode link design.

Most of the BeamPROP simulation parameters should be set in the index file, specified via the parameter indexFile. However, a number of additional parameters are available within OptSim. First, to enforce that the grid spacing of the fields sent to BeamPROP are not larger than a quarter wavelength, grid_correction should be set to yes. Second, the flag reduce_field allows the user to direct the coupler to make the output field as compact as possible. When reduce_field=yes, OptSim will strip off any extraneous zero values at the boundaries of the output spatial field from BeamPROP, thereby creating a smaller object that makes the simulation of subsequent spatial components more efficient. Third, the user may specify the type of BPM simulation to perform via the parameter vector_mode. If this parameter is set to none, then a scalar BPM simulation is performed. If it is set to semi, then a semi-vectorial simulation is done. Finally, the parameter cleanup_files allows the user to control the deletion of the BeamPROP input- and output-field data files generated by the model. Setting this parameter to no prevents these files from being deleted.

Properties



Parameter Values Name Type Default Range Units indexFile string none

grid_correction enumerated yes yes, no none

reduce_field enumerated yes yes, no none

vector_mode enumerated none none, semi none cleanup_files enumerated yes yes, no none

Parameter Descriptions IndexFile BeamPROP index file

grid_correction Flag to control enforcement of quarter-wavelength grid spacing

reduce_field Flag that determines whether extraneous zero values should be stripped from field

vector_mode Flag that controls BeamPROP simulation method

cleanup_files Flag that controls deletion of BeamPROP input/output files


Multimode Fiber

This is a general-purpose model for circularly symmetric waveguides (e.g., multimode optical fiber) that can be operated in one of five different configurations. The first four configurations treat the fiber as a spatially dependent component by modeling the transverse field profiles and propagation constants of each optical mode that is supported by the fiber. The fifth configuration ignores the fiber’s spatial attributes and instead utilizes a high-level transfer-function-based approach.

Library Configuration The first configuration is library-based and models the fiber’s modal attributes using a pre-constructed database of spatial information. Basically, the user employs a device-level simulator to simulate a fiber with a specific refractive index profile and uses the results to produce a library of spatial mode profiles and propagation constant data over a specified range of wavelengths. OptSim then uses this library to simulate the system-level attributes of the multimode fiber. This enables multimode studies without the need to solve Maxwell’s equations, resulting in greater computational efficiency and faster execution. RSoft Design Group’s BeamPROP tool is ideally suited for this task, though other simulators can also be used provided that they produce output in the same format (see the Appendix). Because device simulators accept arbitrary refractive index profiles, the library-based approach enables the analysis of nonideal profiles, manufacturing variations, and similar effects. It is also ideal for generating manufacturer libraries for proprietary fiber structures. The biggest drawback to this approach is the need to generate a separate library for each fiber configuration anytime any physical attribute of the fiber (e.g., radius, peak index) changes. However, this is an unfortunate trade-off that must be made between flexibility and speed.

A sample library for a parabolic, graded-index fiber is included with this release as an example (Figure 1) and is contained in the folder named parabolic_multimode_fiber_library in the manufacturerlibrary folder in the OptSim installation directory. This library contains one file for each spatial mode at each wavelength of interest. It also contains a single file effective_indices that contains all of the information about the modal propagation constants over the wavelength range of interest. Additionally, it contains some header files that provide information about the library itself. Details of the library generation are contained in the Appendix to this section.

Figure 1: Ideal, parabolic refractive index profile.

This profile is described analytically as:


[ ]

>∆−

≤≤

∆−

=

arn

ararn

rn

21

021)(

21

2212 2

1

22

21

2nnn −=∆

Here, 1n is the peak index at the fiber center, 2n is the refractive index in the cladding, r is the radial distance from the fiber axis, a is the radius of the fiber core, and ∆ is a parameter quantifying the index step. For the sample library parabolic_multimode_fiber_library, 414.11 =n , %1=∆ , and

25=a µm ( 414.11 =n was chosen to correspond to a numerical aperture of 2.0=NA and 2n can be calculated from ∆ ). As indicated previously, if any of these parameters changes, a new library will have to be generated.

Once the library has been generated, there is very little work to simulate it in OptSim. There are three parameters for the library configuration: mode_directory (which is parabolic_multimode_fiber_library in this case), effective_index_filename (which is effective_indices here) and delay_method (to be discussed).

Modal spatial properties As mentioned above, the spatial multimode fiber model supports only circularly symmetric waveguides. Mathematically, the field is separated into independent radial and azimuthal components with the azimuthal portion assumed to vary sinusoidally [1] – [4]:

( ) ( ) ( )( )

φφ

⋅=φll

rErEsincos

,

The fiber library contains only the radial dependence of the mode E(r) for each spatial profile. It should be noted that each ( )φ,rE profile is normalized so that its total power is 1.0. The angular dependence is implicit; OptSim multiplies the radial profile by the appropriate sine or cosine terms during the simulation. Here, l is the azimuthal index number; the OptSim convention is that l denotes the cosine term while –l denotes the sine term.


Figure 2: Radial mode profile E(r) from library (top), cosine dependence (bottom left) sine dependence (bottom right).

In [1]-[4], it is shown that the cosine and the sine field profiles are degenerate modes; that is, ( ) ( )φ⋅ lrE cos and ( ) ( )φ⋅ lrE sin have identical propagation properties. Consequently, the propagation constant that is stored in the fiber library is the same for both of these degenerate modes. Since each fiber mode can exist in either the x or y polarization, there is actually a four-fold degeneracy. Note that when the azimuthal index is 0, the sine form of the fiber mode vanishes; since only the cosine form exists, there is only polarization degeneracy for l=0.

Consider the simple topology of Figure 3 in which a VCSEL emits an optical field that contains only the x polarization. The spatial portion of this field is also shown in Figure 3.

Figure 3: Simple spatial multimode fiber topology (left) VCSEL output beam (right).

For this simulation, a wavelength of 820 nm is chosen. When the collection of spatial fiber modes stored in the fiber library forms a complete basis, an arbitrary spatial field can be decomposed into a weighted sum of these modes:


( ) ( )∑=

φ=φn

iiiin rEcrE

1

,,

Here, n denotes the number of modes supported by the fiber and ( )φ,rEi is the spatial profile (including either the sine or cosine factor). It is important to note that the above relation is valid only when the fiber modes ( )φ,rEi form a complete basis. If the basis is incomplete, then the above relation will only be approximately true. OptSim models only the guided multimode fiber modes; unguided modes such as radiation and leaky modes, are neglected. This is true for all operational_mode configurations. When the multimode fiber input falls well within the fiber core, all of its power is guided; thus, the above relation is valid. As the input beam moves closer and closer to the core/cladding interface, more of its power gets coupled into unguided fiber modes. Thus, the relation below is more accurate:

( ) ( ) ( )∑∑==

φ+φ=φm

jjunguidedjunguided

n

iiguidediguidedin rEcrEcrE

1,,

1,, ,,,

Here, ( ) ( )φ=φ ,,, rErE iiguided from above. Since the unguided modes are, by definition, not guided, they propagate only a minimal distance before being attenuated. Thus, the assumption in OptSim is that by the time the energy reaches the fiber output, only the guided modes survive. If significant power is coupled into leaky modes and if the fiber length is very short, this may not be a good assumption. However, it is usually satisfactory for practical systems.

At 820 nm, there are 100 radial modes in the library; when the sine/cosine terms are both accounted for, a total of n=190 transverse modes result. The multimode fiber model computes the coupling coefficient

ic between the input spatial field and each of the spatial fiber modes according to:

( ) ( ) φφφ= ∫ ∫π ∞

rdrdrErEc iini ,, *2

0 0

Based on the above two equations, it is clear that the square of the coupling coefficient determines how much power gets coupled into each of the fiber modes. As seen from this analysis, it is clear that a single input field excites every fiber mode to a varying degree. The multimode fiber models this by internally generating n time-domain optical signals (one for each fiber mode) and attaching a different fiber mode profile to each. Each of these time-domain optical signals is identical to the time-varying portion of the input signal except that each of these signals is scaled by its respective coupling coefficient. This can be represented mathematically as:

( ) ( ) ( ) ( )[ ] ( )∑=

φ=φ=φn

iiiniininin rEtEctErEtrE

1

,,,,

Thus, a single incident field produces n signals just inside the fiber. The effects of dispersion, delay, and attenuation are then applied to each of these signals individually. The summary produced by the spatial analyzer at the fiber output is shown in Figure 4 and verifies that the output of the multimode fiber consists of 190 OpSigs. Following the OptSim convention, the multimode fiber model outputs the 190 optical signals as a linked list.

At the end of a simulation, the user can view a listing of the coupling coefficients by double clicking on the multimode fiber icon. The coupling coefficients are indexed by their azimuthal and radial indices and are outputted in (real, imaginary) form. Since the ci terms describe the degree of field coupling, the power coupling is described by *

ii cc ⋅ . The user can also view a list of the power coupling coefficients indexed by degenerate mode group number (DMG). The DMG is defined to be the sum of the azimuthal index plus twice the radial index and is a quantity that will be discussed in subsequent sections.


OpSig Spatial Field Summary---------------------------Input is a linked list of 190 OpSigs.

OpSig #1 has an Ex time-domain field and a spatial field in the X direction. Average power in the x-polarization is 0 wavelength=820 nm startTime=1.41517838374e-006 sOpSig #2 has an Ex time-domain field and a spatial field in the X direction. Average power in the x-polarization is 0 wavelength=820 nm startTime=1.415180304569e-006 s

Figure 4: Summary of fiber output.

Following the preceding discussion, it is clear that if the VCSEL produced light in both the x and the y polarizations, the multimode fiber would generate 2n optical signals internally (just inside the fiber):

( ) ( ) ( ) ( )[ ] ( )∑=

φ=φ=φn

iixxixxinxin rEtEctErEtrE

1,, ,,,,

( ) ( ) ( ) ( )[ ] ( )∑=

φ=φ=φn

iiyyiyyinyin rEtEctErEtrE

1,, ,,,,

Extending this to the general case, if the fiber input is a multimode optical signal that consists of q different modes in p different polarizations, the fiber model will generate qpn ⋅⋅ signals internally. Going one step farther, because each fiber mode profile is a function of wavelength, if the multimode input consists of different modes at r different wavelengths, then the fiber model will produce rqpn ⋅⋅⋅ signals internally.

Modal delay properties Each radial mode of a multimode fiber is characterized by its own specific delay time. Because each fiber mode travels at a different speed, a compact optical beam at the fiber input results in a broadened beam at the fiber output (Figure 5). This phenomenon is known as modal dispersion [1] – [4] and is the main speed limiter in multimode fibers. Recall that degenerate fiber modes possess the same modal delay; for example, the four fiber modes corresponding to the x and y polarizations of the sine and cosine fields all travel down the fiber at the same speed.

Figure 5: Modal dispersion due to varying modal delays.

The library configuration supports modal delay through a database of effective index information that is generated in a manner similar to the one used to generate the database of radial mode profiles. This is stored in a file effective_indices in the parabolic_multimode_fiber_library folder in the manufacturerlibrary folder of the OptSim installation directory. The effective index is proportional to the propagation constant:


0

, kN i

ieffβ

=

Here, 0k is the free-space wave number ( )02 λπ and iβ is the modal propagation constant. The group velocity igv , of each fiber mode is inversely related to the derivative of the propagation constant with

respect to the optical frequency; the delay of the mode iτ is directly proportional to this derivative.

i

ig ddvβω=,

ωβ

⋅==τdd

Lv

L i

igi

,

Here L is the fiber length. In this manner, each fiber mode possesses the same temporal attributes as the input field that stimulated it (scaled by the coupling coefficient) except that it is delayed by iτ . Furthermore, the optical carrier for each mode incurs an additional longitudinal phase delay proportional to the modal propagation constant:

( ) ( ) ( ), , , , ij zout i i in i iE r t c E t E r e βφ τ φ= − ⋅

As before, the total output field is obtained by combining all of the fields generated by each fiber mode:

( ) ( ) ( )1

, , , i

nj z

out i in i ii

E r t c E t E r e βφ τ φ=

= − ⋅ ∑

Thus, Figure 5 schematically represents the temporal output of a fiber that has three modes. The output pulses arrive at different times (corresponding to 31 τ−τ ) and have different heights (corresponding to

31 cc − ). Note that dispersive effects other than modal have been neglected in order to focus on the delay aspects of the model; these effects will be discussed in a later section.

To demonstrate the simulation of modal delays, consider the topology of Figure 3. A simulation is run with the VCSEL beam set to be a zero-order Gaussian beam offset from the fiber axis by 20 µm. The resulting time-domain signals are depicted in Figure 6. The single-pulse input from the VCSEL produces multiple signals at the fiber output. Although not discernable from Figure 6, there is a total of 190 pulses. Each output pulse is characterized by an amount of time that is characterized by its group velocity as defined above. As mentioned previously, degenerate modes share the same group velocity; thus, while there are 190 pulses, there are only 100 distinct delay times.

Figure 6: Multimode fiber time-domain input (left) output (right).


The delays are calculated by differentiating the effective index/propagation constant as described previously. Another technique is to use the Wentzel-Kramers-Brillouin (WKB) method which assumes that all of the optical power in the fiber modes is confined to the core [1], [2], [6]-[8]. Strictly speaking, this is not a valid assumption since a small amount of power tends to extend evanescently into the core for higher-order modes.

The previous example exhibited 100 different fiber delays, one for each distinct fiber mode ),( ml± , where l is the azimuthal index number and m is the radial index number. One of the by-products of the WKB assumption is that there is a unique modal delay not for each ),( ml± , but for each degenerate mode group (DMG), | l | + 2m. Clearly, the degeneracy comes about from the fact that different sets of ),( ml± can produce the same value of | l | + 2m; for example, )2,3( and )0,7( both possess the same degenerate mode group number 7 and thus possess the same modal delay. The running example of this section possesses 19 such degenerate mode groups.

The degenerate mode group concept is often used in system analysis to simplify the treatment of delay. The user can select which delay method is used in OptSim by setting the delay_method parameter to either normal or to WKB. The mathematical difference between the two delay methods is shown below. (Note that while the group velocity was previously defined in terms of optical frequency, the effective index in the library is actually stored as a function of wavelength).

λλ−=τ

oo d

dNN

cL eff

effnormal

+=

1

11

2 nN

Nn

cLn eff

effWKBτ

Here, c is the speed of light in vacuum and all other quantities are as previously defined. Please note that the WKB functionality in OptSim as stated above is not general and is valid only for parabolic refractive index profiles. Applying it to non-parabolic profiles will produce nonphysical results.

At the end of a simulation, the user can view a plot of the modal delays, scaled by the coupled power into each mode, by double clicking on the multimode fiber icon. The plot that corresponds to the simulation of Figure 6 is shown on the left of Figure 7. The numerical values of the delays can be discerned by using multimode fiber test function. The same simulation run with delay_method=WKB is shown on the right of Figure 7.

Figure 7: Modal delay plot produced by fiber model. Normal method (left), WKB method (right).


The other output plot generated at the end of simulation shows relative delay (in unit of s/m) and coupling coefficients (aka mode partition distribution - MPD) as a function of DMG. Figure 8 shows this type of plot corresponding to delay given in Figure 7 with WKB method.

Figure 8: Modal delay and MPD vs. DMG for WKB method.

To use the library configuration, the parameter operational_mode must be set to library and the field spatial_effects should be set to on.

Parabolic Configuration When using the library-based configuration, the user must generate a new database of fiber attributes whenever any physical property of the fiber changes. For example, even minute changes in the fiber’s physical properties, such the core radius or peak index, require an entirely new library to be generated. The fact that a new library is required for each different refractive index profile goes without saying. The main advantage of this approach is that a very wide range of fiber types can be simulated.

The documentation of the previous section contained a running example of a fiber library that was created using the parabolically graded refractive index profile. In practice, this is one of the most widely studied profiles in systems analysis. For this reason, the multimode fiber model also contains an analytical representation of the modes and propagation constants of this type of fiber. No library is needed and the fiber’s physical parameters can be modified arbitrarily. The main advantage of this model is the flexibility that it provides in changing all parameters except for the shape of the index profile. The main disadvantage is the inflexibility that it provides in changing the shape of the index profile.

The parabolic profile has been researched extensively and there are numerous analytical approximations to the field profiles and propagation constants [1] – [5], [9], [10]. In OptSim, the analytical field solutions in the core are described by:

φφ

ρρ=φρ−

)cos()sin(

)(),( 22, ll

eLErE ml

l

corelm o a

rnk ∆=ρ

221o

while the field solutions in the cladding are given by:

( )2 2 2, 2

sin( )( , ) ( 1)

cos( )m

lm clad l lm o

lE r E K r k n

lφ

φ βφ

= − −

o

In these expressions, Eo is a scaling factor that is used to match boundary conditions at the core/cladding interface and L(·) and K(·) are the associated Laguerre polynomials and the modified Bessel functions,


respectively. As with the library configuration, the parabolic configuration power normalizes the field profiles to 1.0.

The propagation constant lmβ is approximated as:

lmlm Qnk −=β 11o ank

mlQlm1

8)12(

o

∆++=

The corresponding group velocities are:

1

,1

2 12

lmlmg lm

lm

Qd cvd n Qβω

− − = = −

Notice that under the assumptions employed in this analytical approach, the Qlm factor is not a function of ),( ml± ; rather, it is a function of | l | + 2m . This is basically the degenerate mode group concept discussed

in the previous section. Thus, the analytical implementation of modal delays that is used in OptSim treats the delays in mode groups rather than as individual modes.

Note that because of the approximations that are made in the analytical formulation, high-order modes extremely close to cut-off sometimes go undetected. However, since there is usually minimal-to-negligible power coupled into these modes, the analytical method is still a very useful configuration. When it is important to capture each and every transverse mode, either the library or numerical configurations should be used.

There are three parameters that can be used to tailor the analytical description of the parabolic-index fiber: core_radius (a), peak_index ( )1n , and delta ( )∆ . These parameters were described in detail in the previous section.

The analytical formulation can detect a large number of modes depending on the settings of the parameters. If for the range of inputs (wavelength being considered an input), the number of calculated modes exceeds available memory, an error is issued and execution is halted. The user can do one of the following to alleviate the situation: he can decrease the core radius, decrease the index step, increase the wavelength, or adjust the Spatial parameters accordingly as described below.

To use this analytical configuration, the parameter operational_mode must be set to parabolic and the field spatial_effects should be set to on.

As a final note, OptSim requires that each spatial field object contain parameters describing the domain over which the field is defined and the grid spacing between field data points. The reason for this is intuitive when the spatial field is described numerically as in the case of the library configuration, where these parameters are automatically defined. Their purpose when using the parabolic configuration is not as obvious. When OptSim performs numerical calculations such as integrations, plots, and the like, the analytical field must be numerically sampled over a finite spatial grid. In a few special cases, these calculations can be performed analytically, obviating the need for numerical sampling. However, in the vast majority of cases (e.g., calculation of the coupling coefficient between a numerical input field and analytical fiber modes), sampling is necessary. The mode_default_grid_spacing parameter in the Spatial menu allows the user to decide whether to let OptSim choose an appropriate grid spacing. If the user wishes to set the numerical sampling rate higher, he can do so by lowering the mode_dx and mode_dy parameters in the Spatial menu. For example, if the coupling coefficients do not appear to have been calculated accurately enough, the user can reduce the spacing between points to force a more accurate numerical integration. In another case, the contour plot produced by the spatial analyzer may appear “grainy.” Reducing the grid spacing in the analytical field will force subsequent models to reduce their numerical sampling intervals. The user should be aware, however, that reducing the grid spacing will increase the simulation time and the memory requirements. On the other hand, if simulations are taking too long to complete, the user may wish to lower the accuracy of the simulation and plots by increasing the grid spacing. Experimentation is often necessary to determine the optimum settings. Similarly, the domain over which the fiber modes are defined is set by the mode_default_radius and mode_radius parameters in


the Spatial menu. By default, all analytical fiber modes are defined over a range that is two times the core_radius.

Numerical Configuration The library configuration is useful when the user wishes to employ a device simulator to calculate the fiber modes and propagation constants. The parabolic configuration is useful when the user wishes to study the popular parabolic graded-index profile. There are occasions when a user would like to study the properties of a fiber with an arbitrary refractive index profile but either does not own a device simulation tool or does not wish to construct the required fiber library. For this reason, OptSim also offers a configuration that determines the fiber mode profiles and propagation constants for an arbitrary refractive index profile. This configuration employs a numerical simulation algorithm that generates solutions to the scalar radial Helmholtz equation [4]:

( ) ( ) ( ) ( ) 012

2222

2

2=

−β−++ rE

rlrnk

drrdE

rdrrEd

For a given azimuthal index l, the mode solver will generate solutions for each radial index m. As described in previous sections, the solution Elm(r) is then combined with either a sine or cosine term to incorporate the mode’s azimuthal dependence.

As with most numerical processes, appropriate bounds and tolerances must be established. A number of parameters are provided for this purpose. The radial mode solver will increment over successive values of the azimuthal and radial indices during its attempt to solve for the fields and propagation constants. Unfortunately, there is no way for the mode solver to know if it has found solutions to all of the possible guided modes without numerically iterating ad infinitum. To place bounds on this process, OptSim provides the parameters mode_limiting, l_max, m_max, and dmg_max. When mode_limiting is set to lm, OptSim only solves for modes with values for l and m no larger than l_max and m_max, respectively. When mode_limiting is set to dmg, OptSim only solves for modes whose mode group numbers are no larger than dmg_max. Once OptSim reaches these limits, it will stop the mode solving process. This is particularly useful in a scenario where hundreds or thousands of modes may exist but perhaps only a handful of them have meaningful amounts of power coupled into them. As an example, the parabolic profile described in previous sections had azimuthal numbers ranging from 0 ≤ l ≤ 18 and radial indices ranging from 0 ≤ m ≤ 9. Thus, in this case, mode_limiting should be set to lm, l_max should be set to 18, and m_max should be set to 9. In general, it is difficult, if not impossible, to know these maximum values in advance. In this case, the user should choose an initial value and let OptSim search for all of the modes (for example by running the output_type=report test in the Test menu). For example, if the maximum (l, m) mode indices for which OptSim finds guided modes are equal to (l_max, m_max), it is likely that more modes exist beyond (l_max, m_max). The user can then increase (l_max, m_max) until no further values are found. If (l_max, m_max) are set too high, there is no harm done; however, the larger these values are, the longer the simulation will take. If the user knows that only the first handful of modes are important, he can set (l_max, m_max) to low values, thus saving OptSim the time and effort of solving for unnecessary modes. In general, the maximum values for l and m increase with increasing core radius and with increasing index contrast between different regions of the fiber (for example, ∆ in the parabolic case).

The next numerical parameter is lambda_step. The solver calculates modal delay by differentiating the effective index over a range of wavelengths. The lambda_step parameter enables the user to select the range over which the group velocity is computed. If this parameter is too large, OptSim will have difficulty computing the mode velocities and may produce unexpected or nonphysical results (e.g., negative group velocity, infinite delay, and the like). Users are advised to maintain the default value of lambda_step = 1e-6, which should be sufficient for most applications. If specific applications are producing questionable results, users may choose to reduce lambda_step below 1e-6 to enhance simulation accuracy.

The final numerical parameter is mode_set. The mode solver operates by searching over effective index intervals spanning a specific range. If the user feels that OptSim is missing some of the modes, mode_set can be increased. The default value of 1000 is generally adequate for most applications. Increasing


mode_set will reduce the search interval and enhance simulation accuracy. However, this accuracy does not come without a price as the simulation will take much longer.

File-Based Index Profile The numerical mode of operation has three different configurations. The first configuration enables the user to specify the refractive index profile in a text file. This configuration is activated by setting spatial_effects=on and operational_mode=file. Once this is done, the user must enter the name of the index file in the index_file field under the Index tab of the parameter dialog box. OptSim will first check for the data file in the present working directory; if it is not located there, it will look in the directory C:\RSoft\products\OptSim\manufacturerlibrary. If the specified file is found in neither directory, an error message will be issued.

The format of the first line of this file is as follows: <NR> <RMIN> <RMAX> <0> <OUTPUT_REAL>

Here, NR is the number of data points in the file, RMIN is the minimum radial value in microns, RMAX is the maximum radial value in microns, 0 is simply a placeholder that must be there and OUTPUT_REAL specifies that the field data that follow will be in real format. Currently OptSim only accepts real values for the refractive index, though in future releases, complex values will be accommodated. For proper operation, RMIN should always be 0. As an example, the following are the first few lines of the file dip.ipf which is supplied in the OptSim multimode examples directory: 501 0 50 0 OUTPUT_REAL 1.4094487400000 1.4096817530000 1.4099147660000 . . .

The first line specifies that the file contains data at 501 radial points, that it begins at 0=r and ends at 50=r µm. Based on this information, it can be deduced that the radial increment is 1.0=∆r µm. Note

that the actual r values are not included in the file since they can always be calculated from the header information.

Function-Based Index Profile For simple core-cladding fiber geometries, users can also specify the refractive index profile as an analytical function. Two fields are provided for this purpose in the Index menu: core_index_profile and cladding_index_profile. The required syntax for these functions follows the standard OptSim conventions as described in the Parameter Expressions section of Chapter 3 of the OptSim User Guide (“Using the OptSim GUI”). Returning to our running example, the parabolic refractive index can be specified numerically through the following settings:

core_index_profile = n1*sqrt(1-2*delta*(r/radius)^2)

cladding_index_profile = n1*sqrt(1-2*delta)

Here, n1, delta, and radius represent the peak value of the index in the core, the index step, and the core radius, respectively. Following standard OptSim syntax conventions, these values must be established and assigned in the symbol table. The variable r is a special one and is reserved for describing the radial dependence of the index profile. Users should not define r in the symbol table as it will be automatically recognized by OptSim. To enable this capability, users should set spatial_effects=on and operational_mode=function.

Alpha Profile The parabolic refractive index profile is a special case of the more general alpha profile:


( )[ ]

>∆−

≤≤

∆−

=

α

arn

ararn

rn

21

021

21

212

When α = 2, several analytical approximations exist for the mode profiles and propagation constants as we described in previous sections. When α ≠ 2, however, analytical approximations do not exist and the fiber properties must be solved numerically. We could easily implement the alpha profile using the function-based index described in the previous section:

core_index_profile = n1*sqrt(1-2*delta*(r/radius)âlpha)

cladding_index_profile = n1*sqrt(1-2*delta)

Clearly, a new variable alpha would have to be established in the symbol table. Because the alpha profile is studied fairly often, this profile can be simulated in OptSim directly by setting spatial_effects=on and operational_mode = alpha. The effect is identical to manually setting the core and cladding index profiles as described above; this feature has been added mainly for the convenience of the end user. Along with this option the user must set the peak_index, delta, and alpha parameters in the Index menu.

Step-Index Configuration While communications-grade multimode fiber is typically of the graded-index variety, there may be cases where a basic step-index fiber is warranted (including large-core fiber applications). For these cases, by setting spatial_effects=on and operational_mode = step, the user may specify a step-index fiber via the parameters core_radius, peak_index (corresponding to the core index), and delta (where delta is the percent difference between the core and cladding indices, relative to the core). LP fiber modes are assumed, as described in [1], [3], and [9].

Differential Mode Attenuation Generally speaking, modes in a fiber will experience differential mode attenuation (DMA), wherein each mode experiences different amounts of loss. The model accounts for DMA by treating it as a function of wavelength λ and mode group number, which we earlier defined to be | l | + 2m, where l and m are the azimuthal and radial mode indices, respectively. Based on work in [15], we provide two approaches.

In the first case, strictly only valid for alpha index profiles, diff_mode_atten is set to alpha_fit, and we model the attenuation γ for a given mode as [15]:

22

01

(| | 2 ) 212

l mIan

αα

ρλ αγ γ η

π α

+

+ ⋅ + = ⋅ + ⋅ ⋅ ∆

where γ0 (attenuation) is the basic attenuation seen by all modes, Iρ is the ρth-order (dma_bessel_order) modified Bessel function of the first kind, η (dma_scaling_factor) is a scaling factor, n1 is the core peak index, a is the core radius, ∆ is the profile delta, and α is the profile alpha parameter. This function is an empirical relationship to be used when fitting experimental data.

More generally, the user may set diff_mode_atten to file, and use a data file (dma_filename) to specify scaling factors by which to scale the basic attenuation γ0 for each mode. Typically, measured values for DMA are determined as a function of some parameter which is directly proportional to either (| | 2 1)l m λ+ + ⋅ [16] or (| | 2 )l m λ+ ⋅ [17]. The user should specify which approach is used via the parameter dma_file_xconversion, with type1 used for the (| | 2 1)l m λ+ + ⋅ case, and type2 used for the (| | 2 )l m λ+ ⋅ case. In either case, the user must also specify a factor (dma_file_xfactor) by which to scale the x values in the data file in order to obtain corresponding values for (| | 2 1)l m λ+ + ⋅ or (| | 2 )l m λ+ ⋅ .


The y values in the data file are automatically normalized such that the first data point in the file corresponds to a scaling factor of 1. Thus, the resulting data is treated as measurements of scaling factor versus (| | 2 1)l m λ+ + ⋅ or (| | 2 )l m λ+ ⋅ . The format for the data file is:

DMAFormat

<num_pts>

<x-value 1> <DMA 1>

<x-value 2> <DMA 2>

<x-value 3> <DMA 3>

<x-value 4> <DMA 4>

...

Example:

DMAFormat

5

1 1.0

2 1.0

3 1.1

4 1.2

5 1.3

Corrections to the Modal Delay Due to Dispersion In order to properly calculate modal delays, the model should account for the effects of material dispersion on group velocity. In the case of the Library Configuration, the user may do so by including material dispersion when calculating the library data. Alternatively, if the index profile is of the alpha variety, the user may have the model automatically adjust the normally calculated delay values to include the effects of dispersion. In this case, the group velocity vg is adjusted according to the following equations, based on the analysis in [15]:

2

1

1 2

412

(4 )12

g gonv vN y

αα

αα

ξα

ξα

+

+

∆−+= ⋅ ⋅

+ ∆−+

where 2 2

2 21

2 ( 2 1)(2 )

l ma n

α λξα π+ + += ⋅

∆

In the above equations, vgo is the unadjusted group velocity, n1 is the core peak index, y (profile_disp_param) is the profile dispersion parameter, N1 is a user specified material group index (N1), a is the core radius, ∆ is the profile delta, α is the profile alpha parameter, l is the azimuthal mode index, and m is the radial mode index. To activate the above correction, the parameter dispersion_delay_mode should be set to alpha.

Alternatively, a Taylor-series-based expansion may be used instead [18]:


222 2

122

1 2 2

2 3 212 2 2

2 3 2 212 2 2

g gonv vN y y

α αα α

α αα α

α αξ ξα α

α αξ ξα α

+ +

+ +

− ∆ −+ ∆ ⋅ + ⋅+ += ⋅ ⋅

− − ∆ − −+ ∆ ⋅ + ⋅+ +

In this case, profile_disp_param should be set to alpha_taylor.

Note that the above expressions are strictly only valid for alpha profiles. For more complicated index profiles, a custom library should be generated.

Linear Configuration This fifth configuration analyzes the multimode fiber model from a non-spatial perspective. The linear model does not treat each fiber mode separately as do the other two modes of operation. Rather, it treats the individual fiber modes together, as an aggregate.

The linear multimode fiber configuration is implemented as a standard lowpass filter at baseband in input and output optical power with a Gaussian exponential decay transfer function [11] – [13]. This model assumes the fiber has sufficient mode mixing as described by Personick [11]. In the model, attenuation, connector efficiency, coupling loss, pulse broadening, and propagation delay are taken into account. For parallel fiber ribbons, the additional skew in the fiber ribbon due to the different laser wavelengths in each of the channels is also taken into account.

To represent multimode optical signals, this configuration modifies the interpretation of OptSim’s standard optical signal representation. The optical phase and frequency chirp information present in the optical signal is ignored and not propagated to the output of the model. This because the linear model operates on the optical power rather than on the field [11], [12] (here, * denotes complex conjugate):

( ) ( ) ( ) ( )[ ]fEfEfHfE ininout*=

In cases where it is important to preserve the phase of the input field, the user can set the flag ignore_phase to no. In this case, the phase of the input field is stored before the above calculations are made. At the end of the simulation, this phase is added to the output field.

The power transfer function for the fiber model is given by

( ) ( )[ ]dftjffH πσ 2)(2lnexp 2 −−=

where σ is the rms impulse response width, dt is the time delay of the fiber. The scaling factor of ln(2) is used in the exponential so that the magnitude of the transfer function is 0.5 (-3 dB) when f = 1/σ. When the cutback factor = 1.0 and modal dispersion is the dominant dispersion mechanism (which is often the case), 1/σ corresponds to the modal bandwidth of the fiber. The total rms impulse response width σ is

222modmat σ+σ=σ where the pulse broadening matσ due to material dispersion and the pulse broadening

modσ due to the modal dispersion are given by

LDsrcmat σ=σ cutback

im

mod

LB

γ

⋅

=σ1

1

Here L is the fiber length. The dispersion D and the attenuation are discussed in the next section. The linewidth srcσ is not a model input parameter; rather, it is a property that is added either directly by an optical source model or by the linewidth adder model.


The parameter imB is the intermodal bandwidth of the fiber and cutbackγ is the cutback factor that takes into account the variation of modal bandwidth with length. When the bandwidth of a multimode fiber is measured, it is clearly done on a piece of fiber that has a given length. Although modal bandwidth is expressed in units of MHz-km, in reality it is not always valid for all lengths of fiber (i.e., it can be non-scalable). Sometimes the bandwidth is measured on very short lengths of fiber; in this case, the bandwidth obtained by dividing imB by a large L may be too low. The converse can also be true when the bandwidth is measured on long lengths of fiber. The physical basis for this has to do with the effects of fiber length on differential mode attenuation and can be difficult to quantify. When this effect is significant, a fiber manufacturer should provide a table of bandwidths for several different lengths [14]. In this case, the user should be able to empirically determine what the cutback factor should be. When this effect is negligible, the cutback factor can simply be set to 1.0.

The time delay dt has two main components: the fixed delay propt as a result of the propagation delay in

the fiber, and the variable delay skewt that represents the fiber skew. The variable delay due to the wavelength is included in the chromatic dispersion. While the fixed delay has a direct correlation with the fiber length, the variable delay is generally treated as a random variable with mean value of zero and whose maximum value is determined by the skew in the ribbon. The skew variable delay is specified as either a fixed value in nonstatistical simulations or as a statistically varied value in the statistical simulation mode. The expression for the time delay td is given by

( )Lttt skewpropd +=

In addition to modeling the fiber itself, the linear multimode fiber model block also models the connector and coupling losses at each end of the fiber ribbon in the optical bus. These losses are specified explicitly as the coupling efficiency and connector loss parameters.

To use the linear configuration, the field spatial_effects should be set to off.

General There are several parameters that are common to all of the multimode fiber configurations. The first parameter spatial_effects can be set on or off as described by each configuration above. The second parameter, operational_mode, determines which spatial configuration is used; the choices are parabolic, file, function, and library. This option is only active when spatial_effects=on. The third parameter is the fiber’s radius, and the fourth parameter is the fiber length, while the fifth parameter specifies the attenuation. The next parameter, dispersion_mode, specifies how the chromatic dispersion is to be modeled. When dispersion_mode=specify, the user must enter a value in the dispersion field. When dispersion_mode=calculate, the model uses the zero-dispersion slope S0 and the zero-dispersion wavelength lambda0 to calculate the dispersion according to:

( )

λ

λ−λ=λ

4

400 1

4S

D

Finally, when spatial_effects=on, you may want to ignore modes into which coupling is negligible. In this case, you should set the parameter mode_drop_threshold to a value between 0 and 1. Any modes with power coupling coefficients below this value are dropped from the simulation. By default, this value is 0, ensuring that all modes are accounted for.

Test The multimode fiber model has several test functions for its various configurations.


Fiber mode plot When spatial_effects are on, the user can plot one of the fiber modes by specifying values for plot_azimuthal and plot_radial. Negative values for the azimuthal number are acceptable. Additionally, the user must specify the test_wavelength at which the plot is to be made. As usual, the user has the plot_mode choice of displaying the plot in magnitude, magnitude/phase, or real/imaginary format. If the specified mode does not exist (i.e., is cut-off) at the target wavelength, an error message will be returned.

Radial field plot This option is similar to the fiber mode plot except that it plots only the radial dependence of the fiber mode. Recall that the radial dependence is the only aspect of the field that is explicitly stored by OptSim. The azimuthal dependence is implied rather than stored. Consequently, this test function is useful for verifying either the contents of individual files in the fiber library or the results of radial mode calculations performed while using the analytical configuration. Because it is the radial field that is plotted, the same graph will be displayed regardless of the plot_azimuthal field.

Overfilled field plot The overfilled launch condition is defined as a spatial field that equally excites all fiber modes. Based on the definition of the coupling coefficient provided in the library configuration section of this model documentation, an overfilled launch profile can be constructed by superimposing all n of the fiber modes:

( ) ( )∑=

φ=φn

jjoverfilled rEErE

1

,, o

The scaling factor oE is chosen so that the total power contained in ( )φ,rEoverfilled is 1. Because the fiber

modes are orthogonal, the coupling coefficient ic will be zero whenever ji ≠ and constant when ji = .

( ) ( ) ( ) ( ) φφφ=φφφ= ∫ ∫ ∑∫ ∫π ∞

=

π ∞rdrdrErEErdrdrErEc i

n

jjioverfilledi ,,,, *

2

0 0 1

*2

0 0o

Because of the unity-power normalization of ( )φ,rEoverfilled and of each ( )φ,rEi , all coupling coefficients should be identical and equal to

nn

Pc totalin

i1, ==

The overfilled launch for the default fiber library included with OptSim is depicted in Figure 9.


Figure 9: Overfilled fiber launch.

All 190 coupling coefficients have a value that is approximately 0.0725. As usual, the user has the plot_mode choice of displaying the plot in magnitude, magnitude/phase, or real/imaginary format. Also, the overfilled launch plot function works equally well for all spatial operational_mode settings.

This test function can be used to save the overfilled launch profile to a disk file. This file can then be loaded into any of the spatial optical source models or the Spatial Adder model and subsequently launched into the fiber using the x _single_mode_type (y _single_mode_type ) = file option under the X _Spatial (Y _Spatial) tab. Indeed, this is how Figure 9 was generated. To save the overfilled launch profile to a file, simply enter the desired filename in the overfilled_filename field in the Test tab.

Delay plot When spatial_effects are on, the user can plot the power coupled into each spatial mode as a function of the individual modal delays. The assumption is that all modes are excited equally so that the power coupled into each mode is 1/n. Also, test plot function works equally well for both spatial operational_mode settings. When operational_mode=library, the user can select either delay_method.

Figure 10: Plots of delay_plot test function. Analytical configuration (left) library configuration (right).


Index Profile Since the multimode fiber model has many different ways to specify the refractive index profile, it is often useful to verify that the profile is correct before embarking on a lengthy simulation. The index_profile test option was created for this purpose. Regardless of the method of index description (parabolic, alpha, function, file, library), this test function will plot the refractive index as a function of the radial distance.

Figure 11: Index profile test plot.

Test Report The report test function produces a text-based summary of various attributes of the fiber. Figure 13 is an example of a report generated using the library configuration:

Multimode Fiber Test Report =========================== mmf_1 is described by: Mode library: C:\usr\jim\dev\src\LinkSIM\distrib\fiber_library Effective index file: C:\usr\jim\dev\src\LinkSIM\distrib\effective_indices Modal delays were calculated using the standard method. There were a total of 190 modes found at a wavelength of 8.2e-007 m. (azimuthal #, radial #) delay ----------------------- --------------------- (0,0) 1.41517838374e-006 s (0,1) 1.415180304569e-006 s (0,2) 1.41518413672e-006 s (0,3) 1.415189867185e-006 s

· · ·

Figure 12: Report generated by test function.


Linear Test Function When spatial_effects are off, the linear test mode is the only one that can be activated. In this case, the test function plots the magnitude and phase of the power transfer function described in the documentation of the linear configuration. Recall that the linewidth is embedded in the fiber model’s input signal. Because there are no input signals when using the test function, the user must specify a test value for the linewidth. The user can specify the range of frequencies to be plotted through the test_fmax parameter.

Figure 13: Results of linear test function.

References [1] M. J. Adams, Introduction to Optical Waveguides. New York, NY: John Wiley & Sons, Inc., 1981.

[2] D. Gloge and E. A. J. Marcatili, “Multimode theory of graded-core fibers,” Bell System Technical Journal, vol. 52, no. 9, pp. 1563-1578, 1973.

[3] B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics. New York, NY: John Wiley & Sons, Inc., 1991.

[4] J. Gowar, Optical Communication Systems. New York, NY: Prentice Hall, 1984.

[5] D. G. Cunningham and W. G. Lane, Gigabit Ethernet Networking. Indianapolis, IN: Macmillan Technical Publishing, 1999.

[6] C. Pask, “Exact expressions for scalar modal eigenvalues and group delays in power-law optical fibers,” Journal of the Optical Society of America, vol. 69, no. 11, pp. 1599-1603, 1979.

[7] D. Krumbholz, E. Brinkmeyer, and E. G. Neumann, “Core/cladding power distribution, propagation constant and group delay: Simple relation for power-law graded-index fibers,” Journal of the Optical Society of America, vol. 70, no. 2, pp. 179-183, 1980.

[8] K. Kurokawa, “Group delay in multimode optical fiber,” IEEE Proceedings, vol. 65, pp. 1217-1218, 1977.

[9] M. S. Sodha and A. K. Ghatak, Inhomogeneous Optical Waveguides. New York, NY: Plenum Press, 1977.

[10] A. R. Michelson, Guided Wave Optics. New York, NY: Van Nostrand Reinhold, 1993.


[11] S. D. Personick, “Baseband linearity and equalization in fiber optic digital communication systems,” Bell System Technical Journal, vol. 52, pp. 1175-1195, 1973.

[12] D. G. Duff, “Computer-aided design of digital lightwave systems,” IEEE Journal on Selected Areas in Communications, vol. SAC-2, no. 1, pp. 171-185, 1984.

[13] B. K. Whitlock, et al., “Computer modeling and simulation of the Optoelectronic Technology Consortium (OETC) optical bus,” IEEE Journal on Selected Areas in Communications, vol. 15, no. 4, pp. 717-730, 1997.

[14] H. J. R. Dutton, Understanding Fiber Optics. Upper Saddle River, NY: Prentice Hall, 1998.

[15] G. Yabre, “Comprehensive theory of dispersion in graded-index optical fibers,” Journal of Lightwave Technology, vol. 18, no. 2, pp. 166-177, 2000.

[16] M. J. Yadlowsky and A. R. Mickelson, “Distributed loss and mode coupling and their effect on time-dependent propagation in multimode fibers,” Applied Optics, vol. 32, no. 33, pp. 6664-6677, 1993.

[17] R. Olshansky and S. M. Oaks, “Differential mode attenuation measurements in graded-index fibers,” Applied Optics, vol. 17, no. 11, pp. 1830-1835, 1978.

[18] J. A. Buck, Fundamentals of Optical Fibers. New York, NY: John Wiley & Sons, Inc.

Properties

Inputs #1: Optical Signal with or without spatial object

Outputs #1: Optical Signal with or without spatial object

Parameter Values Name Type Default Range Units spatial_effects enumerated on on, off operational_mode enumerated parabolic parabolic, alpha, file, function,

library, step

length double 300 [0, 1e32] m

attenuation double 0 [0, 1e32] dB/km

diff_mode_atten enumerated none none, alpha_fit, file

dma_bessel_order integer 1 [1, 1000] dma_scaling_factor double 1 [0, 1e32]

dma_filename string

dma_file_xconversion enumerated type1 type1, type2 dma_file_xfactor double 1.0 [0, 1e32]

dispersion_mode enumerated specify specify, calculate

dispersion double 0 [-1e32, 1e32] ps/nm/km

S0 double 0.101 [0, 1] ps/nm/nm/km

lambda0 double 1.310e-6 [100e-9, 10e-6] m dispersion_delay_mode enumerated none none, alpha, alpha_taylor

N1 double 1.4142 [0, 100]

profile_disp_param double 0.0 [-1e32, 1e32]


mode_drop_threshold double 0 [0, 1]

mode_directory string parabolic_multimode_fiber_library

effective_index_filename string effective_indices

delay_method enumerated Normal normal, WKB

core_radius double 25 [1, 50] microns peak_index double 1.4142 [1, 2]

alpha double 2 [0, 1e32]

delta double 1 [0.1, 3] percent

index_file string core_index_profile string

cladding_index_profile string

mode_limiting enumerated lm lm, dmg

l_max integer 20 [0, 500] m_max integer 10 [0, 500]

dmg_max integer 40 [0,3000]

lambda_step double 1e-6 [1e-9, 1e-1] microns

mode_set double 1000 [100, 1000000] mode_default_grid_spacing enumerated yes yes, no

mode_dx double 0.25 [0.01, 10] microns

mode_dx double 0.25 [0.01, 10] microns

mode_default_radius enumerated yes yes, no mode_radius double 25 [0.01, 250] microns

Bim double 160 [0, 1e32] MHz.km

CE double 64 [0, 100] percent

conn_loss double 2 [0, 1e32] dB gammacb double 0.75 [0, 1e32]

t_skew double 0 [-1e32, 1e32] ps/m

t_prop double 5e-6 [0, 1e32] s/km

ignore_phase enumerated yes yes, no output_type enumerated fiber_mode_plot fiber_mode_plot,

radial_field_plot, overfilled_plot, delay_plot, index_profile, report

plot_azimuthal integer 0 [-200, 200]

plot_radial integer 0 [0, 200] plot_mode enumerated magnitude magnitude, mag_phase,

real_imag

test_wavelength double 850e-9 [500e-9, 2e-6] m

test_linewidth double 5e-9 [0, 1e-3] m

test_fmax double 5 [0.1, 25] GHz overfilled_filename string



General Parameters spatial_effects Specifies whether to use spatial or nonspatial multimode fiber model operational_mode If spatial_effects=on, determines whether to use parabolic, alpha, file,

function, library or step configurations core_radius The radius of the fiber core in microns length Length of the fiber in meters attenuation Attenuation in dB/km diff_mode_atten Specifies whether differential mode attenuation should be included dma_bessel_order Order of modified Bessel function of the first kind, when numerical DMA model

is used dma_scaling_factor Scaling factor used by numerical DMA model dma_filename Name of data file used by file-based DMA model dma_file_xconversion Specifies type of conversion used on x values in DMA data file dma_file_xfactor Factor by which to scale x values in DMA data file dispersion_mode Denotes if the dispersion is to be specified by the user or calculated by the model dispersion Contains the value of the dispersion in ps/nm/km if dispersion_mode=specify S0 The zero-dispersion slope, used to determine the dispersion if

dispersion_mode=calculate lambda0 The zero-dispersion wavelength, used to determine the dispersion if

dispersion_mode=calculate dispersion_delay_mode Controls whether OptSim should adjust the modal delays to account for material

dispersion (alpha profiles only) N1 Material group index profile_disp_param Profile dispersion parameter mode_drop_threshold Power coupling coefficient threshold for dropping modes

Spatial Parameters mode_default_grid_spacing Enables the user to specify whether OptSim should decide the grid spacing or

whether the grid spacing should be specified manually. mode_dx If mode_default_grid_spacing is no, the user must set the grid spacing in the

x direction here. mode_dy If mode_default_grid_spacing is no, the user must set the grid spacing in the

y direction here. mode_default_radius Enables the user to specify whether OptSim should decide the fiber simulation

domain or whether the domain should be specified manually. mode _radius If mode_default_radius is no

Index Parameters

peak_index The value of the refractive index at the center of the fiber delta The index step between the core and the cladding alpha The value of the alpha exponent when simulating alpha index profiles index_file The name of the data file containing the refractive index profile to be simulated core_index_profile Analytical function describing the refractive index profile in the core cladding_index_profile Analytical function describing the refractive index profile in the cladding mode_limiting Controls the method by which the mode solver limits the number of solved modes l_max Upper limit of the azimuthal index used by the mode solver


m_max Upper limit of the radial index used by the mode solver dmg_max Upper limit of the mode group number used by the mode solver lambda_step Wavelength range used to compute group velocities mode_set Effective index range used to calculate propagation constants

Library Parameters mode_directory The name of the directory that contains the fiber library database of files. This is

assumed to reside in the OptSim installation directory. effective_index_filename The name of the effective index library; this is assumed to be placed in

mode_directory delay_method If normal, the delays are determined by differentiating the effective indices. If

WKB, the delays are calculated using the Wentzel-Kramers-Brillouin method.

Linear Parameters Bim Intermodal bandwidth of fiber (MHz*km) CE Connector efficiency (%) conn_loss Optical loss at connectors (dB) gammacb Gamma cutback parameter t_skew Time skew with length (ps/m) t_prop Optical propagation rate in fiber (s/km) ignore_phase Flag to instruct model to either ignore or keep the phase of the input field

Test Parameters output_type Chooses the type of test function to be performed plot_azimuthal If a mode plot is selected, this specifies the azimuthal number of the mode plot_radial If a mode plot is selected, this specifies the radial number of the mode plot_mode The format of the output plot test_wavelength Wavelength used for the test functions test_linewidth Linewidth used for the linear test function test_fmax Highest frequency displayed in linear test function overfilled_filename Name of file to be used to store the overfilled launch profile


Appendix: Library Generation To use the library configuration, the user must first generate a library of radial modes for a fiber with a given refractive index profile and specific physical attributes. The library consists of a collection of data files representing the spatial and propagation attributes of the fiber over a range of wavelengths. A sample library parabolic_multimode_fiber_library is included in the manufacturerlibrary folder of OptSim distribution directory as described in the library configuration section of the documentation for this model.

Library mode format Each file that represents a spatial field profile must be named: azX_lamYY.mZZ

Here, X represents the azimuthal index number of the fiber mode. The single letter X is used to denote that no leading zeros are necessary for single-digit indices. The code ZZ denotes the radial index of the fiber mode; the double letter ZZ notation is used to indicate that single-digit indices must be preceded by a zero (i.e., radial mode 1 is represented as .m01). Because the mode profiles are wavelength dependent, it is necessary to specify in the filename the wavelength at which the mode was generated. Here we specify a wavelength index YY to again denote that single-digit indices must be preceded by a leading zero. The index YY is translated to an actual number in meters based on information in the library header files (to be discussed shortly). All indices (az, lam, m) start at zero. Thus, a file containing data for the radial field with azimuthal index #4, radial index #6 and the third wavelength index is named az4_lam02.m06

The data in each of these files must adhere to a specific format. The first line of the file indicates the field format and the radial range over which the field is defined. The field data can be specified in different formats, depending on the desired application: <NR> <RMIN> <RMAX> <0> <OUTPUT_REAL_IMAG>

or

<NR> <RMIN> <RMAX> <0> <OUTPUT_REAL>

Here, NR is the number of data points in the file, RMIN is the minimum radial value in microns, RMAX is the maximum radial value in microns, 0 is simply a placeholder that must be there, and OUTPUT_REAL_IMAG or OUTPUT_REAL specify that the field data that follow will be in either real/imaginary format or real format, respectively. For the fiber library, RMIN must always be 0; if it is not, an error will be generated. As an example, the following are the first few lines of the file az4_lam02.m06: 501 0 50 0 OUTPUT_REAL 0 2.82566e-006 4.8975e-005 0.000231438

The first line specifies that the file contains data at 501 radial points, that it begins at 0=r and ends at 50=r µm. Based on this information, it can be deduced that the radial increment is 1.0=∆r µm. Note

that the actual r values are not included in the file since they can always be calculated from the header information. Three lines of sample field data are depicted. When the OUTPUT_REAL_IMAG format is used, clearly two columns of field data will be required.

A note to users of BeamPROP: The OptSim multimode fiber model can also accept files in the format of BeamPROP output files as seen below. The extra information is simply ignored. /rn,a,b/nx0/ls1 501 0 50 0 OUTPUT_REAL 1.401604716 -2.312887006e-015 0


2.82566e-006 4.8975e-005 0.000231438

If OptSim needs field information at a wavelength for which a field does not exist in the library, it will attempt to interpolate between the fields that do exist.

To summarize, the fiber library must contain one file for each radial mode that the fiber can support for a given wavelength. The filename and format of the data must be as described above.

In addition to the individual radial field data files, a mode header file is required. In the sample fiber library that is included with OptSim, this file is named libraryHeader.table. The first line of this file contains the following fields: <maxAz> <maxRad> <maxLam> <noModes> <lamStart> <lamStep> <coreRadius>

Here, maxAz is the maximum azimuthal index number that exists in the entire fiber library, regardless of the radial index number or the wavelength. Similarly, maxRad is the maximum radial index number and maxLam is the maximum wavelength index number. These first three fields must be less than or equal to 500. The field noModes indicates the total number of mode files in the library. If there were no modes cut-off, noModes would be equal to

(maxAz+1) × (maxRad+1) × (maxLam+1)

However, since not all modes exist at all wavelengths, noModes will generally be less than this. The field lamStart is the actual value in meters of the lowest wavelength stored in the table while lamStep is the value in meters of the wavelength increment. Currently, OptSim’s library configuration supports wavelengths in the range of 6101.0 −× m to 6102 −× m. Finally, coreRadius is the value in microns of the radius of the fiber core; currently values from 1 µm to 75 µm are supported.

The sample header file libraryHeader.table has the following as its first line: 18 9 10 1011 8e-007 1e-008 25

The maximum wavelength index number is 10. Recall that all indices start at 0; this means that there are data for 11 wavelengths in the table. Based on the values of lamStart and lamStep, it becomes apparent that the wavelength indices correspond to a wavelength range of 800 nm – 900 nm in 10 nm increments. In other words, wavelength index 0 corresponds to 800 nm, wavelength index 1 corresponds to 810 nm, and so on. Apart from the first line of the file, all other lines follow the same format: <azNo> <radNo> <lamNo> <flag>

Here, azNo denotes azimuthal index number, radNo denotes radial index number and lamNo denotes wavelength index number. Basically, there is a line in the file for every combination of these three parameters. The field flag is either a 1 or a 0. If the mode (azNo, radNo) exists at the wavelength index lamNo, flag is 1; otherwise (i.e. if it is cut-off) it is 0. Following the example above, the total number of lines in the header file after the first one is:

(maxAz+1) × (maxRad+1) × (maxLam+1) = 2090

However, only noModes=1011 of them actually have flag values of 1. The first 5 lines of the header file are shown below: 18 9 10 1011 8e-007 1e-008 25

0 0 0 1

0 0 1 1

0 0 3 1

0 0 4 1

To summarize, the fiber library must also contain a mode header file that describes the minimum/maximum ranges for the azimuthal, radial, and wavelength parameters as well as other pertinent physical information.


Library propagation constant format While the transverse field profiles need individual files for each mode at each wavelength, information about the fiber’s propagation constants are stored in a single file called effective_indices. The first line of this file contains critical header information that enables OptSim to parse the rest of the file. The format of this header line is: <maxLam> <maxModes> <lamStart> <lamStep> <maxAz> <maxRad> <peakIndex>

The definitions of maxAz, maxRad, maxLam, lamStart, and lamStep are the same as before. The parameter maxModes describes the maximum number of modes that exists at a given wavelength and peakIndex is simply the refractive index at the center of the fiber 1n . Below is the first line of the effective index library included with the OptSim release: 101 100 8.00E-07 1.00E-09 18 9 1.4142

Clearly, the database contains effective index information for 101 wavelengths starting at 800 nm and extending to 900 nm in 1 nm increments. Notice that the wavelength increment is smaller that it was for the mode library. This is done for two reasons. The first is that since the effective index is only a single number, storing its value at a smaller wavelength increment does not affect the size of the effective index library as much as it would affect the mode library. Second, since the variation of modal delays for a typical multimode fiber application is in the picosecond range, a higher degree of accuracy is needed in their determination. Storing the effective indices at a smaller wavelength will result in less interpolation and thus higher accuracy.

The highest azimuthal and radial numbers are the same as before and the peak index in the core is 1.4142. The fact that the maximum number of modes is 100 means that for any given wavelengths in the library, no more than 100 modes exist. Note that this is the maximum number of modes; in general, the actual number of modes will be less. For example, in the sample library, at 820 nm 100 modes exist while at 850 nm, only 91 exist (9 modes are cut-off).

Each subsequent line contains a list of the effective indices for a given (azimuthal #, radial #) pair: <azNo> <radNo> <Eff_Ind#1> <Eff_Ind#2> <Eff_Ind#3> ...

Thus, the first two entries on each line list the azimuthal and radial numbers of the mode. The next entry is the effective index at the first wavelength (800 nm in this case). The next entry is the effective index at the second wavelength (801 nm) and so on. When a mode is cut-off, the Eff_Ind field is simply left blank; it is not set to zero. The first 4 lines of the sample library are shown below: 101 100 800e-9 1e-9 18 9 1.4142 0 0 1.4134787135 1.4134778127 1.4134769119 ... 0 1 1.4120371121 1.4120344070 1.4120317019 ... 0 2 1.4105943795 1.4105898664 1.4105853533 ...

To summarize, the library of effective indices consists only of a single file. The first line contains pertinent header information. Each subsequent line contains two mode identifiers (azimuthal and radial numbers) followed the effective index value at each specified wavelength.

Library Radial Field Parameters NR Number of radial data points RMIN Minimum value of the radial coordinate in microns – Must be 0 RMAX Maximum value of the radial coordinate in microns data type Two choices: OUTPUT_REAL, OUTPUT_REAL_IMAG

Library Mode Header Parameters maxAz The largest azimuthal index number in the entire database maxRad The largest radial index number in the entire database maxLam The largest wavelength index number in the entire database


noModes The total number of extant (i.e., non-cut-off) modes lamStart The first wavelength in the database expressed in meters lamStep The wavelength increment expressed in meters coreRadius The radius of the fiber core in microns azNo A specific azimuthal index number radNo A specific radial index number lamNo A specific wavelength index number flag Parameter in header file that indicates the presence/absence of a given mode

Library Effective Index Parameters maxAz The largest azimuthal index number in the entire database maxRad The largest radial index number in the entire database maxLam The largest wavelength index number in the entire database maxModes The maximum number of modes that exists for any given wavelength lamStart The first wavelength in the database expressed in meters lamStep The wavelength increment expressed in meters peakIndex The refractive index at the center of the fiber


Spatial Aperture

The spatial aperture model provides a means for the user to pass a spatial field object through a circular, square, or ring-shaped window. The topology of Figure 1 depicts a typical situation.

Figure 1: Use of spatial aperture model.

The spatial coupler model in Figure 1 is used to offset the VCSEL spatial field by 10 µm in both the x and y directions. In this simulation, the spatial aperture is set to be a 30-µm-wide square opening. The field both before and after the aperture are shown in Figure 2. Also depicted in Figure 2 is same simulation except with a 50-µm-diameter circular aperture.

Figure 2: Spatial field before (top) and after square aperture (bottom left) and circular aperture (bottom right).


The aperture model performs a simple truncation of the incident spatial field; inside the aperture, the light is allowed to pass through ideally while outside the aperture the field is reduced to zero. Note that despite the fact that the aperture can be made arbitrarily small, diffractive effects are not modeled.

In addition to truncating the optical field, the spatial aperture model attenuates the time-domain portion of the signal ( )(tEx , )(tE y ) to reflect the amount of power lost in the aperturing process. The model then power-normalizes the truncated spatial field to 1.0.

Properties


Outputs #1: Optical signal with spatial object

Parameter Values Name Type Default Range Units aperture_type enumerated round round, square

dimension double 10 [ 0, 1000 ] microns

inner_dimension double 5 [ 0, 1000 ] microns

Parameter Descriptions aperture_type Allows the user to choose either a square or round aperture dimension Specifies the aperture width. In the case of a circular aperture, this parameter denotes the

diameter. For ring apertures, it denotes the outer diameter. inner_dimension For ring apertures only. Specifies the inner diameter of the ring.


Spatial Compound Optical Receiver

This is the spatially enabled version of the compound optical receiver model. The main difference between this model and its nonspatial counterpart is that spatial effects have been included in the photodetector element. In all other respects, the two models are identical.

This models a spatial optical receiver and all its standard parts. The OptSim photoreceiver model is composed of several individual building blocks: the photodetector, the preamplifier, and the postamplifier/filter complex:

Figure 1: Basic components of an optical receiver

Each block is a separate entity complete with its own input parameters and options. The spatial photodetector model converts an optical input signal to an electrical current. The documentation for the spatial photodetector explains the spatial behavior of this element in detail. This photocurrent is then passed to the preamplifier model which converts it to a voltage. Finally, the postamplifier model contains a set of baseband filters that shape the output waveforms. The model also computes the photoreceiver noise components.

In fact the receiver model is implemented directly in terms of the three stand-alone models the PIN/APD Photodetector, Electrical Amplifier and Electrical Filter model described elsewhere in this manual and all the parameters of each of those models are also parameters of the monolithic receiver model. In other words the two configurations shown in the topology in Figure 2 serve equivalent functions.

There are two motivations for providing this additional receiver model. As a matter of convenience and topology compactness, some users may find it useful to represent the receiver configuration with a single block. However, this function could also have been achieved by creating a super-block containing the three individual models. Rather, the principal reason for providing a combined model is to allow a “Quasi-Analytic” (QA) treatment of the receiver noise. Both the photodetector and electrical amplifier model a number of noise sources – shot noise, dark current noise, signal-spontaneous emission beta noise, thermal noise in the pre-amp transistor etc. In these two models, the noise is added directly as a stochastic contribution to the electrical signal – the so-called Monte-Carlo (MC) picture.

Figure 2: Two configurations for modeling an optical receiver.

However, as explained in the chapter on signal representations, electrical noise in OptSim may also be represented as a time-dependent vector of the standard deviation of a Gaussian white noise source. This representation is especially useful in bit error rate calculations, since it allows the contribution of each bit to the BER to be evaluated separately and


trivially accounts correctly for intersymbol interference (“pattern-dependent”) effects. This is referred to as the QA method for BER calculation. As discussed in the BERTester documentation, extracting the BER from a signal with Monte-Carlo noise is considerably more involved, particularly if the signal exhibits strong pattern dependence. Thus, provided the receiver components are the dominant source of noise in the system, the QA approach is preferred due to increased accuracy and speed. On the other hand, if ASE is the dominant source of noise, and the propagation through the fiber is strongly nonlinear, then the noise properties are non-Gaussian and the MC approach is unavoidable. Which of these two cases is true is determined by experimentation with simulations.

So the representation of noise as a separate standard deviation is necessary to allow BER calculations with the QA method, but as we explore below in the section QA noise expressions, this is only possible if all stages of the receiver from photodetection from filtering can be considered together because the variances for each noise source depend on parameters from at least two of the sub-models.

Model Parameters Since the receiver is constructed from the photodetector, electrical amplifier and electrical filter models, it includes all the physics and essentially all the parameters of those three models. For the sake of space and clarity, we will not repeat the explanations of the deterministic parts of the models – definition of photodetector quantum efficiency, amplifier spectral response, filter bandwidth, etc – which are identical for this model. Moreover, we will not repeat the discussions of these many parameters here and the reader should consult the specific documentation for each of the other models. The parameters have the same names in the monolithic receiver as in the individual models with the addition of prefixes denoting which part of the system the parameter belongs to. The prefixes are pd_ for the photodetector parameters, fe_ (front end) for the amplifier parameters and flt_ for the filter parameters. So for example, the dark current in the photodetector is set by pd_darkCurrent, the transimpedance of the amplifier by fe_tZ and the filter bandwidth by flt_bandwidth. While not discussed in detail, all the parameters are listed in the tables at the conclusion of the receiver documentation.

Since the calculation of noise effects involves several models, some of the noise parameters do not have any prefixes. So the spectrum of thermal noise is specified by n_a0, n_a2, n_a4, n_a6 exactly as in the electrical amplifier model.

Noise Representation and Effects The choice between the quasi-analytic and Monte-Carlo treatments of noise is set with the parameter n_representation. With n_representation=MC, the Monte-Carlo approach is used, and the model is exactly equivalent to the concatentation of the three component models.

The quasi-analytic treatment (n_representation=QA) is identical with respect to the deterministic parts of the component models. That is, the generated electrical signal is identical to the signal that would be generated by concatenating the three models and disabling all noise terms. The noise terms are calculated quite differently as we see shortly. It is important to note that the QA representation of noise incurs approximations in the spectral features of the noise. Any noise source which has a non-flat spectrum such as the thermal noise or the spontaneous-spontaneous beat noise which typically has a triangular shape is replaced by a flat spectrum of an equivalent noise density. This is an unavoidable consequence of treating the noise strictly as variances in the time domain.

We now discuss the expressions for the various noise sources and how they are combined. Each noise source may be separately disabled using the parameters beginning include_. Note that when the Monte-Carlo noise treatment is used, the spontaneous emission noise is controlled via the parameter include_SE_noise, whereas when the quasi-analytic treatment is used, the signal-spontaneous and spontaneous-spontaneous ASE beat noise are included separately via the parameters include_sigspon and include_sponspon, respectively.

Preamplifier Noise Parameters OptSim accounts for the following types of noise in the receiver model: circuit or thermal noise, shot noise due to both the detector dark current and the received signal, signal-spontaneous beat noise (when ASE noise is present in the received signal), spontaneous-spontaneous beat noise, APD excess noise (for APD receivers), and relative intensity noise (when RIN is specified in the transmitter). In the literature, noise is described mathematically as the variance ( 2

nσ or


>< 2ni ) of the photogenerated current. In the receiver, the noise is amplified along with the signal; therefore, the noise

contained in the electrical signal output from OptSim’s receiver model is represented as the standard deviation of the amplified voltage signal coming out of the receiver. The standard deviation is simply:

><σ 2nn i or

The spectral density is also a commonly used noise representation. This quantity describes the amount of noise power per unit frequency HzA2 and is related to the noise variance by:

dffSi in )(

0

2 ∫∞

=

Many noise sources are described in the literature in terms of their spectral density. It is common practice to denote the effective bandwidth of the photoreceiver by Beff (as computed from the receiver front end amplifier and filter response)

∫∞

=0

2)( dffHBeff

where the combined filter response )( fH is assumed to vanish at sufficiently high frequencies. In numerical modeling, there is a necessary cutoff to the response at the sampling frequency, so for a flat response tBeff ∆= 21 .

The current variance is then

effiin BfSffSi )()(2 =∆=

The photocurrent in the detector is described by:

P

hcPq

I ℜ=ηλ

= 0

where q is the charge of an electron, η is the quantum efficiency of the photodetector, λ is the wavelength of the received optical signal, P is the optical power of the received optical signal, h is Planck’s constant, c is the speed of light, and ℜ is the responsivity. The following noise expressions are relative to the photocurrent, so they are called input-referred noise expressions. For APD photodetectors, the noise expressions are relative to the photogenerated current after multiplication in the avalanche region, or MI .

OptSim represents the circuit or thermal noise as a power series expansion of frequency. The total noise power per Hz bandwidth at the input can be expressed as:

6

64

42

20, )( fafafaafS circuiti +++= This is a more general form of the commonly accepted expression:

22

,)2(

44)( fgCkT

RkTfS

m

T

fcircuiti

πΓ+=

which describes the thermal contribution of the feedback resistor in the transimpedance amplifier and the thermal channel noise in the preamplifier input transistor. In this expression, k is Boltzmann’s constant, T is the temperature,

fR is the amplifier feedback resistance, mg is the transconductance of the preamplifier input transistor, Γ is the excess

channel noise factor, and TC is the total input capacitance. The generalized polynomial representation is chosen to allow the user to tailor the noise spectral density as he sees fit. It is also useful when actual noise spectra are available since it allows the noise to be represented by simply fitting the polynomial coefficients to measured data. To model white noise, simply set the coefficients 2a , 4a and 6a to zero.


The photoreceiver is characterized by a transimpedance response; in other words, the transfer function of the preamplifier converts an input current to an output voltage. The relationship between the output noise voltage and the input noise current (or equivalently the input spectral density) is:

∫∞

=0

22 )()( dffSfHv iout

Random fluctuations in current are characterized by shot noise which is described in general by:

qIfSqIBi ieffshot 2)(22 =⇔=

Here q is the electronic charge and I is the current under consideration. In the OptSim photoreceiver model, shot noise due to the photoreceiver dark current and the photogenerated signal current are considered. Consequently, the shot noise due to the dark current is represented by:

effdarkdarkshot BFIqMi 22

, 2=

where M is the gain of the APD (1 for PIN devices), darkI is the dark current, effB is the effective receiver bandwidth), and F is defined as follows:

)12)(1( MkkMF −−+= where k is the APD ionization coefficient, and the other terms are as previously defined. Similarly, the shot noise due to the photogenerated signal itself is represented by:

effsignalshot FIBqMi 22

, 2=

The signal-spontaneous and spontaneous-spontaneous beat noise sources are derived from standard approaches in the literature and describe the effects of amplified spontaneous emission (ASE) on the noise performance of the photoreceiver. These analyses have their roots in the fact that the optical field can be analytically decomposed into two parts, the signal sE and the noise nE :

( ) ( ) ( )tEtEtE ns += Since the optical power is defined to be the intensity of the electric field, when light hits a detector, the resulting photocurrent can be described loosely by:

++ℜ=+ℜ=ℜ= 2222 )()()(2)()()()()( tEtEtEtEtEtEtEtI nnssnsph

As seen by the last term on the right, the square-law detection process performed by the detector causes the various components of the optical field to multiply, or “beat,” with each other. The first term on the right represents the photocurrent signal itself sPℜ . The second term models the effect of the signal field beating with the noise field while the third term represents the beating of the noise with itself.

The variance due to signal-spontaneous ASE beat noise is modeled in the literature essentially by multiplying the spontaneous emission at every frequency by the optical signal:

( ) )(42222 fPPBFMi n

endf

startffseffsps ∑

=− ℜ=

Here, startf and endf represent the beginning and end frequencies of the ASE noise spectrum. In general, the frequency range startend ff − will be much larger than the photoreceiver bandwidth; thus, the signal-spontaneous beat noise is effectively truncated when it is passed through the receiver transfer function.

Similarly, the variance due to spontaneous-spontaneous ASE beat noise is:


( ) )()(4 2

1 2

12222 fPfPBFMi n

endf

startff

endf

startffneffspsp ∑ ∑

= =− ℜ=

Here, a double summation is required because the ASE noise at each frequency beats with the ASE noise that exists at every other frequency in the noise bandwidth. Again, the ASE spectral density is captured from the power spectrum portion of the spectrum analyzer measurement tool and will generally be truncated by the finite receiver bandwidth.

The RIN noise is represented by the following expression:

effRINRIN BNFIMi 222 )(=

where RINN is the RIN noise parameter defined in the transmitter and the other terms are as previously defined.

The total input-referred noise expression in the receiver is then:

222222RINspspspsshotcircuitn iiiiii ++++= −−

Calibrating receiver sensitivity The bit error rate for Gaussian systems can be roughly described by:

=

221BER Qerfc

20

21

01

nn ii

IIQ

+

−=

where 1I and 0I are the photocurrents in the one and zero states and the standard deviation terms represent the noise in the respective states. The complementary error function is represented by erfc. Thus, for a desired BER, the Q factor can be calculated by inverting the complementary error function; although this cannot be performed analytically, it is a simple matter to either perform it numerically or look it up in tables.

Using the equations given in this documentation, it is thus possible to set the receiver parameters to model a specific device. It is important to model the receiver’s frequency response as accurately as possible using the photodetector, front end amplifier, and filter parameters; then using the receiver’s sensitivity based on a given BER with a given average input optical power and difference between a 1 level and a 0 level, set the noise parameters so that the model’s sensitivity matches the device being modeled. Note that it is very important that the noise parameters be set properly to represent the receiver you are using if you wish the simulation results to correlate with a particular device. Also, note that simply changing the default parameters for the APD multiplier and ionization coefficient will not convert an optimal PIN receiver into an optimal APD receiver, as there are other parameters that will vary between PIN receivers and APD receivers since their designs generally differ from one another.

The documentation for the amplifier describes how to set the frequency response of the preamplifier to match a 3-dB frequency that is either measured or found on a data sheet. As a further example, suppose the desired BER for a particular receiver is 1210 − ; the corresponding value of Q can be easily determined from tables to be approximately 7. Once the value of Q is known, some further design guidelines can be established. One parameter that is often found on photoreceiver data sheets is the sensitivity or, more accurately, the minimum average received power. If we neglect noise terms that depend explicitly on the signal level (usually a good assumption), this quantity can be approximated as:

211

nirrQP

−+

ℜ=

Here, r is the extinction ratio of the signal, namely 10 II . Thus, if the BER (and hence Q), the detector responsivity, the signal extinction ratio, and the receiver sensitivity are known, it is possible to estimate the standard deviation of the noise that is needed to model such a device.


Test display A number of spectral responses are available as test functions, selected with test_output. The exact format for the display of the response functions is controlled with test_display. The available settings for test_output and their meanings are elec_filter_spectrum (electrical filter response function), photodetector_resp (photodetector response function), front_end_resp (amplifier response function), cumulative_resp (combined response of the detector, amplifier and filter) and noise_spectrum (thermal noise spectrum of the amplifier).





[5] N. A. Olsson, “Lightwave systems with optical amplifiers,” IEEE Journal of Lightwave Technology, vol. 7, no. 7, pp. 1071-1082, 1989.

Properties



Parameter Values Name Type Default Range Unit

n_representation enumerated QA QA, MC

pd_APD_Multiplier double 1.0 [1, 1e32]

pd_ionizationCoef double 1.0 [0, 1]

pd_QEmethod enumerated Defined Defined, Computed

pd_quantumEff double 0.8 [0, 1]

pd_layerThickness double 0.5e-6 [0, 1e32] m

pd_absorptionCoeff double 0.68e6 [0, 1e32] 1/m

pd_reflectivity double 0.04 [0, 1]

pd_detect enumerated false false, true

pd_modeltype enumerated empirical empirical, intrinsic

pd_loadResistance double 50.0 [0, 1e32] Ohm

pd_seriesResistance double 5.0 [0, 1e32] Ohm

pd_deviceCapacitance double 50e-15 [0, 1e32] F

pd_electronVelocity double 6.5e6 [0, 1e32] m/s

pd_holeVelocity double 4.8e6 [0, 1e32] m/s


pd_lossGain double 0.0 < 0 for loss, > 0 for gain

dB

pd_respfp double 52.259e9 [0, 1e32] Hz

pd_respfo double 25.679e9 [0, 1e32] Hz

pd_respg double 0.5874 [0, 1e32]

pd_darkCurrent double 1e-6 [0, 1e32] A

pd_spatial_effects enumerated on on, off

pd_aperture_type enumerated round round, square

pd_aperture_dimension double 10 [0, 1000] microns

pd_aperture_inner_ dimension

double 5 [0, 1000 ] microns

fe_modeltype enumerated defined defined

fe_filename string

fe_tZ double 1.0 [0, 1e18] Ohm

fe_zero double 0.00 [0, 1e18] Hz

fe_pole double 1e18 [0, 1e18] Hz

fe_lo_trunc enumerated extend extend, zero

fe_hi_trunc enumerated extend extend, zero

n_a0 double 5.4617e-23 [0, 1e32] HzA2

n_a2 double 2.924e-43 [0, 1e32] 32 HzA

n_a4 double 1.1118e-63 [0, 1e32] 52 HzA

n_a6 double 0 [0, 1e32] 72 HzA

flt_type enumerated LPbessel LPbutterworth, LPchebyshev, LPbessel, LPideal, HPbutterworth, HPchebyshev, HPbessel, HPideal, BPbutterworth, BPchebyshev, BPbessel, BPideal

flt_preserve_alignment enumerated YES NO, YES

flt_bandwidth double 10e9 [0, 1e18] Hz

flt_order integer 4 [0, 128]

flt_lossGain double 0.0 [-1e32, 1e32] dB

flt_passbandRipple double 0.0 [0, 1e32] dB

flt_geometricCenter double 0.0 [0, 1e32] Hz


include_thermal enumerated YES NO, YES

include_sigshot enumerated YES NO, YES


include_sigspon enumerated YES NO, YES

include_sponspon enumerated YES NO, YES


include_rin enumerated YES NO, YES

monte_seed integer 0 [-1e8, ≤1]


test_output enumerated elec_filter_spectrum, elec_filter_spectrum, photodetector_resp, front_end_resp, cumulative_resp, noise_spectrum


General Receiver Parameters n_representation: Select quasi-analytic or Monte-Carlo treatment of noise

include_thermal: Toggle inclusion of thermal noise

include_sigshot: Toggle incluson of signal shot noise

include_darkshot: Toggle inclusion of dark current shot noise

include_sigspon: Toggle inclusion of signal-spontaneous beat noise in quasi-analytic treatment of noise

include_sponspon: Toggle inclusion of spontaneous-spontaneous beat noise in quasi-analytic treatment of noise

include_SE_noise Toggle inclusion of ASE beat noise in Monte-Carlo treatment of noise

include_rin: Toggle inclusion of relative intensity noise

monte_seed: Random number seed for Monte-Carlo noise

test_display: Format for display of response functions

test_output: Select response function for display

Photodetector Parameters pd_APD_Multiplier: APD multiplier value (1.0 for PIN detector)

pd_ionizationCoef: APD ionization coefficient (1.0 for PIN detector)

pd_QEmethod: If Computed, the quantum efficiency is computed by the model. If Defined, the entered value is used.

pd_quantumEff: Quantum efficiency.

pd_layerThickness: Thickness of the active region

pd_absorptionCoeff: Absorption coefficient

pd_reflectivity: Reflectivity at the photodiode


pd_detect: If true then use a PD frequency response model, else assume it is included in front end response

pd_modeltype: intrinsic or empirical models for frequency response

pd_loadResistance: Load resistance

pd_seriesResistance: Series resistance

pd_deviceCapacitance: Device capacitance

pd_electronVelocity: Electron saturated velocity

pd_holeVelocity: Hole saturated velocity

pd_lossGain: Gain or loss of the photodetector response (only used if pd_detect is set true)

pd_respfp: Parasitic frequency of the empirical frequency response

pd_respfo: Resonance frequency of the empirical frequency response

pd_respg: Gamma of the empirical frequency response fit

pd_darkCurrent: Dark current for dark current noise computation

pd_spatial_effects: Turns spatial effects on or off

pd_aperture_type: If pd_spatial_effects=on, this specifies the shape of the detector

pd_aperture_dimension: If pd_spatial_effects=on, this specifies the size of the detector

pd_aperture_inner_dimension: If pd_spatial_effects=on, this specifies the inner diameter of a ring-shaped detector.

Preamplifier Parameters fe_modeltype: Select parameterized or user-defined transfer function

fe_filename: Filename for user-defined transfer function

fe_tZ: Transimpedance coefficient

fe_zero: Frequency of transfer function zero

fe_pole: Frequency of transfer function pole

fe_lo_trunc: Low frequency truncation behavior for user-defined transfer function

fe_hi_trunc: High frequency truncation behavior for user-defined transfer function

n_a0: Thermal noise coefficient




Filter Parameters flt_type: Filter type

flt_preserve_alignment: Preserve alignment of incoming bits

flt_bandwidth: Filter 3dB bandwidth

flt_order: Geometric center frequency for bandpass filters

flt_lossGain: Order of the filter

flt_passbandRipple: Filter gain or loss

flt_geometricCenter: Passband ripple for Chebyshev filter


Spatial Photodetector

This component models spatial photodetectors – either a PIN photodiode or an avalanche photodiode – converting an optical signal into an electrical current. Noise in this model is treated stochastically. For an analytic treatment of the noise effects, refer to the spatial compound optical receiver model. The main difference between this model and its nonspatial counterpart is that the spatial effects of the detector geometry are accounted for. In all other respects, the two models are identical.

Spatial Effects The spatial version of the photodetector model allows the user to specify the geometry of the detector’s active region and apertures the incident optical fields accordingly. This enables the user to model the effects of misalignment between the photodetector and its preceding element. Currently, round, square, and ring photodetectors are supported. The model performs a simple windowing of the incident spatial field; thus, if the incident field falls entirely within the active area, it is ideally detected. Any portion of the field that falls outside the active area is truncated. Diffractive effects are not modeled.

In accordance with the OptSim spatial field convention, it is assumed that if multiple optical fields impinge upon the receiver simultaneously, their spatial fields are mutually orthogonal. In other words, the spatial receiver assumes that coherence effects between multiple optical signals are negligible, an assumption that is only valid if these signals are orthogonal.

Detection Process This model acts as the first stage of an optical receiver assembly to convert optical signals to an electrical current. Both PIN and avalanche photodiode (APD) type detectors can be modeled. Typically the output of the photodiode would be connected to an electrical amplifier and then to a filter. If a BER analysis is to be performed, at the very least an amplifier must be placed somewhere between the photodetector and BER Tester to convert the current output of the photodetector to the voltage source expected by the BER Tester.

The quantum efficiency is a measure of the percentage of incident photons that are successfully converted to electrons. This quantity may be input directly by the user as a parameter quantumEff; alternatively, the user can allow OptSim to calculate the quantum efficiency internally from the user-specified absorption layer thickness L (layerThickness), absorption coefficient (absorptionCoeff), and reflectivity of the photosensitive area of the detector Rf (reflectivity) according to:

( )( )L

fq eR α−−−=η 11

To input the quantum efficiency directly, set the QEmethod parameter to Defined and enter the desired value in the quantumEff field. To have the model compute it for you, set the QEmethod parameter to Computed and assign values to layerThickness, absorptionCoeff, and reflectivity.

Regardless of the method chosen to specify the quantum efficiency ηQ, the responsivity is computed as:

hce Qλη

=ℜ

where e is the electron charge, λ is the wavelength of the input optical signal, h is Planck’s constant, and c is the speed of light. The responsivity has units of A/W and gives the quantity of electrical current generated for each watt of incident optical power P. That is, the generated photocurrent obeys the square-law relation


( ) ( ) ( )( ) ( )

( ) ( ) ( )tNtEtE

tNtPtNtItI

jyjxj +

+ℜ=

+ℜ=+=

∑ 22

o

where the sum is over all optical channels in the incoming signal. (Recall that in OptSim, the square of the electric field for an optical signal has units of Watts). The term N(t) includes all noise effects and is discussed below.

In an avalanche photodiode, the photocurrent is multiplied by successive ionization and the final current is modified to

( ) ( ) ( ) ( ) ( )tNtPMtNtMItI +ℜ=+= o

where the avalanche factor M is specified with the parameter APD_Multiplier. The behavior of APD detectors also differs in the noise response (see below).

There is no specific switch to convert from a PIN model to an APD model. Simply setting APD_Multiplier>1, and the noise-related parameter ionizationCoefficient (see below), produces APD behavior. Note however, that simply changing the default parameters for the APD multiplier and ionization coefficient will not convert an optimal PIN receiver into an optimal APD receiver, as there are other parameters which will vary between PIN receivers and APD receivers since their designs generally differ from one another.

Frequency Response If desired, the photodetector may be given a frequency response that acts by filtering the photocurrent I(t). In most cases it may be more convenient to model the total electrical response of the receiver assembly together in the subsequent electrical amplifier or electrical filter models. For that approach, set the parameter detect=false.

To control the photodetector response directly, set detect=true, and configure the modeltype parameter to use one of the following frequency responses. This may be useful, for example, if the photodetector is simply connected to a 50 Ω amplifier.

• Intrinsic Response (modeltype=intrinsic) The frequency response is described by:

( )( ) ( )

ωτ−α−+

ωτ−+

ωτ−+

α+ωτ−

−

+ω+=ω

αωτ−α−

ωτ−ωτ−α−

α−ωτ−

α−

h

LhjL

h

hj

e

ejL

e

Lej

Lpsd

G

jLeee

je

jee

Ljee

eCRRjii

1111

11

1110

020

In this formulation, the electron and hole transit times are given by ee vL=τ and

hh vL=τ , pC is the parasitic capacitance of the photodetector, dR is the load resistance,

sR is the series resistance, and the other symbols are as previously defined. The user should enter values for seriesResistance, loadResistance, deviceCapacitance, electronVelocity, holeVelocity, layerThickness and absorptionCoeff and lossGain (G).

• Empirical Response (modeltype=empirical) The response is described by:


( )( ) ( )

20

2 2 22 2 2 2 20 0

10 10 1 1 4 /

G

p

ii f f f f f f

ω

γ

= + − +

where 0,, ffG p and γ are empirically determined by the user and supplied to the model in the form of the parameters lossGain, respfp, respfo and respg, respectively. The response can be multiplied by a gain or loss constant by setting the lossGain parameter to a nonzero value. This representation is particularly useful when the inductance of the bond wire dominates the photodetector response. In this situation, the frequency response is often modeled as the second order response depicted above.

Noise Response The photodetector model allows modeling of the following noise effects: shot noise of the signal, shot noise associated with the dark current of the device, RIN noise and both signal-spontaneous and spontaneous-spontaneous beat noises. Each of these effects may be separately disabled or enabled with the parameters include_sigshot, include_darkshot, include_rin and include_SE_noise respectively.

In this model, all these noise sources appear as stochastic terms added directly to the sampled electrical signal. The monolithic receiver model should be used if an analytic treatment of noise is desired, (typically to use the Quasi-Analytic approach to bit error estimation).

Spontaneous Emission Noises The two spontaneous emission noise effects – signal-spontaneous (sig-spon) and spontaneous-spontaneous (spon-spon) beat noise arise from the action of the square law detection mixing the deterministic signal and random noise. When include_SE_noise=YES, before applying the square-law detection to generate the photocurrent, the model first converts any power in ASE noise bins in the optical signal into a stochastic time series which is added to the sampled optical signal. (See the NoiseAdder documentation for details on this procedure). The spontaneous emission noises thus arise naturally in the detection calculation. Note that any ASE power outside the simulation bandwidth of the sampled signal is dropped. Since an electrical filter with a bandwidth narrower than the simulation bandwidth will normally be included later in the topology, this is not a significant restriction.

Due to the direct inclusion of ASE as stochastic noise, it is not possible to separately control the inclusion of sig-spon and spon-spon noise.

Other Noise Sources The remaining noise sources (shot noise, dark current shot noise and RIN), are treated by calculating noise variances for each effect and then adding a Gaussian random variable to each sample of the photocurrent:

( ) 222ˆ

RINdarkshotii iiitN ++ξ=

where iξ is a Gaussian random variable of zero mean and unit variance. The seed for the random variable is set with random_seed. This variable obeys the standard OptSim convention for the meanings of negative, zero and unity seeds. Since the spectrum of each noise source is assumed to be white (a reasonable approximation for electrical bandwidths), it is necessary to choose an effective bandwidth for the noises. In each case the effective bandwidth is taken to be the numerical one-sided bandwidth ( )tBeff ∆= 21 , where t∆ is the sampling rate. Below we give the expressions for each noise

variance and the associated spectral density functions ( )fS . The two are related through the expression

( )∫=

effB

nn dffSi0

2


Shot noise The shot noise derives from the random distribution in arrival times of photons at the photodetector. The expressions are

( ) ( ) ( )tqIfSBtqIi effshot 222 =⇔=

where q is the charge on the electron. Note that the noise variance depends on the photocurrent, but that for electrical frequencies the noise can be considered “locally” white.

Dark current noise The dark current noise is shot noise associated with leakage currents in the active region of the photodetector which flow even in the absence of incident optical poser, and is described by

( ) FMqIfSFBMqIi darkeffdarkdark

222 22 =⇔=

Here the (time-independent) dark current darkI is entered as the parameter dark_current. For PIN diodes, both M and F are unity. For an APD, M is again the multiplier factor, while

)12)(1( MkkMF −−+= is an excess noise factor associated with the ratio βα=k of the electron and hole ionization coefficients. The ratio k is determined by ionizationCoef.

RIN The RIN variance is given by

( ) ( ) ( ) RINeffRINRIN NFIMfSBNFIMi22222 =⇔=

where RINN is the RIN parameter specified in the source models.

Test display The model's test functions displays the current frequency response of the detector if detect=true. The format for display of the complex response can be controlled through test_display.





[5] N. A. Olsson, “Lightwave systems with optical amplifiers” IEEE Journal of Lightwave Technology, vol. 7, no. 7, pp. 1071-1082, 1989.


Properties


Outputs #1: Electrical signal (current)

Parameter Values Name Type Default Range Unit

APD_Multiplier double 1.0 [1, 1e32]

ionizationCoef double 1.0 [0, 1]

QEmethod enumerated Defined Defined, Computed

quantumEff double 0.8 [0, 1]

absorptionCoeff double 0.68e6 [0, 1e32] 1/m

layerThickness double 0.5e-6 [0, 1e32] m

reflectivity double 0.04 [0, 1]

detect enumerated false false, true

modeltype enumerated empirical empirical, intrinsic

loadResistance double 50.0 [0, 1e32] Ohm

seriesResistance double 5.0 [0, 1e32] Ohm

deviceCapacitance double 50e-15 [0, 1e32] F

electronVelocity double 6.5e6 [0, 1e32] m/s

holeVelocity double 4.8e6 [0, 1e32] m/s

lossGain double 0.0 < 0 for loss, > 0 for gain

dB

respfp double 52.259e9 [0, 1e32] Hz

respfo double 25.679e9 [0, 1e32] Hz

respg double 0.5874 [0, 1e32]

darkCurrent double 1e-6 [0, 1e32] A

include_sigshot enumerated YES NO, YES



include_rin enumerated YES NO, YES

random_seed double 1.0 [-1e8, 1]


aperture_type enumerated round round, square

aperture_dimension double 10 [0, 1000] microns

aperture_inner_dimension double 5 [0, 1000 ] microns

test_display: enumerated norm_phase norm_phase, h


norm_phase_wrap, dB_phase, dB_phase_wrap, real_imag

Parameter Descriptions APD_Multiplier: APD multiplier value (1.0 for PIN detector)

ionizationCoef: APD ionization coefficient (1.0 for PIN detector)

QEmethod: If Computed, the quantum efficiency is computed by the model. If Defined, the quantum efficiency defined above is used instead.

quantumEff: Quantum efficiency.

layerThickness: Thickness of the active region

absorptionCoeff: Absorption coefficient

reflectivity: Reflectivity at the photodiode

detect: If true then use a PD frequency response model, else assume it is included in front end response

modeltype: intrinsic or empirical models for frequency response

loadResistance: Load resistance

seriesResistance: Series resistance

deviceCapacitance: Device capacitance

electronVelocity: Electron saturated velocity

holeVelocity: Hole saturated velocity

lossGain: Gain or loss of the photodetector response (only used if pd_detect is set true)

respfp: Parasitic frequency of the empirical frequency response

respfo: Resonance frequency of the empirical frequency response

respg: Gamma of the empirical frequency response fit

darkCurrent: Dark current for dark current noise computation

include_sigshot: Enable/disable optical shot noise

include_darkshot: Enable/disable dark current shot noise

include_SE_noise: Enable/disable spontaneous emission noise from ASE spectrum

include_rin: Enable/disable RIN noise

random_seed: Seed for random number generation for noise sources

spatial_effects: Turns spatial effects on or off

aperture_type: Specifies the shape of the detector

aperture_dimension: Specifies the size of the detector

aperture_inner_dimension: Specifies the inner diameter of a ring-shaped detector.

test_display: Display format for test function of photodetector response


Spatial Analyzer

The Spatial Analyzer is a general-purpose measurement tool that produces plots and reports about optical signals with attached spatial fields. It can be thought of as the spatial-domain analog of the Signal Analyzer (time domain) or the Spectrum Analyzer (frequency domain). Consider the topology of Figure 1 in which a spatial analyzer has been placed at the multimode fiber output:

Figure 1: Spatial analyzer sample topology.

The Spatial VCSEL has been configured to add spatial fields in both the x and y polarizations at a wavelength of 820 nm and the multimode fiber has 190 spatial modes in this example. The analyzer allows the user to display information about the input signal’s x polarized fields, its y polarized fields, or both. This is achieved by setting the polarization parameter to Ex, Ey, or Both.

The analyzer basically produces two types of outputs: a summary report of spatial information or plots of the spatial fields. These are chosen by setting the parameter output_type to Summary, Plots, or Both. The summary report produced for the above example is depicted below:

OpSig Spatial Field Summary --------------------------- Input is a linked list of 190 OpSigs. OpSig #1 has both Ex and Ey time-domain fields and both X and Y spatial fields. Average power in the x-polarization is 6.4542e-010 Average power in the y-polarization is 0 wavelength=820 nm startTime=1.415179223321e-006 s OpSig #2 has both Ex and Ey time-domain fields and both X and Y spatial fields. Average power in the x-polarization is 6.44688e-010 Average power in the y-polarization is 0 wavelength=820 nm startTime=1.415180771222e-006 s • • •

Figure 2: Summary output option of spatial analyzer.

In addition to the summary report, the spatial analyzer offers the user a range of plotting options. The first choice, plot_type=Individual, simply generates a contour plot for each spatial field that the analyzer detects. Continuing the example of Figure 1, the first three of the 190 fiber modes are depicted in Figure 3. The second choice, plot_type=Total, creates a superposition of all of the detected spatial fields. Spectral coherence is accounted for when necessary. While the plot_type=Total option creates a simple superposition, the third choice, plot_type=Weighted, creates a superposition of all the detected spatial fields weighted by their time-domain values. To better understand this option, recall that the spatial optical signal is defined to be

( ) ),()(,, yxitiEtyxiE ψ=


The plot_type=Total option simply plots the superposition of ),( yxiψ for all fields i. The plot_type=Weighted option plots the superposition of ( ) ),()(,, yxitiEtyxiE ψ= for all i.

Figure 3: Contour plots of first three spatial modes of the fiber.

Since it would be computationally prohibitive to plot the weighted quantity for all time t, the user must set the parameter plot_time to a time value at which to plot the weighted profile. This time should be specified relative to the start time of the first signal at the analyzer input. Note that if the user sets plot_time to be negative, then the analyzer automatically uses the first time point from each signal. In this case, the analyzer requires that the time-varying input be a continuous-wave quantity; thus, when using the plot_type=Weighted option with a negative plot_time, the user should run the simulation in CW-mode (Figure 4). This can be achieved by using the Spatial CW Laser model, which is the preferred method, or by using a time-varying optical source whose output signal is not modulated.

Figure 4: Spatial analyzer run with continuous-wave input.

As an example of the application of the plot_type=Weighted option, consider either topology of Figure 4. The VCSEL output, and the analyzer output using both the plot_type=Total and plot_type=Weighted options are depicted in Figure 5. The difference between the two analyzer outputs basically amounts to the difference between:

),(),(1

yxyxEn

iitotal ∑

=

ψ= (plot_type=Total) and ),(),(1

yxEyxEn

iiitotal ∑

=

ψ= (plot_type=Weighted)


Here, Ei is the CW value of the time-domain portion of each optical signal i and n denotes the number of spatial fields input to the analyzer. In the context of Figure 4, the plot_type=Total plot basically corresponds to a nonweighted superposition of all of the normalized fiber modes (i.e., an overfilled launch).

Figure 5: VCSEL output (left), plot_type=Total (center), plot_type=Weighted (right).

When plot_method=coherent, both the plot_type=Total and plot_type=Weighted functions add the fields as if they were perfectly coherent. This means that for fields at the same wavelength, the composite field is determined by adding individual fields, thus preserving the optical phase. Fields occurring at different wavelengths, however, are added incoherently; in other words, the relative optical phase between fields is neglected. This is accomplished by adding powers rather than fields as shown below. When plot_method=incoherent, all signals are added incoherently (i.e., square root of the sum of the powers rather than the sum of the fields).

( ) ( )∑=

ψ=n

iicoherent y,xy,xE

1

( ) ( ) ( )∑=

ψψ=n

i

*iiincoherent y,xy,xy,xE

1

The radial_plot parameter enables the user to plot the normalized optical power along a ray that starts at the origin and that lies at an angle plot_angle degrees with respect to the x-axis. For example, Figure 6 shows a contour plot on the left and the radial plot that results from setting plot_angle to 45 degrees.

Figure 6: Demonstration of the radial_plot feature.

The spatial analyzer creates its plots by sampling the spatial field data. If the sampling is performed over a very fine grid, the resulting plots will appear smooth, but the simulation will take a long time. Conversely, if the sampling is performed over a coarse grid, the plots will appear jagged, but the simulation will execute quickly. OptSim allows the user to make his own trade-off between speed and accuracy through the accuracy parameter, which can be set to low, medium, or high. The final field to be discussed is plot_mode. This flag enables the user to display either the magnitude of the spatial field, the magnitude and phase (mag_phase) of the fields, or the real and imaginary parts (real_imag).


Properties


Outputs None

Parameter Values Name Type Default Range Units polarization enumerated Ex Ex, Ey, Both None output_type enumerated Summary Summary, Plots, Both None

plot_type enumerated Individual Individual, Total, Weighted

None

plot_mode enumerated magnitude magnitude, mag_phase, real_imag

None

accuracy enumerated medium low, medium, high

plot_method enumerated coherent coherent, incoherent

radial_plot enumerated No Yes, No

plot_angle double 0 0-360 degrees plot_time double -1 [-1, 1e32] s

Parameter Descriptions polarization Determines which polarization(s) the tool will analyze output_type Designates whether to prepare a report, plots, or both plot_type Allows the user to choose whether individual mode plots, total field plots, or weighted

fields should be displayed plot_mode

Toggles plots between the three supported formats: magnitude, magnitude/phase, and real/imaginary

plot_method Allows the user to choose whether to add fields coherently or incoherently radial_plot Determines whether the analyzer will produce a radial plot plot_angle Angle at which the radial plot is created accuracy Determines the precision of the sampling used to produce contour plots plot_time Time at which to calculate the weighted field plot


Encircled Flux Analysis Tool

It has been well documented that the position of the laser relative to the fiber has a significant impact on multimode system bandwidth [1]-[5]. The fiber modal delays can typically span a range of several nanoseconds; clearly, if all of these modes are excited equally, the system will not function at high speeds. The relative position of the optical source causes different multimode fiber modes to be excited preferentially. Thus, in order for a multimode system to meet a given bandwidth requirement, the attributes of the optical source must be restricted. This can be achieved by imposing a requirement on the number, size, and type of spatial modes emitted by the source and by establishing rigid transceiver packaging tolerances. While this is certainly the most intuitive type of specification, it creates a manufacturing nightmare for the transceiver vendor. Different vendors use different types of lasers that produce different mode profiles. Different connector and packaging technologies have different positional tolerances.

To address this issue, the multimode community has established a new specification called encircled flux [6]. Encircled flux is a measurement that quantifies the percentage of optical power, for a given transceiver/fiber pair, that falls within a given radial distance from the center of the fiber (Figure 1). When combined with a detailed knowledge of the dependence of the fiber’s modal delays on the source position (see Differential Mode Delay model), the encircled flux measurement enables a prediction of the overall multimode system bandwidth [7].

Figure 1: Sample encircled flux boundary.

Figure 1 depicts the two-dimensional intensity profile of a beam launched by an optical transceiver just inside a 25-µm core radius multimode fiber. Also shown is a circular boundary of radius r. In this case, r is about 15 µm. The encircled flux calculation determines the fraction of optical power that is contained within a given radial distance r relative to the guided power contained in the entire fiber. Mathematically this is expressed as [6]:

∫∫

∫ ∫∫ ∫

∫ ∫∫ ∫

∫ ∫∫ ∫

′′′π

′′′π

=π

θ′′θ′

πθ′′θ′

=π

θ′

πθ′

=π

θ′

πθ′

=maxmaxmaxmax r

rdrIr

rrdrIr

rdrdrrI

rdrdrrI

rdArI

rdArI

rrdP

rrdP

rEF

0)(2

0)(2

2

0 0),(

2

0 0),(

2

0 0),(

2

0 0),(

2

0 0),(

2

0 0),(

)(

Here, r is the distance from the fiber center and rmax is nominally defined to be the radius of the fiber core. In practice, rmax is usually set to a number slightly higher than the core radius (such as 1.15 times the core radius in [1]) for


measurement reasons. In the above equation, the denominator represents the total guided power in the fiber. If the total power (denominator) is defined to be EFmax, then:

∫ ′′′π⋅=r

rdrIrEF

rEFmax 0

)(21)(

Notice that the encircled flux is a one-dimensional quantity because the angular dependence of the intensity is integrated out in the form of a π2 scaling factor. Because the integrations above are performed numerically, the grid spacing dr will greatly affect the accuracy of the result. This can be controlled through parameter accuracy which can be set to low, medium, or high to allow the user to make his own trade-off between simulation speed and numerical accuracy.

As an example, consider a small, tightly focused optical beam that strikes the fiber at its center. The top half of Figure 2 depicts this optical input along with the resulting encircled flux. Because the majority of the power is near the fiber center, the encircled flux increases rapidly near r = 0. Consider the same simulation with the input shifted 10 µm (Figure 2, bottom). In this case, the encircled flux increase shifts towards the vicinity of the 10 µm point and reaches only about 82% at 10 µm.

Figure 2: Fiber input (left), resulting encircled flux (right).

Another quantity of interest in multimode system design is the average radial intensity. This is defined as the scaled derivative of the encircled flux:

)()(2 rEFdrdEFrrI max ⋅=π


The average radial intensity is also plotted by the encircled flux analysis tool. Examples for the two previous test cases (no offset and 10 µm offset) are depicted in Figure 3.

Figure 3: Average radial intensity for no offset (left) and 10 µm offset (right).

Note that there is no time dependence in the encircled flux definition; the dependent variable is radial position. In view of this fact, it is most appropriate for the user to drive the fiber with a continuous wave source (Figure 4). This can be achieved by using the Spatial CW Laser model, which is the preferred method. Alternatively, a time-varying optical source can be used provided that in the Electrical Signal Generator, the parameter Vmin is set to the same value as Vmax. If the encircled flux analyzer detects a non-CW signal, it will still attempt to perform the calculation; however, the results may not be what the user expects. In addition, OptSim will issue a simulation warning.

Figure 4: Encircled flux simulation with spatial CW laser source (left), with spatial DM laser source (right).

It is important to note that the encircled flux analyzer requires the radius of the fiber that it is analyzing as one of its input parameters. The reason for this is clear from the mathematical definitions presented previously; this quantity is needed to determine rmax. The other two parameters, plot_EF and plot_radial_intensity, basically allow the user to choose which plots the analyzer displays. Because the analyzer’s main function is to calculate the encircled flux integrations defined previously, it will accept any optical input signal that has an attached spatial field. In this case, the radius parameter simply defines an arbitrary rmax. However, this tool is clearly intended to be connected directly to the output of a multimode fiber.

The model also allows adding a mask on encircled flux plot to see if computed encircled flux satisfies design requirements. For example, IEEE 802.3ae Standard for 10 Gigabit Ethernet sets following requirements for transmitter encircled flux: “The encircled flux at 19 µm shall be greater than or equal to 86% and the encircled flux at 4.5 µm shall be less than or equal to 30%”. Figure 4 shows an example of plot of encircled flux with mask with requirement mentioned above. In this case one can see that encircled flux data satisfy the mask requirements.

In order to activate mask feature one has to set parameter add_EF_Mask to Yes and then specify the values for lower limit radius and encircled flux value at this radius (lower_limit_radius, lower_limit_EF) and the same for upper limit radius and encircled flux (upper_limit_radius, upper_limit_EF).


Encircled Flux vs. r EF mask: EF<30% at 4.5 um and EF>86% at 19 um

r (microns)0 10 20 30

Enc

ircle

d Fl

ux

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 4: Encircled flux with a mask.

References [1] D. G. Cunningham and W. G. Lane, Gigabit Ethernet Networking. Indianapolis, IN: Macmillan Technical

Publishing, 1999.

[2] J. Gowar, Optical Communication Systems. Hertfordshire, UK: Prentice Hall International, 1993.

[3] D. Marcuse, Theory of Dielectric Optical Waveguides. New York, NY: Academic Press, 1974.

[4] S. E. Golowich, P. F. Kolesar, A. J. Ritger, and P. Pepeljugoski “Modeling and Simulations for 10 Gb Multimode Optical Fiber Link Component Specifications,” Optical Fiber Communications Conference Technical Digest, Paper #WDD57, 2001.

[5] Enhanced bandwidth performance over laser-based multimode fiber local area networks. TIA/EIA Telecommunications Systems Bulletin, TSB 62-20, February 2001.

[6] FOTP-203, Launched Power Distribution Measurement Procedure for Graded-Index Multimode Fiber Transmitters, TIA Standard #TIA/EIA-455-203, March 2000.

[7] FOTP-220, Differential Mode Delay Measurement of Multimode Fiber in the Time Domain, TIA Standard #TIA/EIA-455-220, 2002.

[8] IEEE Std 802.3ae™-2002 (Amendment to IEEE Std 802.3, 2002 Edition).

Properties


Outputs None


Parameter Values Name Type Default Range Units radius double 25 [1, 500] microns

plot_EF enumerated Yes Yes, No plot_radial_intensity enumerated Yes Yes, No

accuracy enumerated medium low, medium, high

add_EF_Mask enumerated No Yes, No

lower_limit_radius double 4.5 [1, 500] microns lower_limit_EF double 0.3 [0, 1]

upper_limit_radius double 19 [1, 500] microns

upper_limit_EF double 0.86 [0, 1]

Parameter Descriptions radius Radius of the fiber being analyzed plot_EF Allows user to choose whether to plot the encircled flux plot_radial_intensity Allows user to choose whether to plot the average radial intensity accuracy Determines the numerical precision of the encircled flux and the radial intensity calculations add_EF_Mask Allows user to choose whether to plot the encircled flux mask lower_limit_radius Lower limit radius for encircled flux mask lower_limit_EF Encircled flux reqirement ar lower limit radius upper_limit_ radius Upper limit radius for encircled flux mask upper_limit_EF Encircled flux reqirement ar upper limit radius


Differential Mode Delay Analysis Tool

This analysis tool is used to plot and measure the differential mode delay (DMD) of a multimode fiber. DMD has been adopted by the multimode fiber community as a means to characterize the transient performance of a multimode fiber. The DMD measurement procedure is described in great detail in [1] – [5] and can be represented schematically by Figure 1.

Figure 1: DMD measurement procedure.

In Figure 1 a narrow-beam optical point source is coupled into the multimode fiber. Although schematically illustrated as a laser, the point source is often generated by connecting a single-mode fiber between the laser and the multimode fiber as a convenient means of forcing the laser field to conform to a predetermined spatial shape required by standards [1], [2]. The DMD measurement procedure involves mounting the input source on an assembly that enables it to be shifted radially in tiny increments (usually 1 µm) across the face of the multimode fiber, with the first position at the center of the fiber. At each position, a short temporal pulse [1] is launched into the multimode fiber and the resulting temporal waveform at the output is measured. Because the extent of the optical input is very small in both space and time, the output waveforms can be used to determine the spatio-temporal impulse response at each position through Fourier or convolution methods. Because the same optical input at different radial offsets will excite the fiber modes differently, each spatio-temporal impulse response will possess a unique modal delay characteristic. The DMD is a means to characterize this delay. A typical delay plot is shown in Figure 2.

Figure 2: DMD plot.

This figure shows that as the fiber excitation moves away from the axis, the resulting output waveforms are delayed by varying degrees. The fiber used to produce the results of Figure 2 was 300 m long and had a core radius of 25 µm. These data are critical to the delay calculations and are specified by the parameters length and core_radius, respectively. Note that the y-axis units are the offset distance from the fiber axis. Although the individual waveforms have units of watts, the actual power levels are not of interest; rather, the important quantity is the relative delay between these fields.


The DMD plotting tool uses the peak power of the zero-offset field as the delay baseline. The powers in fields at all of the other offset points are plotted relative to this. The delay is expressed in units of time/distance as is conventional in the multimode community.

To produce the results of Figure 2, the topology of Figure 3 was simulated in parameter-scan mode. A mode-locked laser was used to generate the optical temporal pulse and a spatial adder was used to generate the optical spatial pulse. A spatial coupler was used to provide the radial offset of the input beam from the fiber axis; the radial offset was the parameter that was scanned.

Figure 3: Sample DMD measurement topology.

The DMD analysis tool is somewhat unique in that unlike many other OptSim analysis tools, it depends on a certain topology to produce the intended results (i.e., those that are described in the standard [1]). For example, the DMD analyzer is intended to be connected to the output of the multimode fiber model. In this configuration, it would not make sense if the input signal to the multimode fiber model were not a spatio-temporal point source nor would the analyzer produce the predicted results if the simulation were not run in scan mode. However, since the DMD tool is at its core a field-delay analyzer, strictly speaking, it is not incorrect to use it in configurations other than the intended one. Thus, there is nothing that prevents the user from attaching the tool to any model that he wishes; however, the results may be unpredictable. Only when using a topology such as the one depicted in Figure 3 will the user generate results similar to those described in the standard [1].

The DMD analyzer produced the plot of Figure 2 by intelligently searching for ranges of the output field power profiles that had nonnegligible magnitudes. This adaptive algorithm can be circumvented by setting the parameter scale_plot to no. In that case, the plotter will display the raw data as computed by the analyzer (Figure 4).

Figure 4: Unscaled DMD delay plot.

Of course, the user can always manually zoom in on the unscaled WinPlot graph himself for a better view. When scale_plot is set to yes, the user has the option of specifying a power_tolerance, below which the power in the field is considered to be (for the purposes of plotting) zero.

If the user runs the DMD analyzer in a non-parameter-scan mode, the analyzer simply plots the temporal field profile of the one signal that is present at its input. In this mode of operation, the DMD analyzer behaves in a manner similar to that of the signal analyzer model.

DMD is defined to be the total spread in time between the earliest rising edge and the latest falling edge of all waveforms in the profile. Here, edge is defined to be the time at which the waveform achieves a value that is 25% of its maximum value (Figure 5).


Figure 5: Definition of differential mode delay.

Using the criteria of Figure 5, the DMD analyzer also provides a plot of delay spread values and pulse widths as a function of radial offset (Figure 6). To calculate the delay spread, the analyzer examines all of the waveforms, starting with the one at the fiber axis and ending at the one at the specified offset. For example, the delay spread at an offset of 10 microns is calculated by comparing the temporal attributes of the eleven waveforms that exist over the offset range of 0 µm -10 µm.

Figure 6: DMD and pulse width plots.

References [1] FOTP-220, Differential Mode Delay Measurement of Multimode Fiber in the Time Domain, TIA Standard #TIA/EIA-455-220, 2002.

[2] FOTP-191, Measurement of Mode-Field Diameter of Single-Mode Optical Fiber, TIA Document #TIA/EIA-455-191. September 1998.

[3] J. B. Schlager and D. L. Franzen, “Differential mode delay measurements on multimode fibers in the time and frequency domains,” NIST Symposium on Optical Fiber Measurements Technical Digest, pp. 127-130, September 1998.

[4] M. J. Hackert, “Characterizing multimode fiber bandwidth for Gigabit Ethernet applications,” NIST Symposium on Optical Fiber Measurements Technical Digest, pp. 113-118, September 1998.


[5] K. Petermann, “Simple relationship between differential mode delay in optical fibres and the deviation from optimum profile,” Electronics Letters, vol. 14, no. 24, pp. 793-794, 1978.

Properties


Outputs None

Parameter Values Name Type Default Range Units length double 300 defined, custom Microns

core_radius double 25 [1, 75] Microns

scale_plot enumerated yes yes, no

power_tolerance double 100e-6 [1e-15, 1 ] Watts

Parameter Descriptions length Length of fiber whose differential mode delay is being analyzed core_radius Radius of the core of the fiber whose differential mode delay is being analyzed scale_plot Flag to either scale the delay plots to a reasonable time scale or to simply plot the raw data power_tolerance If plot scaling is chosen, the level below which the field’s optical power is considered to be zero


Signal Band Converter

This model collapses a linked-list of optical signals into a single OpSig. In the analysis of multimode systems, there are often occasions when large linked lists of spatio-temporal OpSigs are used. When analyzing these signals, it is often much more convenient to treat them as an aggregate group rather than as individual entities. As an example, consider the OptSim multimode fiber model. When passed a single spatio-temporal OpSig as its input, the model produces a linked-list of spatio-temporal OpSigs as its output; the linked list consists of one OpSig for each mode that exists in the fiber. It is often desirable to view the fiber output in the time domain (Figure 1):

Figure 1: Topology demonstrating usage of signal band converter.

The outputs of the signal analyzer with and without the signal band converter are shown in Figure 2.

Figure 2: Signal analyzer without (left) and with (right) signal band converter.

The converter does not include any spatial information at its output, so it should only be used before time-domain analyzers, such as in the example above. Also, the converter makes the standard OptSim assumption that the spatial fields in the linked list are orthogonal. Thus, there is no phase information in the resulting output signal. This enables the conversion to take place without the need to consider coherence effects. If this condition is not met, the converter will produce inaccurate results.

Properties




Parameter Values None

Parameter Descriptions None


Spatial Gridded Field Converter

The Spatial Gridded Field Converter takes as input an optical signal, and converts its spatial mode profiles (if any exist) into two-dimensional gridded representations. This capability can be useful when you want to represent an analytical mode profile such as a Laguerre-Gaussian mode as a set of numerical values on a two-dimensional grid.

By default, this model automatically determines the grid spacing to use when converting a spatial mode profile into a gridded mode. However, by setting default_grid_spacing to no, a user may specify the grid spacing on both the X and Y axes via dx and dy, respectively.

The model also does not convert an input mode profile if it is already a gridded mode. However, by setting default_grid_spacing to no and override_if_gridded to yes, a user may force these gridded modes to be resampled over a new grid spacing, as specified by dx and dy.

Properties



Parameter Values Name Type Default Range Units default_grid_spacing enumerated yes yes, no none

dx double 0.1 [ 0, 1e32 ] µm dy double 0.1 [ 0, 1e32 ] µm

override_if_gridded enumerated no yes, no none

Parameter Descriptions default_grid_spacing Switch to override default grid spacing.

dx X grid spacing override.

dy Y grid spacing override.

override_if_gridded Switch to override even if input profile is already gridded.

OptSim Models Reference: Block Mode Chapter 14: Predefined Compound Components •••• 611

Chapter 14: Predefined Compound Components

This chapter describes predefined compound components (superblocks):

This chapter describes predefined superblocks:

A. Optical Components: • Dual-Arm Mach-Zehnder Modulator

model dual-arm MZ interferometer

• Mach-Zehnder Delay Interferometer

model ideal dual-arm Mach-Zehnder interferometer

• Tunable Mach-Zehnder Delay Interferometer

model tunable dual-arm Mach-Zehnder interferometer

B. Transmitters:

Non-Return-to-Zero (NRZ) Transmitter

model an NRZ transmitter

Return-to-Zero (RZ) Transmitter

model an RZ transmitter

Chirped Returm-to-Zero (CRZ) Transmitter

model a CRZ transmitter

Carrier Suppressed Return-to-Zero (CSRZ)Transmitter

model a CSRZ transmitter

Soliton Transmitter

model a soliton transmitter

Duobinary Transmitter

model a duobinary transmitter

Differential Phase-Shift-Keying (DPSK) Transmitter

612 •••• Chapter 14: Predefined Compound Components OptSim Models Reference: Block Mode

model a DPSK transmitter

Return-to-Zero DPSK Transmitter

model an RZ DPSK Transmitter

WDM Transmitter

model 8- and 16-channel WDM transmitters

C. Transmission Channels

Fiber Link 1

model an amplified fiber link

Fiber Link 2

model a dispersion compensated fiber link

Fiber Link 3

model a dispersion slope matched fiber link

Free-Space Optical (FSO) Channel

model a weak-turbulence free-space optical channel

D. Receivers:

DPSK Receiver

model a DPSK balanced receiver

WDM Receiver

model 8- and 16- channel WDM receivers


Dual-Arm Mach-Zehnder Modulator

This superblock models dual-arm Mach-Zehnder interferometer. The inside topology is the following:

i.e. has three inputs – two electrical and one optical. The difference from regular MZ modulator is that two electrical signals, VA and VB , drive the modulator. The response function of the dual-arm MZ modulator is:

⋅

−+=

π

φ

π

ππVV

jV

VVVEE offsetbiass

in

out

2exp

2sin

where

2

and ,2

BABAs

VVVVVV +=−= φ

The 2x1 Expression Operators generate driving signals VS and φV from input signals VA and VB . The first Modulator in MZ regime driven by VS provides intensity modulation. The second Modulator in phase modulation regime driven by φV provides phase shift.

Properties


#2: Optical signal




Parameter Values Name Type Default Range Units Chirp double 0 [ -1e32, 1e32 ]

ExtRatio double 0 [ 0, 1e6 ] dB

insertionLoss double 0 [ 0, 1e6 ] dB

vBias double 0 [ -1e32, 1e32 ] Volts vOffset double 0 [ -1e32, 1e32 ] Volts

vPi double 1 [ -1e32, 1e32 ] Volts

Parameter Descriptions Chirp Chirp parameter for modulator ExtRatio Extinction or on-off ratio insertionLoss Insertion loss vBias Bias voltage of the modulator vOffset Offset voltage of the modulator vPi vPi of the modulator


Mach-Zehnder Delay Interferometer (MZDI)

This superblock models ideal dual-arm Mach-Zehnder interferometer. The inside topology is the following:

i.e. includes Splitter, Fiber Delay, and Coupler models. Time delay and Insertion Loss are external parameters of MZDI Superblock. Coupling coefficients in Coupler model are hard-coded so that upper output node is Constructive port and lower node is Destructive port.

Properties



#2: Optical Signal

Parameter Values Name Type Default Range Units loss double 0 [ 0, 1e32 ] dB

timeDelay double 0 [ 0, 1e32 ] seconds

Parameter Descriptions loss Insertion loss timeDelay Time delay


Tunable Mach-Zehnder Interferometer (Tunable_MZI)

This superblock models tunable dual-arm Mach-Zehnder interferometer. The inside topology is the following:

i.e. it is similar to ideal Mach—Zehnder Interferometer except that here we add a Phase Shift block. This block applied additional phase difference between two arms of interferometer and that difference depends on the tuning frequency of the device and incoming signal frequency. One can derive the transfer functions for both output ports of the interferometer as:

[ ][ ])(cos)(

)(sin)(22

2

221

tune

tune

fffH

fffH

−=

−=

πτ

πτ

where τ is time delay (parameter timeDelay), f is signal frequency, and ftune is tuning frequency (defined by parameter tuning_wvl). From here we can derive the phase difference for the Phase Block as:

−⋅=∆

tune

cλλ

πτφ 112

Parameter loss specifies the Insertion Loss of the device. Coupling coefficients in Coupler model are hard-coded so that upper output node is Constructive port and lower node is Destructive port.

Properties



#2: Optical Signal


Parameter Values Name Type Default Range Units timeDelay double 1e-10 [ 0, 1e32 ] Seconds

tuning_wvl double 1.55e-6 [ 0, 1e32 ] m signal_wvl double 1.55e-6 [ 0, 1e32 ] m


Parameter Descriptions timeDelay Time delay tuning_wvl Tuning wavelength signal_wvl Wavelength of input signal loss Insertion loss


Non-Return-to-Zero (NRZ) Transmitter (NRZ_TX)

It models non-return-to-zero (NRZ) modulated signal. The inside topology is the following:

i.e. includes CW Laser, Electrical Signal Generator, External Modulator, and Optical Normalizer. Superblocks parameters are wavelength and RIN (defined in CW Laser), power (defined by Normalizer), raise/fall time for NRZ signal and sampling rate (pointsPerBit defined in Electrical Signal Generator), chirp and extinction ratio (defined in Modulator). The input to the superblock should be a Bit Pattern Generator, and output is the optical signal to be launched to fiber or other optical component.

Properties



Parameter Values Name Type Default Range Units Chirp double 0 [ -1e32, 1e32 ] ExtRatio double 0 [ 0, 1e6 ] dB

RIN double -150 [ -1e32, 1e32 ] dBm/Hz

pointsPerBit double 5 [ 1, 27 ]

power double -3 [ -1e3, 1e3 ] dBm risefallTime double 1e-011 [ 0, 100 ] seconds

wavelength double 1.55e-006 [ 0, 1e18 ] meters

Parameter Descriptions Chirp Chirp parameter for modulator ExtRatio Extinction or on-off ratio RIN Relative intensity noise of laser pointsPerBit Number of sampling points per bit in electrical signal


power Average output optical power risefallTime Rise and Fall time of electrical signal wavelength Laser wavelength


Return-to-Zero (RZ) Transmitter (RZ_TX)

It models return-to-zero (RZ) modulated signal. The inside topology is the same as for NRZ signal. The difference in setting is that Electrical Signal Generator is set to RZ RaisedCosine format and pulse width is defined by parameter dutyCycle instead of raisefallTime. All other parameters are the same as for NRZ_TX.

Properties



Parameter Values Name Type Default Range Units ExtRatio double 0 [ 0, 1e6 ] dB

Chirp double 0 [ -1e32, 1e32 ]

RIN double -150 [ -1e32, 1e32 ] dBm/Hz dutyCycle double 0.5 [ 1e-32, 0.5 ]


power double -3 [ -1e3, 1e3 ] dBm


Parameter Descriptions ExtRatio Extinction or on-off ratio Chirp Chirp parameter for modulator RIN Relative intensity noise of laser dutyCycle Duty cycle of the RZ signal pointsPerBit Number of sampling points per bit in electrical signal power Average output optical power wavelength Laser wavelength


Chirped Return-to-Zero (CRZ) Transmitter (CRZ_TX)

It models chirped return-to-zero (CRZ) modulated signal. The inside topology is the following:

i.e., CW signal passes though 2 Modulators. First Modulator is set to Amplitude Modulation (rather than Mach-Zehnder like in case of NRZ and RZ) with driving Electrical Signal set to RZ RaisedCosine modulated signal (with dutyCycle parameter defining pulse width). The second Modulator is set to PM (Phase Modulation) and applies chirp to the signal. Parameter PMcoef defines the amplitude (zero-to-peak) of PM in radians.

Properties



Parameter Values Name Type Default Range Units ExtRatio double 0 [ 0, 1e6 ] dB PMcoef double 1 [ -1e32, 1e32 ] radians


dutyCycle double 0.5 [ 1e-32, 0.5 ]

pointsPerBit double 5 [ 1, 27 ] power double -3 [ -1e3, 1e3 ] dBm



Parameter Descriptions ExtRatio Extinction or on-off ratio PMcoef Amplitude of phase modulation RIN Relative intensity noise of laser dutyCycle Duty cycle of the RZ signal pointsPerBit Number of sampling points per bit in electrical signal power Average output optical power wavelength Laser wavelength


Carrier-Suppressed Return-to-Zero (CSRZ) Transmitter (CSRZ_TX)

It models carrier-suppressed return-to-zero (CSRZ) modulated signal. The inside topology is the following:

Here in order to create carrier-suppressed RZ signal, first the NRZ signal is generated by Mach-Zehnder modulator and the output is sent to the other MZ Modulator block which is driven by sinusoidal signal. The drive frequency is a half of the bitrate and the amplitude is πV2 .

Properties




RIN double -150 [ -1e32, 1e32 ] dBm/Hz bitrate double 1e+010 [ 1, 1e15 ] bps

patternLength double 7 [ 1, 27 ]


power double -3 [ -1e3, 1e3 ] dB wavelength double 1.55e-006 [ 0, 1e18 ] meters


Parameter Descriptions ExtRatio Extinction or on-off ratio RIN Relative intensity noise of laser bitrate The bit rate of the binary sequence generated patternLength The number of bits in the generated bit sequence is 2x where x is the parameter value pointsPerBit Number of sampling points per bit in electrical signal power Average output optical power wavelength Laser wavelength


Soliton Transmitter (Soliton_TX)

It models soliton-shape modulated signal. The inside topology is the following:

Here the difference with RZ transmitter is that instead of CW Laser, the Mode-Locked Laser is used as optical source for with the pulse shape selected as type sech to provide a hyperbolic secant shape typical for optical solitons.

Properties



Parameter Values Name Type Default Range Units Bitrate double 1e+010 [ 1, 1e15 ] bps

Chirp double 0 [ -1e32, 1e32 ] ExtRatio double 30 [ 0, 1e6 ] dB




pulsewidth double 2e-011 [ 0, 1e32 ] seconds


Parameter Descriptions Bitrate The bit rate of the binary sequence generated Chirp Chirp factor ExtRatio Extinction or on-off ratio


RIN Relative intensity noise of laser patternLength The number of bits in the generated bit sequence is 2x where x is the parameter value pointsPerBit Number of sampling points per bit in electrical signal power Average output optical power pulsewidth Pulse width wavelength Laser wavelength


Duobinary Transmitter (Doubi_TX)

It models duobinary modulated signal. The inside topology is the following:

To generate duobinary signals an ideal NRZ electrical signal is passed through a delay-and-add circuit followed by an bandlimiting filter of bandwidth (bitrate)/2. Delay is equal to bit duration and electrical filter’s shape is low-pass 4th order Bessel.

Properties





bitrate double 1e+010 [ 0, 1e15 ] bps



Parameter Descriptions ExtRatio Extinction or on-off ratio RIN Relative intensity noise of laser bitrate The bit rate of the binary sequence generated pointsPerBit Number of sampling points per bit in electrical signal power Average output optical power wavelength Laser wavelength


Differential Phase-Shift-Keying Transmitter (DPSK_TX)

It models differential phase-shift-key (DPSK) modulated signal. The inside topology looks the same as for NRZ signal with the only difference that the Modulator is set to Phase Modulation with phase shift of 180 degrees.

Properties



Parameter Values Name Type Default Range Units Chirp double 0 [ -1e32, 1e32 ] RIN double -150 [ -1e32, 1e32 ] dBm/Hz




Parameter Descriptions Chirp Chirp factor RIN Relative intensity noise of laser pointsPerBit Number of sampling points per bit in electrical signal power Average output optical power wavelength Laser wavelength


Return-to-Zero DPSK Transmitter (RZ-DPSK_TX)

It models return-to-zero differential phase-shift-key (RZ-DPSK) modulated signal. The inside topology is the following:

Here the difference from DPSK transmitter is that instead of CW Laser, the Mode-Locked Laser is used as optical source for RZ signal. RaisedCosine shape of the signal is used with parameter dutyCycle defining its width.

Properties




Chirp double 0 [ -1e32, 1e32 ]


dutyCycle double 0.5 [ 1e-32, 0.5 ] patternLength double 7 [ 1, 27 ]




Parameter Descriptions Bitrate The bit rate of the binary sequence generated


Chirp Chirp factor RIN Relative intensity noise of laser dutyCycle Duty cycle of the RZ signal patternLength The number of bits in the generated bit sequence is 2x where x is the parameter value pointsPerBit Number of sampling points per bit in electrical signal power Average output optical power wavelength Laser wavelength


Differential Phase-Shift-Keying Transmitter – Advanced (DPSK_TX_adv)

This compound component model implements a detailed Differential Phase-Shift-Keying (DPSK) transmitter which takes as input a differentially encoded binary signal and produces a DPSK optical output signal. Figure 1 depicts the model’s schematic.

Figure 1: DPSK_TX_adv compound component schematic.

The model is composed of the following components:

• NRZ_driver

This component converts the input binary signal into an electrical waveform. It is configured via the parameters Vdrive_lo – the signal value for binary zeroes; Vdrive_hi – the signal value for binary ones; and pointsPerBit – the number of samples per bit.

• LPF_TX

This component implements a low-pass Bessel filter for modeling the non-ideal characteristics of the binary-to-electrical conversion. It is configured via the parameters filter_poles – the order of the Bessel filter; and filter_BW_GHz – the filter bandwidth.

• Laser_source

This component models the transmitter’s cw laser source. It is configured via the parameters wavelength_nm – the laser’s center wavelength; power_dBm – the cw laser power; and RIN – the laser’s relative intensity noise.

• MZM_phase

This component models a Mach-Zehnder modulator used to create the transmitter’s phase-modulated output signal. It is configured via the parameters MZM_Vpi – Vπ of the modulator; MZM_Von – the modulator’s offset voltage; MZM_Vbias – the modulator’s bias voltage; MZM_excess_loss_dB – the modulator’s insertion loss; MZM_extinction_ratio_dB – the modulator’s extinction ratio; and MZM_chirp_factor – the modulator’s chirp factor.


Properties



Parameter Values Name Type Default Range Units Vdrive_lo double 5.0 [ -1000, 1000 ] Volts

Vdrive_hi double -5.0 [ -1000, 1000 ] Volts

pointsPerBit integer 5 [ 1, 27 ] filter_poles integer 1 [ 0, 128 ]

filter_BW_GHz double 10.0 [ 0, 1e9 ] GHz

wavelength_nm double 1550.0 [ 0, 1e27 ] nm

power_dBm double 0.0 [ -1e32, 210 ] dBm RIN double -150.0 [ -1e32, 1e32 ] dB/Hz

MZM_Vpi double 5.0 [ -1e32, 1e32 ] Volts

MZM_Von double 5.0 [ -1e32, 1e32 ] Volts

MZM_Vbias double 5.0 [ -1e32, 1e32 ] Volts MZM_excess_loss_dB double 0.0 [ 0, 1e6 ] dB

MZM_extinction_ratio_dB double 1000.0 [ 0, 1e6 ] dB

MZM_chirp_factor double 0.0 [ -1e32, 1e32 ]

Parameter Descriptions Vdrive_lo Minimum value of the driver electrical signal (binary zero value)

Vdrive_hi Maximum value of the driver electrical signal (binary one value)

pointsPerBit Number of sampling points per bit in driver electrical signal

filter_poles Order of the low-pass Bessel filter filter_BW_GHz Bandwidth of the low-pass Bessel filter

wavelength_nm Wavelength of the cw optical source

power_dBm Output power of the cw optical source

RIN RIN of the cw optical source MZM_Vpi Vπ of the Mach-Zehnder Modulator MZM_Von Offset voltage of the Mach-Zehnder Modulator MZM_Vbias Bias voltage of the Mach-Zehnder Modulator

MZM_excess_loss_dB Insertion loss of the Mach-Zehnder Modulator

MZM_extinction_ratio_dB Extinction ratio of the Mach-Zehnder Modulator

MZM_chirp_factor Chirp factor of the Mach-Zehnder Modulator


Return-to-Zero Differential Phase-Shift-Keying Transmitter – Advanced (RZ_DPSK_TX_adv)

This compound component model implements a detailed Return-to-Zero (RZ) Differential Phase-Shift-Keying (DPSK) transmitter which takes as input a differentially encoded binary signal and produces an RZ-DPSK optical output signal. The RZ carving signal should be provided via the model’s electrical input. Figure 1 depicts the model’s schematic.

Figure 1: RZ_DPSK_TX_adv compound component schematic.


• NRZ_driver

This component converts the input binary signal into an electrical waveform. It is configured via the parameters Vdrive_lo – the signal value for binary zeroes; Vdrive_hi – the signal value for binary ones; and pointsPerBit – the number of samples per bit.

• LPF_TX

This component implements a low-pass Bessel filter for modeling the non-ideal characteristics of the binary-to-electrical conversion. It is configured via the parameters filter_poles – the order of the Bessel filter; and filter_BW_GHz – the filter bandwidth.

• Laser_source

This component models the transmitter’s cw laser source. It is configured via the parameters wavelength_nm – the laser’s center wavelength; power_dBm – the cw laser power; and RIN – the laser’s relative intensity noise.

• MZM_phase

This component models a Mach-Zehnder modulator used to create the transmitter’s phase-modulated output. It is configured via the parameters MZM_Vpi – Vπ of the modulator; MZM_Von – the modulator’s offset voltage; MZM_Vbias – the modulator’s bias voltage; MZM_excess_loss_dB – the modulator’s insertion loss; MZM_extinction_ratio_dB – the modulator’s extinction ratio; and MZM_chirp_factor – the modulator’s chirp factor.

• MZM_carver


This component models a Mach-Zehnder modulator used to create the transmitter’s pulsed output (i.e., it generates the RZ output pulses). It is configured via the parameters MZMcarver_Vpi – Vπ of the modulator; MZMcarver_Von – the modulator’s offset voltage; MZMcarver_Vbias – the modulator’s bias voltage; MZMcarver_excess_loss_dB – the modulator’s insertion loss; MZMcarver_extinction_ratio_dB – the modulator’s extinction ratio; and MZMcarver_chirp_factor – the modulator’s chirp factor.

Properties




Parameter Values Name Type Default Range Units Vdrive_lo double 5.0 [ -1000, 1000 ] Volts Vdrive_hi double -5.0 [ -1000, 1000 ] Volts

pointsPerBit integer 5 [ 1, 27 ]

filter_poles integer 1 [ 0, 128 ]

filter_BW_GHz double 10.0 [ 0, 1e9 ] GHz wavelength_nm double 1550.0 [ 0, 1e27 ] nm

power_dBm double 0.0 [ -1e32, 210 ] dBm


MZM_Vpi double 5.0 [ -1e32, 1e32 ] Volts MZM_Von double 5.0 [ -1e32, 1e32 ] Volts

MZM_Vbias double 5.0 [ -1e32, 1e32 ] Volts

MZM_excess_loss_dB double 0.0 [ 0, 1e6 ] dB

MZM_extinction_ratio_dB double 1000.0 [ 0, 1e6 ] dB MZM_chirp_factor double 0.0 [ -1e32, 1e32 ]

MZMcarver_Vpi double 5.0 [ -1e32, 1e32 ] Volts

MZMcarver_Von double 5.0 [ -1e32, 1e32 ] Volts

MZMcarver_Vbias double 5.0 [ -1e32, 1e32 ] Volts MZMcarver_excess_loss_dB double 0.0 [ 0, 1e6 ] dB

MZMcarver_extinction_ratio_dB double 1000.0 [ 0, 1e6 ] dB

MZMcarver_chirp_factor double 0.0 [ -1e32, 1e32 ]

Parameter Descriptions Vdrive_lo Minimum value of the driver electrical signal (binary zero value) Vdrive_hi Maximum value of the driver electrical signal (binary one value)

pointsPerBit Number of sampling points per bit in driver electrical signal

filter_poles Order of the low-pass Bessel filter

filter_BW_GHz Bandwidth of the low-pass Bessel filter



power_dBm Output power of the cw optical source

RIN RIN of the cw optical source MZM_Vpi Vπ of the Mach-Zehnder Modulator MZM_Von Offset voltage of the Mach-Zehnder Modulator

MZM_Vbias Bias voltage of the Mach-Zehnder Modulator


MZM_extinction_ratio_dB Extinction ratio of the Mach-Zehnder Modulator MZM_chirp_factor Chirp factor of the Mach-Zehnder Modulator

MZMcarver_Vpi Vπ of the carving Mach-Zehnder Modulator MZMcarver_Von Offset voltage of the carving Mach-Zehnder Modulator

MZMcarver_Vbias Bias voltage of the carving Mach-Zehnder Modulator

MZMcarver_excess_loss_dB Insertion loss of the carving Mach-Zehnder Modulator

MZMcarver_extinction_ratio_dB Extinction ratio of the carving Mach-Zehnder Modulator MZMcarver_chirp_factor Chirp factor of the carving Mach-Zehnder Modulator


Differential Quadrature Phase-Shift-Keying Transmitter (DQPSK_TX)

This compound component model implements a detailed Differential Quadrature Phase-Shift-Keying (DQPSK) transmitter which takes as input encoded in-phase (P) and quadrature (Q) binary signals and produces a DQPSK optical output signal. Figure 1 depicts the model’s schematic.

Figure 1: DQPSK_TX compound component schematic.

The model is primarily composed of the following components:

• NRZ_driver_P and NRZ_driver_Q

These components convert the input binary signals into electrical waveforms. They are configured via the parameters Vdrive_lo – the signal value for binary zeroes; Vdrive_hi – the signal value for binary ones; and pointsPerBit – the number of samples per bit.

• LPF_TX_P and LPF_TX_Q

These components implement low-pass Bessel filters for modeling the non-ideal characteristics of the binary-to-electrical conversion. They are configured via the parameters filter_poles – the order of the Bessel filters; and filter_BW_GHz – the filter bandwidths.

• Laser_source

This component models the transmitter’s cw laser source. It is configured via the parameters wavelength_nm – the laser’s center wavelength; power_dBm – the cw laser power; and RIN – the laser’s relative intensity noise. The component OpSplit divides the output of the laser source between the P and Q modulators.

• MZM_P and MZM_Q

These components model Mach-Zehnder modulators used to create the transmitter’s phase-modulated output signal. They are configured via the parameters MZM_Vpi – Vπ of the modulators; MZM_Von – the modulators’ offset voltages; MZM_Vbias – the modulators’ bias voltages; MZM_excess_loss_dB – the modulators’ insertion losses; MZM_extinction_ratio_dB –


the modulators’ extinction ratios; and MZM_chirp_factor – the modulators’ chirp factors. The component Phase_modulator provides the additional phase shift required for the quadrature signal, while the component OpComb combines the two modulator outputs into a final DQPSK output signal.

Properties


#2: Binary Signal



Vdrive_hi double -5.0 [ -1000, 1000 ] Volts PointsPerBit integer 5 [ 1, 27 ]

filter_poles integer 1 [ 0, 128 ]

filter_BW_GHz double 10.0 [ 0, 1e9 ] GHz

wavelength_nm double 1550.0 [ 0, 1e27 ] nm power_dBm double 0.0 [ -1e32, 210 ] dBm


MZM_Vpi double 5.0 [ -1e32, 1e32 ] Volts MZM_Von double 5.0 [ -1e32, 1e32 ] Volts


MZM_excess_loss_dB double 0.0 [ 0, 1e6 ] dB

MZM_extinction_ratio_dB double 1000.0 [ 0, 1e6 ] dB MZM_chirp_factor double 0.0 [ -1e32, 1e32 ]


Vdrive_hi Maximum value of the driver electrical signal (binary one value)

PointsPerBit Number of sampling points per bit in driver electrical signal filter_poles Order of the low-pass Bessel filter



power_dBm Output power of the cw optical source RIN RIN of the cw optical source

MZM_Vpi Vπ of the Mach-Zehnder Modulator MZM_Von Offset voltage of the Mach-Zehnder Modulator




MZM_extinction_ratio_dB Extinction ratio of the Mach-Zehnder Modulator



Return-to-Zero Differential Quadrature Phase-Shift-Keying Transmitter (RZ_DQPSK_TX)

This compound component model implements a detailed Return-to-Zero (RZ) Differential Quadrature Phase-Shift-Keying (DQPSK) transmitter which takes as input encoded in-phase (P) and quadrature (Q) binary signals and produces an RZ-DQPSK optical output signal. The RZ carving signal should be provided via the model’s electrical input. Figure 1 depicts the model’s schematic.

Figure 1: RZ_DQPSK_TX compound component schematic.

The model is primarily composed of the following components:

• NRZ_driver_P and NRZ_driver_Q

These components convert the input binary signals into electrical waveforms. They are configured via the parameters Vdrive_lo – the signal value for binary zeroes; Vdrive_hi – the signal value for binary ones; and pointsPerBit – the number of samples per bit.

• LPF_TX_P and LPF_TX_Q

These components implement low-pass Bessel filters for modeling the non-ideal characteristics of the binary-to-electrical conversion. They are configured via the parameters filter_poles – the order of the Bessel filters; and filter_BW_GHz – the filter bandwidths.

• Laser_source

This component models the transmitter’s cw laser source. It is configured via the parameters wavelength_nm – the laser’s center wavelength; power_dBm – the cw laser power; and RIN – the laser’s relative intensity noise. The component OpSplit divides the output of the laser source between the P and Q modulators.

• MZM_P and MZM_Q

These components model Mach-Zehnder modulators used to create the transmitter’s phase-modulated output. They are configured via the parameters MZM_Vpi – Vπ of the modulators; MZM_Von – the modulators’ offset voltages; MZM_Vbias – the modulators’ bias voltages;


MZM_excess_loss_dB – the modulators’ insertion losses; MZM_extinction_ratio_dB – the modulators’ extinction ratios; and MZM_chirp_factor – the modulators’ chirp factors. The component Phase_modulator provides the additional phase shift required for the quadrature signal, while the component OpComb combines the two modulator outputs.

• MZM_carver

This component models a Mach-Zehnder modulator used to create the transmitter’s pulsed output (i.e., it generates the RZ output pulses). It is configured via the parameters MZMcarver_Vpi – Vπ of the modulator; MZMcarver_Von – the modulator’s offset voltage; MZMcarver_Vbias – the modulator’s bias voltage; MZMcarver_excess_loss_dB – the modulator’s insertion loss; MZMcarver_extinction_ratio_dB – the modulator’s extinction ratio; and MZMcarver_chirp_factor – the modulator’s chirp factor.

Properties


#2: Binary Signal




Vdrive_hi double -5.0 [ -1000, 1000 ] Volts

pointsPerBit integer 5 [ 1, 27 ]

filter_poles integer 1 [ 0, 128 ] filter_BW_GHz double 10.0 [ 0, 1e9 ] GHz

wavelength_nm double 1550.0 [ 0, 1e27 ] nm

power_dBm double 0.0 [ -1e32, 210 ] dBm

RIN double -150.0 [ -1e32, 1e32 ] dB/Hz MZM_Vpi double 5.0 [ -1e32, 1e32 ] Volts

MZM_Von double 5.0 [ -1e32, 1e32 ] Volts


MZM_excess_loss_dB double 0.0 [ 0, 1e6 ] dB MZM_extinction_ratio_dB double 1000.0 [ 0, 1e6 ] dB

MZM_chirp_factor double 0.0 [ -1e32, 1e32 ]

MZMcarver_Vpi double 5.0 [ -1e32, 1e32 ] Volts

MZMcarver_Von double 5.0 [ -1e32, 1e32 ] Volts MZMcarver_Vbias double 5.0 [ -1e32, 1e32 ] Volts

MZMcarver_excess_loss_dB double 0.0 [ 0, 1e6 ] dB

MZMcarver_extinction_ratio_dB double 1000.0 [ 0, 1e6 ] dB

MZMcarver_chirp_factor double 0.0 [ -1e32, 1e32 ]



Vdrive_hi Maximum value of the driver electrical signal (binary one value) pointsPerBit Number of sampling points per bit in driver electrical signal

filter_poles Order of the low-pass Bessel filter


wavelength_nm Wavelength of the cw optical source power_dBm Output power of the cw optical source

RIN RIN of the cw optical source

MZM_Vpi Vπ of the Mach-Zehnder Modulator MZM_Von Offset voltage of the Mach-Zehnder Modulator


MZM_excess_loss_dB Insertion loss of the Mach-Zehnder Modulator MZM_extinction_ratio_dB Extinction ratio of the Mach-Zehnder Modulator


MZMcarver_Vpi Vπ of the Mach-Zehnder Modulator MZMcarver_Von Offset voltage of the carving Mach-Zehnder Modulator

MZMcarver_Vbias Bias voltage of the carving Mach-Zehnder Modulator

MZMcarver_excess_loss_dB Insertion loss of the carving Mach-Zehnder Modulator MZMcarver_extinction_ratio_dB Extinction ratio of the carving Mach-Zehnder Modulator

MZMcarver_chirp_factor Chirp factor of the carving Mach-Zehnder Modulator


WDM Transmitters

The four superblocks for multichannel transmitters included are the 8-channel and 16-channel transmitters with NRZ and RZ modulated signals.

This superblock uses inside superblocks for RZ and NRZ modulated signals. For example, 8-channel NRZ transmitter has following inside topology:

User specifies bitrate, first channel wavelength, channel spacing, power per channel, extinction ratio, chirp, pre- and post-bits, and bits shift. In the case of RZ signals user also specifies dutyCycle parameter that


controls the pulse width. Multiplexer is set to MultiBand representation by default but can changed by user to SingleBand.

Properties

Inputs None

Outputs #1-8 or #1-16: Optical signal

Parameter Values (NRZ Transmitter) Name Type Default Range Units Bitrate double 1e+010 [ 0, 1e15 ] bps

BitsShift double 0 [ 0, 1e3 ] bits

Chirp double 0 [ -1e32, 1e32 ] ExtRatio double 0 [ 0, 1e3 ] dB


ch_spacing_Hz double 1e+011 [ 0, 1e15 ] Hz

lambda_first double 1.55e-006 [ 0, 1e18 ] meters patternLength double 7 [ 1, 27 ]


postBits double 0 [ 0, 1e3 ] bits

powerPerCh_dBm double -3 [ -1e3, 1e3 ] dBm preBits double 0 [ 0, 1e3 ] bits

Parameter Descriptions (NRZ Transmitter) Bitrate The bit rate of the binary sequence BitsShift The number of bits to shift each successive output’s binary sequence relative to previous

output. This applies to second and subsequent outputs. Chirp Chrip factor ExtRatio Extinction ratio of the modulator RIN Relative intensity noise of the laser ch_spacing_Hz Inter-channel spacing in Hz lambda_first Wavelength of the first channel patternLength The number of bits in the generated bit sequence is 2x where x is the parameter value pointsPerBit Number of sampling points per bit in electrical signal postBits The number of zero bits at the end of the sequence powerPerCh_dBm Per channel power (dBm) preBits The number of zero bits at the start of the sequence

Parameter Values (RZ Transmitter)



Bitrate double 1e+010 [ 0, 1e15 ] bps

BitsShift double 0 [ 0, 1e3 ] bits Chirp double 0 [ -1e32, 1e32 ]

ExtRatio double 0 [ 0, 1e3 ] dB


ch_spacing_Hz double 1e+011 [ 0, 1e15 ] Hz dutyCycle double 0.5 [ 1e-32, 0.5 ]

lambda_first double 1.55e-006 [ 0, 1e18 ] meters


pointsPerBit double 5 [ 1, 27 ] postBits double 0 [ 0, 1e3 ] bits

powerPerCh_dBm double -3 [ -1e3, 1e3 ] dBm

preBits double 0 [ 0, 1e3 ] bits

Parameter Descriptions (RZ Transmitter) Bitrate The bit rate of the binary sequence BitsShift The number of bits to shift each successive output’s binary sequence relative to

previous output. This applies to second and subsequent outputs. Chirp Chrip factor ExtRatio Extinction ratio of the modulator RIN Relative intensity noise of the laser ch_spacing_Hz Inter-channel spacing in Hz duty_cycle Duty cycle of RZ signal lambda_first Wavelength of the first channel patternLength The number of bits in the generated bit sequence is 2x where x is the parameter value pointsPerBit Number of sampling points per bit in electrical signal postBits The number of zero bits at the end of the sequence powerPerCh_dBm Per channel power (dBm) preBits The number of zero bits at the start of the sequence


Fiber Link 1

It models regular amplified fiber link. The inside topology is the following:

Here the fiber link is defined as a number of fiber spans followed by optical amplifier. User can decide on the total length of the link by changing the number of spans and fiber span length (default SpanLength=50 km). EDFA Black Box model is used for amplifier with Gain calculated from a fiber length and attenuation to compensate fiber loss. User can modify amplifier’s noise figure (AmpNF), saturation power (AmpSaturPower), noise bandwidth (AmpBW) and center (noiseCenter). For fiber model the parameters from ManufacturerLibrary for Corning Submarine LEAF fiber are used.

Properties



Parameter Values Name Type Default Range Units AmpBW double 3e-008 [ 0, 1e6 ] meters

AmpNF double 5 [ 3, 1e32 ] dB

AmpSaturPower double 18 [ 0, 1e32 ] dBm

NumberOfSpans double 1 [ 0, 1e18 ] SpanLength double 50000 [ 0, 1e32 ] meters

noiseCenter double 1.55e-006 [ 1e-32, 1e32 ] meters


Parameter Descriptions AmpBW Noise bandwidth of the amplifier AmpNF Noise figure of the amplifier AmpSaturPower Saturation power of the amplifier NumberOfSpans Number of fspans SpanLength Length of each span noiseCenter Noise center of the amplifier


Fiber Link 2

It models dispersion-compensated fiber link. The inside topology is the following:

Here one loop consists of six regular fiber spans and one dispersion compensation fiber span. Number of loops is defined by parameter NumperOfSpans. Hence, in this case the total length will be (7 x NumperOfSpans x SpanLength). For regular transmission fiber spans the fiber model for Corning Submarine LEAF from Manufacturer Library is used. For dispersion compensating fiber span the fiber model for Corning Submarine SMF28+ is used. To get symmetric dispersion map and to minimize dispersion spread the compensating span is positioned in the middle of a loop. The accumulated negative dispersion from regular spans will be fully compensated by dispersion compensated fiber at wavelength 1550 nm. The amplifier setting is the same as for Fiber Link 1.

Properties




AmpNF double 5 [ 3, 1e32 ] dB AmpSaturPower double 18 [ 0, 1e32 ] dBm

NumberOfSpans double 1 [ 0, 1e18 ]

SpanLength double 50000 [ 0, 1e32 ] meters


Parameter Descriptions AmpBW Noise bandwidth of the amplifier AmpNF Noise figure of the amplifier


AmpSaturPower Saturation power of the amplifier NumberOfSpans Number of fspans SpanLength Length of each span noiseCenter Noise center of the amplifier


Fiber Link 3

It models dispersion-slope-matched (or dispersion-flattened) fiber link. The inside topology is the following:

Here the difference with Fiber Link 1 is that fiber section of the span consists of 2 fiber spans where fiber parameters are selected such that both dispersion and dispersion slope are fully compensated. For fiber models the fiber parameters form Manufacturer Library are used - first fiber is Corning Vascade L1000 with positive dispersion parameter and the second one - Corning Vascade S1000 with negative dispersion parameter. User can decide on the total length of the link by changing the number of spans and fiber span length (combined for two fiber sections). The amplifier setting is the same as for FL 1.

Properties




AmpNF double 5 [ 3, 1e32 ] dB AmpSaturPower double 18 [ 0, 1e32 ] dBm

NumberOfSpans double 1 [ 0, 1e18 ]

SpanLength double 50000 [ 0, 1e32 ] meters


Parameter Descriptions AmpBW Noise bandwidth of the amplifier


AmpNF Noise figure of the amplifier AmpSaturPower Saturation power of the amplifier NumberOfSpans Number of fspans SpanLength Length of each span noiseCenter Noise center of the amplifier


Free-Space Optical Channel (Weak Turbulence)

This simple free-space optical-channel model takes into account the effects of weak turbulence and background radiation on the free-space transmission of an optical signal. The model is based on the work presented in [1].

The model treats attenuation (specified as a negative number) as a combination of geometric and environmental contributions, the latter of which is expected to vary statistically due to weak turbulence. For a beam divergence angle Θ (beam_divergence in degrees), a propagation distance L (prop_distance in meters), and a receiver area A (area in cm2), the geometric attenuation in dB is:

3 2 2π10log12.96geom

LA

α Θ= −

If we represent the additional attenuation in dB as αadd (atten_add), then for an input signal power Pi, the output power Po is:

1010geom add

o iP Pα α+ = ⋅

Weak turbulence is expected to lead to a stochastic variation in αadd. Because of the long-time scales involved in this variation, the model treats αadd as a statistical parameter with a normal distribution. The user specifies the standard deviation (in dB) via the parameter sigma_add.

Finally, background radiation can be added to the signal via the specification of the total background radiation power (p_back), its center wavelength (p_back_center), its bandwidth (p_back_bw), and the desired numerical resolution (p_back_res).

References [1] G. Hansel, E. Kube, J. Becker, J. Haase, P. Schwarz, “Simulation in the design process of free space optical transmission systems,” Proc. 6th Workshop, Optics in Computing Technology, 2001.

Properties



Parameter Values Name Type Default Range Units beam_divergence double 1 [ 0, 90 ] ° prop_distance double 100 [ 0, 1e32 ] m

atten_add double 0 [ -1e16, 0 ] dB


sigma_add double 0 [ 0, 1e16 ] dB

area double 100 [ 1e-32, 1e32 ] cm2

p_back double 1e-9 [ 0, 1e32 ] W

p_back_center double 1.55e-6 [ 1e-32, 1e6 ] m

p_back_bw double 1e-8 [ 0, 1e6 ] m

p_back_res double 1e-11 [ 1e-32, 1e6 ] m

Parameter Descriptions beam_divergence divergence angle of optical beam during free-space propagation

prop_distance free-space propagation distance

atten_add additional attenuation sigma_add standard deviation of the additional attenuation

area area of the receiver

p_back total power of the background radiation

p_back_center center wavelength of the background radiation p_back_bw bandwidth of the background radiation

p_back_res resolution of the background radiation spectrum during simulation


Differential Phase-Shift-Keying Receiver (DPSK_RX)

It models balanced Receiver with DPSK decoding. The inside topology is the following:

Here the DPSK signal is first decoded by delay interferometer. Time delay is set to the bit duration. Then outputs from Coupler for constructive and destructive ports are sent to balanced receiver built by two Compound Receiver models. The sign of electrical signal from destructive port generated by lower-arm Receiver is inverted and then signals from both Receivers are added up by Electrical Adder.

Properties




ElecFilterBitrateRatio

double 0.7 [ 0, 1e32 ]

Parameter Descriptions Bitrate The bit rate of the binary sequence generated ElecFilterBitrateRatio Ratio of filter bandwidth to the bit rate


Differential Phase-Shift-Keying Receiver – Gaussian, Advanced (DPSK_RX_gauss_adv)

This compound component model implements a detailed Differential Phase-Shift-Keying (DPSK) receiver which takes as input a DPSK or RZ-DPSK optical signal and produces an electrical output signal. A Gaussian optical filter is used in the receiver. Figure 1 depicts the model’s schematic.

Figure 1: DPSK_RX_gauss_adv compound component schematic.


• OpFilt_gauss

This component implements a Gaussian optical filter. It is configured via the parameters ofilter_order – the order of the filter; ofilter_wavelength_nm – the filter’s center wavelength; and ofilter_BW_GHz – the filter’s bandwidth.

• MZI_RX

This component implements a Mach-Zehnder Interferometer. It is configured via the parameters MZI_bitrate_Gbps – the bit-rate of the optical channel being detected; and MZI_signal_wavelength_nm – the wavelength of the optical channel being detected.

• PIN1 and PIN2

These components implement the receiver’s photodetectors. They are configured via the parameters PIN_qe – the detectors’ quantum efficiencies; and PIN_Id_nA – the detectors’ dark currents.

• SUB

This component calculates the difference between the two photodetector output signals, and produces the receiver’s final electrical output signal.

Properties




Parameter Values Name Type Default Range Units ofilter_order integer 1 [ 1, 128 ]

ofilter_wavelength_nm double 1550.0 [ 1e-32, 1e32 ] nm ofilter_BW_GHz double 50.0 [ 1e-32, 1e32 ] GHz

MZI_bitrate_Gbps double 10.0 [ 1e-32, 1e32 ] Gbps

MZI_signal_wavelength_nm double 1550.0 [ 0, 1e32 ] nm

PIN_qe double 0.7 [ 0, 1 ] PIN_Id_nA double 0.1 [ 0, 1e32 ] nA

Parameter Descriptions ofilter_order Order of the Gaussian optical filter

ofilter_wavelength_nm Center wavelength of the Gaussian optical filter

ofilter_BW_GHz Bandwidth of the Gaussian optical filter MZI_bitrate_Gbps Bit-rate of the optical channel to detect

MZI_signal_wavelength_nm Wavelength of the optical channel to detect

PIN_qe Quantum efficiency of the PIN detector

PIN_Id_nA Dark current of the PIN detector


Differential Phase-Shift-Keying Receiver – Raised Cosine, Advanced (DPSK_RX_rcos_adv)

This compound component model implements a detailed Differential Phase-Shift-Keying (DPSK) receiver which takes as input a DPSK or RZ-DPSK optical signal and produces an electrical output signal. A Raised-Cosine optical filter is used in the receiver. Figure 1 depicts the model’s schematic.

Figure 1: DPSK_RX_rcos_adv compound component schematic.


• OpFilt_rcos

This component implements a Raised-Cosine optical filter. It is configured via the parameters ofilter_rolloff – the rolloff parameter of the filter; ofilter_alpha – the alpha parameter of the filter; ofilter_wavelength_nm – the filter’s center wavelength; and ofilter_BW_GHz – the filter’s bandwidth.

• MZI_RX

This component implements a Mach-Zehnder Interferometer. It is configured via the parameters MZI_bitrate_Gbps – the bit-rate of the optical channel being detected; and MZI_signal_wavelength_nm – the wavelength of the optical channel being detected.

• PIN1 and PIN2


• SUB

This component calculates the difference between the two photodetector output signals, and produces the receiver’s final electrical output signal.

Properties




Parameter Values Name Type Default Range Units ofilter_rolloff double 0.5 [ 0, 1 ]

ofilter_alpha double 1.0 [ 0, 1e32 ] ofilter_wavelength_nm double 1550.0 [ 1e-32, 1e32 ] nm

ofilter_BW_GHz double 50.0 [ 1e-32, 1e32 ] GHz


MZI_signal_wavelength_nm double 1550.0 [ 0, 1e32 ] nm PIN_qe double 0.7 [ 0, 1 ]

PIN_Id_nA double 0.1 [ 0, 1e32 ] nA

Parameter Descriptions ofilter_rolloff Rolloff parameter of the Raised-Cosine optical filter ofilter_alpha Alpha parameter of the Raised-Cosine optical filter

ofilter_wavelength_nm Center wavelength of the Raised-Cosine optical filter

ofilter_BW_GHz Bandwidth of the Raised-Cosine optical filter

MZI_bitrate_Gbps Bit-rate of the optical channel to detect

MZI_signal_wavelength_nm Wavelength of the optical channel to detect PIN_qe Quantum efficiency of the PIN detector



Differential Quadrature Phase-Shift-Keying Receiver – Gaussian (DQPSK_RX_gauss)

This compound component model implements a detailed Differential Quadrature Phase-Shift-Keying (DQPSK) receiver which takes as input a DQPSK or RZ-DQPSK optical signal and produces decoded in-phase (P) and quadrature (Q) electrical output signals. A Gaussian optical filter is used in the receiver. Figure 1 depicts the model’s schematic.

Figure 1: DQPSK_RX_gauss compound component schematic.

The model is composed primarily of the following components:

• OpFilt_gauss

This component implements a Gaussian optical filter. It is configured via the parameters ofilter_order – the order of the filter; ofilter_wavelength_nm – the filter’s center wavelength; and ofilter_BW_GHz – the filter’s bandwidth. The output of the filter is split via the component OpSplit.

• MZI_RX_P and MZI_RX_Q

These components implement Mach-Zehnder Interferometers. They are configured via the parameters MZI_bitrate_Gbps – the bit-rate of the optical channel being detected; and MZI_signal_wavelength_nm – the wavelength of the optical channel being detected.

• PIN_P_1, PIN_P_2, PIN_Q_1, and PIN_Q_2


• SUB_P and SUB_Q


These components calculate the output signal differences for each photodetector pair, thereby producing the receiver’s final decoded in-phase and quadrature electrical output signals.

Properties




Parameter Values Name Type Default Range Units ofilter_order integer 1 [ 1, 128 ]

ofilter_wavelength_nm double 1550.0 [ 1e-32, 1e32 ] nm

ofilter_BW_GHz double 50.0 [ 1e-32, 1e32 ] GHz MZI_bitrate_Gbps double 10.0 [ 1e-32, 1e32 ] Gbps


PIN_qe double 0.7 [ 0, 1 ]

PIN_Id_nA double 0.1 [ 0, 1e32 ] nA

Parameter Descriptions ofilter_order Order of the Gaussian optical filter

ofilter_wavelength_nm Center wavelength of the Gaussian optical filter

ofilter_BW_GHz Bandwidth of the Gaussian optical filter

MZI_bitrate_Gbps Bit-rate of the optical channel to detect MZI_signal_wavelength_nm Wavelength of the optical channel to detect

PIN_qe Quantum efficiency of the PIN detector



Differential Quadrature Phase-Shift-Keying Receiver – Raised Cosine (DQPSK_RX_rcos)

This compound component model implements a detailed Differential Quadrature Phase-Shift-Keying (DQPSK) receiver which takes as input a DQPSK or RZ-DQPSK optical signal and produces decoded in-phase (P) and quadrature (Q) electrical output signals. A Raised-Cosine optical filter is used in the receiver. Figure 1 depicts the model’s schematic.

Figure 1: DQPSK_RX_rcos compound component schematic.

The model is composed primarily of the following components:

• OpFilt_rcos

This component implements a Raised-Cosine optical filter. It is configured via the parameters ofilter_rolloff – the rolloff parameter of the filter; ofilter_alpha – the alpha parameter of the filter; ofilter_wavelength_nm – the filter’s center wavelength; and ofilter_BW_GHz – the filter’s bandwidth. The output of the filter is split via the component OpSplit.

• MZI_RX_P and MZI_RX_Q

These components implement tunable Mach-Zehnder Interferometers. They are configured via the parameters MZI_bitrate_Gbps – the bit-rate of the optical channel being detected; and MZI_signal_wavelength_nm – the wavelength of the optical channel being detected.

• PIN_P_1, PIN_P_2, PIN_Q_1, and PIN_Q_2


• SUB_P and SUB_Q


These components calculate the output signal differences for each photodetector pair, thereby producing the receiver’s final decoded in-phase and quadrature electrical output signals.

Properties




Parameter Values Name Type Default Range Units ofilter_rolloff double 0.5 [ 0, 1 ]

ofilter_alpha double 1.0 [ 0, 1e32 ]

ofilter_wavelength_nm double 1550.0 [ 1e-32, 1e32 ] nm ofilter_BW_GHz double 50.0 [ 1e-32, 1e32 ] GHz



PIN_qe double 0.7 [ 0, 1 ] PIN_Id_nA double 0.1 [ 0, 1e32 ] nA

Parameter Descriptions ofilter_rolloff Rolloff parameter of the Raised-Cosine optical filter

ofilter_alpha Alpha parameter of the Raised-Cosine optical filter

ofilter_wavelength_nm Center wavelength of the Raised-Cosine optical filter ofilter_BW_GHz Bandwidth of the Raised-Cosine optical filter

MZI_bitrate_Gbps Bit-rate of the optical channel to detect

MZI_signal_wavelength_nm Wavelength of the optical channel to detect

PIN_qe Quantum efficiency of the PIN detector PIN_Id_nA Dark current of the PIN detector


WDM Receivers

The four superblocks for multichannel receivers included are the 8-channel and 16-channel receivers with MC and QA methods of noise treatment in receiver model. For example, 8-channel receiver has following inside topology:

User specifies bitrate, first channel wavelength, channel spacing, and the bandwidth of optical filter in demultiplexer. For filter shape at demultiplexer the 4th order Gaussian is used. For electrical filter at receiver the 4th order low pass Bessel filter is used with 3dB-bandwidth set to 0.64 x Bitrate.


Properties


Outputs #1-8 or #1-16: Electrical signal

Parameter Values (Receiver with MC noise treatment)

Name Type Default Range Units Bitrate double 1e+010 [ 0, 1e15 ] bps

OptFilterBW double 6e-010 [ 0, 1e32 ] meters



Parameter Descriptions (Receiver with MC noise treatment) Bitrate The bit rate of the binary sequence OptFilterBW Bandwidth of the optical filter ch_spacing_Hz Inter-channel spacing in Hz lambda_first Wavelength of the first channel

Parameter Values (Receiver with QA noise treatment) Name Type Default Range Units Bitrate double 1e+010 [ 0, 1e15 ] bps

OptFilterBW double 6e-010 [ 0, 1e32 ] meters



Parameter Descriptions (Receiver with QA noise treatment) Bitrate The bit rate of the binary sequence OptFilterBW Bandwidth of the optical filter ch_spacing_Hz Inter-channel spacing in Hz lambda_first Wavelength of the first channel

OptSim Models Reference: Block Mode Index •••• 665

Index

1 1x1 Expression Signal Operator 38

2 2x1 Expression Signal Operator 39

A Amplitude Modulator 116 Analog Sine Generator 27 Analytical Configuration of Multimode Fiber 557 Analyzers 309 Appendix: File formats 457 Asymmetric Mach-Zehnder Filter (AMZ) 364

B BeamPROP Interface 549 BER Calculation 344 BER Estimation Algorithm 342 BER Estimation Techniques 335 Bi-directional Nonlinear Fiber (Raman Amplifier) 149 Binary Differential Receiver 366 Bit Error Rate Tester 335 Bit Shift Function 66 Black Box Optical Amplifier 170 Boolean Operator 69

C Calibrating Receiver Sensitivity 296 Carrier-Suppressed Return-to-Zero (CSRZ) Transmitter

(CSRZ_TX) 623 Chirped Return-to-Zero (CRZ) Transmitter (CRZ_TX)

621 Coherent Solution Calculation 157 Compact Transient EDFA 425 Compatibility with LinkSIM version 2.1 VCSEL Model

83

Compound Optical Receiver 292 Confidence Limits for the BER and Q 341 Constellation Diagram Analyzer 386 Construction of Eye Diagram and Eyelids 343 Controlled Optical Coupler (2x2) 263 Controlled SOA 224 Corrections to the Modal Delay Due to Dispersion 562 Crosstalk Block 61 Custom ASE Noise Model 171 Custom Gain Model 170 Custom Signal Generator 24 CW Laser 91

D D Flip-Flop 70 Data Storage and Meta Blocks 407 Decision-Feedback Equalizer (DFE) 48 Defined ASE Noise Models 170 Defined Gain Model 170 Definitions of Filter Types 42 Delay Block 415 Detection Process 302, 588 Differential Mode Attenuation 561 Differential Mode Delay Analysis Tool 603 Differential Phase-Shift Keying (DPSK) 364 Differential Phase-Shift-Keying Receiver – Gaussian,

Advanced 655 Differential Phase-Shift-Keying Receiver – Raised

Cosine, Advanced 657 Differential Phase-Shift-Keying Receiver (DPSK_RX)

653 Differential Phase-Shift-Keying Transmitter –

Advanced 631 Differential Phase-Shift-Keying Transmitter

(DPSK_TX) 628 Differential Polarization-Shift Keying (DPolSK) 364 Differential Quadrature Phase-Shift Keying (DQPSK)

364 Differential Quadrature Phase-Shift-Keying Receiver –

Gaussian 659 Differential Quadrature Phase-Shift-Keying Receiver –

Raised Cosine 661 Differential Quadrature Phase-Shift-Keying Transmitter

636 Direct Addition of Noise 238 Direct Modulated Laser 76 DQPSK Precoder 72 DQPSK Receiver 367 Driving Source 76, 99, 110 Dual-Arm Mach-Zehnder Modulator 613 Duobinary 364 Duobinary Transmitter (Doubi_TX) 627 Dynamic Optical Switch (2x2) 464

666 •••• Index OptSim Models Reference: Block Mode

E EDFA 173 Electrical Amplifier 55 Electrical Eye Diagram 343 Electrical Filter 42 Electrical Gain 54 Electrical Integrate And Dump 62 Electrical Modules 37 Electrical Monitor 318 Electrical Noise Adder 58 Electrical Power 318 Electrical Signal Generator 18 Electrical Signal Resampler 73 Electroabsorption Modulator 119 Electronic Dispersion Compensation (EDC) 48 Encircled Flux Analysis Tool 598 Expression Signal Generator 34 EYCDFA 194 Eye Diagram 343, 387 Eye Diagram Analyzer 387 Eye Mask 349 Eyelids 343

F Fabry Perot CW Laser 95 Feed-Forward Equalization 47 Fiber Bragg Grating Filter 268 Fiber Delay 167 Fiber Link 1 645 Fiber Link 2 647 Fiber Link 3 649 Fork 416 Forward Error Correction 345 Free-Space Optical Channel 651 Frequency Comb Generator 27 Frequency Sweep Generator 32

G Gain/NF Analyzer 325 General Multiport Optical Device (NxM and WDM)

283

H Hierarchical Input Signal Port Block 418 Hierarchical Output Signal Port Block 419

I Ideal Frequency Converter 255 Intensity-Modulation Direct-Detection (IMDD) 364 Interferometric Fiber Optic Gyroscope (I-FOG) 257 Interior Property Map 139, 311

J Jones Matrix Transfer Function 286

K Karhunen-Loeve BER Estimation 364 Karhunen-Loeve Technique (KLT) 368

L Laser Cavity 77, 479 LED Electrical Model 110, 531 LED Optical Response 109, 530 Library Configuration of Multimode Fiber 550 Library Generation for Multimode Fiber Model 573 Liekki LAD Interface 245 Light Emitting Diode (LED) 109 Linear Configuration of Multimode Fiber 563 Linewidth 91, 110, 505, 531 Linewidth Adder 243 Logical Models 65

M Mach-Zehnder 115 Mach-Zehnder Delay Interferometer 615 Minimum Mean Square Error (MMSE) 48 Mixer 41 Mode Types 472 Mode-Attachment Styles 471 Mode-Locked Laser 87 Modulator 115 Modulator-Driving S Block 52 Monte Carlo DPSK BER Estimation Technique 376 Monte Carlo DPSK BER Estimator 376 Monte-Carlo BER Estimation 339 Multi-Line Output 22, 81, 88, 91, 104, 111 Multimode Fiber 550 Multimode File Format 473 Multimode Modules 469 Multiplot 397

N Noise 138 Noise Filtering 44 Noise Representation and Effects 293, 580 Nonlinear Fiber 124 Non-Return-to-Zero (NRZ) Transmitter (NRZ_TX) 618 Null Signal Block 421 Numerical Configuration of Multimode Fiber 559

O Optical Add Multiplexer 276 Optical Add/Drop Multiplexer 280

OptSim Models Reference: Block Mode Index •••• 667

Optical Amplifiers 169 Optical Attenuator 248 Optical Autocorrelator Analyzer 395 Optical Binary Differential Receiver 366 Optical Components 247 Optical Coupler (2x2) 261 Optical DeMultiplexer (1xN DEMUX) 274 Optical Drop Multiplexer 278 Optical Eye Analyzer 332 Optical Fibers 123 Optical Filter 251, 264 Optical Frequency/Wavelength Chirp Analyzer 392 Optical Monitor 320 Optical Multiplexer (Nx1 MUX) 272 Optical Noise Adder 238 Optical Parameters 22 Optical Phase Shift 250 Optical Power Normalizer 249 Optical Receivers 291 Optical Signal Generator 21 Optical Signal Resampler 74 Optical Sources and Modulators 75 Optical Splitter (1xN) 260

P Parasitics 77, 100, 479, 515 Performance Budget 347 Phase Modulator 116 Photodetector 302 Physical EDFA 173 Physical EYCDFA 194 Poincaré Sphere Plot 329 Polarization 22, 80, 88, 91, 104, 111, 483, 496, 505,

519, 532 Polarization Transformer 252 PRBS Pattern Generator 16 PreBits and PostBits 16 Property Map 315

Q Quasi-Analytical BER Estimation 336

R Relative Intensity Noise (RIN) 100 Repeat Loop 413 Return-to-Zero (RZ) Transmitter (RZ_TX) 620 Return-to-Zero Differential Phase-Shift-Keying

Transmitter – Advanced 633 Return-to-Zero Differential Quadrature Phase-Shift-

Keying Transmitter 639 Return-to-Zero DPSK Transmitter (RZ-DPSK_TX) 629

S Sagnac Effect Model 257 Save and Load Signal To/From File 408 Sawtooth Generator 31 Semi-Analytical Technique 367 Semiconductor Optical Amplifier (SOA) 218, 226 Shift Function 66 Shift Signal 66 Signal Analyzer 384 Signal Band Converter 607 Signal Generators 15 Signal Spectrum Analyzer 389 Sine Wave Generator 27 Skew 139 Soliton Transmitter (Soliton_TX) 625 Spatial Adder 471 Spatial Analyzer 594 Spatial Aperture 577 Spatial BeamPROP Interface 549 Spatial Compound Optical Receiver 579 Spatial Coupler 545 Spatial CW Laser 505 Spatial Direct Modulated Laser 478 Spatial Effects 483, 496, 506, 519, 532, 588 Spatial Gridded Field Converter 609 Spatial Light Emitting Diode (LED) 530 Spatial Mode-Locked Laser 495 Spatial Photodetector 588 Spatial VCSEL 514 Special Functions 65 Standard S-Parameter Block 50 Static Optical Switch (2x2) 460 Step-Index Multimode Fiber 561 Stochastic Representation of Noise 239 Summer 40

T T Flip-Flop 71 Test Curves for Multimode Fiber Model 564 Thin Lens 541 Time Delay Function 66 Time Shift Function 66 Transfer Function Analysis Tool 403 Transient EDFA 442 Transient Modules 423 Transient Optical Switch (2x2) 462 Transient Plotter 466 Transient Pulse Generator 424 Tunable Mach-Zehnder Interferometer 616 Typed Fork 417 Typed Repeat Loop 413

668 •••• Index OptSim Models Reference: Block Mode

U User-Specified Profiles 153

V VCSEL 99 VCSEL Cavity 100, 515 Vortex Lens 543

W WDM Components 259 WDM Receivers 663 WDM Transmitters 642 Word Error Rates 345 Write Once Read Many (WORM) Block 420

X XY-Plotter 381

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