Research Collection

Doctoral Thesis

Advanced indium-phosphide waveguide Mach-Zehnderinterferometer all-optical switches and wavelength converters

Author(s): Leuthold, Juerg

Publication Date: 1998

Permanent Link: https://doi.org/10.3929/ethz-a-002052325

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ETH Library

Diss. ETH No. 12991

Advanced Indium-Phosphide WaveguideMach-Zehnder Interferometer

All-Optical Switches and

Wavelength Converters

A dissertation submitted to the

SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICHfor the degree of

Doctor of Natural Sciences

by

JUERG LEUTHOLDdipl. Physicist ETH

Born on July 11th, 1966Citizen of Nesslau SG, Switzerland

Submitted on the recommendation ofProf. Dr. H. Melchior, examinerProf. Dr. H. Jäckel, co-examiner

Prof. Dr. F.K. Kneubühl, co-examiner

1998

Jürg Leuthold

Advanced Indium-Phosphide WaveguideMach-Zehnder Interferometer

All-Optical Switches and

Wavelength Converters

Hartung-Gorre Verlag Konstanz, Germany1998

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“It is the honour of God to conceal things,but the honour of kings to explore things.”

(King Solomon, ~990 - 930 B.C.)

To my parentsand

to my wonderful wife Barbara

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Table of Contents

Table of Contents iiiAbstract 1

1 Introduction 71.1 All-Optical Components in Future Terabit-Per-Second Telecommunication Systems 8

1. Progress towards Terabit-Per-Second Transmissions 82. WDM/OTDM Enabling Technologies 103. Conclusions 12

1.2 From Early to State-of-the-Art All-Optical Devices 131. Different Nonlinear Materials 132. Techniques to Exploit the Nonlinearities 153. Configurations 18

1.3 Outline 241.4 References 26

2 Theory of MZI Based All-Optical Devices 352.1 Optical Nonlinearities 36

1. The Wave Equation for Nonlinear Optical Media 362. Solutions of the Nonlinear Wave Equation 373. Kramers-Krönig Relations in Nonlinear Optics 394. Nonlinear Effects 405. Time Regimes of Nonlinear Effects 566. Qualitatively Calculated Refractive-Index Changes under Control-Signal Injec-

tions597. Concluding Remarks on XPM All-Optical Devices 60

2.2 The Semiconductor Optical Amplifier Equations 611. Introductory Definitions 612. The Propagation of an Amplified Signal in an SOA 653. A More Complete Solution 68

2.3 The MZI-SOA Transfer Functions 712.4 References 75

3 Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers 813.1 Material Gain 82

1. Introduction 822. Structures and Devices 833. Material-Gain Characterisation 844. Comparison of Experiment with Theory 915. Differential Material Gain dgm/dN and dgm/dT 92

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3.2 Refractive-Index Changes 951. Measurement of the Refractive-Index Changes 952. Effective Group Refractive Index 102

3.3 Alpha Factors 1031. Introduction 1032. Experiments on αN 1043. Experiments on αT 106

3.4 Parametrizations of the Material Parameters 1081. Material-Gain Parametrization 1082. Internal-Loss Parametrization 1123. Refractive-Index-Change Parametrization 1134. Alpha Factor Parametrization 114

3.5 Appendix 1151. Temperature Correction 1152. Current <-> Carrier-Density Relations 1173. Output-Power 119

3.6 References 121

4 Modified Multimode-Interference Couplers 125

4.1 Summary of Multimode-Interference Theory 1261. Self Images and the Term Multimode Interference 1262. Classification of MMIs 127

4.2 Design Guidelines for MMIs 1321. MMI Widths and MMI Lengths 1322. Trap Waveguides 135

4.3 Guided Wave 1.30/1.55 μm Wavelength Division Multiplexers based on Multimode Interference 136

1. Introduction 1362. MMI-WDMs 1373. Experiments 1434. Future Applications as MMI-WDM Add-Drop Switches 145

4.4 Multimode Interference Couplers for the Conversion and Combining of Zero and First Order Modes 147

1. Introduction 1472. Principles of MMI-Converter-Combiners 1483. Power-Splitter for Zero and First Order Modes 1574. Experiment 1615. Conclusions 1666. Appendix: Generalisation of the Concept to Converter-Combiner MMIs with K

outputs1674.5 Spatial Mode Filters realized with Multi-Mode Interference Couplers 170

1. Introduction 170

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2. Principle of MMI-Filters 1713. Experiment 1734. Conclusions 176

4.6 Optical Bandwidths and Design Tolerances of Multimode-Interference Converter-Combiners 177

1. Introduction 1772. Theory 1783. Comparison with Mode Analysis and Experiment 1814. Conclusions 1855. Appendix 186

4.7 References 188

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios 193

5.1 Extinction-Ratio Limitations in All-Optical Devices 1945.2 All-Optical Space Switches with Gain and Principally Ideal Extinction-Ratios 196

1. Introduction 1962. Basic MZI-SOA All-Optical Switch 1973. Analysis 1994. Specific MZI-SOA Implementations 2045. Experiments 2146. Conclusions 218

5.3 References 222

6 All-Optical Devices with Integrated Data- and Control-Signal Separation Schemes 225

6.1 All-Optical Mach-Zehnder Interferometer Wavelength Converters and Switches with Integrated Data- and Control-Signal Separation Scheme 226

1. Introduction 2262. Co- and Counter-Propagating Operation with

Versatile All-Optical Devices2283. All-Optical Control- And Data-Signal Separation Schemes 2304. Static and Dynamic Characterizations 2365. Conclusions 248

6.2 Polarization Independent Optical Phase Conjugation with Pump-Signal Fil-tering in a Monolithically Integrated Mach-Zehnder Interferometer Semiconductor Optical Amplifier Configuration 249

1. Introduction 2492. Configuration 2503. Experiments 251

6.3 References 256

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7 Conclusions and Outlook 2617.1 Conclusions 2627.2 Outlook 264

Appendix 265App. A Material Gain 266App. B Nonlinear Gain Compression 273App. C InGaAsP Material Parameters 276App. D Material Parameters Used in the Calculations 280App. E Useful Formulas 281App. F List of Symbols and Acronyms 283

Acknowledgements 289List of Publications 291Curriculum Vitae 293Index 295

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Abstract

To meet the future telecommunication-bandwidth requirements, optical-fibre com-munication systems will be an absolute need. In order to exploit the terahertz band-width of optical fibres, generally both of the two current primary techniques for data multiplexing i.e. wavelength-division multiplexing (WDM) and time-division mul-tiplexing (TDM) - are used. While components necessary for implementing a WDM point-to-point link are already on the market, components for TDM links are still in research state. However, the situation tends to change with the occurrence of all-op-tical components that have the potential for 100 GHz operation. All-optical com-ponents enable highest switching speeds as needed in TDM systems and in addition, they allow all-optical regeneration of pulse streams and the implementation of trans-parent networks, in which the signals traverse a network consisting of optical fibres and optical nodes without being converted into electrical current, except at the final destination.This thesis introduces novel concepts for all-optical devices based on a Mach-Zeh-nder interferometer (MZI) configuration with semiconductor optical amplifiers (SOAs) on the MZI arms for high-speed TDM telecommunication. New concepts are applied to provide high extinction ratios and integrated data- and control-signal sep-aration schemes. Beyond, our devices feature high crosstalk, large optical band-widths and principally polarization insensitivity. The devices are multi-functional, since they may be used for switching, multiplexing, demultiplexing and wavelength conversion. For all-optical processing we focused our attention on the improvement and devel-opment of devices based on the cross-phase modulation (XPM) technique. In XPM devices small refractive-index changes induced in a nonlinear medium by a control signal are exploited in a Mach-Zehnder-interferometer configuration to switch a data signal from one output port to the other. Alternative techniques to perform all-optical switching are based on four-wave mixing (FWM) and cross-gain modulation (XGM). Devices based on the XPM technique offer advantages such as high extinc-tion ratio, switching with moderately high control-signal powers and low chirp in comparison with the alternative techniques.The XPM all-optical devices proposed in this thesis use semiconductor optical am-plifiers (SOAs) as a nonlinear medium. We apply semiconductor material equations to quantify the different contributions to the nonlinear effects of our devices. As oth-er authors, we find that the dominant nonlinear mechanisms are the bandfilling effect due to gain-saturation, the band-gap shrinkage and the plasma effect. Ultrafast non-linear effects such as gain compression from spectral hole burning and carrier heat-

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ing or the optical Kerr effect due to the a.c. quadratic Stark effect and two-photon absorption will only contribute to the nonlinearities when the control pulses length is reduced to the picosecond range and the control-signal powers are increased.Relevant material parameters needed for an optimization of the devices are the ma-terial gain, the refractive-index change and the alpha factor. We have measured these parameters throughout the gain spectrum and for different carrier densities at various temperatures. Beyond, other important material parameters such as the internal loss-es, the carrier densities as a function of the applied current and the internal quantum efficiency have been determined.A key issue of any switching device is the extinction or on-off ratio. We show that asymmetries are needed to overcome the extinction-ratio limitations in MZI-SOA all-optical switches. To obtain high extinctions within an interferometric configura-tion, such as the MZI, both the intensities as well as the phases of the signals have to be properly controlled. However, this is one of the challenges for the all-optical de-vices, since the strong optical control signal that operates the device, not only induc-es phase shifts but also modulates the intensities of the data signal to be switched. These extinction-ratio limitations can be overcome by the introduction of asym-metries. The extent of the required asymmetry depends on the SOA material param-eters as the alpha factor.For the separation of the data and control signal after signal processing high-per-formance wavelength multiplexer or filters are needed. We show three new concepts that allow to perform all-optical operation, while at the same time the data and con-trol signal are separated at the output. The first concept is based on a dual-order-mode configuration, in which the data sig-nal propagates as a zero-order mode and the control signal propagates as a first-order mode. Due to the different symmetries of data and control signal, the first-order mode control signal can be easily separated after signal processing and reused. For the introduction and the conversion of the zero-order mode into a first-order mode control signal, we have invented a new coupler type based on multimode-interfer-ence (MMI). Excellent performances were found for the new MMI couplers both, for the conversion efficiencies as well as the optical bandwidths. A second concept that allows to separate and to reuse the control signal after signal processing is based on interleaved MZI configurations. Additional MZIs are placed on the arms of an exterior MZI to direct the control signal and the data signal by in-terference onto separate outputs. A third concept needs 1.55 mm data signals and 1.3 mm control signals to modulate the all-optical device. In contrast to the previous concept, this concept has been in-troduced by another group. Here we propose compact and simple couplers based on MMI to perform the wavelength multiplexing at the inputs and the outputs. We have realized the first two concepts and have compared them. Depending on the

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application, each of the concepts has its advantages for specific applications. A re-cent implementation of our dual-order mode concepts by a global telecommunica-tion company shows the high interest for new solutions in the field of optical high-speed communication.

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ZusammenfassungUm die steigenden Kapazitätsanforderungen in der Telekommunikation bewältigen zu können, werden immer mehr Glasfaserkommunikationssysteme eingesetzt, denn die Signalbandbreite von optischen Glasfasern reicht in den Terahertzbereich. Damit man diese Bandbreite auch ausschöpfen kann, wird sowohl die Technik des Wellen-längenmultiplexens (WDM) als auch die Technik des Zeitmultiplexens (TDM) an-gewendet. Währenddem Komponenten, welche für das WDM benötigt werden, bereits kommerziell erhältlich sind, befinden sich die Komponenten, welche für das TDM gebraucht werden, noch immer im Forschungsstadium. Die Situation könnte sich jedoch mit dem Aufkommen von optisch-optischen Bauteilen verändern. Diese können mit Schaltgeschwindigkeiten von weit über 100 GHz betrieben werden. Ausserdem erlauben sie die optisch-optische Signalaufbereitung und die Installation von transparenten Netzwerken. In transparenten Netzwerken werden die Datensi-gnale ausschliesslich via Glasfasern und rein optische Schalterzentralen übertragen, ohne dass sie unterwegs in elektrische Signale umgewandelt werden.In der vorliegenden Dissertation werden neue Konzepte von Schaltern, welche auf optisch-optischen Mach-Zehnder-Interferometern (MZI) basieren und über optische Halbleiterlaserverstärker (SOA) verfügen, eingeführt und getestet. Neue Konzepte werden benötigt, um bessere Schaltverhältnisse zu erzielen und um die zum Betrieb verwendeten Kontrollsignale bereits innerhalb des Schalters von den Datensignalen zu trennen. Darüber hinaus verfügen die neuen Schalter über eine grosse optische Bandbreite und erlauben einen polarisationsunabhängigen Betrieb. Ausserdem sind die Bauteile auf vielfältige Weise einsetzbar, da sie sowohl zum Schalten, Multiple-xen, Demultiplexen wie auch für die Wellenlängenkonversion geeignet sind.Zum Betrieb der optisch-optischen Schalter machen wir Gebrauch von der soge-nannten Kreuzphasenmodulations(XPM)-Schalttechnik und konzentrieren unsere Aufmerksamkeit vor allem auf die Entwicklung und Verbesserung dieser Technik. XPM-betriebene Schalter bestehen zur Hauptsache aus einer MZI-Konfiguration mit nichtlinearen Materialien auf den MZI-Armen. Wenn nun ein Kontrollsignal in einem nichtlinearen Bereich auf einem MZI-Arm eine Brechungsindexänderung er-zeugt, können dank der MZI-Konfiguration kleinste Brechungsindexänderungen zum Schalten ausgenutzt werden. Alternative Techniken zum Betrieb von optisch-optischen Schaltern basieren auf der Technik des Vierwellenmischens (FWM) oder auf der Technik der Kreuzverstärkungsmodulation (XGM). Allerdings bieten Bau-teile, die auf der XPM-Technik basieren, gegenüber diesen alternativen Techniken einige wesentliche Vorteile. Zum Beispiel erreicht man mit XPM-betriebenen Bau-teilen bessere Schaltverhältnisse. Ausserdem erlauben sie das Schalten mit kleineren Kontrollsignalstärken und kleinerer Frequenzverschiebung.

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Die auf XPM basierenden optisch-optischen Schalter, wie sie in dieser Dissertation eingesetzt werden, verwenden optische Halbleiterverstärker als nichtlineares Mate-rial. Wir gebrauchen die optischen Halbleiterverstärkergleichungen, um in unseren Bauteilen die verschiedenen nichtlinearen Anteile am Schaltvorgang zu quantifizie-ren. Wie bereits andere Autoren vor uns haben wir gefunden, dass die dominanten nichtlinearen Effekte von der Ladungsträgersättigung, der Verschiebung der Ener-giebandkante des Halbleiters und vom Plasmaeffekt herrühren. Ultraschnelle nicht-lineare Effekte wie “Spektrales-Loch-Brennen” (SHB), Ladungsträgererwärmung (CH) oder der optische Kerr-Effekt aufgrund des quadratischen Stark-Effektes und der Zwei-Photonen-Absorption werden erst für stärkere Kontrollpulse unter ~1ps wirksam.Wichtige SOA-Materialparameter, welche zur Optimierung der Bauteile benötigt werden, sind die Materialverstärkung, die Brechungsindexänderungen und der soge-nannte Alphafaktor. Wir haben diese Parameter in unseren SOA gemessen. Die Mes-sungen wurden über den ganzen verstärkungswirksamen spektralen Bereich, bei verschiedenen Stromstärken und bei verschiedenen Temperaturen durchgeführt. Darüber hinaus wurden auch andere wichtige Parameter bestimmt wie zum Beispiel die internen Verluste, die Abhängigkeit der Ladungsträgerdichte vom angelegten Strom sowie die internen Quanteneffekte.Die erzielbaren An-/Abschaltverhältnisse stellen generell ein wichtiges Kriterium zur Beurteilung der Schaltqualität dar. Wir zeigen, dass man die An-/Abschaltver-hältnisse der auf MZI-SOA basierenden optisch-optischen Schalter durch das Ein-führen von Asymmetrien erhöhen kann. Um mit einer MZI-SOA-Konfiguration gute Schaltverhältnisse zu erzielen, muss man sowohl die Phasenverhältnisse als auch die Intensitäten der Lichtsignale auf den beiden MZI-Armen gegeneinander abstimmen. Aber genau das ist die Schwierigkeit in optisch-optischen Schaltern, denn das starke Kontrollsignal, das zum Schalten benötigt wird, moduliert nicht nur die Phase, son-dern auch die Intensität des zu schaltenden Datensignals. Durch das Einführen von Asymmetrien können derartige Begrenzungen der Schalteigenschaften überwunden werden. Die benötigte Asymmetrie hängt von Materialeigenschaften wie dem Alp-hafaktor ab. Für das Herausfiltern des Kontrollsignals aus dem Datensignalpfad werden effizien-te Wellenlängenfilter oder Wellenlängenmultiplexer benötigt. Wir zeigen drei neue Konzepte, die es erlauben, optisch-optische Schalter zu betreiben und gleichzeitig die Daten- und Kontrollsignale am Ausgang des Bauteiles zu trennen. Das erste Konzept basiert auf einer “Zwei-Ordnungsmoden-Konfiguration”. In die-ser Konfiguration werden die Kontrollsignale in Moden höher Ordnung umgewan-delt, während man für die Datensignale nach wie vor Moden nullter Ordnung verwendet. Da die Symmetrie der beiden Moden total verschieden ist, kann diese Ei-genschaft ausgenützt werden, um die Moden nach der Signalverarbeitung voneinan-

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der zu trennen. Um die Kontrollsignale in höhere erste Ordungsmoden umzuwandeln und sie dann in den Signalpfad der Datensignale einzukoppeln, haben wir neuartige Koppler erfunden. Diese basieren auf dem Multimode-Interfe-renz(MMI)-Prinzip. Für diese Koppler haben wir hohe Umwandlungs- und Kopp-lungseffizienzen, aber auch grosse optische Bandbreiten gefunden. Ein weiteres Konzept, um Kontroll- und Datensignale zu trennen, basiert auf meh-reren ineinander verschachtelten MZI-Konfigurationen. Dabei wird je eine zusätzli-che MZI-Konfiguration auf den MZI-Armen einer äusseren MZI-Konfiguration plaziert. Diese werden gebraucht, um das Kontroll- und das Datensignal interfero-metrisch auf verschiedene Ausgänge zu führen. Ein drittes Konzept verwendet 1.55 mm Datensignale und 1.3 mm Kontrollsignale zur Modulation des Schalters. Im Gegensatz zu den vorangegangen Konzepten wur-de dieses Prinzip bereits von einer andern Gruppe demonstriert. Wir zeigen im Rah-men dieser Dissertation, dass kompakte, einfache MMI-Koppler existieren, die das Einkoppeln und Multiplexen am Ein- und am Ausgang des Bauteils ermöglichen. Die ersten beiden optisch-optischen Schalterkonzepte wurden in InGaAsP/InP rea-lisiert, getestet und miteinander verglichen. Jedes der beiden Konzepte hat seine Vor- und seine Nachteile für spezifische Anwendungen. Eine erst kürzlich erfolgte erfolg-reiche Realisierung eines unserer Konzepte durch einen globalen Telekommunikati-onsanbieter zeigt das grosse Interesse an neuen Lösungen im Gebiet der optischen Hochgeschwindigkeitstelekommunikation.

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1 Introduction

A short overview on the latest developments enabling terabit-per-second telecom-munication is given. One of the enabling technologies for future terabit-per-second networks will be all-optical devices. We show some of the potential applications for this new class of devices. Special emphasis is put on the all-optical devices. We briefly review different non-linear materials, that allow us to build all-optical components and compare their per-formance. The largest nonlinearities are found in semiconductor optical amplifiers. Various techniques have been suggested to exploit the nonlinearities for all-optical operation. A most promising technique is cross-phase modulation. Important con-cepts based on these techniques are studied in more detail at the end of this chapter.

1 Introduction

8

1.1 All-Optical Components in Future Terabit-Per-Second Telecommunication Systems

High-capacity telecommunication networks will play an important role in the forth-coming information society of the 21st century. The information communications as well as health care, social welfare, energy systems, traffic etc. will depend on relia-ble, high-speed transmission networks. The predicted requirements for the first dec-ade of the next century are transmission rates of 5 TBit/s for the backbone networks and 100 MBit/s for homes as stated in a recent visionary report by Hirahara et al. [1.1]. And indeed, a constant capacity increase of 35-60% on a yearly base is pre-dicted [1.2]. Capacity requirements are boosted by the increasing number of users, a move towards an information-oriented lifestyle and the increased bandwidth de-mand of new services. Especially the real-time communication services pose high requirements on a network, since they need the same high-transmission capacity at all times. If the transmission of a digitally transmitted voice signal requires

, new compressed real-time media demand 128 kBit/s for hi-fi music, 1.5 MBit/s for cumulative video images, 6 MBit/s for commercial TV images and even 30 MBit/s for high-definition TV images [1.1].

1. Progress towards Terabit-Per-Second TransmissionsNecessary technological innovations towards terabit-per-second transmissions in-clude the move from copper to silicon fibres and the transition from space to wave-length multiplexing towards time-division multiplexing (Fig. 1.1). While the former revolution initiated the step from the electrical to the optical domain, the latter might initiate progress from optoelectronic towards all-optical transmission equipment.The replacement of copper by optical fibres closer and closer to the residential cus-tomers creates the conditions required for the coming terabit-per-second age. Only optical fibres offer an unprecedented capacity bandwidth of as much as 25 THz in combination with minimum attenuation losses as low as 0.2 dB/km for wavelengths around 1.55 μm. Alternative transmission means such as the copper coaxial cable and wireless distribution have much smaller transmission capacities. Copper coax cables have losses of ~2.5 dB/km at modulation frequencies of 1 MHz and more than 50 dB/km at 1 GHz [1.3]. Even with the introduction of new copper cables that provide reliable 1 GBit/s transmission as recently announced by Lucent Technolo-gies [1.4], the transmission capacity remains a factor of 25’000 smaller in compari-son with optical fibres. Wireless video-broadcast services suitable for long-distance distributions operate at ‘moderate‘ 2.5 GHz. Faster modulated millimetre-wave sys-tems operating between 28 GHz and 40 GHz have a restricted cell size of 1-5 km and low earth orbit satellites at an altitude of 1500 km offer throughputs of .

64kBit/s

4 5 GBit/s–

1.1 All-Optical Components in Future Terabit-Per-Second Telecommunication Systems

9

The second necessary innovation involves the transition from optical space-division multiplexing (OSDM) to wavelength-division multiplexing (WDM) towards optical time-division multiplexing (OTDM). At the beginning of the optical revolution, OSDM was exploited in order to make use of the space domain by using different fibres each carrying a single wavelength. Meanwhile, WDM techniques have entered the market to exploit the frequency domain by multiplexing different wavelengths over a single fibre. Although the technique is relatively new, all major US long-dis-tance carriers are utilising WDM equipment in their networks since 1997 [1.2]. It is foreseeable, that OTDM will be exploited next. OTDM systems in combination with WDM systems will allow to exploit the available optical bandwidths of silicon fibres and thus satisfy the demand for increased capacities. In addition, OTDM systems will allow a user to burst very high-speed (>100 GBit/s) data packets onto the net-work whenever the network traffic allows it. This is in contrast to a WDM system where each user can access only a small portion of the total bandwidth of the network at any given time, since in a WDM system the fibre bandwidth is divided into a large

ElectronicTransmission

OpticalTransmission OSDM

WDM

OTDM

Multiwavelength SourcesMultiwavelength AmplifiersWavelength Mulitplexers (Phased Array)Optical Filters

Innovations Necessary Technological Developments

Optical Transmitter

Optical ReceiverOptical Amplifier

Short Optical Pulse GenerationDispersion CompensationAll-Optical Multiplexing/DemultiplexingHigh Speed Clock Extraction

Fig. 1.1 Progress towards terabit-per-second networks goes along with innova-tions, such as a move from electrical to optical transmission and a transi-tion from space (OSDM) to wavelength (WDM) to optical time-division multiplexing (OTDM). Each innovation requires some necessary techno-logical developments. After the foreseeable transition from WDM to WDM/OTDM transmission all-optical multiplexing will become a key technology enabling high-capacity transmission.

1 Introduction

10

number of lower-rate channels [1.5].The feasibility of terabit-per-second transmission was for the first time demonstrated in spring 1996 when simultaneously three terabit-per-second transmission experi-ments were reported, see Ref. [1.6-1.8] or [1.9] for a review. WDM/OTDM tech-niques were applied. In the autumn 1996 a group of NEC further expanded the demonstrated transmission capacity to 2.6 TBit/s over 120 km single-mode fibre us-ing 132 channels, each modulated at 20 GBit/s [1.10].All these experiments succeeded in high-speed point-to-point WDM/OTDM trans-mission. In practice, however, networks will be required. For this purpose high-speed switches, multiplexers, demultiplexers and wavelength converters will be needed. All-optical components have the potential to work beyond the limits en-countered by electronics and will most probably be a prerequisite in future high-speed WDM/OTDM systems.

2. WDM/OTDM Enabling TechnologiesGenerally, innovations in high-tech markets are based on one or more technological breakthroughs. In Fig. 1.1 we have listed representative necessary developments en-abling new multiplexing technologies.Enabling technologies for WDM systems were the development of the erbium-doped fibre amplifier (EDFA) and the invention of the phased array providing a practicable wavelength multiplexer.The exploitation of the low-loss regions of the silica fibres relies on the availability of high-performance amplifiers in the respective wavelength regions. Nowadays mainly the lowest-loss 1.55 μm range of the silica fibre is exploited. This has been made possible by the invention of the erbium-doped fibre amplifier (EDFA) in 1987, which made the 1.53 to 1.56 μm region accessible [1.11]. EDFAs routinely exceed amplification of 40 dB with a noise figures of 4-5 dB. The 1.53 to 1.56 μm window corresponds to a bandwidth of 3-4 THz, which was nearly completely exploited by the previous mentioned 2.6 THz experiment [1.10]. Lately, new gain-shifted EDFAs make the 1.57 to 1.60 μm region accessible [1.12]. These amplifiers provide a net gain of 38 dB with noise figures below 4.5 dB [1.13]. More recently, the well-known SOAs seem to be rediscovered especially for exploiting the 1.31 μm transmission window [1.14]. Optical fibre-to-fibre gains of typically 30 dB and intrinsic noise fig-ures of 6-8 dB are achieved [1.15-1.17]. Overall noise figures at the gain peak are around 11 dB [1.15,1.16]. In principle, SOAs can be built for use throughout the 1.00 to 1.60 μm range. Raman amplifiers also address the 1.30 μm window. They allow gain levels of 40 dB with noise figures of 4.3 dB [1.18, 1.19]. Other amplifiers work-ing around 1.30 μm encompass fibre amplifiers with active rare-earth ions like the Praseodymium-Doped Fluoride Fibre Amplifier (PDFFA) [1.20].

1.1 All-Optical Components in Future Terabit-Per-Second Telecommunication Systems

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Key technologies needed for the current transition from WDM to WDM/OTDM sys-tems are short-pulse generation, dispersion-slope compensation, all-optical multi-plexing and clock extraction from high-speed OTDMs [1.3,1.21]. Other developments such as wavelength converters are desirable but not necessary.Several methods are studied to generate short pulses that are synchronized with a master clock. Examples are the mode-locked laser diode [1.22], external modulation of continuous wave (cw) light by an electro-absorption modulator [1.23] and har-monic mode-locking of Er-doped fibre ring lasers [1.24].Dispersion-slope compensation techniques become essential at highest bitrates, since dispersion tolerance together with fibre-nonlinearity effects decreases as

increases [1.25]. Table 1.2 gives an idea of the tolerances at different bit rates.

In high-speed OTDM networks, all-optical devices are prerequisite for multiplexing, demultiplexing and various signal processing functions [1.26]. In the operation range between 10 and future 100 GBit/s all-optical devices have already proved that they

Table 1.1: High-Performance Amplifiers

Type Wavelength Range [μm]

Fiber-Fiber Gain [dB]

Intr. Noise Figure [dB]

Saturation Point [dBm]

EDFA 1.53 - 1.56 40 4 - 5

Gain-Shifted EDFA 1.57 - 1.60 38 4.5

SOA 1.551.30

and others

29 30

6 - 86 - 8

7.410

Raman Amplifier 1.30 - 1.32and others

40 4.3

PDFFA 1.28 - 1.32 30

Table 1.2: Simulated Dispersion Tolerance (5% Eye Merging Closure), cf. [1.25]

Bit Rate Dispersion Tolerance

10 GBit/s < 450 ps

20 GBit/s < 200 ps

40 GBit/s < 60 ps

BitRate( )∼ 1,4

1 Introduction

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can overcome the bandwidth bottlenecks of electronic circuitry. They exploit ultra-fast nonlinear effects to control light by light. An overview of the different tech-niques will be given in the next section. The main advantages of all-optical processing in comparison with electronic processing are that it• enables ultrafast operation: Currently all-optical devices have performed switch-

ing windows as short as 200 fs at 10.5 GHz [1.27] and 100:10 GBit/s multiplex-ing [1.28, 1.29].

• allows wavelength conversion [1.32] besides switching [1.30] and multiplexing [1.31], if adequate configurations and techniques are used.

• permits multiplexing in combination with clock recovery [1.33, 1.34].• enables an implementation of all-optical 3R regeneration techniques [1.35-1.38].

The expression “3R” refers to amplification, re-shaping and re-timing.• permits operation of transparent networks. I.e., networks in which a signal is not

converted into an electronic signal during transmission. In contrast, present non-transparent networks stop an optical signal in its travel along a fibre by converg-ing it into an electronic signal. They then perform switching and use the electric signal to drive a semiconductor laser, which reproduces the original signal. If there were no physical limitations, such a network could be transparent to channel wavelength and digital-modulation format as well as being independent of bit rates [1.39].

• allows the combination of high-speed multiplexing with high-speed optical logic [1.40-1.42].

• permits signal processing with storage functions [1.43].

3. ConclusionsPresumably, the technological progress in the next few years will mainly concentrate on two fields. On one hand the usable bandwidth of optical fibres will be further ex-panded. On the other hand OTDM techniques will mature.OTDM systems are already in an experimental stage. The necessary technological innovations are under the way. All-optical devices as proposed in this thesis may be the key enabling technologies for practical applications of multiplexing. Successful field trials with all-optical components have already been performed [1.44]. The ex-tension of the usable optical fibre-bandwidth will open new opportunities to apply OTDM techniques.

1.2 From Early to State-of-the-Art All-Optical Devices

13

1.2 From Early to State-of-the-Art All-Optical Devices

The intention of this section is to highlight the technological landmarks leading to practical devices exploiting the optical nonlinearities for ultra-fast high-speed all-op-tical switching. It is useful to first classify the different concepts and to treat each of the classes separately. The classification is depicted in Fig. 1.2. We can distinguish the different concepts with respect to the nonlinear materials (Kerr, SOA, etc.), to the techniques used to exploit the nonlinearity (XPM, XGM, FWM, DFG) and, last but not least, to the configurations (fibre loops, MZI, MI, single SOA).

1. Different Nonlinear MaterialsMainly three different materials provide optical nonlinearities sufficient for all-opti-cal operation: Kerr based optical fibres, SOAs and materials with a strong second-order nonlinear susceptibility . Nonlinear effects in optical fibres are due either to changes in the refractive index by intense light or to Brillouin and Raman scattering. The power dependence of the re-

All-Optical

Gating Methods New Wave Generation

Fibres with Kerr Effect SOA χ(2) Materials

FibreLoops

MZI MZI

XPM XPM XGM FWM DFG

MI SingleSOA

SingleSOA

SingleDevice

Fig. 1.2 Classification of all-optical concepts. Devices based on three different ma-terials are employed, primarily. There are Kerr materials, SOA materials, and materials with a strong second-order nonlinearity. The techniques to exploit the nonlinearities are based on cross-phase modulation (XPM), cross-gain modulation (XGM), four-wave mixing (FWM) or difference-fre-quency generation (DFG). They are applied in different configurations.

χ 2( )

1 Introduction

14

fractive index is of particular interest, since it can be exploited for switching in an interferometric configuration. Phenomena, where the refractive index changes are caused by an electric field, i.e. a light intensity , are generally referred as Kerr effects, described by:

(1.1)

with the second-order nonlinear refractive index . A typical value of in a non-dispersion-shifted single-mode fibre for 1.55 μm radiation is given in Table 1.3. Kerr-based nonlinearities in fibres are extremely fast.However, it was recognized that the use of active nonlinear elements like SOAs pro-vides much larger nonlinearities [1.45]. SOAs exhibit at least three different nonlin-ear regimes. The strongest nonlinearities are due to carrier-related refractive-index changes from gain saturation. In order to compare the Kerr effect in optical fibres with the refractive-index changes in SOAs we represent the SOA nonlinearity in terms of in Table 1.3. It shows that the carrier-related refractive-index changes in SOAs are by orders of magnitude larger than in fibres. We emphasize that this val-ue may be used only for comparison. The real value depends on the wavelength, the material composition etc. Other nonlinearities in SOAs are either due to gain com-pression (Spectral Hole Burning, Carrier Heating, ...) or due to SOA related Kerr ef-fects. These effects are usually weaker.Recently, AlGaAs waveguide devices were produced with alternating crystal orien-tation such that Type-II phase matching for difference-frequency generation (DFG) was obtained [1.46]. The published related experiments require strong pump pulse powers, e.g. 65 mW at 0.78 μm, and exhibit conversion efficiencies of -16 dB.

Table 1.3: Comparison between Fiber and SOA Nonlinearities at 1.55 μm

[cm2/W] Relaxation Times [ps]

Opt. Kerr Effectof a Standard Fibre Ref.: [1.47-1.48]

< 1

SOA(at

band-gap)

Carrier EffectRef.: [Chapter 6.1]

50 - 10’000

GainCompression Ref.: [Chapter 2.1]

< 1

Kerr EffectRef.: [Chapter 2.1]

< 1

Δn'I

Δn' I n'2⋅=

n'2 n'2

n'2

n′2

n′2 2,2 3,4– 10 16–⋅=

n′2 1,0 10 9–⋅=

n′2 6,0 10 10–⋅=

n′2 1,0 10 12–⋅=

1.2 From Early to State-of-the-Art All-Optical Devices

15

2. Techniques to Exploit the NonlinearitiesIn the previous subsection it was outlined that SOAs have largest nonlinearities. In this subsection we review the three most promising techniques to exploit these non-linearities [1.49,1.50].In the cross-phase modulation (XPM) technique, an optical control signal is used to modulate the phase of another signal within an interferometric configuration. These configurations have the advantage that they permit to use even small phase changes for switching. In our case, the nonlinear media are SOAs. Among the interferometric configurations we mention the Mach-Zehnder interferometer with two SOAs on the arms as depicted in Fig. 1.3(a). Without control signal , an input signal is di-rected towards the cross output port. When a control signal is introduced into one

(a) Cross-Phase Modulation (XPM)

SOA

SOA

PC

Pin

Px

P=Control Power PCO

utpu

t-Pow

er Px

P=

(b) Cross-Gain Modulation (XGM)

SOAPC

Pin

(c) Four-wave Mixing (FWM)

SOAPC

PinPconj

Wavelength

Out

put-P

ower

Pconj

PC

Pin

Fig. 1.3 Different techniques to exploit the SOA nonlinearities. (a) Cross-phase modulation: The phase shift induced by the optical control signal PC is ex-ploited in an interferometric configuration for switching. (b) Cross-gain modulation: A control signal PC is used to modulate the carrier density of the SOA and thus to modulate an input signal Pin. (c) Four-wave mixing: the SOAs are applied to generate a new signal wave Pconj.

Pout

~

Wavelength Filter50:50 Splitter

~~~~~

~~~

~~~

Operation Power

Control Power PC

Out

put-P

ower

Pout

Ope

ratio

nPo

wer

PC Pin

1 Introduction

16

arm of the MZI, the phase relations within the SOA on the respective MZI arm change and the input signal is directed from the cross into the bar output port. For complete switching, a phase shift of π must be attained by choosing the control-sig-nal power correspondingly. XPM modulated devices allow lossless, polarization-in-sensitive high-speed multiplexing [1.51] with weak control-signal powers and high extinction ratios that are only limited by noise [1.52]. The 3 dB optical bandwidth for switching an input signal is about 60 nm wide [1.53]. Advantageous features of the XPM devices are the extinction ratio enhancement [1.54] and possibly some noise reductions [1.55] which are due to the interferometric transfer functions. In ad-dition, the chirp is weak since in this concept an intrinsic phase shift of is induced. New configurations allow to operate the device even at the wavelength of the pump signal, when pump and input signal copropagate [1.56]. The different characteristics are summarized in Table 1.4.An extremely simple technique is based on cross-gain modulation (XGM). In XGM devices a strong control pulse is used to modulate the carrier density of an SOA and thereby to modulate a second signal, according to Fig. 1.3(b). Hitherto this tech-nique, which inverts the pulse shapes, has found applications mainly for wavelength conversion. The scheme is surprisingly fast. Nearly penalty-free conversion at 40 GBit/s has been reported for this scheme [1.57]. Problems associated with XGM are extinction-ratio degradation [1.58] and chirp produced by the phase modulation that necessarily accompanies gain modulation in a SOA [1.59,1.60], as shown in Ta-ble 1.4.Four-wave mixing (FWM) is a coherent wave-generation technique where intraband nonlinearities are exploited to generate a new wave from an input signal and a strong pump signal. The principal setup is depicted in Fig. 1.3(c). FWM is extremely fast [1.28] and transparent to signal formats. In addition, it allows multi-channel opera-tion [1.61] and provides phase conjugation, which is an efficient method to solve the dispersion problem in fibres [1.60-1.63]. However, for FWM in SOAs, polarization sensitivity, wavelength dependence of the conversion efficiency, amplified sponta-neous emission (ASE) noise from the SOA and the low conversion efficiencies are issues. Part of the problems can be solved with proper arrangements. For example, one can achieve polarization-insensitive operation by using two pump sources that are orthogonally polarized [1.64]. Pump- and input-signal filtering has become pos-sible with interleaved MZI configurations [1.65]. Recently, conversion efficiencies above 0 dB were achieved by using long SOA cavities [1.66]. However, signal-to-noise ratio issues and power requirements remain significant.In conclusion, the XPM technique is probably the most promising method to exploit SOA nonlinearities, since it combines important advantages such as high speed, low penalty, low power requirements, polarization insensitivity and high extinction ra-tios with various other desirable features.

π

1.2 From Early to State-of-the-Art All-Optical Devices

17

Table 1.4: Comparison of Different Techniques to Exploit SOA Nonlinearities

XPM XGM FWM

Fastest Experiment 80:10 GBit/s [1.51]

40 GBit/s[1.57]

100 GHz[1.28]

Gain dyn. 0 dB [1.69],

-1.5 dB [1.57]

normally -10 dB (0 dB [1.66])

Pump Power weak moderate strong

Polarization Independence

yes yes noyes with 2 pumps

Extinction Ratios ideal [1.52] limited [1.58] ideal

Reshaping yes [1.54] yes [1.67]

Noise Reduction yes, small [1.55] no degradation

Chirp weak high phase conjuga-tion [1.63]

3 dB Wavelength Dependency of Pin

~ 60 nm[1.53]

~ 60 nm ~ 2 nm,

Transparent to Modulation Format

no no yes

Multi-Channel Operation

no no yes

Bidirectional Operation

yes yes no, but with efforts [1.68]

Coprop. Without External Filters

no (yes with[1.69][1.56])

no (yes with[1.69][1.56])

no (yes with [1.65])

Same Signal and Pump Wavelength Possible

counterprop: yescopropag with

[1.56]

counterprop: yescopropag with

[1.56] trick

no

Complexity moderate low low

1 Introduction

18

3. ConfigurationsAs outlined above, XPM is generally the most efficient technique to make use of the strong SOA nonlinearities. In this section we give an overview on various interfero-metric configurations and show the different advantages and disadvantages. For his-torical reasons we start with interferometric configurations based on the Kerr effect before studying SOA interferometric configurations.Early all-optical devices were primarily using Kerr-based optical nonlinearities in optical fibres [1.70-1.72]. To exploit the small intensity-dependent refractive-index changes of the Kerr effect, interferometric configurations such as a Sagnac interfero-meter (Nonlinear Optical Loop Mirror, NOLM) [1.72-1.74] or Mach-Zehnder inter-ferometers [1.70,1.75] were used. In the NOLM, high-speed multiplexing has been demonstrated by introducing an optical control signal. The scheme of Fig. 1.4(a) elu-cidates the operation principle. The configuration as depicted, was for the first time realized in this complete form by Andrekson [1.74]. In this device an input pulse

(a) Nonlinear Optical-Loop Mirror (NOLM)

PC

Pin

SOA ΔL

PC

Pin

(b) Terahertz Optical Asymmetric Demultiplexer (TOAD)

Fig. 1.4 All-optical demultiplexing of input pulses (Pin) (a) in a nonlinear optical-loop mirror (NOLM) configuration by exploiting Kerr-based fibre nonlin-earities induced by a strong optical control signal (PC) and (b) in a tera-hertz optical asymmetric demultiplexer (TOAD) by exploiting carrier-related refractive index changes in a SOA induced by gain saturation from strong control pulses.

Pout

Fibre(Several km)

Midpoint

1.2 From Early to State-of-the-Art All-Optical Devices

19

train is split up into a clockwise and counter-clockwise propagating pulse train. In case where a clock signal is introduced in clockwise propagating direction, it co-propagates with the clockwise input pulse and changes its phase. If the control-signal power is sufficiently high, the pulses that are propagating with the control sig-nal experience a phase shift of π and couple into the output port. When there is no control signal, the input signals are mirrored back into the input port.It was soon recognized that the use of active nonlinear elements such as SOAs could reduce the required optical power and the optical loop lengths considerably [1.45]. A 50 GBit/s demultiplexing experiment with an asymmetrically positioned SOA in a nonlinear loop was described in 1993 [1.76]. Fig. 1.4(b) shows the configuration, which was called by its inventors “terahertz optical asymmetric demultiplexer” (TOAD). In the TOAD only the clockwise propagating input signal experiences the phase shift from the control signal in the SOA, since it enters the SOA with the con-trol pulse, whereas the counter-propagating input signal has already passed the SOA a time span before.However, MZIs [1.77] and Michelson-interferometer configurations (MI) [1.78]with SOAs on the arms were found to be more generally applicable than the TOAD concept. MZI based devices can e.g. be used as wavelength converters [1.77] as well as all-optical switches (AOS) [1.30], multiplexers [1.79], signal regenerators [1.37]. In addition, the MZI or MI configuration allows monolithic integration, which yields more stable wavelength converters [1.80] and more stable all-optical switches or de-multiplexers [1.30].One of the challenges concerning SOAs as nonlinear elements is the trade off be-tween large nonlinearity and speed (Table 1.3). The nonlinearity induced by the res-onant excitation is efficient, yet the carrier excitation results in slow relaxation time. To overcome the speed limitations of SOA-based MZI all-optical switches, an ar-rangement with two consecutive optical control pulses can be applied [1.75]. There-by a first control signal PC1 is introduced into SOA1 to induce the necessary π-phase shift for switching, as shown in Fig. 1.5(a). The second control signal is then intro-duced with a time delay into SOA2. It is used to induce the same phase shift in the second MZI arm and thereby resets the switch. In this way the switch-off as well as the switch-on time is controlled by the fast optical excitation process. Another solu-tion consists in displacing the SOAs along the propagation direction of the light and using counter-propagating control and input signals [1.81]. In this way, the control signal induces the refractive index changes at different times in the MZI arms, there-by setting a phase-shift difference of π with the first control signal and resetting it with the second control signal. We have depicted the configuration in Fig. 1.5(b). Unfortunately, this switching scheme allows only counter-propagating operation, which limits the switching speeds (see Chapter 6.1). A further solution with only 1 SOA is presented in Ref. [1.82]. However, this method is intrinsically polarization

PinPC

Δt c ΔL⁄=

1 Introduction

20

sensitive. Progress has also been obtained in the improvement of the on-off ratios [1.52]. To obtain high on-off ratios within an interferometric configuration, both the intensities as well as the phases of the signals have to be properly controlled within the config-uration. However, in all-optical devices the strong optical signal that controls the de-vice, induces phase shifts and modulates the intensity of the data signal to be switched. It was shown in Ref. [1.52, chapter 5.2], that asymmetries are needed to overcome the extinction-ratio limitations in MZI-SOA all-optical switches. Two ex-amples for such asymmetries are given in Fig. 1.6(a) and (b). The first configuration contains asymmetrically biased SOAs and a phase shifter to compensate the phase offset from the asymmetrically biased SOAs. When the control signal is intro-duced into SOA1, the amplifier saturates. But since a higher current bias was applied relative to SOA2, the saturation of the two SOAs relative to each other remains small and higher extinctions are obtained. In addition, the second configuration contains

P=

PX

PC2

PC1t

t

t

t

SOA

SOA

PC1

Pin Px

P=

PC2

(a) Consecutive Optical Control Pulses

(b) Asymmetrically Displaced SOAs

Fig. 1.5 Device configurations for switching on and off with fast resonant excita-tion. (a) Two consecutive optical control signals are introduced with a time delay. The first control signal (PC1) sets a π phase shift. The second (PC2)resets the phase difference by introducing the same phase shift in the second SOA. (b) The SOAs are displaced within the MZI cavity. Then the control signal PC is introduced in counter-propagating direction, such that the res-onant excitation takes place at different times.

SOASOA Px

P=

ΔL

PC

Pin

PC1

1.2 From Early to State-of-the-Art All-Optical Devices

21

asymmetric splitters and is asymmetric biased. A device with two asymmetries can in principle attain ideal extinction ratios. Details are explained in chapter 5.2.New devices with an integrated input- and control-signal separation scheme offer important features. They allow operation in the fast co-propagative direction at any wavelength within the gain spectrum, even when control- and data signal have the same wavelength. Besides they enable bidirectional operation without external wavelength filters and in the future they will allow cascadation of several devices on a single chip. Even the possibility to circumvent external wavelength filters itself is of relevance, since these expensive external filters introduce losses, modify the pulse form of short signals, lack integration possibilities. Three possibilities to separate data- and control signals have been presented. A first concept is based on a dual-order mode configuration, in which the input signals prop-agate as zero-order modes and the control signals propagate as first-order modes [1.83, 1.56, 1.53, chapter 6.1]. Due to the different symmetries of data and control signal, the first-order mode control-signal can be easily separated after signal-processing and reused. This configuration is depicted in Fig. 1.7(a). A second con-

Fig. 1.6 MZI-SOA all-optical switches with integrated asymmetries to obtain high extinction ratios. (a) Asymmetrically applied current biasing allows switch-ing with high extinctions. (b) When two asymmetries are introduced, e.g. an asymmetric splitter and asymmetric current bias, even ideal extinctions may be obtained.

SOA1

SOA2

Pc1

Pin

I1

I2

Pc2

45:55 Splitter

SOA1

SOA2

Pc1

Pin

Px

P=

I1

I2

Pc2

Δφ=π/2

Px

P=

Δφ=π/2

(a) Asymmetrically Biased

(b) Asymmetric Splitter and Asymmetrically Biased

1 Introduction

22

SOA2

SOA1 P=

PX

MMI-converter-combiner

PC1

PC2

SOA1

Δφ=π/2

PC2

Pin

PC1

(a) Dual-Order Mode (DOMO) Configuration

(b) Interleaved MZI ConfigurationΔφ=π

SOA4 Δφ=π

Δφ=π/2SOA2SOA3

PC2

Pin

PC1

P=

PX

PC1

PC2

1.3 μm SOA

(c) 1.30/1.55 μm All-Optical Configuration

1.3 μm SOA

1.30/1.55 mm MMI-WDM

1.55 μm Pin

1.30 μm PC1

1.30 μm PC2

P=PX

PC1

PC2

Fig. 1.7 Configurations with a separation scheme for the control (PC1, PC2) and in-put signal (Pin). (a) Dual-order mode configuration, where control and in-put signal propagate as first- and zero-order modes. (b) Interleaved MZI configuration, where control- and input-signal are separated interferomet-rically. (c) 1.30/1.55 μm configuration, where control and input signal have different wavelengths.

1.2 From Early to State-of-the-Art All-Optical Devices

23

cept that allows to separate and to reuse the control signal after signal processing is based on interleaved MZI configurations [1.69, chapter 6.1]. The method uses addi-tional MZIs that are placed on the arms of an exterior MZI to direct the control signal and the data signal by interference to separate outputs as shown in Fig. 1.7(b). A third concept requires 1.55 μm data signals and 1.3 μm control signals to modulate the all-optical device [1.84,1.85]. For multiplexing the 1.3 μm and the 1.55 μm pulses, new and simple multimode-interference couplers can be used, as shown in chapter 4.3. This 1.30/1.55 μm all-optical device possesses an advantage when it is used with a 1.30 μm SOA and 1.30 control pulses. It is of advantage that there is practically no gain saturation for the 1.55 μm signal since the alpha factor is very high. On this con-dition asymmetries to obtain high extinctions are not needed anymore [1.52]. As dis-advantage the switch needs higher control-signal powers and can not anymore be used for wavelength conversion in the EDFA window.Finally, an interesting version of a XPM all-optical wavelength converter, requiring only one SOA was recently published [1.86]. This configuration is shown in Fig. 1.8. It consists only of one SOA and a split-delay interferometer. It works as follows. An incoming signal modulates gain and phase of the SOA. The rise time is deter-mined by the input-pulse width, while the fall time is determined by the relatively long carrier-recovery time. The copropagating cw signal experiences these phase shifts before it is guided into the MZI. There it is split and directed into the straight and delayed MZI arms. When the two signals recombine at the output, the two sig-nals interfere constructively until the pulse which is delayed by arrives carrying the same phase shift as the signal from the straight MZI arm. The advantage of the concept is, that it is simple and that it allows switching with the fast transients. The disadvantage is, that it permits only wavelength conversion of return-to-zero (RZ) pulses.

P1

P2

Δt c ΔL⁄=

SOAP1P2 cw

ΔL

Fig. 1.8 Delayed-interference wavelength converter (DISC). Wavelength conver-sion of an incoming signal P1 at λ1 with fast transients onto a new wave-length λ2.

P2 Modulatedλ2

λ1λ2

Delayed Arm

Straight Arm

1 Introduction

24

1.3 Outline

This thesis describes novel all-optical device concepts based on Mach-Zehnder interferometer semiconductor optical amplifier configurations for high-speed oper-ation in optical telecommunication. The main achievements are all-optical devices with high extinction ratios and separation schemes for control and input signals. This study has resulted in several papers already published. Together with additional in-formation, eight of the published papers are reproduced in this thesis. References are given at the end of each chapter.The motivation for this work, key issues addressed and the state of the art of all-op-tical components have been presented in chapter 1.Chapter 2 presents the physical background required for understanding the opera-tion principles of MZI-SOA all-optical devices. In a first section we discuss and quantify the different optical nonlinearities that occur in SOAs. In a second section we describe how these nonlinearities modify amplitude and phase of a signal in the SOA. Finally, the MZI-SAO transfer characteristics are presented by taking into ac-count the nonlinear effects in a single SOA as described in the previous section.In chapter 3 we devote an entire section to each of the relevant material parameters such as material gain, refractive-index change and alpha-factor. Since the parameters are needed for an optimization of the devices, we have measured them for the com-plete gain spectrum and for different carrier densities at different temperatures.New couplers based on the multimode-interference principle are discussed in chapter 4. We start with some general design principles and then introduce a first MMI type that acts as a wavelength multiplexer (MMI-WDMs) for signals at 1.30/1.55 μm wavelength. A second MMI type, called MMI converter combiner (MMI-cc), is introduced afterwards. It allows us to combine a zero-order optical mode with a first-order optical mode into the same waveguide, whereas the conversion of the zero- into the first-order mode is performed within the same MMI. For the extraction of the first-order mode a similar new device is introduced. To determine the optical bandwidth of the new MMI-cc a model is derived. We have compared the predic-tions of this theory with measured optical bandwidths.Near-to-ideal extinction ratios are key issues of any switching device. In chapter 5we show that by introducing appropriate asymmetries into the MZI-SOA-based all-optical switches, the carrier-related extinction-ratio degradations can be eliminated.New configurations with monolithically integrated control- and input-signal separa-tion mechanisms are introduced in chapter 6. We discuss two concepts that permit a mode separation. On one hand we present a concept dealing with modes of differ-ent orders using a so-called dual-order mode configuration, and on the other hand we demonstrate a concept that is based on interleaved MZIs to separate the control and

1.3 Outline

25

input signals. We show that these concepts can be used for all-optical switching and filtering within cross-phase modulated and four-wave mixing all-optical devices.Chapter 7 concludes this thesis with a summary of the important achievements and some on future investigations.

1 Introduction

26

1.4 References

[1.1] K. Hirahara, T. Fujii, K. Ishida, S. Ishihara; “Optical Communications Tech-nology Roadmap”; IEICE Trans. Electron., vol. E81-C, no. 8, pp. 1328-1341, Aug. 1998

[1.2] P. Lagasse, P. Demeester, A. Ackaert, W. Van Parys, B. Van Caenegem, M. O’Mahony, K. Stubkjaer, J. Benoit; “Roadmap towards the optical Commu-nication Age”; A European view by the ACTS Photonic Domain; June 1998 draft Edition

[1.3] Alan Eli Willner; “Mining the optical bandwidth for a terabit per second”; IEEE Lasers and Electrooptic Society Spectrum, April 1997

[1.4] Lucent Technologies, Murray Hill, NJ, USA; announcement in IEEE LEOS (Lasers and Electrooptic Society) Spectrum, pp. 32-41, April 1997

[1.5] K.L. Hall, K.A. Rauschenbach, S.G. Finn, J.D. Moors; “100GBit/s Multi-Access Networks”; IEEE LEOS NewsLetter, pp. 3-7, Aug. 1998

[1.6] H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimoto, T. Chikama; “1.1 Tb/s WDM Transmission over a 150 km 1.3 μm zero-dispersion single mode fiber”; in Proc. Optical Fiber Communication Conf. 1996 (OFC’96), San Jose, USA, postdeadline paper 19.1-19.5, Feb. 1996

[1.7] A.H. Gnauck, A.R. Chraplyvy, R.W. Tkach, J.L. Zyskind, J.W. Sulhoff, A.J. Lucero, Y. Sun, R.M. Jopson, F. Forghieri, R.M. Derosier, C. Wolf, A.R. Mc-Cormick; “One terabit/s transmission experiment”; in Proc. Optical Fiber Communication Conf. 1996 (OFC’96), San Jose, USA, postdeadline paper 20.1-20.5, Feb. 1996

[1.8] T. Morioka, H. Takara, S. Kawanishi, O. Kamatani, K. Takiguchi, K. Uchi-yama, M. Saruwatari, H. Takahashi, M. Yamada, T. Kanamori, H. Ono; “100 Gbit/s x 10 channel OTDM/WDM Transmission using a single super-continum WDM source”; in Proc. Optical Fiber Communication Conf. 1996 (OFC’96), San Jose, USA, postdeadline paper 21.1-21.5, Feb. 1996

[1.9] A.R. Chraplyvy, R.W. Tkach; “Terabit/Second transmission experiments”; J. of Quantum Electron., vol. 34, no. 11, pp. 2103-2108, Nov. 1998

[1.10] Y. Yano, T. Ono, K. Fukuchi; T. Ito, H. Yamazaki, M. Yamaguchi, K. Emura, “2.6 terabit/s WDM transmission experiment using optical duobinary cod-ing”; in Proc. 22nd European Conference on Optical Communication (ECOC’96), vol. 5, postdeadline paper ThB 3.1

1.4 References

27

[1.11] R.J. Mears, L. Reekie, I.M. Jauncey, D.N. Payne; “Low-noise erbium-doped amplifier operating at 1.54 μm”; Electron. Letters, vol. 23, no. 19, pp. 1026-1028, Sept. 1987

[1.12] J.F. Massicott, J.R. Armitage, R. Wyatt, B.J. Ainslie, S.P. Craigryan; “High-gain, broad-band, 1.6 μm Er3+ doped silica fiber amplifier”; Electron. Lett; vol. 26, no. 20, pp. 1645-1646, Sept. 1990

[1.13] M. Jinno. T. Sakamoto, J. Kani, S. Aisawa, K. Oda, M. Fukui, H. Ono, K. Oguchi; “First demonstration of 1580 nm wavelength band WDM transmis-sion - for doubling usable bandwidth and suppressing FWM in DSF”; 2nd Optoelectronics & Communications Conference (OECC’97), Seoul, Korea, pp. 406-407, July 1997

[1.14] J.J.E. Reid, L. Cucala, M. Settembre, R.C.J. Smets, M. Ferreira, H. Haun-stein; “An international field trial at 1.3 μm using an 800 km cascade of semi-conductor optical amplifiers”; Proc. European Conference on Optical Communication (ECOC’98), Madrid, Spain, vol. 1, invited paper pp. 567-568, Sept. 1998

[1.15] J. Eckner; “Semiconductor optical amplifiers: Optimization of polarization and monolithical integration in ridge waveguide bulk active InGaAsP/InP”; Diss. ETH No. 12792 (1998)

[1.16] Ch. Holtmann; “Polarization insensitive semiconductor optical amplifiers in InGaAsP/InP for 1.3 mm wavelengths exploiting bulk ridge-waveguide structure”; Diss. ETH 12195, Hartung-Gore Konstanz, 1997

[1.17] G. van den Hoven; “Applications of semiconductor optical amplifiers”; in Proc. European Conference on Optical Communication (ECOC’98), Madrid, Spain, vol. 2, invited paper pp. 5-6, Sept. 1998

[1.18] G.A. Koepf, et al.; “Raman amplification at 1.118 μm in single mode fibre and it‘s limitations by Brillouin scattering”; Electron. Letters, vol. 18, pp. 942-943, 1982

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[1.20] D. N. Payne; “Extended wavelength and second window optical amplifiers”; Proc. European Conference on Optical Communication (ECOC’98), Madrid, Spain, vol. 3, invited paper pp. 9-13, Sept. 1998

[1.21] S. Kawanishi; “Ultrafast all-optical TDM transmission technology”; 2nd Op-toelectronics & Communications Conference (OECC’97), Seoul, Korea, pp. 12-13, July 1997

1 Introduction

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[1.32] T. Durhuus, B. Mikkelsen, C. Joergensen, S.L. Danielsen, K.E. Stubkjaer, “All-optical wavelength conversion by semiconductor optical amplifiers”; J. of Lightwave Technol., vol. 14, no. 6, invited paper, pp. 942-954, June 1996

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[1.36] K.S. Jepsen, A.T. Clausen, B. Mikkelsen, H.N. Poulsen, K.E. Stubkjaer; “All-optical network interface for bit synchronisation and regeneration”; in Proc. European Conference on Optical Communication (ECOC’97), Eding-burgh, Great Britain, pp. 89-92, Sept. 1997

[1.37] J.C. Simon, L. Billes, A. Dupas, B. Kowalski, M. Henry, B. Landousies; “All-optical regeneration”; in Proc. European Conference on Optical Com-munication (ECOC’98), Madrid, Spain, invited paper, pp. 467-469, Sept. 1998

[1.38] G. Morthier, J. Sun, T. Gyselings, R. Baets; “A novel optical decision circuit based on a Mach-Zehnder interferometer and gain-clamped semiconductor optical amplifiers”; Photon. Technol. Letters, vol. 10, no. 8, pp. 1162-1164, Aug. 1998

[1.39] Lars Gillner; “Transmission limitations in the all-optical network”; in Proc. European Conference on Optical Communication (ECOC’96), Oslo, Swe-den, pp. 2.39-2.43, Sept. 1996

[1.40] N.S. Patel, K.A. Rauschenbach, K.L. Hall; “40 GBit/s cascadable all-optical logic using an ultrafast nonlinear interferometer”; in Proc. Optical Fiber Communication Conf. 1996 (OFC’96), San Jose, USA, postdeadline paper 14.1-14.5, Feb. 1996

[1.41] A. Sharaiha, H.W. Li, F. Marchese, J. Le Bihan; “All-optical logic NOR gate using a semiconductor laser amplifier”; El. Letters, vol. 33, no. 4, pp. 323-325, Feb. 1997

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[1.42] D. Nesset, M.C. Tatham, L.D. Westbrook, D. Cotter; “Degenerate wave-length operation of an ultrafast all-optical AND gate using four wave mixing in a semiconductor laser amplifier”; El. Letters, vol. 30, no. 23, pp. 1938-1940, Nov. 1994

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[1.44] M.W. Chbat, A. Leclert, E. Jones, B. Mikkelsen, H. Février, K. Wünstel, N. Flaarønningen, J. Verbeke, M. Puleo, H. Melchior, P. Demester, J. Mørk, T. Olsen, D.R. Hjelme”; The OPEN (Optical Pan-European Network) ACTS Project: early achievements and perspectives” in Proc. of European Confer-ence on Optical Communication (ECOC’96), Oslo, Norway, vol. 4, paper WeP.11, Sept. 1996

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[1.51] R. Hess, M. Caraccia-Gross, W. Vogt, E. Gamper, P.A. Besse, M. Dülk, E. Gini, H. Melchior, B. Mikkelsen, M. Vaa, K.S. Stubkjaer, S. Bouchoule; “All-optical demultiplexing of 80 to 10 Gb/s signals with monolithic inte-grated high-performance Mach-Zehnder Interferometer”, Photon. Technol. Lett., vol. 10, no. 1, pp. 165-167, Jan. 1998

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[1.52] J. Leuthold, P.A. Besse, J. Eckner, E. Gamper, M. Dülk, H. Melchior; “All-optical space switches with gain and principally ideal extinction ratios”; IEEE J. of Quantum Electronics, vol. 34, no. 4, invited paper, pp. 622-633, April 1998

[1.53] J. Leuthold, E. Gamper, M. Dülk, P.A. Besse, J. Eckner, R. Hess, H. Mel-chior; “Cascadable all-optical space switch with high and balanced extinc-tion ratios”; 2nd Optoelectronic and Communications Conference (OECC’97), Seoul, Korea, pp. 184-185, July 97

[1.54] W. Idler, M. Schilling, K. Daub, D. Baums, U. Korner, E. Lach, E. Laube, K. Wünstel; “Signal quality and BER performance improvement by wave-length conversion with an integrated three-port Mach-Zehnder interferom-eter”; Electron. Lett, vol. 31, no.6, pp. 454-455, March 1995

[1.55] B. Mikkelsen, S.L. Danielsen, C. Joergensen, R.J.S. Pedersen, H.N. Poulsen, K.E. Stubkjaer; “All-optical noise reduction capability of interferometric wavelength converters”; Electron. Lett., vol. 32, no. 6, pp. 566-567, 1996

[1.56] J. Leuthold, P.A. Besse, E. Gamper, M. Dülk, St. Fischer, H. Melchior; “Cas-cadable dual-order mode all-optical switch with integrated data- and control-signal separators”; Electron. Lett., vol. 34, no. 16, pp. 1598-1600, Aug. 1998

[1.57] S.L. Danielsen, C. Joergensen, M. Vaa, K.E. Stubkjaer; “Bit error rate assess-ment of a 40 GBit/s all-optical polarization independent wavelength convert-er”; in Proc. Optical Fiber Communication Conf. 1996 (OFC’96), San Jose, USA, postdeadline paper 12.1-12.5, Feb. 1996

[1.58] J.M. Wiesenfeld; “Wavelength Conversion at 10 GBit/s using a semiconduc-tor optical amplifier”; Photon Technol. Lett., vol. 5, no. 11, pp. 1300-1303, 1993

[1.59] J.S. Perino, J.M. Wiesenfeld, G.Glance; “Fibre transmission of 10 GBit/s sig-nals following wavelength conversion using a travelling-wave semiconduc-tor optical amplifier”; Electron. Lett., vol. 30, no. 3, pp. 256-258, 1994

[1.60] L. Occhi, F. Girardin, M. Dülk, S. Fischer, G. Guekos; “Chirp induced by cross-gain modulation in bulk semiconductor optical amplifiers used for wavelength conversion”; ETH measurement, winter 1998

[1.61] S. Watanabe, S. Takeda, T. Chikama; “Interband wavelength conversion of 320 Gb/s (32x10 Gb/s) WDM signal using a polarization-insensitive fiber four-wave mixer”; Proc. European Conference on Optical Communication (ECOC’98), Madrid, Spain, postdeadline paper pp. 85-86, Sept. 1998

[1.62] D.F. Geraghty, R.B. Lee, M. Verdiell, M. Ziari, A. Mathur, K.J. Vahala; “Time-resolved spectral analysis of wavelength conversion by four-wave mixing in semiconductor optical amplifiers”; Photon. Technol. Lett., vol. 10,

1 Introduction

32

no. 1, pp. 69-71, Jan. 1998[1.63] D.D. Marcenac, D. Nesset, A.E. Kelly, M. Brierley, A.D. Ellis, D.G. Moodie,

C.W. Ford, “40 Gbit/s transmission over 406 km of NDSF using mid-span spectral inversion by four-wave-mixing in a 2 mm long semiconductor opti-cal amplifier”; Electron. Lett., vol. 33, no. 10, pp. 879-880, May 1997

[1.64] T. Hasegawa, K. Inoue, K. Oda; “Polarization independent frequency con-version by fiber four-wave mixing with a polarization diversity technique”; Photon. Technol. Letters, vol. 5, no. 8, pp. 947-949, Aug. 1993

[1.65] J. Leuthold, F. Girardin, P.A. Besse, E. Gamper, G. Guekos, H. Melchior; “Polarization independent optical phase conjugation with pump-signal filter-ing in a monolithically integrated Mach-Zehnder interferometer semicon-ductor optical amplifier configuration”, Photon. Technol. Letters, vol. 10, no. 11, pp. 1569-1571, Nov. 1998

[1.66] A. D‘Ottavi, F. Martelli, P. Spano, A. Mecozzi, S. Scotti, R. Dall‘Ara, J. Eckner, G. Guekos; “Very high efficiency four-wave mixing in a single semiconductor travelling-wave amplifier”; Appl. Phys. Lett., vol. 68, no. 16, pp. 2186-2188, April 1996

[1.67] S. Chelles, F. Devaux, D. Meichenin, D. Sigone, A. Ougazzaden, A. Caren-co; “Polarization insensitive wavelength conversion by cross gain modula-tion in a strained MQW optical amplifier”; in Proc. European Conference on Optical Communication (ECOC’96), Oslo, Norway, pp. 4.53-4.57, Sept. 1996

[1.68] A. Buxens, H.N. Poulsen, A.T. Clausen, K.S. Jepsen, K.E. Stubkjaer; “New bi-directional mid span spectral inversion using bi-directional four-wave mixing in semiconductor optical amplifiers”; in Proc. European Conference on Optical Communication (ECOC’98), Madrid, Spain, postdeadline paper pp. 97-98, Sept. 1998

[1.69] J. Leuthold, P.A. Besse, E. Gamper, M. Dülk, W. Vogt, H. Melchior; “Cas-cadable MZI-all-optical switch with separate ports for data- and control-sig-nals”; in Proc. European Conference on Optical Communication (ECOC’98, Spain, Madrid, pp. 463-464, Sept. 1998

[1.70] H. Kawaguchi; “Proposal for a new all-optical waveguide functional de-vice”; Optics Letters, vol. 19, no. 8, pp. 411-413, April 1985

[1.71] S.R. Friberg, A.M. Weiner, Y. Silberberg, B.G. Sfez, P.S. Smith; “Femtosec-ond switching in a dual-core-fibre nonlinear coupler”; Optics Letters, vol. 10, no. 13, pp. 904-907, Oct. 1988

[1.72] M.C. Farries, D.N. Payne; “Optical fiber switch employing a Sagnac inter-ferometer”; Appl. Phys. Lett., vol. 55, no. 1, pp. 25-26, July 1989

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[1.73] B.P. Nelson, K.J. Blow, P.D. Constantine, N.J. Doran, J.K. Lucek, I.W. Mar-shall, K. Smith; “All-optical GBit/s switching using nonlinear optical loop mirror”; Electron. Lett., vol. 27, no. 9, pp. 704-705, April 1991

[1.74] P.A. Andrekson, H.A. Olsson, J.R. Simpson, D.J. Digiovanni, P.A. Morton, T. Tanbun-Ek, R.A. Logon, K.W. Wecht; “64 Gb/s All-optical demultiplex-ing with the nonlinear optical-loop mirror”; Photon., Technol. Lett., vol. 4., no. 6, pp. 644-647, June 1992

[1.75] K. Tajima; “All-optical switch with switch-off time unrestricted by carrier lifetime”; Jpn. J. Appl. Phys. vol. 32, no. 12A, pp. L1746-L1749, Dec. 1993

[1.76] J.P. Sokoloff, P.R. Prucnal, I. Glesk, M. Kane; “A terahertz optical asym-metric demultiplexer (TOAD)”; Photon. Technology Lett., vol. 5, no. 7, pp. 787-790, July 1993

[1.77] C. Joergensen, T. Durhuus, B. Mikkelsen, K.E. Stubkjaer; “Wavelength con-version at 2.5 GBit/s using a Mach-Zehnder interferometer with SOA’s”; in Proc. Optical Amplifiers and their Applications Conference, Yokahama, Ja-pan, vol. 14, OSA Technical Digest Series, pp. 154-157, July 1993

[1.78] B. Mikkelsen, T. Durhuus, C. Joergensen, R.J.S. Pedersen, C. Braagaard, K.E. Stubkjaer; “Polarization insensitive wavelength conversion of 10 GBit/s signal with SOA’s in a Michelson interferometer”; Electron. Letters, vol. 30, no. 3, pp. 260-261, Feb. 1994

[1.79] K.I. Kang, I. Glesk, T.G. Chang, P.R. Prucnal, R.K. Boncek; “Demonstration of all-optical Mach-Zehnder demultiplexer”; Electron. Lett., vol. 31, no. 9, pp. 749-750, May 1995

[1.80] N. Vodjdani, F. Ratovelomanana, A. Enard, G. Glastre, D. Rondi, R. Blon-deau, T. Durhuus, C. Joergensen, B. Mikkelsen, K.E. Stubkjaer, P. Pagnod, R. Baets, G. Dobbelare; “All-optical wavelength conversion at 5 GBit/s with monolithic integration of semiconductor optical amplifiers in a passive asymmetric Mach-Zehnder interferometer”; in Proc. European Conference on Optical Communication (ECOC’94), Firenze, Italy, vol. 4, pp. 95-98, Sept. 1994

[1.81] K. I. Kang, T.G. Chang, I. Glesk, P.R. Prucnal, R.K. Boncek, “Demonstration of ultrafast, all-optical, low control energy, single wavelength, polarization independent, cascadable, and integrateable switch”, Appl. Phys. Lett., vol. 67, no. 5, pp. 605-607, July 1995

[1.82] K. Tajima, S. Nakamura, Y. Sugimoto, “Ultrafast polarization-discriminating Mach-Zehnder all-optical switch”, Appl. Phys. Lett., vol. 67, no. 25, Dec. 1995, pp. 3709-3711

1 Introduction

34

[1.83] J. Leuthold, J. Eckner, P.A. Besse, G. Guekos, H. Melchior, “Dual-order (DOMO) all-optical space switch for bidirectional operation”, in Proc. of the Conference on Optical Fiber Communication OFC’96, San Jose, pp. 271-272, Feb. 1996

[1.84] J.P.R. Lacey, G.J. Pendock, R.S. Tucker, “All-optical 1300 nm to 1550 nm wavelength conversion using cross-phase modulation in a semiconductor optical amplifier”; Photon. Technol. Lett., vol. 8, no. 7, pp. 885-887, July 1996

[1.85] R. Ludwig, S. Diez, H.G. Weber, “A novel all-optical switch for demultiplex-ing in OTDM-Systems demonstrated in a 640 Gbit/s WDM/TDM experi-ment”, Optical Fiber Communications Conference OFC’98, San Jose, USA, postdeadline paper 22, Feb. 1998

[1.86] Y. Ueno, S. Nakamura, K. Tajima, S. Kitamura; “New wavelength converter for picosecond RZ pulses”; in Proc. European Conference on Optical Com-munications (ECOC’98), Madrid, Spain, pp. 657-658, Sept. 1998

35

2 Theory of MZI Based All-Optical Devices

This chapter concerns the theoretical framework of the MZI-based all-optical devic-es discussed subsequently.In a first section we investigate the different optical nonlinearities that occur in SOAs. We identify and quantify the dominant nonlinear effects. In our MZI-SOA de-vices the bandfilling, the plasma, the bandgap shrinkage and gain-compression ef-fects are by orders of magnitude stronger than e.g. the Kerr effects such as the a.c. Stark effect or two-photon absorption.In a second section we describe how these nonlinearities modify amplitude and phase of a signal in the SOA. Simple descriptions including the relevant nonlineari-ties in terms of the alpha factor are derived.Finally, the MZI-SOA transfer-characteristics are presented under consideration of the nonlinear effects as described for a single SOA in the previous section. Equations are derived that include all relevant nonlinearities in the alpha-factor material param-eter. These equations remain valid, if the modal gain within the amplifier changes along the propagation direction of the amplified signals.

2. Theory of MZI Based All-Optical Devices

36

2.1 Optical Nonlinearities

The term “all-optical” implies, that light is controlled by light. This requires mate-rials which change their properties in the presence of light such that a second optical signal is modified or created.This is essentially the topic of nonlinear optics, which is by definition is the “study of phenomena that occur as a consequence of the modification of the optical prop-erties of a material system by the presence of light” [2.1]. Practically, only laser light is sufficiently intense to modify the optical properties of a material. There exists a vast variety of nonlinear effects which has caused considerable con-fusion in the nomenclature. It is the goal of this section to attempt a classification from the phenomenological point of view and to define the effects, which are of rel-evance in XPM all-optical switches. This attempt must necessarily be incomplete but it shall motivate the subsequent chapters.

1. The Wave Equation for Nonlinear Optical MediaThe nonlinear wave equations are derived from the Maxwell equations

, (2.1)

with the electric field , the polarization vector , the velocity of light and . This equation is derived on the assumption that we consider

nonmagnetic regions of space that contain no free charges and no currents

, , . (2.2)

These assumptions are valid for most of our integrated-optics devices under consid-eration. They are also valid for semiconductor optical amplifiers. Because the cur-rents applied to these devices are never so large, that they could considerably influence the electric field. There are also no free charges within a SOA, since non-intentionally doped materials are used in general.In addition, we can assume in Eq. (2.1), that the second term on the right-hand side vanishes. It always vanishes when the electric field propagates in the form of a trans-verse infinite plane wave. This means that the vector components are decoupled. This statement is equivalent to the requirement of a homogenous medium . This can be demonstrated by transforming the last term of Eq. (2.1) into [2.2]

. (2.3)

ΔE 1c2----- ∂2

∂t2-------E⋅– μ0∂2

∂t2-------P ∇ ∇ E⋅⎝ ⎠⎛ ⎞+=

E P cμ0 4π 10 7– Vs/Am⋅=

ρ 0= j 0= B μ0H=

∇ε 0=

∇ ∇ E⋅⎝ ⎠⎛ ⎞ ∇ ∇ ε( )ln E⋅⎝ ⎠

⎛ ⎞–=

2.1 Optical Nonlinearities

37

This expression vanishes if . In integrated optics, there are only few cases where the term (2.3) and the inhomogenity in becomes relevant. Examples are de-vices, that turn the polarization of light [2.3].The polarization can often be expressed as a power series of the field

, (2.4)

where is the free-space permittivity constant and , and indicate lin-ear-, the second- and third-order optical susceptibilities. In the most general case they are second-, third- and forth-order tensors.It is convenient to use the permittivity tensor

(2.5)

to split off the linear part from the right hand side of Eq. (2.4) and to rewrite Eq. (2.5)

. (2.6)

In we can include effects, that can neither be described by linear nor nonlinear sus-ceptibilities. An example, where an expansion into a power series of is impossible is absorption saturation ([2.1], chapter 5: Two-level approximation).

2. Solutions of the Nonlinear Wave EquationEq. (2.6) is a nonlinear partial differential equation. Predominantly, the nonlineari-ties are identified with the contributions due to the terms on the right hand side of Eq. (2.6). However this is only partly true, the equation remains only linear as long as remains a constant or varies with the space coordinates. But Eq. (2.6) is also a nonlinear differential equation if varies with the electric field , the carrier density or the temperature.The solutions of Eq. (2.6) represent nonlinear waves. They should be compared with the characteristics of linear waves. For comparison we repeat the three equivalent re-quirements which characterize linear waves [2.4]. Linear waves

• represent solutions of linear partial differential equations• obey the superposition principle of Huygens• propagate without perturbation by other excitations or waves

Superposition principle and unperturbed propagation will generally be lost for non-linear waves.However, one can consider nonlinear waves as linear waves

∇ε 0=ε

P t( ) E t( )

P ε0 χ 1( )E t( ) χ 2( )E2

t( ) χ 3( )E3

t( ) …+ + +[ ]=

ε0 χ 1( ) χ 2( ) χ 3( )

ε 1 χ 1( )+=

ΔE ε

c2----- ∂2

∂t2-------E⋅– ε0μ0∂2

∂t2------- χ 2( )E2

t( ) χ 3( )E3

t( ) …+ +[ ]=

εE

εε E

2. Theory of MZI Based All-Optical Devices

38

• If a nonlinear perturbation acts during an extremely short time interval, then one can restrict the description to the situation before and after the nonlinear pertur-bation. E.g. a data-pulse train of return-to-zero signals is demultiplexed by a strong control signal. The control-signal excites the material in the time intervals when the data pulses are zero.

• If one linearizes the nonlinearity with respect to the field of interest. This is usually possible if the field of interest is not too strong. This becomes ob-vious if we insert the electric field into Eq. (2.4), where and

represent two electric fields of angular frequency and . Expansion of in Eq. (2.4) provides polarisation perturbations in various new frequencies. If we are interested in polarisation perturbations at and reject all other terms we ob-tain

. (2.7)

The contribution can be neglected if the field is not too strong. The ex-pression is then linear in and an effective susceptibility can be intro-duced [2.1]. Equivalently one can introduce in the Helmoltz equations an effective permittivity .

In all these cases we write the solutions of Eq. (2.6) as linear fields in the form of

, (2.8)

with the static electric field , the wavenumber , the space vector , the angular frequency , the time and the phase offset . The wavenumber is defined by

, (2.9)

where is the vacuum wavelength and is the complex refractive index which is composed of the real refractive index and the intensity absorption coefficient

. (2.10)

When using the usual equation

, (2.11)

insertion of Eq. (2.8) in Eq. (2.6) under consideration of Eq. (2.7) delivers the rela-

E EΩ Eω+= EΩEω Ω ω E

ω

P ε0 χ 1( ) χ+2( )

EΩ34---χ

3( )Eω

22 EΩ

2+( )+ Eω ε0χeffEω≡=

Eω2

EωEω χeff

εeff 1 χeff+=

E x t,( ) Eo ei knx ωt– φ+( )⋅=

Eo kn xω t φ

kn2π

λvac---------- n⋅=

λvac nn′ α

n n' iλvac4π

----------α+=

n2 εeff 1 χeff+= =

2.1 Optical Nonlinearities

39

tions of and with the real and complex part of the effective susceptibility

(2.12)

. (2.13)

These approximations are valid for non-absorbing media, if .

3. Kramers-Krönig Relations in Nonlinear OpticsThe real and imaginary part of the first order optical susceptibility are related by the Kramers-Krönig (KK) relations. In terms of the absorption and the re-fractive index it gives [2.5, 2.6]

, (2.14)

where P indicates taking the principal value integral and and are the frequen-cies of the optical fields. For the derivation of the equation the very general assump-tions of linearity between the electric field and the polarisation., time independency and causality were used. Causality corresponds to the statement that no output can occur before an input.Causality holds for nonlinear as well as linear systems. However, the question arises under what conditions the equations may be applied in nonlinear optics.The Kramers-Krönig relations can be applied both in the presence and in the absence of a perturbation and difference can be taken between the two cases. That is how the Kramers-Krönig relations can be applied onto the real and imaginary part of the optical susceptibilities in the presence of a nonlinear perturbation. Doing this, one can derive a modified form of the Kramers-Krönig relation [2.7]

, (2.15)

where denotes the perturbation, which must be independent of the frequency of observation, , in the integral (i.e. the excitation must be held constant). This equa-tion is applicable to any perturbation, whether it is optically generated or not. And it

n′ α

n′1 χ′eff+

2--------------------

1 χ′eff+2

--------------------⎝ ⎠⎛ ⎞

2 χ″eff2

-----------⎝ ⎠⎛ ⎞

2++

1 2⁄

1 12---χ′eff+≅=

α kvacχ″eff

n′----------- kvak

χ″eff

1 12---χ′eff+

-----------------------≅=

χ″eff 1«

χ′ 1( ) χ″ 1( )

n′ ω( ) 1– cπ--- P α ω′( )

ω′2 ω2–--------------------- ω′d

0

∞

∫=

ω ω′

χ′eff χ″eff

Δn′ ω ξ,( ) cπ--- P Δα ω′ ξ,( )

ω′2 ω2–------------------------ ω′d

0

∞

∫=

ξω′

2. Theory of MZI Based All-Optical Devices

40

may be applied for resonant and nonresonant optical nonlinearities as long as the op-tical perturbation is another optical field. That means, we have to restrict ourselves to nondegenerate optical nonlinearities, i.e. pump-probe spectra, with the pump at a fixed frequency. This again is due to the restriction, that the perturbation has to re-main constant over the integration. The integral in Eq. (2.15) is difficult to compute because of the singularity at

. This singularity can be avoided by transforming the integral into [2.8]

. (2.16)

Now the integrand is no more singular, since the term in the brackets is the derivation of .

4. Nonlinear EffectsWhen two or more waves propagate inside a nonlinear medium, they interact with each other by a variety of nonlinear phenomena. An appropriate phenomenological classification is possible:• On the one hand new waves can be generated through processes such as second-

harmonic generation, sum- and difference-frequency generation, third-harmonic generation and fourwave mixing. These processes involve the decay of two or more photons and the excitation to higher virtual energy levels in the medium. The virtual states remain excited for times of the order , according to He-isenberg’s uncertainty relation and then decay into new waves. The process is only possible, if energy and momentum are conserved.

• On the other hand nonlinearities in materials can couple incident waves by a phe-nomena called cross-phase modulation (XPM) and cross-gain modulation(XGM). The effects occur because the effective refractive index of a wave de-pends on the intensity of that wave as well as on the intensity of other waves. One refers to XPM when the dominant effect involved in the application concerns a phase-shift encountered by the wave of interest while one speaks of XGM when the dominant effect concerning the wave of interest is a gain or intensity modu-lation. In any cases the two processes always occur together as is implied by the Kramers-Krönig relations. XPM is always related with self-phase modulation(SPM), since an intensity modulation of a wave not only modulates other waves but also the phase of the wave itself.

• In addition, scattering processes such as stimulated Raman, stimulated Brillouin and Rayleigh scattering may transfer the energy of incident waves onto other waves. Raman scattering describes energy transfer with the help of optical pho-

ω ω′=

Δn′ ω ξ,( ) cπ--- P 1

ω′ ω+---------------- Δα ω′ ξ,( ) Δα ω ξ,( )–

ω′ ω–------------------------------------------------------⎝ ⎠

⎛ ⎞ ω′d

0

∞

∫=

Δα

h ΔE⁄

2.1 Optical Nonlinearities

41

nons, Brillouin scattering assigns light scattering with sound waves and Rayleigh scattering addresses scattering of light with nonpropagating density fluctuations.

Subsequently we describe some of the more important nonlinear phenomena, which are relevant in the following all-optical SOA devices or at least important for under-standing. For a quantitative estimate we need the real as well as the imaginary parts of the optical susceptibilities, respectively the real refractive-index changes and absorption changes induced by the nonlinearity.

4.1 Nonlinearities related to the permittivity or the linear optical susceptibilityNonlinear effects usually described in terms of the effective permittivity:

• bandfilling effects: gain saturation, absorption bleaching, ...• gain compression: from spectral hole burning, carrier heating and

two-photon absorption• band-gap effects: bandgap shrinkage, ...• plasma effect

Subsequently we discuss these effects in more detail.

4.1.1 Bandfilling effectsMainly two effects are identified if an optical signal enters a semiconductor and its wavelength corresponds a photon energy equal or larger than the bandgap energy of the semiconductor:• A change of transmission intensity due to carrier-density population changes.

This is designated as absorption saturation (absorption bleaching) when the in-tensity increases due to stimulated absorption and gain saturation when the inten-sity decreases due to stimulated emission.

• A shift of the absorption energy respectively the gain peak. The effect is known as Burstein-Moss effect [2.9, 2.10]. This effect too, has its origin in carrier-density population changes.

All these effects occur independent of the origin of the carrier-density population changes (bandfilling). They may be induced by a strong optical field, by current car-rier injection or doping. Doping and carrier injection are equivalent, except that in-jection results in bandfilling effects from both, the electrons and, the holes.Since bandfilling gives the largest contribution to XPM and XGM in the MZI-SOA all-optical devices, we have implemented a program that calculates this effect. Due to the Kramers-Krönig relations, the effect is sufficiently described if we have either an expression for the absorption or for the refractive index change . Subsequently we consider the optical absorption in a semiconductor and the opti-cal-material gain related by

. (2.17)

Δn'Δα

Δα Δn′α

gm

gm α–=

2. Theory of MZI Based All-Optical Devices

42

The material gain is equivalent to the fractional increase of photons per unit length. It can be calculated by Fermi’s golden rule by taking into account the dipole transitions between the conduction and valence bands [2.11, 2.12]

(2.18)

where is the energy of the light, the transition ener-gy, corresponds to the effective group index of the waveguide and is the electron-hole transition wave vector. The terms in the integral are: the squared dipole matrix element, the bulk density of states in -space, and

the Fermi functions for the conduction- and valence-band states and the linewidth-broadening function, that takes into account the finite lifetime of the car-riers due to scattering effects. The details of the calculation and a more precise de-scription of the terms is found in App. A.Calculated material-gain curves for In0.60Ga0.40As0.85P0.15 are depicted in Fig. 2.1.The gain region occurs at 1.50 μm. The curves have been calculated for current-in-jection densities ranging from N=0.5·10+24m-3 to 3.0·10+24m-3 in steps of 0.5·10+24m-3. We have assumed that the carrier- and hole-carrier injection currents are equal, i.e. . The parameters used in the calculation have been derived from our experiments described in chapter 3 and from literature as summarized in the Appendix. The precise values are listed in Table D.1 of the Appendix.

gm

gm Eϖ( ) 1Eϖ------- e2πh

ε0cm0---------------

n'g,eff

n'2-------------=

Mave0

∞

∫×2ρ ktr( ) fc Ev ktr( )( ) fv Ec ktr( )( ) 1–+[ ]L Eϖ Ecv ktr( )–( )dktr

Eϖ hϖ= Ecv Ec Eg Ev+ +=n'g,eff ktr

Mave2

ρ ktr( ) ktr fcfv L( )

Fig. 2.1 Calculated material gain curves for different carrier current densi-ties of 1.55 μm-InGaAsP.

gm

Wavelength [μm]

Mat

eria

l-Gai

ng m

[cm

-1]

0

-10’000

-20’000

N=0.5·1024m-3

N=3.0·1024m-3

0.800 1.000 1.200 1.400 1.600 1.800

5000

-5‘000

-15‘000In0.60Ga0.40As0.85P0.15

N P=

2.1 Optical Nonlinearities

43

By writing rather than we emphasize, that Eq. (2.18) represents the material gain at thermal equilibrium. Non-thermal equilibrium terms have to be considered separately.Calculated refractive-index changes, for an increasing of the carrier densities to N=3.0·10+24m-3 with respect to a reference carrier density at N=0.5·10+24m-3 are shown in Fig. 2.2. The refractive index corresponds to the Kramers-Krönig transfor-mation performed over the spectral material gain curves shown in Fig. 2.1.Besides the interband transitions between conduction and valence band, one should include the so-called intervalence-band contributions from heavy-hole to light-hole, heavy-hole to spin-orbit and light-hole to spin-orbit. However, only heavy-hole to spin-orbit transitions will contribute to the refractive-index variation [2.13]. Its con-tribution is estimated to be no more than 10-20% of the plasma effect discussed be-low [2.14]. For that reason we neglect this effect.

4.1.2 Bandgap EffectsPerturbations of the bandgap may be due to

• carriers (Bandgap shrinkage)• temperature and • pressure.

Here we can restrict our considerations to the first two effects since we do not change the pressure within the device. The knowledge of these energy gap shifts is crucial for refractive-index calculations, because it affects a region of the spectrum where absorption and gain is high.Thus it is of considerable weight in the Kramers-Krönig integral.

gm gm

Wavelength [μm]0.800 1.000 1.200 1.400 1.600 1.800

0.000

0.020

-0.020

-0.040

Ref

ract

ive

Inde

xC

hang

eΔn

‘

Fig. 2.2 Calculated refractive-index changes due to bandfilling. The changes are due to a carrier density increase from N=0.5·10+24m-3 (n‘ref) onto N=1.0, 1.5, 2.0, 2.5 and 3.0·10+24m-3.

Δn‘=n‘(1.0·1024)-n‘ref

Δn‘=n‘(3.0·1024)-n‘ref

2. Theory of MZI Based All-Optical Devices

44

Bandgap Shrinkage:The energy-bandgap shift due to carrier-concentration changes is usually called bandgap shrinkage or bandgap renormalization (BGR). Bandgap-shrinkage is an-other manifestation of the many-body nature in semiconductor physics.The basic mechanism is occupation of states at the bottom of the conduction band by injected electrons. If the concentration is sufficiently large, the electron wave functions will overlap, forming a gas of interacting particles. The electrons will repel each other by Coulomb forces and occupy new states of lower energy. In addition, electrons can occupy a lower energy state by avoiding electrons of the same spin. The net result is a screening of electrons and a decrease of their energy, thus lowering the energy of the conduction band edge. A similar effect for holes increases the en-ergy of the valence band edge. At last but not at least the direct gaps of semiconduc-tors are also governed by excitonic effects. One can interpret these excitonic effects as Coulomb like interactions between electrons and holes [2.15].An expression for the energy-bandgap shift originally derived by Wolff [2.16]is

(2.19)

with the free electron and hole carrier concentrations and and the relative static permittivity function of the semiconductor. The application of this formula to our 1.55-μm InGaAsP device discussed in chapter 3 yields for

. The experiments in chapter 3 do not confirm such a high band-gap shrinkage. On the contrary, the shrinkage presented in the textbook of Agrawal [2.17] is confirmed by our experiments. Agrawal estimates for InGaAsP materials

. (2.20)

For GaAs semiconductor devices good agreement with Eq. (2.20) was found [2.13]. Modifications of Eq. (2.20) based on experimental measurements in GaAs systems were proposed by Benett et al. [2.14] and used also by other authors [2.18]. However when applying these modified models to our InGaAsP devices the bandgap-shrink-age is considerably to low.The modification of the material-gain by bandgap-shrinkage is demonstrated in Fig. 2.3. The calculation has been performed with the value obtained from Eq. (2.20) at carrier densities of N=0.1, 1.0, 2.0 and 3.0·10+24m-3.Refractive index changes due to bandgap shrinkage are depicted in Fig. 2.4. The re-

ΔEg

ΔEge2

2πε0εs-----------------

⎝ ⎠⎜ ⎟⎛ ⎞ 3

π---⎝ ⎠

⎛ ⎞ 1 3⁄N1 3⁄ P1 3⁄+( )–

dEgdN--------- N1 3⁄ P1 3⁄+( )= =

N Pεs

dEg dN⁄2,35– 10 10– eVm⋅

dEg dN⁄

dEg dN⁄ 1,6– 10 10– eV m⋅[ ]⋅=

2.1 Optical Nonlinearities

45

fractive-index changes are the differences between a refractive index calculation with and without the bandgap shrinkage at the respective current density.Temperature induced Change of the Bandgap:Besides changes of the band structure induced by thermal expansion of the lattice, the temperature variation of is mainly due to the electron-phonon interactions. Near and above room temperature, varies approximately linearly with the

Fig. 2.3 Red-shift of the material-gain curves due to the carrier related bandgap shrinkage (BGR). The corresponding curves with and without BGR were calculated at the carrier densities: N=0.1, 1.0, 2.0 and 3.0·10+24m-3.

Wavelength [μm]1.200 1.300 1.400 1.500 1.600

Mat

eria

l-Gai

ng m

[cm

-1]

0

-4’000

-8’000

2‘000

-2‘000

-6‘000

with BGR

without BGR

Fig. 2.4 Refractive-index change contribution from bandgap shrinkage effect at the respective current densities N=1.0, 1.5 2.0 2.5 and 3.0·10+24m-3.

Wavelength [μm]0.800 1.000 1.200 1.400 1.600 1.800

0.000

0.020

0.010

-0.010Ref

ract

ive

Inde

xC

hang

eΔn

‘ 0.030

N=1.0·1024m-3

N=3.0·1024m-3

EgEg TL( )

2. Theory of MZI Based All-Optical Devices

46

lattice temperature. In In1-xGaxAsyP1-y materials one finds [2.19]

(2.21)

Other references use for in InGaAsP lattice matched to InP [2.20] or -0.325 meV/K [2.17].

4.1.3 Plasma Effect (Free-carrier absorption)Free carriers can absorb a photon and move to higher energy states within a band. In the Drude model, this intraband free-carrier absorption (FCA) is modelled as being directly proportional to the concentration of the electrons and holes. Since the light-induced carrier changes within the conduction and valence band can reach consider-able values, we have to consider this effect [2.13, 2.14].Within the Drude model of free electrons, the equations of motion solved under con-sideration of an oscillating electric field excitation and a linear damping one obtains for the linear susceptibilities [2.21]

and (2.22)

, (2.23)

where N is the carrier concentration. In the limit of weak damping and high oscilla-tion frequencies ( ) [2.21], one finds for the corresponding change in refractive index under consideration that both electron and valence band carriers contribute to the plasma effect with Eq. (2.12) and (2.22)

, (2.24)

where is the conduction effective mass and is the hole effective mass with contributions from the light and heavy holes

. (2.25)

Calculated refractive-index changes due to the plasma effect are given in Fig. 2.5. These changes were evaluated for 1.55 μm In0.60Ga0.40As0.85P0.15/InP at carrier concentrations between N=0.5·1024m-3 and 3.0·1024m-3 in steps of 0.5·1024m-3.

dEg dTL⁄ 0,40– 0,03y+( ) meV/K=

dEg dTL⁄0,348 meV/K–

χ′ 1( ) Nq2–ε0m------------- 1

ω2 γ2–( )----------------------=

χ″ 1( ) Nq2

ε0m--------- γ

ω ω2 γ2–( )---------------------------=

ω γ»

Δn 12n------ q2

ε0ω2------------ Nmc------ P

mh------+⎝ ⎠

⎛ ⎞–=

mc mh

1mh------

mhh1 2⁄ mlh

1 2⁄+

mhh3 2⁄ mlh

3 2⁄+-------------------------------=

2.1 Optical Nonlinearities

47

4.1.4 Nonlinear-Gain CompressionWhen strong optical fields are introduced to the semiconductor, there may occur hot-carrier distributions within the conduction and valence band due to spectral hole burning and carrier heating. These non-equilibrium distributions lead to a gain satu-ration. This is usually referred to as gain compression effect. The gain compression recovers quickly, since a new thermalized distribution is achieved via intraband car-rier-carrier scattering and carrier-phonon scattering. In contrast to the section on the bandfilling effect, where we considered interband effects under thermal equilibrium conditions, we now are dealing with intraband carrier-relaxation effects under non-thermal equilibrium.For better information we briefly survey the different gain compression effects:• Spectral Hole Burning (SHB) is the generation of a dip in the fermi energy distri-

bution of electrons and holes due to stimulated recombination or absorption. The thermalized distribution is re-established through carrier-carrier scattering with lifetimes of 100 fs or less [2.22]. The SHB effect can be distinguished from other effects because at transparency, the net transitions are zero, so that SHB is absent.SHB is centred around the signal wavelength. Yet because of the large spectral width of about 50 nm, due to the short intraband relaxation time, it modifies the whole gain spectrum independent of the wavelength [2.23].

• Carrier Heating (CH) occurs because stimulated recombinations eliminate carri-ers cooler than the average, i.e. those with an energy below the quasi-Fermi level. Since non-thermal carrier distributions are immediately restored, the loss of these ’cold’ carriers increases the average temperature. For the same reason, “hot” car-

Fig. 2.5 Refractive-index changes due to plasma effect for carrier concentrations between N=0.5·1024m-3 and 3.0·1024m-3.

Wavelength [μm]0.800 1.000 1.200 1.400 1.600 1.800

-0.020

0.000

-0.010

-0.030Ref

ract

ive

Inde

xC

hang

eΔ

n‘ 0.010N=0.5·1024m-3

N=3.0·1024m-3

2. Theory of MZI Based All-Optical Devices

48

riers that are added by absorption (FCA) or two-photon absorption (TPA) also in-crease the temperature. The non-equilibrium carrier-distribution relaxes to the lattice temperature via car-rier-phonon scattering. Typical carrier-phonon scattering times in 1.55-μm InGaAsP are in the range of 250 fs-700 fs [2.22]. Carrier heating occurs in the absorption as well as in the gain spectrum.CH has a full half-maximum of about 150 nm wavelength. Some experiments show, that the maximum occurs at wavelengths below the gain maximum, namely near the maximum of spontaneous emission [2.23].

For a while gain compression has been included by the phenomenological expres-sion [2.24, 2.25]

, (2.26)

where corresponds to the linearized equilibrium material-gain from Eq. (2.18). The parameter is the nonlinear gain suppression parameter

, (2.27)

with contributions to gain-compression from SHB and CH.The intracavity photon density is related to the power of a signal measured at the facet by

, (2.28)

with the effective area of the waveguide, the waveguide width and the thickness of the waveguide.Eq. (2.26) is the new material gain expression and replaces the equilibrium material gain when gain compression becomes relevant and when the pulse widths are at least in the picosecond range.The phenomenological approach was later on explained by Agrawal and Olsson [2.26] by the depletion of carrier density by stimulated emission. However, they ne-glected the energy distribution of carriers. Subsequently, the same authors also pro-posed a modified theory [2.27] for high-power injections suggesting a form of

. However, the phenomenological approach based on exper-iments has found wide acceptance. A good theoretical description of the gain com-pression, including SHB and CH has recently been made by Mecozzi and Mørk [2.28]. Their approach is briefly outlined in App. B.

gmgm N TL,( )1 ∈totS+-------------------------=

gm∈tot

∈tot ∈SHB ∈CH+=

S

Pout hωAeffvgS=

Aeff 1 Γ⁄ wd⋅= wd

gm

g gL 1 ∈totS+( ) 1 2⁄–=

2.1 Optical Nonlinearities

49

Typical values of for bulk InGaAsP are given in Table 2.1.

Expected refractive-index changes from gain compression are depicted in Fig. 2.6.Typical photon densities for switching have been determined from experiments as described in chapter 6. For a dual-order mode experiment Eq. (2.28) and the corre-sponding Aeff=(3.5·10-5 · 0.23·10-6m2/0.6) yield a photon density S =0.4·1021m-3. Thus we can conclude, that gain-compression effects do not significantly contribute to the refractive index changes. Our calculation was performed with the gain-com-

Table 2.1: Characteristic Values of Gain-Compression Factor

Experiment & Structure / Calculation Value Ref.

Exp: 500 μm, 1.54 bulk amplifier(for input-powers from 1.5-3mW)

Exp: 1.51 bulk amplifierExp: 1.30 bulk FP LaserExp: 1.30 bulk DFB LaserExp: 1.55 strained QW DFB LaserCalculation for bulk 1.55

6·10-17-14·10-17cm3

2.0 ·10-17cm3

5.4 ·10-17cm3

2.8 ·10-17cm3

3.0 ·10-17cm3

1.45·10-17cm3

[2.29]& [2.30][2.23][2.31][2.31][2.32][2.33]

Exp: 1.55 bulk laser diodeCalculation

1.1 ·10-17cm3

0.79·10-17cm3[2.34][2.33]

Exp: 1.55 bulk laser diodeCalculation

1.6 ·10-17cm3

0.66·10-17cm3[2.34][2.33]

∈tot

∈tot

∈

∈tot

∈CH

∈SHB

N=1.0e24 m3

N=2.0e24 m3

N=3.0e24 m3

N=1·1024m-3

N=2·1024m-3

N=3·1024m-3

Wavelength [μm]1.200 1.300 1.400 1.500 1.700 1.800

-0.004

0.004

-0.000

-0.008Ref

ract

ive

Inde

xC

hang

eΔ

n‘ 0.008

1.600

S=10·1021m-3

S=1·1021m-3

Fig. 2.6 Refractive-index changes due to gain compression at two different photon densities. Significant refractive-index changes require photon densities that larger than 1·1021 m-3. The calculations were performed for three dif-ferent carrier densities.

2. Theory of MZI Based All-Optical Devices

50

pression factors =8.4·10-23 m-3 and =5.6·10-23m-3 as measured by Girar-din et al. in [2.29] for high photon densities. The signal wavelength at which the depletion is centred was chosen approximately at 1.55 μm. We restricted the mate-rial gain-compression effect as described with Eq. (2.26) for SHB onto a spectral range of 50 nm and for CH onto a spectral range of 120 nm around the centre wave-length. The values of 50 and 120 nm are given in Ref. [2.23].

4.2 Second Order NonlinearitiesThe second order susceptibility allows to generate refractive-index changes through a static or slowly varying electric field. The effect is known as:

- Linear electrooptic effect (Pockels effect)In addition, since two electric fields interact with each other new waves can be gen-erated. The new waves are termed according to the generation process as:

- Second Harmonic Generation (SHG)- Sum Frequency Generation (SFG) or- Difference Frequency Generation (DFG)

Subsequently we discuss the linear electrooptic effect and the processes leading to new waves separately.

4.2.1 The linear electrooptic effectWe first discuss the linear electrooptic effect. To relate the effect of the static electric field with the refractive-index change we linearize the second order polarisation with respect to the electric field , which is of inter-est [2.1]

, (2.29)

and thus obtain with Eq. (2.12)

. (2.30)

It is convenient to describe the Pockels effect in terms of the tensor and in terms of , i.e. the unperturbed real refractive index. Due to the crystal symmetry of the InGaAsP materials (Appendix B) there are only six nonvanishing elements which all have identical values

. (2.31)

This implies that the Pockels effect depends on the polarization. In typical ridge waveguide structures with the electric contacts on top and bottom of the device, where the static field is parallel to the TM-mode, the TM-mode is not influent.

∈SHB ∈CH

EΩ EΩ 1, EΩ 2, EΩ 3,, ,( )= ΔnEω

P2( )

t( ) ε0χ 2( )EΩEω ε0χeff2( )Eω= =

Δnij12---χeff

2( )≅ 12---n′0

3rijkEΩ k,–=

rijkn′0

r r123≡ r213 r132 r312 r231 r321= = = = =

2.1 Optical Nonlinearities

51

Measured values in 1.24 μm InGaAsP lattice matched to InP at 1.53 μm yield r=1.3 pm/V [2.35].

4.2.2 SHG, SFG, DFGNext we discuss the generation of new waves due to second order nonlinearities. New waves may occur when one or two strong electric fields are introduced into a nonlinear media. In this situation the electric field can be represented by

(2.32)

and the second-order nonlinear polarisation by

, (2.33)

which can be expanded as

(2.34)

The last term represents the optical rectification (OR).Eq. (2.34) shows, that four different nonzero-frequency components occur. Howev-er, to generate one of these frequency components, the phase matching condition

(2.35)

has to be satisfied and a centrosymmetric crystal is needed [2.1]. In our bulk quarter-nary InGaAsP layers neither of these conditions is satisfied.Recently DFG experiments have been performed in III/V semiconductors [2.36]. They may become important for wavelength conversion. However, considerable technological efforts are necessary to grow structures that can fulfil the phase-match-ing conditions.

E t( ) 12--- Eo 1, e

iω1t–Eo 2, e

iω2t–c.c.+ +( )=

P2( )

t( ) χ 2( )E2

t( )=

P2( )

t( ) 12---ε0χ

2( ) 12--- Eo 1,

2e

2ω1tEo 2,

2e

2ω2tc.c.+ +( ) (SHG)=

ε0χ 2( ) 12--- Eo 1, Eo 2, e

ω1 ω2+( )tc.c.+( ) (SFG)+

ε0χ 2( ) 12--- Eo 1, Eo 2,

*e

ω1 ω2–( )tc.c.+( ) (DFG)+

12---ε0χ

2( )Eo 1,

2Eo 2,

2+( ) . (OR)+

Δkω3c------n′ ω3( )

ω1c------n′ ω1( )

ω2c------n′ ω2( )–– 0= =

2. Theory of MZI Based All-Optical Devices

52

4.3 Third-Order NonlinearitiesNonlinear third-order effects can manifest by a field dependence of the refractive in-dex. Such effects are the

• electrooptic Kerr effect, i.e. a refractive-index change due to a static electric field. Occasionally the effect is termed quadratic-electrooptic effect [2.13], electrorefractive effect [2.37] or d.c. stark effect• optical-Kerr effect, i.e. refractive-index change proportional to the intensity of an

optical field. This the effect is also termed a.c. stark effectDue to the Kramers-Krönig relation the changes of the refractive index cause chang-es of gain or absorption. A well-known effect associated with the electrooptic Kerr effect is the

• electroabsorption effect, i.e. also known as Franz-Keldysh effect, due to the in-tensity dependent tilt of the band edges in presence of a static electric field

New waves are generated through processes such as• third-harmonic generation• degenerate fourwave mixing• nondegenerate fourwave mixing etc.

The mechanisms leading to third-order nonlinearities are manifold. There are con-tributions from the electronic polarization, the molecular orientation of the crystal, electrostriction, i.e. the tendency of the material to become more dense in regions of high optical fields, thermal effects, the photorefractive effect etc.In order to describe the optical Kerr effect and the related generation of new waves we consider a media with three optical fields

. (2.36)

Third-order polarisation

(2.37)

includes terms, which yield an optical Kerr-effect due to self-phase modulation (SPM)

, (2.38a)

where assigns permutations over all fields. Furthermore it contains terms, that in-

E t( ) 12--- Eo 1, e

iω1t–Eo 2, e

iω2t–Eo 3, e

iω2t–c.c.+ + +( )=

P3( )

t( ) ε0χ 3( )E3

t( )=

P3( )

t( ) 34---ε0χ

3( )Eo 1,

2E1 …+∴+=

∴

2.1 Optical Nonlinearities

53

duce an optical Kerr effect due to cross-phase modulation (XPM)

. (2.38b)

Other terms are responsible for the generation of new waves via third harmonic gen-eration (THG)

, (2.38c)

or degenerate four-wave mixing (FWM)

(2.38d)

and nondegenerate four-wave mixing

(2.38e)

Our interest is focused on terms that change the refractive index of existing fields rather than generating new fields. Such terms are the SPM and XPM terms. At first one notices that XPM is twice as effective as SPM, since there is a factor 6 instead of a factor 3 in the corresponding equation. In the presence of a strong field the SPM and XPM terms can be linearized in the field of interest. In this case an effective third-order susceptibility can be intro-duced

. (2.39)

It relates the field-dependent refractive-index changes to the refractive index by the

+ 64---ε0χ

3( )Eo 2,

2Eo 3,

2+( )E1 …+∴+

+ 14---ε0χ

3( ) 12--- Eo 1,

2e

3ω1tc.c.+( )⎝ ⎠

⎛ ⎞ …+∴+

+ 34---ε0χ

3( ) 12--- Eo 1,

2Eo 2, e

2ω1 ω2+( )tc.c.+⎝ ⎠

⎛ ⎞⎝ ⎠⎛ ⎞ …+∴+

+ 34---ε0χ

3( ) 12--- Eo 1,

2Eo 2,

*e

2ω1 ω2–( )tc.c.+⎝ ⎠

⎛ ⎞⎝ ⎠⎛ ⎞ …,+∴+

+ 64---ε0χ

3( ) 12--- Eo 1, Eo 2, Eo 3,

*e

ω1 ω2 ω3–+( )tc.c.+⎝ ⎠

⎛ ⎞⎝ ⎠⎛ ⎞ …+∴+

+ 64---ε0χ

3( ) 12--- Eo 1, Eo 2, Eo 3, e

ω1 ω2 ω3+ +( )tc.c.+⎝ ⎠

⎛ ⎞⎝ ⎠⎛ ⎞ .

EiEj

P3( )

t( ) 34---ε0χ

3( )Eo j,

22 Eo i,

2+( )Ej ε0χeff

3( )Ej= =

2. Theory of MZI Based All-Optical Devices

54

definitions

and (2.40)

, (2.41)

where indicates the second-order index of refraction. The electric field can be replaced by the intensity as follows

. (2.42)

The two refractive indices are related by

, (2.43)

where is the wave impedance. This yields the second-order of refraction from SPM and XPM

and . (2.44)

In analogy one defines the second-order of absorption as

. (2.45)

is related to terms of the imaginary part of the third-order susceptibility

and . (2.46)

In the next subsections we consider mechanisms, responsible for an intensity-de-pendent refractive-index change quantify the importance for our XPM-SOA all-op-tical switches.

Intensity dependent refractive index from a.c. quadratic Stark EffectIn the presence of an electric field, the band edges of conduction and valence band are disturbed. This changes the absorption of the material. Expressions for the absorption change have been derived with the aid of the tunnel-ling theory based on an A·p perturbation in a two-band model of a semiconductor [2.38]. The nondegenerate a.c. Stark absorption characterized by an intensity-de-

n'2 1 χeff3( )+=

n′ n′0 n'2 E2

t( )⟨ ⟩+≡ n′012---n'2 Eo

2+=

n'2 EoI

n′ n′0 n′2Ii+=

n'21

cε0n'0---------------n'2 Z n'2⋅= =

Z

n'21

cε0n'02--------------- 3

4---χ′ 3( )⋅= n'2

1cε0n'0

2--------------- 64---χ′ 3( )⋅=

α α0 α2Ii+=

α2

α232---

ωiIi

ε0c2n′02------------------χ″ 3( )= α2

62---

ωiIi

ε0c2n′02------------------χ″ 3( )=

2.1 Optical Nonlinearities

55

pendent perturbation by a strong pump field interacting with a weaker probe sig-nal is

, (2.47)

with

. (2.48)

is the Kane-dipole matrix element as defined by Eq. (A.4) and K represents the material-independent parameter defined in [2.39]. It was applied as fit parameter to match the experiments and theory. The result was GW-1eV5/2 [2.40]. The factor “2” in Eq. (2.47) takes into account the nondegeneracy between the pump and probe field.The wavelength dependence of the second-order nonlinear absorption due to the quadratic a.c. Stark effect is shown in Fig. 2.7. The effect is most pronounced near the band-edge, which experiences a blue shift due to the Stark effect. Since the band-edges are shifted towards higher energies, the absorption decreases.The second-order refractive index changes due to the Stark effect have been de-

E2E1

Δα ω1 ω2,( ) 2KEp

1 2⁄

n1n2Eg3------------------F2

hω1Eg

----------hω2Eg

----------,⎝ ⎠⎜ ⎟⎛ ⎞

=

F2 x1 x2,( ) 1210x1x2

2 x1 1–( )1 2⁄---------------------------------------------- 1

x1 x2–---------------- 1

x1 x2+----------------+⎝ ⎠

⎛ ⎞–=

Ep

K 3100=

α2

Wavelength [μm]

Abs

orpt

ion

α2

[m-1

·10-1

2 ]0

-6

-101.200 1.300 1.400 1.500 1.600

2

-4

-8

Fig. 2.7 The second order nonlinear absorption due to the a.c. Stark effect. The ab-sorption is negative (i.e. the absorption decreases) due to the quadratic a.c. Stark effect, which causes a blue shift of the absorption edge. The ef-fect is most pronounced near the band edge.

Δn'2

2. Theory of MZI Based All-Optical Devices

56

rived from a Kramers-Krönig transformation over the absorption curve of Fig. 2.7.Of interest is the large amplification near the bandedge, which might be of practical importance. Nevertheless, these calculated refractive-index changes might be too small. We can compare them with more elaborate calculations performed for GaAs semiconductors [2.41], which yield values around 1·10-16 m2/W near the bandedge. The calculation in [2.41] included the quadratic a.c. Stark effect and other nonline-arities. An idea of the quantitative influence is obtained by calculating the refractive-index change with typical control-signal intensities as needed in all-optical devices for switching. Such control-signal intensities have been experimentally determined as described in chapter 6. E.g. for a dual-order mode configuration all-optical MZI switch we needed 14 dBm control-signals. The power was measured in the fibre. These 14 dBm correspond to an average control-signal intensity of ~2·109W/m2 in the device. This demonstrates, that the corresponding refractive index changes do not contribute noticeably to our nonlinear effects, even when the higher second-or-der refractive index changes as reported in [2.41] are taken into account.

5. Time Regimes of Nonlinear EffectsThe high-speed switching dynamics of the all-optical devices are determined by the carrier dynamics of the material. The cutoff frequency for a given device is limited by the relaxation lifetime of the underlying process by [2.42]

. (2.49)

(A Fourier transform over an exponentially decaying function delivers a Loretzian spectrum with Eq. (2.51).) One distinguishes between two time regimes.

Wavelength [μm]1.200 1.300 1.400 1.500 1.600N

onl.

Ref

ract

.n2

[m2 /

W·1

0-18 ]

0

-2

2

-1

Fig. 2.8 Second-order nonlinear refraction due to the a.c. quadratic Stark effect.

ΩCut-offτ

ΩCut-off1

2πτ---------=

t τ⁄–( )exp

2.1 Optical Nonlinearities

57

On the one hand there is the long-time regime governing changes of gain and absorp-tion of the semiconductor caused by interband transitions. On the other hand, intra-band nonlinear processes are governed by shorter relaxation times.

Interband-Relaxation TimesNonlinear effects governed by the interband-relaxation time regime are bandfilling, bandgap shrinkage and plasma effect. The relaxation time in this regime is deter-mined by the current injection density, the spontaneous emission (ASE) [2.42] and the stimulated emission [2.43] according to the formula for the effective carrier life-time in SOAs [2.43]

. (2.50)

The four terms of this equation will be discussed in more detail in subsection 3. of Chapter 2.3. They can be added according to the law of addition of reciprocal life-times. The contribution from the first three terms is the differential carrier lifetime

. (2.51)

The different contributions to the carrier lifetimes can be deduced by differentiating the steady-state carrier rate equation (2.86) with respect to the carrier densities.Eq. (2.50) yields guidelines to shorten carrier relaxation times. These demand, that the SOAs have to be operated with [2.42, 2.44]

• large current injections in odrer to obtain large powers in ,• high optical powers because of the large

Furthermore the SOAs should exhibit• large SOA lengths because of the larger ASE and strong signal amplification,• large optical confinement factors because of the large and ,• large differential gains (according the derivation of after in Eq. (2.50) )

Finally, the carrier lifetime may be influenced by the doping densities [2.45].In Table 2.2 we have depicted some typical relaxation times and - if at disposition - the experimental conditions. Measured effective carrier relaxation times range from ~80 fs to 10 ns. The letter L in this table refers to the amplifier length, I to the bias current and Pin to the probe power. As expected the differential carrier relaxa-tion time decreases with increasing driving currents, amplifier lengths and signal power. Experiments described in Ref. [2.42] demonstrate the role of the amplifier length and the influence of the signal power on the device speed. When no probe pulse is injected and the amplifier is operated at maximum current injection, they find for a 500, 750, 1000 and 1500 μm long 1.55 μm bulk InGaAsP amplifiers ex-

1τe----

dRtotdN

------------ A BN CN2 ddN------- vg g

m ±,∑ SASE m,

± vggmSSig+⎝ ⎠⎜ ⎟⎛ ⎞

+ + += =

τd

1τd----- A BN CN2+ +=

NSSig

N SSigg N

τe

2. Theory of MZI Based All-Optical Devices

58

hibit decreasing differential relaxation times of 480, 270, 170 and 100 ps respective-ly. Under injection of an additional 0.7 mW probe signal the respective relaxation times decrease further to 320, 210, 150 and 90 ps. Disadvantages of long SOAs, that must be taken into consideration are [2.43]• shrinkage of the spectral bandwidth• large transit times of the signals and• gain clamping due to lasing of the ASE from small facet reflectivities at low car-

rier densities

Intraband-Relaxation TimesThe shorter intraband nonlinear processes are partly due to the carrier-carrier scat-tering processes with relaxation times of 60-100 fs. This intraband carrier relaxation time determines the broadening of the material gain curve according to Eq. (2.18)and also governs the spectral hole burning (SHB).Hot-carrier distributions in the conduction and valence bands relax somewhat slow-er. Measurements yield typical carrier relaxation times of about 600 fs in proc-esses which cause carrier heating (CH).

Table 2.2: Carrier Relaxation Times in InGaAsP Lattice Matched to InP

Relaxation Process Structure / Experimental Conditions Relaxation

Time Ref.

Differential Carrier

Lifetimes

1.55 bulk, L=250 μm, I=0.1-11 kA/cm2]1.55 bulk, L=500-1500 μm, I=17 kA/cm2

1.55 bulk, L=1500 μm, I=17 kA/cm2, Pin=1 mW

9-1 ns500-100 ps100-90 ps

Ch. 3[2.42][2.42]

Carrier-Carrier

Scattering

1.55 bulk1.30 bulk1.55 bulk

78 fs95 fs70 fs

[2.34][2.34][2.46]

Carrier-Phonon

Scattering

1.50 bulk (Estimated)1.50 bulk L=475 μm, (Experiment)1.50 MQW, L=900 μm, (Experiment)1.55, bulk, L=500 μm, 1.55, bulk, L=320 μm, I=10 mA

250 fs600 fs

1000 fs650 fs700 fs

[2.22]""

[2.47][2.48]

τ2

τh

2.1 Optical Nonlinearities

59

6. Qualitatively Calculated Refractive-Index Changes under Control-Signal Injections

Finally, we apply the calculated refractive-index change to a SOA, whose carriers are depleted by the injection of a control pulse. It is obvious, that the following discus-sion can only give an indication of the effects that take place since the underlying model is simple. In addition, signal-intensity and carrier-density dependencies along the propagation direction of the light are neglected.

Wavelength [μm]0.800 1.000 1.200 1.400 1.600 1.800

-0.002

0.002

0.0

0.004

Ref

r.In

dex

Cha

nge

Δn‘

Fast Gain-Compression

S=0.4·1021m-3S=1.0·1021m-3S=2.0·1021m-3

All “Slow” Effects

Wavelength [μm]0.800 1.000 1.200 1.400 1.600 1.800

-0.002

0.002

0.0

-0.004Ref

r.In

dex

Cha

nge

Δn‘ 0.004 All Effects

Plasma-EffectBandfillingBandgap-Shrinkage

(a)

(b)

Fig. 2.9 Qualitative calculation showing the refractive index changes due to a control-signal injection into a 500 μm long SOA. (a) The long lasting car-rier related nonlinear effects. (b) Comparison of the refractive index changes due to the slower interband carrier effects with the faster gain compression effect from intraband carrier effects. The depicted photon densities are needed for switching with the slower effects when the control pulses are 1.5, 3, and 8 ps long.

Effects

2. Theory of MZI Based All-Optical Devices

60

The induction of a phase shift of in a 500 μm long SOA with light at 1.55 μm re-quires a refractive-index change of

, (2.52)

if the confinement factor is ~0.4. The qualitative refractive-index change calculated under the assumption that a con-trol signal induces a carrier depletion of to is shown in Fig. 2.9(a). The total contributions from the long-lasting effects such as plasma, bandfilling and bandgap-shrinkage effects is depicted as a solid line in Fig. 2.9(a). From experiments we know, that the refractive-index changes have a maxi-mum near 1.50 μm for 1.55 μm-InGaAsP amplifiers. Therefore, the total calculated refractive-index change should vary more around 1.5 μm. To estimate the contribu-tions of the ultrashort intraband-relaxation effects we have also calculated the refrac-tive-index changes due to the gain compression effects as shown in Fig. 2.9(b). The photon density of S=0.4·1021 m-3 is typical for our all-optical 10 GHz experiments with control pulses having a FWHM of 8 ps (see chapter 6). The effect becomes more pronounced when the FWHM is reduced to 3 and 1.5 ps and when the corre-sponding photon density is increased to S=1.0·1021m-3 and S=2.0·1021m-3 in order to maintain the energy of the control-pulse. However, in contrary to the effects illus-trated in Fig. 2.9(a) their lifetime is mainly limited by the duration of the control-pulses.

7. Concluding Remarks on XPM All-Optical DevicesIn this section was demonstrated that the relevant XPM optical nonlinearities in SOAs are due to:

• bandfilling• bandgap-shrinkage• plasma-effect• gain-compression.

The intensity dependent third-order nonlinear Kerr effects require signal powers several orders of magnitude stronger and can therefore be neglected in general.All of these effects can be described with the effective permittivity so that the wave-equation to be solved simplifies from Eq. (2.6) onto

. (2.53)

In the next section we shall use this equation to describe and understand the nonlin-

π

Δn' 1 Γ⁄ λ2L------⋅ 0,04≅=

N 3,0 1024 m 3–⋅= N 2,75 1024 m 3–⋅=

εeff

ΔEεeff

c2-------- ∂2

∂t2-------E⋅– 0=

2.1 Optical Nonlinearities

61

ear effects in a SOA.

2. Theory of MZI Based All-Optical Devices

62

2.2 The Semiconductor Optical Amplifier Equations

This section is devoted to the effect of the nonlinearities on the phase and amplitude of a signal in a SOA. The main interest is on the change of the refractive index caused by the gain change induced by a strong signal.

1. Introductory Definitions1.1 Guided-Wave OpticsSince we are now considering light guided as optical modes in waveguide structures we have to adjust the concept of refractive indexes and material gain expressions.Instead of the refractive index we now use an effective refractive index . It is some sort of weighted average of the various refractive indexes of the waveguide within the optical mode propagates [2.49]. In addition, in a dispersive medium an optical mode propagates with the group velocity [2.4]

. (2.54)

This yields an effective group refractive index

, (2.55)

where is the vacuum wavelength and . The last term becomes relevant for media with a strong absorption, or a strong material gain . In our InGaAsP diodes operated in the gain regime the term can be neglected.Analogously the imaginary part of the refractive index, i.e. the material gain has to be modified, since now only the active layer contributes to the gain. The confine-ment factor gives this contribution of the mode in the active region

, (2.56)

n′ n′eff

vg

vg Re dωdkneff

------------⎝ ⎠⎛ ⎞≡ c n′eff λ

dn′eff λ( )dλ

----------------------–

λ4

16π2------------ Γdαdλ-------⎝ ⎠

⎛ ⎞ 2

n′eff λdn′eff λ( )

dλ----------------------–

------------------------------------------+

⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

1–

=

n′eff g,

n′eff g,cvg-----≡ n′eff λ( ) λ

dn′eff λ( )dλ

----------------------–

λ4

16π2------------ Γdαdλ-------⎝ ⎠

⎛ ⎞ 2

n′eff λdn′eff λ( )

dλ----------------------–

------------------------------------------+=

λ kneff2π λ⁄( )neff=

α gm–=

gm

Γ

F x y,( ) 2 xd yd0

d

∫0

w

∫

F x y,( ) 2 xd yd∞–

∞

∫∞–

∞

∫------------------------------------------------------=

2.2 The Semiconductor Optical Amplifier Equations

63

when represents the waveguide-mode distribution and and the dimen-sions of the active regions on these conditions. The corresponding optical modal gain

is

. (2.57)

denotes the internal-absorption losses, with contributions from the active medi-um and the cladding medium according to .

1.2 The Alpha-Factor ConceptSince the refractive-index changes and the absorption coefficients are related by the Kramers-Krönig relations, it is useful to introduce the so-called alpha- or linewidth-enhancement factor. The alpha factor is defined as the ratio between small refractive-index changes and small gain changes [2.50]

, (2.58)

where N is the carrier density and the vacuum wavelength. The left-hand side is the alpha factor with respect to an optical mode within a given waveguide structure. It is often identical with the alpha factor in the active waveguide layer, since

. (2.59)

However, this approximation is wrong in all those cases, where the confinement fac-tor considerably changes due to the carrier-related refractive-index changes. In such cases the two definitions yield different expressions [2.51].The value of the alpha factor depends on wavelength , carrier density , tem-perature and the power coupled into the active cavity. We have investigat-ed in chapter 3.3 and have found for 1.55 μm bulk InGaAsP, that

• increases with the wavelength (Fig. 3.17)• is independent of the carrier density for small wavelengths

but increases with carrier density for longer wavelengths (Fig. 3.17)• does not change with temperature around the gain peak (Fig. 3.17)

In addition, we can predict a power dependence of the alpha factor due to gain com-pression. This implies that

• increases with the photon density after

. (2.60)

F x y,( ) w d

g

g Γgm αint–=

αintαint Γαact 1 Γ–( )αclad+=

αN 4πλ

------n′eff∂ N∂⁄

g∂ N∂⁄------------------------–≡ 4π

λ------

n′act∂ N∂⁄gm∂ N∂⁄

-------------------------– αn≡≅

λ

n′eff∂ N∂⁄g∂ N∂⁄

------------------------Γ n′act∂ N∂⁄Γ gm∂ N∂⁄

-----------------------------≅n′act∂ N∂⁄gm∂ N∂⁄

-------------------------=

αN λ NTcav Pin

αN λ N Tcav Pin, , ,( )

αNαN

αN

αN S

αN λ N Tcav S, , ,( ) αN λ N Tcav, ,( ) 1 ∈totS+( )=

2. Theory of MZI Based All-Optical Devices

64

The formula is derived by insertion of Eq. (2.26) into the definition (2.58). We be-lieve that Eq. (2.60) is a good approximation, since carrier density changes due to gain compression have on a narrow spectral range, a strong effect on the material gain. However, carrier density changes due to gain compression have a small effect onto the refractive index changes, since refractive index changes arise from a Kram-ers-Krönig transformation of the material gain over the whole spectrum. This does not significantly change. Thus, for finding the power dependence of the , given by the fraction in Eq. (2.58), one has mainly to consider the power dependence of the gain and not the power dependence of the refractive index. Eq. (2.60) is similar to the expression derived by Mortier et al. in Ref. [2.52].The Kramers-Krönig relation suggest that there exists an analogous parameter relat-ing small variations of the real and imaginary parts of the refractive index due to changes of the carrier temperature for constant carrier injection [2.53-2.55]

. (2.61)

The alpha factor is a key parameter for high-speed modulation and high-power ap-plications, since it determines the frequency chirp due to fluctuations of the carrier densities [2.56, 2.26]

. (2.62)

is the modal gain integrated over the whole laser cavity defined later by Eq. (2.82). is the frequency chirp of the input-pulse and the frequency chirp of the output-pulse.Moreover, the alpha factor describes the fundamental linewidth enhancement of modes in semiconductor lasers [2.50, 2.17]

. (2.63)

Expressions for the rate of spontaneously emitted photons coupling into a lasing mode are given below. Eq. (2.63) postulates two contributions to the linewidth. On the one hand instantaneous fluctuations result from each spontaneous photon emis-sion. On the other hand each emitted photon changes the emitted power, which changes gain and carrier density. In turn this affects the refractive index and conse-quently the optical phase according to the alpha factor. The resulting delayed phase fluctuation causes a broadening of the peak.In semiconductor lasers small alpha factors are required to keep chirping small, since

αN

αT 4πλ

------n′eff∂ T∂⁄

g∂ T∂⁄-----------------------–≡ 4π

λ------

n′act∂ T∂⁄gm∂ T∂⁄

------------------------– αt≡≅

Δνout Δνin1

2π------

τ∂∂φ– Δνin

αN4π-------

τ∂∂h+= =

h τ( )Δνin Δνout

ΔvModevgRSp

2πSASE------------------ 1 αN

2+( )=

RSp

2.2 The Semiconductor Optical Amplifier Equations

65

the chirp causes signal degradation and limits modulation speed and transmission distances in optical communication systems. Furthermore small alpha factors permit smaller linewidths, which allow to transmit more signal channels within a given spectrum.In contrast we shall show in chapter 5.2, that high alpha-factor MZI-based all-optical switches are advantageous. It is important that this will not imply an increased chirp degradation, since the total phase changes for switching an all-optical MZI device must always be π.

1.3 Internal and External Quantum EfficienciesThe internal quantum efficiency is defined as the fraction of injected carriers con-verted into photons [2.17]

. (2.64)

It is the ratio of the fraction of radiatively combining electronic transitions generat-ing photons, such as spontaneous emission and stimulated emission, versus the frac-tion of totally recombining electronic transitions.Above threshold, the fraction of injected carriers converted into photons and coupled out through the laser facets is called the external quantum efficiency [2.17]

, (2.65)

where is the power coupled out through one laser facet.External and internal quantum efficiency are related by

. (2.66)

is the mirror losses averaged over the whole device and the internal losses of the device under test.

ηiRradiating

Rtot-----------------------=

ηext2qhν------

dPoutdI-------------⋅=

Pout

ηext ηiαm

αm αi+-------------------⋅=

αm αi

2. Theory of MZI Based All-Optical Devices

66

2. The Propagation of an Amplified Signal in an SOAThe propagation of the electromagnetic field inside the amplifier is governed by the wave equation (2.53).

, (2.67)

where the effective dielectric function contains the contributions from nonlinear ef-fects as discussed in the previous section

. (2.68)

It is useful to express the nonlinear refractive-index change in terms of the alpha fac-tor and to neglect all higher-order terms [2.26]

. (2.69)

Equations (2.67) and (2.69) provide a general theoretical framework of the propaga-tion of optical pulses in semiconductor laser amplifiers. In practice, one can assume, that one deals with ideal travelling-wave amplifier without any back reflections at the facets. Under these conditions one can write the electric field inside the amplifier as

, (2.70)

where is the polarization unit vector, is the normalized-mode distribution and the slowly-varying envelope associated with the optical pulse. If one substitutes Eq. (2.70) into (2.67), neglects the second derivatives of and in-tegrates over the transverse dimensions, one finds

, and (2.71)

. (2.72)

The solution of Eq. (2.71) yields the transverse distribution and the effective

ΔEεeff

c2-------- ∂2

∂t2-------E⋅– 0=

εeff x y,( ) n2 n' x y,( ) Δn' x y,( ) igm x y,( )

2kvac--------------------–+⎝ ⎠

⎛ ⎞2

= =

α

εeff x y,( ) n'2 x y,( ) n' x y,( )kvac

----------------- α i+( )gm x y,( )

2kvac--------------------–≅

E x t,( ) 12---e F x y,( )A z t,( ) i kvakneffz ωt–( )[ ] c.c.+exp⋅{ }=

e F x y,( )A z t,( )

A z t,( )

x2

2

∂

∂ Fy2

2

∂

∂ F n2 x y,( ) neff2–( )kvak

2F+ + 0=

zddA 1

vg----- td

dA+ i12---Γ α i+( )gmA=

F x y,( )

2.2 The Semiconductor Optical Amplifier Equations

67

mode index . Eq. (2.72) governs the evolution of the pulse amplitude along the propagation direction of the light. To gain information on the phase evolution of the electromagnetic field within the amplifier one can separate the amplitude and phase with the aid of

. (2.73)

is the photon density which is related to the electric-field amplitude as described in App. E. This separation results in

, and (2.74)

. (2.75)

In Eqs. (2.74), (2.75) we have replaced by in order to take ac-count of the internal losses. In should be noticed, that the term in Eq. (2.75) was originally introduced in Eq. (2.69) to write the refractive-index changes in a dif-ferent from. Therefore, in the most general form it takes into account refractive-in-dex changes due to carrier- and temperature-related gain changes

. (2.76)

A further simplification can be made by applying the transformation

, (2.77)

where the reduced time is measured in a reference frame moving with the pulse. In summary this yields for and to

, and (2.78)

. (2.79)

The important result for our all-optical devices is, that the induced phase changes are proportional to the alpha factors and to the change of the gain.With these equations we have now tools to study the effect of a control pulse on a

neff

A S e⋅iϕ

∼

S

S∂z∂

----- 1vg----- S∂

t∂-----+ gS=

z∂∂ϕ 1

vg-----

t∂∂ϕ+ 1

2---αg–=

Γgm g Γgm αint–=αg

Δn

αg αN N∂∂g

TΔN αT T∂

∂g

NΔT+ αNg T αT

g∂T∂

------N

ΔT+= =

τ t z vg⁄–=

τS z τ,( ) ϕ z τ,( )

S∂z∂

----- gS=

z∂∂ϕ 1

2---αg– 1

2--- αNg T αT

g∂T∂

------N

ΔT+⎝ ⎠⎛ ⎞= =

2. Theory of MZI Based All-Optical Devices

68

weak input signal before and after the switching processes in an all-optical XPM de-vice. We explicitly exclude the short time interval, where the strong control pulse in-duces time-dependent material-gain changes. This assumption allows us to neglect the time dependence of the modal gain i.e. . A derivation including the time dependence of has been given in Ref. [2.26]. In this derivation one writes an additional differential equation for . The integration of Eq. (2.78) and (2.79) over the amplifier length results in

and (2.80)

, (2.81)

where we have replaced the photon density by the corresponding signal power with the aid of relation (E.8) of App. E. The function is defined by

, (2.82)

where indicates the amplifier length. It represents the integrated gain at each point of the pulse profile.If one assumes in addition, that is constant over the integration interval, or in other words, if one cuts the long amplifier into small discrete intervals wherein the ma-terial gain is constant along the propagation direction of light, i.e. , one finds the familiar single-pass gain equation of an amplifier

with (2.83)

with the single pass gain of the amplifier. The related phase change can be derived from Eq. (2.81)

. (2.84)

These are the basic equations needed for a preliminary calculation of all-optical XPM effects in a multisection-amplifier model. We shall use these equations to study extinction-ratio improvement mechanisms in all-optical switches [Chapter 5.2]. We have also applied them to a multisection model to study the extinction ratios under dynamic conditions [2.57]. The alpha factor includes all nonlinear effects relevant to our all-optical devices, i.e. bandfilling, plasma effect, bandgap-shrinkage. Further more, it includes thermal effects if one takes into account Eq. (2.76).

g z t,( ) g z( )=g

g

Pout τ( ) Pin τ( ) h τ( )[ ]exp⋅=

ϕout τ( ) φin τ( )– 12---αh τ( )–=

S Ph τ( )

h τ( ) g z τ,( ) zd0

L

∫=

L

gΔz

g z( ) g=

Pout Pin G Δz( )⋅= G z( ) egΔz=

G

Δϕ 12---αg Δz⋅–=

α

2.2 The Semiconductor Optical Amplifier Equations

69

3. A More Complete SolutionIn the previous section we used simple formulas to estimate the effect of the optical nonlinearities of an optical control signal on a weak signal. For completeness we briefly outline a more thorough solution [2.58][2.59].The three semiconductor-device equations to be solved are the Poisson equation, which couples the electrical potential with the charge density within the semicon-ductor, and the two carrier continuity equations

(2.85)

(2.86)

(2.87)

with . The current density vectors are

, and (2.88)

(2.89)

with the carrier mobilities and and the drift diffusion coefficients and .

The different recombination terms comprehend• The Schockley-Read-Hall term

(2.90)

For simulation of SOAs it usually suffices to use the approximation on the right-hand side since the intrinsic electron and hole densities and are small in comparison to the injected carriers [2.60].

• the spontaneous band-to-band recombination

(2.91)

ψ

∇ ε∇ψ⎝ ⎠⎛ ⎞⋅ q N P– NA

- ND+ –+( )=

t∂∂N 1

q---∇jN RSRH RSpTot RAuger RASE RSig+ + + +( )–=

t∂∂P 1

q---– ∇jP RSRH RSpTot RAuger RASE RSig+ + + +( )–=

E ∇ψ–=

jN q NμNE DN∇N+⎝ ⎠⎛ ⎞=

jP q NμPE DP– ∇P⎝ ⎠⎛ ⎞=

μN μP DPDN

RSRHNP N1P1–

τp N N1+( ) τp P P1+( )+------------------------------------------------------------ A NP⋅≅=

N1 P1

RSpTot Bsp NP N1P1–( ) B NP⋅≅=

2. Theory of MZI Based All-Optical Devices

70

• the Auger recombination term

(2.92)

• the amplified spontaneous emission (ASE) terms

(2.93)

with the ASE photon densities and coupling into the modes m propagating in the forward and backward direction.

• the terms from the amplified signals, e.g. input-signal, control-signal,...

(2.94)

The optical equations to be solved are:the differential equation for the mode profile

, (2.95)

the differential equations for the photon densities and phases of the amplified signal

, and (2.96)

, (2.97)

as well as the photon-density of the ASE

. (2.98)

In Eq. (2.98) is described by the photon-density rate of Eq. (2.74), whereas the spontaneous-emission term - the origin of the ASE - was added. The differ-ential equation has to be solved under consideration of the boundary conditions posed by the amplifier facets. The term is the spontaneously emitted photon rate

RAuger CA N P+( ) NP N1P1–( )= C NP( )3⋅≅

RASE vg gm SASE,m+ SASE,m

-+( )

mode pol+∑=

SASE,m+ SASE,m

-

RSig vg g⋅ mSSig=

x

2

∂∂ F

y

2

∂∂ F n2 x y,( ) neff

2–( )k2F+ + 0=

S∂z∂

----- 1vg----- S∂

t∂-----+ gS=

z∂∂ϕ 1

vg-----

t∂∂ϕ+ 1

2---αg–=

SASE,m±∂

z∂------------------- 1

vg-----

SASE,m±∂

t∂-------------------+ gSASE,m

± Rsp+=

SASE,m±

RSp

Rsp

2.2 The Semiconductor Optical Amplifier Equations

71

coupling into one lasing mode and is given by

(2.99)

with the population-inversion factor

, (2.100)

where is the photon energy and the quasi Fermi-separation level.

is the product of the spontaneous emission per volume, per time and per wavelength as derived from the Einstein relations [2.61], the coupling factor , which gives the spontaneous emission propagating into one direction and coupling into the waveguide [2.62], and the spectral width over the wavelength range of one mode

, with (2.101)

, (2.102)

and (2.103)

. (2.104)

In the literature a different definition is often used. There the coupling factor rep-resents the rate of the total spontaneous emission coupling into one lasing mode

. (2.105)

This factor has values around 1·10-5.

RSpΓ gm nsp⋅ ⋅2 AeffL( )

--------------------------=

nSp 1 E eV–kBT----------------exp–⎝ ⎠

⎛ ⎞ 1–=

E eV Eg Efc Efv+ +=

Rsp rspβ

λδ

RSp β λδ Γrsp c n'g eff,⁄⋅ ⋅ ⋅=

rsp8πn′2c

λ4----------------- gm nsp⋅ ⋅=

β 12--- λ

n′----⎝ ⎠

⎛ ⎞ 2 14πAeff----------------⋅ ⋅=

λδ λ2

2n'g eff, L---------------------=

β∗

RSp β∗ B NP⋅ ⋅=

2. Theory of MZI Based All-Optical Devices

72

2.3 The MZI-SOA Transfer Functions

In this section we present the transfer functions of an MZI based all-optical switch. As a result we find that the alpha factor is the main material parameter affecting the transfer characteristics. In addition we have come to the conclusion that the derived transfer characteristics remain valid even for long SOAs with varying modal gain along the propagation direction of light. The knowledge of the transfer characteris-tics is important for the design of optimized all-optical devices with near to ideal on-off ratios.A generalized MZI-SOA all-optical device is illustrated in Fig. 2.10. An input signal

or is mapped onto the bar-output ports ( ) or cross-output ports , depending on the phase relations within the MZI cavity. The phases within the MZI cavity are modulated with the control signals and , that ex-ploit the SOAs nonlinear characteristics to induce a phase shift. The control signals are introduced through couplers . The MZI is formed by the two splitters and

. Their splitting ratios may be asymmetric. They are defined as

(2.106)

SA

SOA2

PC2

Pin,1

Pin,2

P=,1

PX,1SB

C

SOA1Inputs

1

2

Outputs

1

2

PC1

PX,2

P=,2

Phase-Shifter 2

Phase-Shifter 1

I1

I2

Δφ1

Δφ2

C

Fig. 2.10 Generalized MZI-SOA all-optical switch with phase shifters to adapt phase-shift offsets and splitters and that allow asymmetric split-ting ratios. Depending on the phase relations in the MZI arms, the data signal is mapped onto the cross and bar-output guide while is mapped onto its respective cross and bar-output guides

. The phase relations within the MZI-arms is changed when a control signal and/or is introduced through the couplers into SOA1 and/or SOA2, respectively.

SA SB

Pin1 PX,1 P=,1Pin2 PX,2

P=,2PC1 PC2 C

Pin 1, Pin 2, P=,i i 1 2,=PX,i

PC1 PC2

C SASB

rAsA

1 sA–--------------≡ rB

sB1 sB–--------------≡

2.3 The MZI-SOA Transfer Functions

73

when and are the bar transmission and and are the cross-trans-mission probabilities. The two SOAs providing the nonlinearities required for switching are placed on the MZI arms. Each provides an amplifier gain of , with

. In addition, two phase-shifters that can induce individually phase-shift off-sets of and are placed on each of the MZI arms. The functions for describing the transfer of the input signals to their respective cross and bar outputs are derived by the transfer-matrix method. A detailed derivation is given in chapter 5.2 under the assumption of a constant material gain over the am-plifier. Here we demonstrated, that the transfer equations of chapter 5.2 remain valid even when the gain varies along the amplifier cavity. This is important, since control and data signals are amplified in the SOA and may thus lead to gain saturation at the SOA output.The relative phase relations between the MZI arms have to be changed for switching a signal from one to the other output port. Contributions due to a relative phase shift in the phase shifters are described by

, (2.107)

where and are the relative phase shifts induced in the respective phase-shifters. Contributions due to a carrier related effect in the SOAs are given by

, (2.108)

where are the carrier-related refractive-index changes due to a variation of the current bias and are the carrier related refractive-index changes due to a control signal in the respective SOAs with indices . However, any phase-shift with-in the SOAs is interrelated with an amplifier-gain change as implied by the alpha fac-tor. The total modal-gain change due to a total phase shift is according to Eq. (2.81)

, (2.109)

which in turn changes the total amplifier gain. The change of the amplifier gain from its original value to the new gain can be evaluated by inserting Eq. (2.109)into (2.80)

. (2.110)

On the basis of these relations interrelating the amplifier-gain changes with the in-

sA sB 1 sA– 1 sB–

Gjj 1 2,=

Δφ1 Δφ2

Δφ Δφ1 Δφ2–=

Δφ1 Δφ2

Δϕ12 Δϕ1I Δϕ1

C+( ) Δϕ2I Δϕ2

C+( )–≡

ΔϕjI

ΔϕjC

j 1 2,=

ΔϕjI C⁄

ΔϕjI C⁄ 1

2---αh τ( )–=

G0 Gj

Gj G0 eh τ( )⋅ G0 e

2ΔϕjI C⁄

α------------------–

⋅= =

2. Theory of MZI Based All-Optical Devices

74

duced phase shifts, we can derive MZI-SOA transfer characteristics which require only the phase shift induced by the control signal as input parameter. The intensity related-amplifier saturation and their effects on the output parameters are deter-mined by the induced phase shift.With the definitions for the bar and cross output powers illustrated in Fig. 2.10, one finds for the transfer function of a data-signal from input-guide 1

(2.111a)

and for a data-signal from input-guide 2

, (2.111b)

with

(2.112a)

(2.112b)

(2.112c)

(2.112d)

where the coupling variable is defined as

. (2.113)

The SOA-induced phase shifts can have their origin in an asymmetric SOA current bias or a control-signal coupled into one of the SOAs

. (2.114)

For a complete description of the MZI-SOA transfer function it suffices to know the alpha factor as the only material parameter including carrier as well as temperature related effects, and to know the configuration parameters. The configuration param-eters are the splitter ratios , , the phase-shift offsets and eventually asym-metrically biased SOA currents described by of Eq. (2.110). The effect of the

P=,1 t112Pin 1,= PX 1, t21

2Pin 1,=

PX 2, t122Pin 2,= P=,2 t22

2Pin 2,=

t112 C 1 2– rArB Δφ Δϕ12+( )e

Δϕ12– α⁄cos rArBe

2Δϕ12– α⁄+⎝ ⎠

⎛ ⎞=

t212 C rB 2 rArB Δφ Δϕ12+( )e

Δϕ12– α⁄cos rAe

2Δϕ12– α⁄+ +( )=

t122 C rA 2 rArB Δφ Δϕ12+( )e

Δϕ12– α⁄cos rBe

2Δϕ12– α⁄+ +( )=

t222 C rArB 2 rArB Δφ Δϕ12+( )e

Δϕ12– α⁄cos– e

2Δϕ12– α⁄+( )=

C

C c2 G2 1 sA–( ) 1 sB–( )⋅⋅=

Δϕ12

Δϕ12 Δϕ12I Δϕ12

C+≡

rA rB ΔφΔϕ12

I

2.3 The MZI-SOA Transfer Functions

75

control signal is then solely described by

(2.115)

with a positive sign, when the control signal is introduced into SOA1 and a negative sign if the control signal is introduced into SOA2.

Δϕ12C

Δϕ12C

0 PC off ,

π PC on ,⎝⎜⎛

±≅

2. Theory of MZI Based All-Optical Devices

76

2.4 References

[2.1] R. W. Boyd; “Nonlinear Optics”; Academic Press Inc., San Diego, Califor-nia, 1992

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[2.3] J.W. Pedersen, J.J.G.M van der Tol, E. G. Metaal, Y. S. Oei, F.H. Groen, and I. Moerman, “Mode converting polarization splitter on InGaAsP/InP”, in Proc. ECOC’94, Firenze, Italy, Sept. 1994, pp. 661-66

[2.4] F. K. Kneubühl; “Oscillations and Waves”; Springer-Verlag Berlin Heidel-berg 1997, chapter 7

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1926[2.7] D.C. Hutchings, M. Sheik-Bahae, D.J. Hagan, E.W. Van Stryland; “Kramers-

Krönig relations in nonlinear optics: Tutorial Review”; Optical and Quantum Electronics, vol. 24, pp. 1-30, 1992

[2.8] C.H. Henry, R.A. Logan, K.A. Bertness; “Spectral dependence of the change in refractive index due to carrier injection in GaAs lasers”; J. Appl. Phys. vol. 52, no. 7, pp. 4457-4461, July 1981

[2.9] E. Burstein; “Anomalous optical absorption limit in InSb”; Phys. Rev., vol. 93, pp. 632-633, 1954

[2.10] T.S. Moss, G.J. Burrell, B. Ellis; “Semiconductor Opto-electronics”; Wiley, New York, pp. 48-94, 1973

[2.11] P.S. Zory; “Quantum Well Lasers”; Academic Press, San Diego, California 1993

[2.12] A. Yariv, “Quantum Electronics” third Ed.; John Wiley & Sons, New York, 1989

[2.13] J. Faist, F.-K. Reinhart; “Phase modulation in GaAs/AlGaAs double heter-ostructures. I. Theory”; J. Appl. Phys., vol. 67, no. 11, pp. 6998-7005, 1990

[2.14] B.R. Benett, R.A. Soref, J.A. D. A. Alamo; “Carrier-induced change in re-fractive Index of InP, GaAs, InGaAsP”; J. Quantum Electron., vol. 26, no. 1, pp. 113-122, 1990

[2.15] S. Schmitt-Rink, D.S. Chemla, D.A.B. Miller; “Linear and nonlinear optical properties of semiconductor quantum wells”; Advances in Physics, vol. 38, no. 2, pp. 89-188, 1989

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[2.16] P.A. Wolff; “Theory of the band structure of very degenerate semiconduc-tors”; Phys. Rev., vol. 126, pp. 405-412, 1962

[2.17] G.P. Agrawal, N.K. Dutta; “Long-wavelength semiconductor lasers”; Van Nostrand Reinhold, 1986

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[2.19] S. Adachi; “Physical Properties of III-V semiconductor compounds”; John Wiely & Sons, New York, 1992

[2.20] A. Katz; “Indium Phosphide and related materials: Processing, technology, and devices”; Artech House, Norwood, USA, 1992

[2.21] J. Schoenes; “Solid State Optics”; ETH lecture notes in quantum electronics[2.22] K.L. Hall, G. Lenz, A.M. Darwish, E.P. Ippen; “Subpicosecond gain and in-

dex nonlinearities in InGaAsP diode lasers”; Optics Communications, vol. 111, pp. 589-612, 1994

[2.23] F. Girardin, G.-H. Duran, P. Gallion; “Experimental investigation of the rel-ative importance of carrier heating and spectral-hole-burning on nonlinear gain in bulk and strained multi-quantum well 1.55 mm lasers”; Appl. Phys. Lett., vol. 67, no. 6, pp. 771-773, Aug. 1995

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[2.25] M.J. Adams, M. Osinski; “Influence of spectral hole burning on quaternary laser transients”; Electron. Lett., vol. 19, pp. 627-629, 1983

[2.26] G.P. Agrawal, N.A. Olsson; “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers”; J. of Quantum Electron., vol. 25, no. 11, pp. 2297-2306, Nov. 1989

[2.27] G.P. Agrawal; “Effect of gain and index nonlinearities on single-mode dy-namics in semiconductor lasers”; J. of Quantum Electron., vol. 26, no. 11, pp. 1901-1909, Nov. 1990

[2.28] A. Mecozzi, J. Mørk; “Saturation induced by picosecond pulses in semicon-ductor optical amplifiers”; J. Opt. Soc. Am. B; vol. 14, no. 4, pp. 761-770, April 1997

[2.29] F. Girardin, St. Pajarola, G. Guekos; “Nonlinear parameters of bulk InGaAsP photonic devices’"; Proc. Materials for Nonlinear Optics, EOS topical meet-ing, 1997

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[2.30] St. Pajarola; “Dual-polarization external cavity diode laser for the optical generation of millimetre waves”; Diss. ETH No. 12633, 1998

[2.31] J. Eom, C.B. Su, J. LaCourse, R.B. Lauger; “Simultaneous measurement of spontaneous emission rate, nonlinear gain coefficient, and carrier lifetime in semiconductor lasers using a parasitic-free optical modulation technique”; Appl. Phys. Lett., vol. 56, no. 6, pp. 518-520, Feb. 1990

[2.32] Z. Toffano, A. Destrez; “New gain compression factor determination by har-monic analysis in semiconductor lasers”; Electron. Letters, vol. 31, no. 3, pp. 202-203, Feb. 1995

[2.33] M. Willatzen, A. Uskov, J. Mørk, H. Olesen, B. Tromborg, A.-P. Jauho; “Nonlinear gain suppression in semiconductor lasers due to carrier heating”; Photon. Technol. Letters, vol. 3, no. 7, pp. 606-609, July 1991

[2.34] R. Frankenberger, R. Schimpe; “Origin of nonlinear gain saturation in index-guided InGaAsP laser diodes”; Appl. Phys. Lett., vol. 60, no. 22, pp. 2720-272, June 1992

[2.35] Gerhard Hagn; “Herstellung und Charakterisierung von integriert optischen InP-Wellenleiterschaltern”; ETH-Diploma Thesis, March 1994

[2.36] S.J.B. Yoo; “Polarisation independent, multi-channel, multi-format wave-length conversion by difference-frequency generation in AlGaAs waveguides”; Proc. ECOC’98, Madrid, Spain, pp. 653-654, Sept. 1998

[2.37] J. G. Mendoza-Alvarez, L.A. Coldren, A. Alping, R.H. Yan, T. Hausken, K. Lee, K. Pedrotti; “Analysis of depletion edge translation lightwave modula-tors”; J of Lightwave Technol., vol. 6, no. 6, pp. 793-807, June 1988

[2.38] M. Sheik-Bahae, D.C. Hutchings, D.J. Hagan, E.W. Van Stryland; J. Quan-tum-Electron, vol. 27, pp. 1296, 1991

[2.39] D.C. Hutchings, E.W. Van Stryland; “Nondegenerate two-photon absorption in zinc blende semiconductors”; J. Opt. Soc. Am. B, pp. 2065-2074, Nov. 1992

[2.40] D.C. Hutchings, M. Sheik-Bahae, D.J. Hagan, E.W. Van Stryland; “Kramers-Krönig relations in nonlinear optics”; Optical and Quantum Electronics, vol. 24, pp. 1-30, 1992

[2.41] D.C. Hutchings, B.S. Wherett; “Theory of the dispersion of ultrafast nonlin-ear refraction in zinc-blende semiconductors below the band edge”; Phys. Rev. B, vol. 50, no. 7, pp. 4622-4630, Aug. 1994

[2.42] F. Girardin, G. Guekos; “Gain recovery of bulk semiconductor optical ampli-fiers”; Photon. Technol. Letters, vol. 10, no. 6, pp. 784-786, June 1998

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[2.43] T. Durhuus, B. Mikkelsen, K.E. Stubkjaer; “Detailed dynamic model for semiconductor optical amplifiers and their crosstalk and intermodulation dis-tortions”; J. Lightwave Technol., vol. 10, no. 8, pp. 1056-1065, Aug. 1992

[2.44] C. Joergensen, S.L. Danielsen, K.E. Stubkjaer, M. Schiliing, K. Daub, P. Doussiere, F. Pommerau, P.B. Hansen, H.N. Pulsen, A. Kloch, M. Vaa, B. Mikkelsen, E. Lach, G. Laube, W. Idler, K. Wunstel; “All-optical wavelength conversion at bit rates above 10 Gb/s using semiconductor optical amplifi-ers”; J. Selected Topics in Quantum Electronics, vol. 3, no. 5, pp. 1168-1179, Oct. 1997

[2.45] R.K. Ahrenkiel, R. Ellingson, S. Johnston, M. Wanlass; “Recombination life-time of InGaAs as a function of doping density”; Appl. Phys. Lett., vol. 72, no. 26, pp. 3470-3472, June 1998

[2.46] K.L. Hall, G. Lenz, E. P. Ippen, J. Lighwave Technol., vol. 10, pp. 616-, 1992[2.47] L.F. Tiemeijer; “Effects on nonlinear gain on four-wave mixing and asym-

metric gain saturation in a semiconductor laser amplifier”; Appl. Phys. Lett, vol. 59, pp. 499-501

[2.48] J. Mark, J. Mørk; “Subpicosecond gain dynamics in InGaAsP optical ampli-fiers: Experiment and theory”; Appl. Phys. Lett, vol. 61, no. 19, pp. 2281-2283, Nov. 1992

[2.49] H. Kogelnik; “Theory of Optical Waveguides”; in T. Tamir (Ed.), “Guided-Wave Optoelectronics”, Springer-Verlag Berlin

[2.50] C.H. Henry; “Theory of linewidth of semiconductor lasers”; J. of Quantum Electron., vol. 18, no. 2, pp. 259-264, Feb. 1982

[2.51] K. Furuya; “Dependence of linewidth enhancement factor a on waveguide structure in semiconductor lasers”; Electron. Letters; vol. 21, no. 5, pp. 200-201, Feb. 1985

[2.52] G. Morthier, P. Vankwikelberge, F. Buytaert, R. Baets; “Influence of gain nonlinearities on the linewidth enhancement factor in semiconductor lasers”; IEE Proc. vol. 137, Pt. J, no.1, pp. 30-32, Feb. 1990

[2.53] K. Kikuchi, M. Kakui, Ch.-En Zah, T.-P. Lee; “Observation of highly nonde-generate four-wave mixing in 1.5 mm travelling wave semiconductor optical amplifiers and estimation of nonlinear gain coefficient”; J. of Quantum Elec-tron., vol. 28, no. 1, pp. 151-156, Jan. 1992

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2. Theory of MZI Based All-Optical Devices

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[2.55] G.E. Shtengel, R.F. Kazarinov, G.L. Belenky, C.L. Reynolds; “Wavelength chirp and dependence of carrier temperature on current in MQW InGaAsP-InP lasers”; J. of Quantum Electron., vol. 33, no. 8, pp. 1396-1401, Aug. 1997

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81

3 Material Parameters of 1.55 μm bulk In-GaAsP Lasers and Amplifiers

This chapter focuses on the material parameters of bulk 1.55 μm InGaAsP laser and amplifiers. We have determined the material gain, the refractive-index change and alpha-factor dependence on wavelength, carrier density and temperature. Special at-tention is paid to temperature effects which strongly influence the refractive index changes. An accurate knowledge of the parameters without disturbances from tem-perature effects is needed for the design of all-optical devices. Furthermore we give simple and reliable parametrizations for the material gain curves and the refractive-index changes as a function of wavelength and carrier density.

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

82

3.1 Material Gain

Material gains of bulk 1.55-InGaAsP were measured from very low up to below-threshold carrier densities in SOAs with active layers at a constant temperature. The material gains were determined by spectral measurement of the spontaneous-emis-sion spectrum with the Hakki-Paoli and Cassidy method. A temperature-correction mechanism was used to keep the diode-cavity at a constant temperature even for changing currents. The theoretical model of the previous chapter confirms the results of the experiments. The material data found here may be used in SOA-simulation tools.

1. IntroductionActive optical device-simulation tools require a detailed knowledge of the material-gain spectrum at all current injection densities [3.1, 3.2]. In addition, the optical gain is a key factor for the design of lasers and amplifiers (e.g. [3.3]) while the differential gain is a key parameter for the dynamic performance.The first meaningful measurements of the optical gain were reported in 1975 by Hakki and Paoli [3.4]. They obtained the gain spectrum of a double-heterostructure laser by measuring the depth of the modulation introduced in the ASE spectrum by the Fabry-Perot resonance. An improvement of the Hakki-Paoli method was achieved by Cassidy [3.5]. He eliminated the measurement inaccuracies by consid-ering a complete free spectral range (FSR) rather than spectral maxima and minima. More recently, Henry et al. [3.6] developed a method to measure gain in a laser struc-ture in which the gain spectra were derived from the spontaneous-emission spectra measured transversely to the laser cavity. This method exploits the relation of the two quantities by a general principle based on detailed balance and thermal equilib-rium. An advantage of the method is that it yields the gain spectra over a wide spec-tral range [3.7, 3.8]. On the other hand it requires further measurements for supplementation, since the spontaneous emission is not measured in terms of abso-lute values.In this study, we determine the material gain by the Hakki-Paoli and Cassidy meth-od, measure the current densities and compare the result with a simple analytical model. A special effort is made to keep a constant temperature within the amplifier cavity at all current densities. In order to gain data for extremely low as well as very high current-injection densities we performed measurements with a straight and a tilted bulk 1.55-InGaAsP diode from the same chip, positioned close to each other and processed in the same run.

3.1 Material Gain

83

2. Structures and DevicesFor the material-gain determination we used a double-heterostructure-laser diode (LD_1) with straight cleaved facets and a second double-heterostructure device (SOA_1) tilted by 10o versus the (100) direction and versus the cleave direction.1The tilted cleave allowed us to use the diode as an amplifier because tilted facets ef-ficiently reduce facet reflectivities below . The layers were grown by MOCVD and the ridge was first dry and then wet etched. Pyralin was applied to isolate the contact pads from the device. The Pyralin was then removed from the ridge to allow contacting with the gold pads. The structures of the ridge-semiconductor laser and amplifier are depicted in Fig. 3.1. The two devices were processed on the same chip in a distance of ~100 μm to each other. To obtain representative data, we addition-ally characterized a second couple of a LD and a SOA also situated close to each oth-er and from the same run. We called them LD_2 and SOA_2 2. Moreover two lasers from another run were characterized by the same method and were given the names LD_3 and LD_4 3.

1. At the Institute of Quantum Electronic the two devices were processed within Run 1483 and got the device number L593 and L593_tilt.2. The devices were processed within Run 1483 and got the device number L591 and L591_tilt3. These LDs were processed within Run 1485 and got the device number L646 and L644

10 4–

p-Doped InP Ridge(Zn: 1018 cm-3)

Quaternary 1.55 μm

Quaternary 1.27 μm

n-Doped InP Bulk(Si: 2·1018 cm-3)

0.15 μm0.145 μm

1.5 μm

3.6 μmL=258 μm

Fig. 3.1 Ridge-waveguide InGaAsP/InP laser and SOA structure as used for the characterisation.

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

84

Subsequently, we present only the results concerning the LD_1and SOA_1, since re-sults of the experiments with the other devices are in agreement with results from these devices. The LD_1’s threshold is at 34 mA while the SOA_1 starts clamping for currents higher than 100 mA when the cavity temperature is near 20oC. This allows us to use device LD_1 to determine material parameters in the current range between 10 and 30 mA and to use device SOA_1 for material-parameter determination in the current range of 30 and 100 mA.Pure single-mode spectra are needed in order to measure the material gain, since the minimum and maximum of the ripples are needed by the Hakki-Paoli method. For that reason a larger number of devices were tested. The two mentioned devices were chosen due to the pure TE single-mode Fabry-Perot spectrum, as shown in Fig. 3.2.In addition, we have chosen small device lengths (of ~250 μm), so that we attained the largest possible free spectral ranges. The 250 μm long devices delivered free spectral ranges for the LD_1 and SOA_1 of ~1.3 nm which exceed the resolution of 0.08 nm of the optical-spectrum analyser allowing precise and fast measurements.

3. Material-Gain CharacterisationTo determine the 1.55 μm-InGaAsP material gain we first experimentally deter-mined the net-modal gain , which corresponds to the modal gain averaged over the whole device [3.9]

, (3.1)

1.45 1.50 1.55 1.60Wavelength [μm]

72

62

64

66

68

70

Spec

tral

Den

sity

[dB

m/A

]

Fig. 3.2 Spectrum of SOA_1 at a current of 60 mA and at a cavity temperature of 20oC.

gmg

g Γgm αtot– Γgm αi– αm–= =

3.1 Material Gain

85

where is the material gain to be determined, the confinement factor as defined in Eq. (2.56), the internal losses and the mirror losses averaged over the whole device of length

. (3.2)

Once we had determined we quantified the different loss terms and and the confinement factor to derive the material gain .

3.1 Determination of Net-Modal Gain Below threshold and under the assumption of single-pass gain amplification, Hakki and Paoli derived the following expression for the net gain [3.4]

(3.3)

in which

, (3.4)

where the subscript “hp” stands for “Hakki-Paoli” and is the ratio of maximal to adjacent minimal intensities of the laser’s Fabry-Perot modes. Our

method relies strongly on the accurately of the measurement of maximum-to-mini-mum ratio of the light intensity can be measured. It fails when the laser is operating near or above the threshold, because then the modes get very narrow and it becomes difficult to determine the maximum precisely enough with an optical-spectrum ana-lyser.A less demanding technique with respect to the resolution of the optical-spectrum analyser, which furthermore has a better signal-to-noise ratio, was proposed by Cas-sidy [3.5]. He integrated the Fabry-Perot mode of the laser spectrum over one FSR and divided the integral by the intensity minimum in the same FSR. This yields

, (3.5)

where

. (3.6)

gm Γαi αm

L

αm1

2L------ 1

R1R2------------ln=

g αi αmΓ gm

g

ghp

ghp λ0( ) 1L---

ρhp λ0( ) 1–

ρhp λ0( ) 1+---------------------------------ln=

ρhp λ0( )Pmax λ0 λΔ–( ) Pmax λ0 λΔ+( )+

2Pmin λ0( )-------------------------------------------------------------------------------=

ρhpPmax Pmin

gcas λ0( ) 1L---

ρcas λ0( ) 1–ρcas λ0( ) 1+------------------------------ln=

ρcas λ0( ) 1λΔ Pmin λ0( )⋅

---------------------------------- P λ( ) λdλ0 λΔ–( )

λ0 λΔ+( )

∫⋅=

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

86

is the output power within the spectrum and the wavelength range of one FSR. For a reliable Fabry-Perot spectrum, the two methods should coincide. Near thresh-old the Hakki-Paoli method yields too small gains and the Cassidy method must be used. If there is a constant underground in the spectrum, the Cassidy method be-comes more inaccurate by yielding too small values. In these cases the Hakki-Paoli method is advantageous. We have taken spectra within a range of 1.460 to 1.640 μm for various currents and for LD_1 as well as SOA_1. All spectra were measured at a diode-cavity tempera-ture of 20oC. Since the cavity temperature would increase with increasing currents,

P λ( ) Δλ

1.45 1.50 1.55 1.60 1.65Wavelength [μm]

0

-50

-100

-150

-200

-250

Γgm

-αto

t [cm

-1]

Fig. 3.3 Net modal gain with total losses of (a) LD_1 for currents between 10 and 30 mA in steps of 5 mA and (b) SOA_1 for currents between 20 mA and 100 mA whereas the current is increased in steps of 10 mA. The SOA_1 cavity temperature is kept at Tc=20oC for all currents. The solid lines are 9th order polynomial fits.

20 mA

100 mA

. . .

1.50 1.60Wavelength [μm]

1.55

Γgm

-αto

t [cm

-1]

0

-50

-100

-200

-150

(a)

(b)

10 mA

30 mA. .

.

3.1 Material Gain

87

we corrected the sensor temperature such that the temperature within the diode cav-ity remained at 20oC. The temperature correction mechanism is outlined in the ap-pendix of this chapter. A typical spectra is shown in Fig. 3.2.The evaluation of the gain spectra at different currents provided the averaged modal gain of SOA_1 and LD_1. The Hakki-Paoli and Cassidy method rendered similar re-sults. We have depicted the net gain curves of the LD_1 and the SOA_1 in Fig. 3.3. The solid curves are 9th order polynomial least-square fits. Measurement errors are below 5 cm-1 at the gain peak and below 15 cm-1 in the lower and upper range of the spectra. The error for the 20 mA SOA_1 net-gain curve is larger, since the ripples of the spectra were extremely weak.

3.2 Determination of Loss TermsInternal losses as well as mirror losses contribute to the total losses of a LD or a SOA diode.Above the bandgap energy the total losses can be measured by the transparency method [3.10, 3.11], and below the bandgap energy they can be determined via the net gain. In the transparency method one takes advantage of the fact that to each wavelength below the bandgap belongs a current bias where the net modal gain dis-appears

. (3.7)

The wavelength and the corresponding current at which the laser or amplifier get transparent are called transparency wavelength and transparency current , re-spectively. To determine the transparency current belonging to a signal at the trans-parency wavelength, the signal is introduced into the diode and its power is modulated. As long as the injection current is smaller or larger than the transparency current, stimulated absorption or stimulated emission is generated, modulating the voltage of the diode. The diode’s modulation can be detected with a Lock-in Ampli-fier. Once a transparency current and its transparency wavelength is found, the total losses are determined by Eq. (3.7). The total losses are identical to the net-modal gains at the transparency wavelength for the respective transparency current. Below the bandgap energy, at longer wavelengths, there is no gain. For that reason the meas-ured net gain must correspond to the total losses. The measured total losses for the LD_1 and the SOA_1 are depicted in Fig. 3.4. The total losses of the LD_1 were measured by the transparency method. The total losses of the SOA_1 were measured with different methods depending on the wavelength region. In the long-wavelength range we have made used that (plus signs). In the short-wavelength range we have applied the transparency method as discussed above (cross signs). In the medium range of our spectrum the transparency method

αi αm

g Γgm λtr Itr,( ) αtot λtr( )– 0= =

λtr Itr

g αtot=

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

88

gave quite inaccurate data for the SOA_1, since the net material gain curves from which the total losses are derived have small currents. But when the currents are small, the ripples of the SOA_1 spectrum disappear and the Hakki-Paoli method be-comes inaccurate. Therefore, we have made use of the fact that below threshold the internal loss and the modal gain of the LD_1 and the SOA_1 should be alike. At first glance there are large differences between these two total losses, but they stem main-ly from the different mirror losses. These differ by a constant offset. This offset can be well determined in the short wavelength range near the bandgap energy. This per-

1.45 1.50 1.55 1.60 1.65Wavelength [μm]

0

-100

-300

-400

-200

Loss

es[c

m-1

]

Fig. 3.4 Internal losses (dashed line), mirror losses (long dashed lines) and totallosses (solid lines) of (a) LD_1 and (b) SOA_1. The total losses (asterix,plus and cross signs) were determined experimentally. The mirror losses

of the LD_1 are calculated. When and are given, the internallosses of the LD_1 are given too. Since should be identical for both theLD_1 and the SOA_1, we also know the different loss contributions ofSOA_1.

αm αtot αmαi

Internal Losses αi

Mirror Losses αmTotal Losses αtot

1.50 1.55 1.60Wavelength [μm]

0

-100

-50

Loss

es[c

m-1

]

Internal Losses αi

Mirror Losses αm

Total Losses αtot

3.1 Material Gain

89

mits to derive the total losses in the medium wavelength range of the SOA_1 spec-trum via the total-loss measurements in the laser. The experimental points in the medium range of the spectrum given in Fig. 3.4(b) (asterix) have been determined by choosing a linear combination of the losses evaluated for the LD_1 as described and the transparency method for the SOA_1. The linear loss contribution of the transparency method for the SOA_1 was increased from 0 to 100% from left to right.The mirror losses of the LD_1 were calculated with a plane wave calculation pro-gram developed at our institute [3.12]. For LD_1 the accuracy of the calculated mir-ror losses is high, since the reflectivities are high (error below 5%). The TE reflectivity of ~38% slightly decreases with increasing wavelength. This reflectivity introduced in Eq. (3.2) yields the mirror loss .The internal losses of the SOA_1 were taken from the internal losses of the LD_1. The internal losses of the two diodes should be alike, since the two structures are identical and since they are from almost the same location on the wafer.The mirror losses of SOA_1 can not be calculated precisely enough, since the reflec-tivities are below 1·10-4. Instead we have used the internal losses and subtracted them from the total losses to get an idea of the mirror losses. Estimated errors for the total losses as measured for LD_1 are below 15 cm-1. For the total losses experimentally determined in SOA_1 the errors are smaller than 15 cm-1 above 1.6 μm, smaller than 20 cm-1 above 1.5 μm and around be-low a wavelength of 1.50 μm.

3.3 Determination of the Confinement FactorThe confinement factor was determined with the help of a finite-element program. We applied femsc, a finite element solver, that was developed at the Institute of Quantum Electronics at ETH Zürich [3.13].The confinement factor varies with wavelength and applied current. We have studied the dispersion of the refractive index of the different layers by using Eq. (C.12) of App. C for the refractive indices . Since this formula applies only for cases where the wavelength is larger than the gap wavelength, we had to extract the refrac-tive index of the active quaternary 1.55 layer differently. The refractive index of the active layer was found by inserting the experimentally determined effective group- indices of the diodes into the differential equation of the effective group index dif-ferential equation and solving for the refractive index of the active layer. Since re-fractive-index changes due to current injections had to be considered, we had to apply the procedure for every current separately.Calculated confinement factors over the gain spectrum and for different currents of the SOA_1 structure are depicted in Fig. 3.5.

αmαi

40 cm 1–

Γ

n' λ( )

Γ

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

90

3.4 Material GainWe can now present the material gain throughout the gain spectrum for the dif-ferent carrier-injection densities. The different contributions needed to derive the material gain from Eq. (3.1) have been plotted in Fig. 3.3 (net gain ), Fig. 3.4 (total losses ) and Fig. 3.5 (confinement factors ). To evaluate the material gain we

Con

fine

mem

ent F

acto

r

1.50 1.55 1.60Wavelength [μm]

32

28

30

Con

finem

entF

acto

r[%

]20 mA

100 mA

. . .

34

Fig. 3.5 Calculated confinement factor for SOA_1 structure under consideration ofthe refractive-index dispersion and carrier induced refractive-indexchanges.

gm

Wavelength [μm]1.45 1.50 1.55 1.60 1.65

Wavelength [μm]

400

000

-200

200

Mat

eria

lgai

ng m

[cm

-1]

600

800

Fig. 3.6 Material gain spectrum of 1.55-quaternary layer as a function of the car-rier injection densities. The active layer temperature is 20oC for all carri-er injections densities. The curves are derived from measurements with LD_1 (10, 20, 30 mA) and with SOA_1 (40,..., 110 mA)

10 mA = 0.80·1024 m-3 ^20 mA = 1.25·1024 m-3 ^30 mA = 1.58·1024 m-3 ^40 mA = 1.88·1024 m-3 ^50 mA = 2.15·1024 m-3 ^60 mA = 2.40·1024 m-3 ^70 mA = 2.67·1024 m-3 ^80 mA = 2.85·1024 m-3 ^90 mA = 3.05·1024 m-3 ^

100 mA = 3.25·1024 m-3 ^110 mA = 3.25·1024 m-3 ^

gαtot Γ

3.1 Material Gain

91

used the least square fitted curves of the experimentally determined and . The relation between the current bias versus the carrier-injection density is elucidated in Fig. 3.26.The material gain for the quaternary 1.55 μm layer is displayed in Fig. 3.6. The ac-tive layer temperature was kept at 20oC for all carrier-injection densities. For smaller carrier densities, i.e. for current biases between 10 and 30 mA, we used the data from LD_1. For higher carrier densities i.e. for current biases above 30 mA, we applied the data from SOA_1 to derive the material gain. We also depicted the material gain curve for a current bias of 110 mA. But, as shown by the figure, current biases above 100 mA do not provide higher material gains, since the material gain is clamped for currents exceeding 100 mA. Measurement errors of the material gains are in the range of 20 cm-1 around the peaks wavelengths and 50 cm-1 at the edges of the spectrum.

4. Comparison of Experiment with TheoryA comparison between the material-gain curves calculated from measurements as shown in Fig. 3.6 with the calculated material gain from Chapter 2.1 Section 2.1 on page 36 is given in Fig. 3.7. We have compared the experimental and theoretical ma-terial gains at three different carrier densities, which belong in our experiment to the 20, 60 and 100 mA curves. For medium carrier densities around the gain peak we find good agreement between experiment (solid lines) and theory (dotted lines). For smaller carrier densities the theoretical model underestimates the material gain and

g αtot

Fig. 3.7 Comparison between experimentally determined material gain (solidlines) with calculated material gains (dotted lines). The material gain iscompared for three different carrier injection densities. Agreement be-tween theory and experiment is good at higher carrier densities.

1.45 1.50 1.55 1.60 1.65Wavelength [μm]

400

000

-200

200

Mat

eria

lgai

ng m

[cm

-1]

600

800 3.25·1024 m-3 (100 mA)2.40·1024 m-3 ( 60 mA)1.25·1024 m-3 ( 20 mA)

1000

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

92

for higher carrier densities it is overestimated. Moreover, we find deviations for short wavelengths, i.e. at higher energies. The deviations at higher energies are most probably due to nonparabolicity in the bandstructure, which become important for higher energies. The theoretical model used in our calculation does only consider a simple first order correction of the conduction band nonparabolicity as described by Eq. (A.10) in App. A.For calculation of the material gain we have used the material gain according to Eq. (2.18) with the Lorentzian lineshape broadening of Eq. (A.17) in App. A and the bandgap shrinkage after Eq. (2.19). Best fit of theory with experiment was obtained for an intraband-relaxation time of 60 fs, which is in agreement with literature values of 70 and 78 fs (see Chapter 2.1, page 58). For the material parameters we used the parameters of In0.6Ga0.4As0.85P0.15. This choice of the material composition then determined all other material parameters according to App. C. A summary of the pa-rameters used in the calculation is given in App. D.

5. Differential Material Gain dgm/dN and dgm/dTAn expression of the differential material gain has been derived based on our measurements and with the help of the polynomial material gain model given in section 3.4. The differential material gain curves are shown in Fig. 3.8.

dgm dN⁄

Fig. 3.8 Differential material gain within the gain spectrum for differentcarrier densities.

dgm dN⁄

1.45 1.50 1.55 1.60 1.65Wavelength [μm]

dgm

/dN

[1·1

018m

-4]

10

8

6

4

2

0

N=1.0·1024 m-3

N=1.5·1024 m-3

N=2.0·1024 m-3

N=2.5·1024 m-3

N=3.0·1024 m-3

3.1 Material Gain

93

The differential gain versus temperature is shown in Fig. 3.9(b). The curves are derived from a gain versus temperature measurement on device SOA_1, as depicted in Fig. 3.9(a). The measurements were performed at a wavelength around the peak gain at 1.535 μm. In order to derive we approximated the measured points with a second order polynomial fit function and got the differential gain by taking the derivative of the polynomial fit function. Errors are smaller than

. The contributions of the curves above 100 mA are mainly due to the shift of the threshold with increasing temperature.

dgm dTcav⁄

dgm dTcav⁄

3cm 1– K 1–±

20 30 40 50 60Cavity-Temperature [oC]

λ=1.535 μm

140 mA

120 mA

100 mA

80 mA

60 mA

40 mA

20 mA

20 30 40 50 60Cavity-Temperature [oC]

140 mA120 mA100 mA

80 mA60 mA40mA20 mA

dgm

/dTca

v[c

m-1

K-1

]

-400

Mat

eria

lgai

ng m

[cm

-1]

0

400

800

0

5

10

λ=1.535 μm

20 mA

140 mA

Fig. 3.9 (a) The material gain decreases with increasing temperature. Derivation of the material gain after the temperature leads to the differential temper-ature gain as shown in (b). Measurements were performed between 20 and 140 mA in steps of 20mA.

(a)

(b)

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

94

The dependence of the differential temperature gain with wavelength is shown in Fig. 3.10. Measurement inaccuracies are quite large. The error is estimated to lie be-low .5cm 1– K 1–±

Fig. 3.10 Dependence of the differential temperature gain on the wavelength. Meas-urements were performed between 20 and 140 mA in steps of 20 mA.

dgm

/dTca

v[c

m-1

K-1

]

0

5

10

15

20

251.520 1.530 1.540 1.550

Wavelength [μm]

20 mA

140 mA. . .

T=25oC

3.2 Refractive-Index Changes

95

3.2 Refractive-Index Changes

Measurements of the 1.55-InGaAsP refractive-index changes due to current injec-tion and temperature variation are reported in the 1.480 to 1.600 μm wavelength range. At gain peak we find for the refractive index due to carrier-density variation

and for the refractive-index change due to temperature variation .

1. Measurement of the Refractive-Index Changes1.1 TheoryThe variation of the refractive index with carrier density and temperature can be de-rived from measurements of the wavelength shifts of the longitudinal mode of LD_1 or SOA_1 when the carrier density and temperature changes [3.14, 3.15]

, (3.8)

where in Eq. (3.8) the partial derivatives after the current and the temperature are re-lated with the respective refractive-index changes. To measure or a specific one picks out one particular longitudinal mode and follows the wavelength shift with an optical-spectrum analyser when changing the carrier density or temperature. Expressions relating measured wavelength shifts with refractive index changes can be derived as follows: To each of the longitudinal modes belongs an absolute phase

. (3.9)

When now varying the current bias or temperature, the wavelength of the longitudi-nal mode will shift by , while the absolute phase of the mode retains the same val-ue. For a varying current this phase conservation condition is described by

. (3.10)

Evaluation of Eq. (3.10) with Eq. (3.9) delivers the partial derivative relating the ex-perimentally determined wavelength shift with the refractive-index change

. (3.11)

dn' dN⁄ 1,7– 10 20– cm 3–⋅=dn' dT⁄ 3,1 10 4– K 1–⋅=

dλI∂

∂λdITcav∂

∂λ dTcav+=

dλ λ∂

ϕ 2π( ) λ⁄ n'g eff, L⋅ ⋅=

λ∂

dϕdI------

Tcav

0=

I∂∂λ λ

n'g eff,--------------

n'g eff,∂I∂

-----------------Tcav

⋅=

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

96

The same method allows to find the relation between wavelength shift and refrac-tive-index change due to cavity-temperature variations

. (3.12)

The second term corresponds to the effect from a thermal expansion of the laser or SOA cavity of length . This term can be neglected since the thermal expansion co-efficients of InP and the copper mount

make the term approximately two decades smaller in comparison with the first term [3.16]. Further, as other authors before us [3.17], we assume that the refractive-index chang-es are mostly due to changes in the material properties and not due to changes in the dispersion relations, so that we may set

. (3.13)

Finally we must take into account the wave confinement factor which dilutes the change of the refractive index. With that we write

. (3.14)

Experimentally we have now two possibilities to determine the refractive-index changes from current variations at constant cavity temperature.We can measure the shift of the wavelength due to carrier-injection changes by ap-plying Eq. (3.11), while the cavity temperature must be adjusted according to Fig. 3.24 for each current

. (3.15)

This method is quite laborious since the temperature has to be readjusted for each current. Alternatively, we can measure the total wavelength shift after Eq. (3.8). In this method the sensor temperature is held constant during the whole measure-ment and the temperature effect is discounted afterwards

(3.16)

Tcav∂

∂λ λn'g eff,--------------

Tcav∂

∂n'g eff,

I

⋅ λL---

Tcav∂

∂L

I

⋅+=

Lα InP( ) 1 L⁄ L∂ T∂⁄⋅ 4,75 10 6–⋅ K 1–= =

α Cu( ) 16,8 10 6–⋅ K 1–=

Δn'eff Δn'g eff,=

Γ

Δn' 1Γ---Δn'

eff≅

n'∂I∂

------- 1Γ---

n'g eff,λ

--------------I∂

∂λ

Tcav

⋅ ⋅=

dλ

n'∂I∂

-------dI 1Γ---

n'g eff,λ

--------------dλ I I dI+ Tsens, ,( )Tcav∂

∂n' dTcav I I dI+ Tsens, ,( )–=

3.2 Refractive-Index Changes

97

with

. (3.17)

Here is the measured wavelength shift due to a current variation between and measured at a sensor temperature of . The partial derivative after

must be measured separately and as well as is determined according to the procedure in the appendix.The temperature-related refractive-index change can be experimentally determined with Eq. (3.12)

. (3.18)

1.2 Carrier-Related Refractive-Index ChangesThe carrier-related refractive-index changes for different wavelengths within the gain spectrum of SOA_1 are shown in Fig. 3.11. We have depicted the total refrac-tive-index change of the active 1.55 μm-quaternary layer with respect to the refrac-tive index at 20 mA. The experiment shows that the refractive-index change decreases with increasing wavelength. The curves were found by applying the meth-od of Eq. (3.15). The confinement factors used in Eq. (3.15) were chosen depending on the wavelength according to Fig. 3.5.

dTcav I I dI+ Tsens, ,( ) Rtherm P2heat I dI+( ) P1

heat I( )–[ ]=

dλ II dI+ Tsens Tcav

Rtherm Pheat

n'∂

Tcav∂------------- 1

Γ---

n'g eff,λ

--------------Tcav∂

∂λ

I

⋅ ⋅=

20 40 100 140Current [mA]

1208060

0.000

-0.010

-0.020

-0.030

Ref

ract

ive

Inde

xC

hang

e

1.585 μm1.560 μm1.535 μm1.510 μm1.490 μm

Fig. 3.11 Carrier induced refractive-index changes for different wavelengths of SOA_1. The measurements were performed for all currents at a cavity lay-er temperature of 20oC. Above the threshold of 100 mA, SOA_1 is clamped. The refractive-index changes are given relative to the refractive index at 20 mA.

Tcav=20oC

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

98

Above 100 mA, where SOA_1 is clamped (i.e. starts lasing and the carrier popula-tion is no more increased), the refractive index does not further change. This is a strong hint that our temperature correction is qualitatively correct. In a reference measurement, without any temperature correction, we found a strongly increasing refractive-index change above threshold. This increasing refractive-index change is due to the temperature, and it increases even further above threshold.The carrier-related refractive-index changes when the SOA_1 cavity is held at dif-ferent temperatures is displayed in Fig. 3.12. It is most interesting to see, that the re-fractive-index change is not significantly modified, if the active-layer cavity temperature changes from 20 to 60oC. The curves were found by applying our meth-od of Eq. (3.16). More points were measured in comparison with the previous figure, since the method is quicker.For modelling purposes and to remain independent of the underlying structure, we give here the refractive-index changes with respect to the carrier densities. Fig. 3.13shows a linear dependence of the refractive-index change against the carrier-density changes. The variation of the refractive index with carrier-density changes decreases with increasing wavelength. The carrier densities for different currents (30, 40, 60, 80, 100 mA) have been determined by applying the current-versus-carrier densities given in the Appendix on page 117. The 20 mA curve with a measured carrier den-sity of 1.25·10-24 m-3 carriers is the reference curve with respect to which the carrier-density changes are derived.From the experiment we can derive quantitative values for the variation of the re-

Tc=20oC

Tc=40oC

Tc=60oC

20 40 100 140Current [mA]

1208060

Ref

ract

ive-

Inde

xC

hang

e 0.000

-0.010

-0.020

-0.030

Tcav=20oCTcav=40oCTcav=60oC

Fig. 3.12 Carrier induced refractive-index changes performed at different active SOA_1 cavity layer temperatures. The figure shows that variation of the active layer temperature does within the measurement tolerances not mod-ify the refractive-index change characteristics.

λ=1.535 μm

3.2 Refractive-Index Changes

99

fractive index with the injected current. A linear least square fit around the peak wavelength gives

, (3.19)

with errors of at μm. In Eq. (3.19) must be given in “μm”. The peak wavelengths for each carrier density can be picked out from Fig. 3.6. Closed expressions for the peak wavelength will be given in section 3.4.

λp

dn'dN------- λ( ) 1,8– 10 20– cm 3–⋅ λ λp–( ) 5 10 20– cm 3– μm

1–⋅ ⋅+=

0.0 1.5 2.0Carrier-Density Changes ΔN [m-3]

1.00.5

Ref

ract

ive-

Inde

xC

hang

e 0.000

-0.010

-0.020

-0.030

Tcav=20oCR

efra

ctiv

e-In

dex

Cha

nge 0.000

-0.010

-0.020

-0.030

1.48 1.50 1.56 1.58 1.60Wavelength [μm]

1.52 1.54

ΔN=0.33·1024 m-3

ΔN=0.63·1024 m-3

ΔN=1.15·1024 m-3

ΔN=1.60·1024 m-3

ΔN=2.00·1024 m-3

(a)

(b)

Fig. 3.13 Variation of the refractive index with carrier-injection changes. (a) With emphasis on the carrier-density changes, and (b) with emphasis on the car-rier-density changes as a function of the wavelength. Note that the curves (a) and (b) are from the same experimental data-set. Only the representa-tion of the values is exchanged.

Tcav=20oC-0.040

-0.040

1.585 μm1.560 μm1.535 μm1.510 μm1.490 μm

0,15 10 20– cm 3–⋅± λ 1,535= λ

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

100

Insertion of the peak wavelengths into Eq. (3.19) delivers at μm re-fractive index changes between and for carrier densities around 1.25·1024 m-3 and 3.25·1024 m-3, respectively.Our values around the peak wavelength have to be compared with those of Bouley et al. [3.18] who reported a value of for 1.5 μm stripe lasers and Manning [3.19], who reported in 1.3 μm InGaAsP broad-area lasers, whereas Westbrook et al. [3.17] reported

in 1.5 μm InGaAsP laser diodes.In many new devices, as for example all-optical devices, the wavelength dependence of the refractive-index change is of special interest. We have depicted the refractive-index variation for a given carrier-density change as a function of the gain spectrum in Fig. 3.13. The curves show that the refractive-index changes are larger, the smaller the wavelength is.The experimentally determined refractive-index changes must be compared with calculated refractive-index changes. For carrier-density variations from

m-3 onto 3.25·1024 m-3, corresponding in our SOA_1 to a current bias increase from 20 to 100 mA, we find the refractive-index changes of Fig. 3.14. Fig. 3.14 was calculated by applying the Kramers-Kronig transformation of Eq. (2.16)onto the material gains as calculated for Fig. 3.7. Additionally, we added in Fig. 3.14according to Eq. (2.24), the refractive-index changes of the Plasma effect. The cal-culation quantitatively confirms the experimental refractive-index changes in the gain window between 1.5 and 1.6 μm of SOA_1. In the experiment we find a de-crease of the refractive index with increasing wavelength. In the calculation, this de-

λp λ 1,535=dn' dN⁄ 2,0– 10 20– cm 3–⋅ 1,4– 10 20– cm 3–⋅

dn' dN⁄ 1,5– 10 20– cm 3–⋅=dn' dN⁄ 2,8– 10 20– cm 3–⋅=

dn' dN⁄ 1,8– 10 20– cm 3–⋅=

1,25 1024⋅

Fig. 3.14 Calculated refractive-index changes due to carrier-density increase. Thevariations are due to a change with respect to a state with a carrier densityof 1.25·10-24 m-3.

1.0 1.2 1.4 1.6 1.8Wavelength [μm]

Ref

ract

ive-

Inde

xC

hang

e

0.000

0.010

-0.010

-0.030

ΔN=0.5·1024 m-3

ΔN=1.0·1024 m-3

ΔN=1.5·1024 m-3

ΔN=2.0·1024 m-3

-0.020

-0.040

3.2 Refractive-Index Changes

101

crease sets in somewhat later - at 1.6 μm. The experimentally determined refractive index change is slightly smaller. This makes sense, since our calculation overesti-mates the material gain change at high carrier densities (as stated in Fig. 3.7).

1.3 Temperature-Related Refractive-Index ChangesThe temperature related refractive-index changes for different current biases at the gain peak around 1.530 μm extracted from the SOA_1, are shown in Fig. 3.15. The temperature related refractive-index change is independent of the current bias and re-mains constant between 20oC and 70oC. Based on our experiments we find for variation of the refractive index caused by temperature for the 1.55 μm-InGaAsP layer at =1.535 μm

. (3.20)

Literature states a value of 0.86·10-4 K-1 for InP measured at room temperature in the mid infrared wavelength range between 5 to 20 μm [3.20]. In order to obtain the curves of Fig. 3.15 we have divided the measured refractive index change by the confinement factor according to the current bias. We have ne-glected the change of due to refractive index changes from temperature effects, since all layers experience the same temperature change. Even if only the active layer would experience a refractive index change due to a temperature variation, would not change much. To be concrete - in such a situation would increase from 0.32 to 0.34 for the values at 20 mA measured at 1.535 μm.

λ

dn'dTcav-------------- 1,0 10 3– K 1– 1,5 10 4– K 1–⋅±⋅=

ΓΓ

ΓΓ

Fig. 3.15 Temperature-related refractive-index change. The measurements were per-formed at different SOA_1 carrier biases for currents between 20 and 140 mA in steps of 20 mA.

20 30 40 50 60 70Cavity-Temperature [oC]

Ref

ract

ive-

Inde

xC

hang

e

0.000

0.010

0.040

λ=1.535 μm

20 mA

140 mA0.020

0.030

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

102

2. Effective Group Refractive IndexThe effective group refractive index can be determined in the usual way from the longitudinal mode spacing (m a number) using the relation

. (3.21)

The group index of the SOA_1 at 100 mA over the gain spectrum is shown in Fig. 3.16. The effective group refractive index decreases with increasing wave-length. At the edges of the gain spectrum, measurement inaccuracies increase, since the ripple in the spectrum become small.

Δλ λm 1+ λm–=

n'g eff,λ2

2L Δλ⋅------------------=

n'g eff,

1.45 1.50 1.55 1.60 1.65Wavelength [μm]

3.7

3.5

3.4

3.6

Eff.

Gro

upR

efra

ct.I

ndex

3.8

3.9

4.0

I=100 mATcav=20oC

Fig. 3.16 The effective group refractive index of the SOA_1 at 100 mA. The smooth line is a fit curve.

3.3 Alpha Factors

103

3.3 Alpha Factors

We report on measurements of the alpha factor of bulk 1.55 μm InGaAsP as a func-tion of the wavelength, the carrier density and the temperature.

1. IntroductionThe alpha factor , also called linewidth-enhancement factor, is a key parameter in semiconductor lasers and amplifiers. It describes the coupling between the in-duced phase changes and amplitude changes of a signal within the material. The fac-tor plays an important role in the spectral-linewidth enhancement in semiconductor lasers as given in Eq. (2.63), in the frequency chirp after Eq. (2.62) and in all-optical switches, where it affects the extinction ratios (see Fig. 5.5 of chapter 5). While the alpha factor should be small in order to keep linewidths narrow and chirps small, a large alpha factor is advantageous for obtaining high extinction ratios in all-optical devices.Since the introduction of the alpha factor in 1982 by Henry [3.21], a variety of the-oretical work has been performed on clarifying various aspects of the alpha factor [3.22-3.24]. Since the alpha factor is small in quantum-well structures, special inter-est have found theoretical [3.24-3.26] and experimental [3.27-3.30] investigations on different issues of the alpha factor in quantum wells. Unfortunately only few lit-erature exists on the alpha factor in bulk InGaAsP. These mostly consider the wave-length dependence [3.17, 3.31, 3.32] or the dependence of the alpha factor on device length [3.32], but temperature effects have - to the best of our knowledge - not yet been considered.Here we put much emphasis on finding the alpha factor as a function of the carrier density , the wavelength and the cavity temperature . The carrier densities are directly related with the currents via Fig. 3.26, and the sensor temperature is re-lated with the cavity temperature in the diode via Eq. (3.35) depicted in Fig. 3.24.In an other subsection we show measurements of the alpha factor . Comparison of our measurements with literature values was nearly impossible, since except of a simulation given in Ref. [3.34] there exists almost no literature on . To the best of our knowledge we are the first who have measured of 1.55 μm InGaAsP.

αN

N λ Tcav

αT

αTαT

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

104

2. Experiments on αN

The alpha factor can be measured by taking the ratio between small refractive-index and gain changes while the current is varied in small steps of . We write with the definition given in Eq. (2.58)

. (3.22)

The two terms in the fraction on the right side of the equation are exactly the terms that have been determined in the previous two sections. Other definitions of the al-pha factor and the validity of the approximations have been discussed in chapter 2.2on page 62.Evaluations of the measured refractive-index and gain changes at defined cavity temperatures have been performed on SOA_1, SOA_2, LD_2 and LD_3. We show the results of measurements on SOA_1 for varying currents, varying wavelengths and varying cavity temperatures in Fig. 3.17(a)-(c)1. From Fig. 3.17(a) we learn that around the gain peak (~1.535 μm) and for short wavelengths, the alpha factor does not change with the current, i.e. the carrier den-sity. For longer wavelengths the alpha factor increases with carrier density. Such a characteristic has theoretically been predicted in Ref. [3.24] for bulk materials. It is different from predicted and experimentally found characteristics in quantum-well structures [3.24, 3.29], where the alpha-factor increases with the carrier density for all wavelengths. The change of the alpha factor with wavelength is displayed in Fig. 3.17(b). The al-pha factor increases with increasing wavelength. This result is both quantitatively and qualitatively in agreement with results of other authors, who reported an alpha-factor of ~7 at gain peak [3.17, 3.31, 3.32]. At longer wavelengths the higher carrier-injection density additionally increases the alpha factor.The change of the alpha factor with temperature is depicted in Fig. 3.17(c). We find, that the alpha factor slightly decreases with increasing temperature for wavelengths above the gain peak, e.g. at 1.55 μm. But it does not change with temperature for wavelengths around the gain peak, i.e. at ~1.535 μm.For short and strong power intensities, we predict that the alpha factor increases due

1. Measurement on SOA_2 delivered qualitatively and quantitatively the same behavior of the alpha factor as found in Fig. 3.17(a)-(c) with SOA_1. Measurement on LD_2 and LD_3 delivered the same results as found with the amplifiers. Since the spectral resolution was better in comparison with the amplifiers, we could measure over a larger spectral range. We found that the alpha factor increases still further in the long wavelength range. For both lasers values up to 40 were found at 1.62 μm [3.33].

ΔI

αN 4πλ

------n′eff∂ N∂⁄

g∂ N∂⁄------------------------–≡ 4π

λ------

n′eff∂ I∂⁄g∂ I∂⁄

----------------------– 4πλ

------Δn′eff ΔI( )

Δg ΔI( )-------------------------–≅=

3.3 Alpha Factors

105

135

105

95

75

55

35

135

105

95

75

55

35

15

1585

1570

1557

1544

1532

1520

1508

1496

1485

Fig. 3.17 Measured alpha factors for SOA_1 (a) versus current bias, (b) versus wavelength and (c) versus cavity temperature.

αN

0

5

10

15

20A

lpha

Fact

orα

N

Current [mA]20

1.48 1.50 1.56 1.58Wavelength [μm]

1.52 1.54

Tcav=20oC

40 8060 100 120 140

0

5

10

15

20

Alp

haFa

ctor

αN

1.585 μm1.570 μm1.557 μm1.544 μm1.532 μm1.520 μm1.508 μm1.496 μm1.458 μm

Tcav=20oC

λ=1.550 μm

135 mA105 mA95 mA75 mA55 mA35 mA15 mA

0

2

4

6

10

Alp

haFa

ctor

αN 8

10 20 30 40 50 60Cavity-Temperature [oC]

135 mA105 mA95 mA75 mA55 mA35 mA

λ=1.535 μm

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

106

to the gain compression effect as described by formula (2.60).Error estimates at not too low currents show that our alpha factor measurements around the peak-wavelength are quite reliable. Errors are below . However, the error increases outside of this range. Nevertheless, qualitative performance will be maintained.

3. Experiments on αT

The alpha factor has been measured as well. relates changes of the refractive index and gain, that are due to changes in the temperature. It is of interest since it allows to estimate phase fluctuations in a device, which are due to temperature changes, see e.g. Eq. (2.79). In all-optical devices part of the noise might be due to temperature fluctuations. In these devices the temperature fluctuations have their or-igin in the short but powerful control-signals, that are needed to control the device.The curves depicted in Fig. 3.18 were obtained by inserting the refractive index and gain changes measured in the previous sections into the definition of . was defined in Eq. (2.61)

. (3.23)

We have evaluated at μm, i.e. near the gain peak, for different cur-rent densities and temperatures of device SOA_1. Fig. 3.18(a) shows significant in-crease of with increasing current. The current density increases from 1.25·1024 m-3 at 20 mA to 3.25·1024 m-3 at about 100 mA. Since all curves have been measured at cavity temperature higher than 20oC, the threshold lies considera-bly higher than 100 mA.A change of the device temperature does not modify the value of . This is shown clearly by the curves of Fig. 3.18(a) that were evaluated at three different device tem-peratures and it becomes even more obvious by Fig. 3.18(b), where we show for an increasing cavity temperature at different currents.The dependence of on the wavelength has been theoretically investigated in [3.34] for 1.5 μm InGaAsP. The authors find for a value around 5 for wave-lengths below and near the gain peak. Above the gain peak increases with wave-length and reaches 40 nm above the gain peak a value of 20.Our measurements are in agreement with the only available literature value of Ref. [3.34]. In Ref. [3.34] is calculated to lie around 5 for the gain-peak of 1.5 μm InGaAsP. That is quantitatively the same value as we have found.Error estimates of our measurements deliver a precision of for the temperature related alpha factor . All experiments in Fig. 3.18 were performed two times. The

1±

αT αT

αT αT

αT 4πλ

------n′eff∂ T∂⁄

g∂ T∂⁄-----------------------–≡

αT λ 1,535=

αT

αT

αT

αTαT

αT

αT

4±αT

3.3 Alpha Factors

107

results are the averaged values of the two experiments.

110

90 m

70 m

50 m

30 m

0

10

20

30

Alp

haFa

ctor

αT

20 30 40 50 60Cavity-Temperature [oC]

Fig. 3.18 Measured values of (a) as a function of current performed at three dif-ferent cavity temperatures and (b) as a function of temperature for threedifferent currents.

αT

(a)

(b)

λ=1.535 μm

I=110 mAI= 90 mAI= 70 mAI= 50 mAI= 30 mA

TCa

TCa

TCa

0

10

20

30A

lpha

Fact

orα

T

20 60 100 120 140Current [mA]

8040

λ=1.535 μm

Tcav=35oCTcav=45oCTcav=55oC

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

108

3.4 Parametrizations of the Material Parameters

In this section we derive new parametrizations of the material gain and refractive-index changes as a function of the wavelength and carrier densities. These analytical expressions excellently fit the measured curves. Such parametrizations of the mate-rial parameters are indispensable in simulation tools, where material gains are need-ed throughout the spectrum and for all carrier densities.

1. Material-Gain ParametrizationMaterial gain curves as a function of the wavelength and carrier densities are excessively used in simulation tools [3.1, 3.2]. However, explicit calculation of the material gain is cumbersome, see e.g. [3.9, 3.35] and match with experi-mental curves is usually moderate. Therefore, different parametrizations of the ma-terial gain have been proposed. Assuming that the SOA is a simple two-level system, Agrawal has modelled the material gain as having a symmetric Lorentzian lineshape [3.36]. But in reality, the carriers in the SOA experience a distribution con-tinuum in both the valence and conduction bands, resulting in the gain coefficient being asymmetric with wavelength. In another attempt the experimentally deter-mined curves were empirically fitted to a quadratic function [3.37]. This formula corresponds only near the centre of the gain well to the real gain coefficient and fails to give adequate correlation especially on the long wavelength side. In order to mod-el more effectively the asymmetric gain, Willner and Shieh proposed a cubic formula [3.38]. However, their parametrization is unsatisfactory in the short wave-length range.We have found a material-gain parametrization that excellently fits the measured curves for all carrier densities and for all wavelengths within the gain spectrum.The parametrization is the sum of a quadratic and a cubic function (see Fig. 3.19)

(3.24)

with

(3.25a)

. (3.25b)

λ N

gm λ N,( )

gm λ N,( )

gmN λ

gm λ N,( )= cN λ λz N( )–[ ]2 dN λ λz N( )–[ ]3+ λ λz N( )<,

0, λ λz N( )≥⎝⎜⎜⎛

cN 3gp N( )

λz N( ) λ– p N( )[ ]2-------------------------------------------=

dN 2gp N( )

λz N( ) λ– p N( )[ ]3-------------------------------------------=

3.4 Parametrizations of the Material Parameters

109

The only fit functions that are needed in Eq. (3.25a-b) are , giving the carrier-density versus material gain dependence at peak wavelength, , describing the carrier dependence of the peak wavelength and determining the begin of the zero-gain region at a given carrier density.The functions have to be chosen such that they fulfil the physically reasonable con-ditions

for all and (3.26a)

for all . (3.26b)

Subsequently, we propose two different parametrizations for these functions. We begin with a parametrization that excellently fits the measured curves within the important carrier density range between the transparency carrier density up to ~4·1024 m-3 but fails to fulfil the conditions (3.26a-b) outside this range

(3.27a)

(3.27b)

. (3.27c)

In these equations is the transparency carrier density at the band edge wavelength of the active layer and assigns the begin of the zero gain region

gp(N)

λp(N) λz(N)

0

λ

gm

gp(N)x

x

Fig. 3.19 The material gain gm(λ,N) is parametrized as the sum of a quadratic and a cubic function with the origins in λz. For determination of the parametri-zation, we only need to know the carrier dependence of λz, λMax and gMax.

gm(λ,N)

Quadratic Fct

Cubic Fct

gp N( )λp N( )

λz N( )

gp N( ) 0> N

λz N( ) λp N( )> N

N0

gp N( ) a0 N N0–( )=

λp N( ) λ0 b0 N N0–( ) b1 N N0–( )2+[ ]–=

λz N( ) λz0z0 N N0–( )–=

N0λ0 λz0

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

110

in the long wavelength direction. The parameters , , and can be deter-mined with a least square fit of the experimentally determined curves of ,

and .A comparison between the parametrized and the experimental curves is shown in Fig. 3.20. We compare experimental curves from 10 to 100 mA as determined for Fig. 3.6 with parametrized curves (solid lines) at the respective carrier densities. An excellent agreement throughout the gain spectrum is found with the coefficients in the first column of Table 3.1. Next we propose a parametrization that delivers reasonable and continuous curves for all values of carrier densities, i.e. in the range [0, ]. Such a parametrization is needed in simulation tools where during the calculation process unrealistic low or high carrier densities may occasionally occur. The parametrization has got the form

with for (3.28a)

(3.28b)

. (3.28c)

In comparison with Eq. (3.27a) we have added in Eq. (3.28a) a term that hinders the peak gain to enter the forbidden negative range when carrier densities de-

a0 b0 b1 z0gp N( )

λp N( ) λz N( )

1.45 1.50 1.55 1.60 1.65Wavelength [μm]

400

000

-200

200

Mat

eria

lgai

ng m

[cm

-1]

600

800

10 mA = 0.80·1024 m-3 ^

40 mA = 1.88·1024 m-3 ^

70 mA = 2.67·1024 m-3 ^

^100 mA = 3.25·1024 m-3

Fig. 3.20 Comparison of experimentally determined material gain (as obtained inFig. 3.6) with parametrized material gain (solid lines). The parametrizedcurves excellently fit the experimental data points throughout the gain spec-trum and for all carrier densities. The parametrized curves were calculatedwith Eq. (3.24) and the coefficients in the first column of Table 3.1.

∞

gp N( ) a0 N N0–( ) a a⋅ 0N0 eN N0⁄–

⋅+= a 1> N 0>

λp N( ) λ0 b0 N N0–( )–=

λz N( ) λz0z0 N N0–( )–=

gp N( )

3.4 Parametrizations of the Material Parameters

111

crease below . This additional term modifies the parametrization only below and around the transparency carrier density . A further modification has been made in Eq. (3.28b), where we use a linear instead of a quadratic fit function. With this linear fit function, condition (3.26b) is not anymore violated for highest carrier densities. In order to fit the material gain in the absorption regime (i.e. at lowest carrier densi-ties) to experimental curves, the parameter can be used as a fit parameter. Howev-er, it’s value must be or else becomes negative for . A further adaptation of the gain curve in the absorption region can be obtained by adaptation of and . Small deviations from the measured positions change the material gain only little at higher carrier densities, but modify the shape of the curve in the absorption region efficiently. Explicit values of parameters fitting the measured curves from Fig. 3.6 are given in the second column of Table 3.1.

In order to get a parametrization that also considers the gain compression effect of strong signal intensities, we combine Eq. (3.24) with Eq. (2.26)

. (3.29)

Table 3.1: Coefficients for In0.6Ga0.4As0.85P0.15 Parametrization a

a. We believe that it suffices to adjust the values of and according to the gap wavelength for use of the parameter values with other compositions of the InGaAsP active quaternary material around 1.5 μm.

Parameters for Parametrizationb

b. The wavelength units are in μm. The carrier densities must be given in units of m-3 and the output units of gm are then given in units of m-1.

with Eq. (3.27a-c) with Eq. (3.28a-c)

Allowed Carrier Density Range [ , ~4·1024 m-3] [0, ]

Comparison with Experiment excellent for all excellent for high good for low

6.5·1023 m-3 6.5·1023 m-3

3.13·10-20 m2 3.13·10-20 m2

- 1.2

1.595 μm 1.575 μm

6.84·10-26 m3μm 3.17·10-26 m3μm

-1.22·10-50 m6μm -

1.650 μm 1.625 μm

5·10-27 m3μm -2.5·10-27 m3μm

N0N0

a1≥ gp N 0=

λz0z0 λz

λ0 λz0

N0 ∞

N NN

N0

a0

a

λ0

b0

b1

λz0

z0

gm λ N S, ,( )gm λ N,( )1 ∈totS+----------------------=

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

112

2. Internal-Loss ParametrizationThe internal loss varies with wavelength. Measured values of our test device are giv-en in Fig. 3.4. The measured values of contain contributions from the active and the cladding layers

. (3.30)

Since the internal losses are needed in simulation tools, we give here a linear least square fit parametrization of

, (3.31)

where is the gap wavelength and are the internal losses at the bandgap of the active layer.The parametrization coefficients are given in Table 3.2.

Often, rather the internal losses of the active layer are needed than the total internal losses of the structure. To get an idea of these parameters, we assume that the internal losses of the cladding layers can be neglected and that all losses arise in the active layer. Dividing the total losses by the confinement factor we find the par-ametrization coefficients of Table 3.3.

Table 3.2: Coefficients for Parametrization of the Structure from Fig. 3.1 ab

a. For InGaAsP with other material compositions than those of our structure given in Fig. 3.1, the gap-wave-length must be adapted according to the bandgap.b. With these values the wavelength must be given in units of μm. The output units of gm are then given in units of m-1.

Parameters to use in combination with Eq. (3.27a-c) Eq. (3.28a-c)

1.595 μm 1.575 μm

3’650 m-1 4’300 m-1

37’000 m-1μm-1 37’000 m-1μm-1

Table 3.3: Estimated Coefficients for Parametrization

Parameters to use in combination with Eq. (3.27a-c) Eq. (3.28a-c)

1.595 μm 1.575 μm

12’000 m-1 13’900 m-1

95’000 m-1μm-1 95’000 m-1μm-1

αi

αi Γαi act, 1 Γ–( )αi clad,+=

αi

αi λ( ) mi λ λ0–( ) αi0–=

λ0 αi0

αi

λ0

λ0

αi0

mi

αi act,

αi act,

λ0

αi0 act,

mi act,

3.4 Parametrizations of the Material Parameters

113

3. Refractive-Index-Change ParametrizationBased on our experiments, we have already given in section 3.2 a closed expression describing the differential refractive index changes as a function of wavelength and carrier density. In that expression the peak wavelength was needed. Since we have got now formulas describing the shift of the peak-wavelength with carrier densities, we can give a parametrized expression for the total refractive-index changes.The suggested parametrization for a refractive index change from carrier density onto a carrier density then becomes

, (3.32)

with the peak wavelength as defined in Eq. (3.28b), and . A refractive index change calculated with Eq. (3.32) and com-

pared with the experimentally determined refractive index changes is given in Fig. 3.21. The parametrization coefficients and are given in Table 3.4 and those of

are given in Table 3.1.

Table 3.4: Coefficients for Refractive-Index-Change Parametrizationa

a. With these values the wavelength must be given in units of μm.

Parameters Values

-1.8·10-26 m-3

5.0·10-26 m-3μm-1

N1N2

Δn' λ ΔN,( ) r0 r1 λ λp N( )–( )⋅+[ ] ΔN⋅=

λp N( ) ΔN N2 N1–=N N1 N2+( ) 2⁄=

r0 r1λp

0.0 1.5 2.0Carrier-Density Changes ΔN [m-3]

1.00.5

Ref

ract

ive-

Inde

xC

hang

e 0.000

-0.010

-0.020

-0.030

Tcav=20oC

-0.040

1.585 μm1.490 μm

Fig. 3.21 Comparison between experimentally determined absolute refractive index changes (dashed lines) and absolute refractive index changes calculated with Eq. (3.32) at two different wavelengths.

N1=1.25·1024 m-3

r0

r1

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

114

4. Alpha Factor ParametrizationThe alpha factor is often used in simulation tools. Therefore we give here a para-metrization derived from our experimental data.The intensity-dependent parametrization of the alpha factor has got the form

, (3.33)

where is the photon density (related with signal intensity after App. E) and is

. (3.34)

The different coefficients are given in Table 3.5 for our InGaAsP active layer structure of Fig. 3.1. The peak wavelength was defined in Eq. (3.28b). In order to get expressions of the alpha factor for other InGaAsP compositions around the wavelength 1.5 μm, one must only adapt the gap wavelength in the term.A comparison between experimental values and our fit is shown in Fig. 3.22.

Table 3.5: Coefficients for Alpha Factor Parametrization

Parameters Values

3.2

3.0

43 μm-1

-8.0·10-25 m-3

αN λ N S, ,( ) αN λ N,( ) 1 ∈totS+( )⋅=

S αN

αN λ N,( ) αN0 αN1 eαN2 λ λp N( )–( ) αN3 N N0–( )+

⋅+=

αN0…N3λp

λ0 λp

105

55

15

Tcav=20oC

Fig. 3.22 Comparison between measured and parametrized alpha factors at threedifferent current densities.

1.48 1.50 1.56 1.58Wavelength [μm]

1.52 1.540

5

10

15

20

Alp

haFa

ctor

αN 105 mA = 3.3·1024m-3

55 mA = 2.3·1024m-3

15 mA = 1.0·1024m-3

αN0

αN1

αN2

αN3

3.5 Appendix

115

3.5 Appendix

This appendix is used to outline the method for adjusting the diode cavity at a given temperature. Moreover, results on the relation between the carrier density versus cur-rent injection of our devices are given here.

1. Temperature CorrectionThe LD_1 or SOA_1 cavity temperature changes when a current is applied. The di-ode-cavity temperature may considerably differ from the value at the copper mount, where a temperature sensor and a Peltier element to control the mount temperature are applied. Since it is our goal to measure the material parameters independent of temperature effects we have to control the diode-cavity temperature via the sensor temperature. The relation between the temperature within the LD_1 or SOA_1 cavity and the measured temperature at the copper mount are related by

, (3.35)

where is the thermal resistance describing the heat dissipation from the LD_1 or SOA_1 cavity into the copper mount. The totally generated power in the diode cavity is given by

, (3.36)

where is the applied voltage and is the applied current on the diode and is the totally emitted optical power through one facet.

In order to use the temperature relation Eq. (3.35), we need to determine . as well as can be measured. A method to measure has been pro-

posed by Dall’Ara [3.39] and Kusenkov [3.40]. In this method, a cw curve is measured at a low sensor temperature . Afterwards a second measurement is performed at a higher temperature with a pulsed current source, i.e. at a re-duced duty cycle . While the current is increased during the measure-ment, the temperature within the cavity increases too. Since the temperature in the cw experiment increases faster, the two curves will have a point of intersec-tion when the cavity temperature is identical in the two experiments. can then be derived by applying Eq. (3.35) onto the two experiments at the intersec-

Tcav

Tsens

Tcav Tsens RthermPheat+=

Rtherm

Pheat I V I( )⋅ 2Pout I( )–=

V I( ) IPout I( )

RthermPheat Tsens Rtherm

Pout I( )T0

sens

T1sens

D1 Pout I( )

Pout I( )Tcav Rtherm

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

116

tion state. Solving the two equations for delivers

, (3.37)

with and the duty cycles within the two experiments and the assumption that .

Measured output-power versus current and voltage versus current curves of the laser diode and the SOA_1 are depicted in Fig. 3.23. At the intersection point we find with

Rtherm

RthermT1

sens T0sens–

D0P0heat D1P1

heat–----------------------------------------------

T1sens T0

sens–

D0 D1–( )Pheat-------------------------------------≅=

D0 D1Pheat P1

heat≡ P0heat≅

Fig. 3.23 Measured output-power versus current and voltage-versus-current curves of (a) the laser diode and (b) the SOA_1. At the intersection point we find with Eq. (3.37) thermal resistances of (a) 129.4 W/K for the LD_1 and (b) 80.6 W/K for the SOA_1.

Current0 50 100 150 200

0

2

4

6

8

Out

put-P

ower

[mW

]

Tcav=20oC, cw-Curve

Tcav=34oC, 1 μs Pulses,

Intersection Point

D1=2%

(b)

0 40 80Current

0

1

Volta

ge[V

]

0 40 80Current

0

1Vo

ltage

[V]

Current0 20 40 60 80

0

2

4

6

8

Out

put-P

ower

[mW

]

Tcav=20oC, cw-Curve

Tcav=27oC, 1 μs Pulses, D1=2%

Intersection Point

(a)

3.5 Appendix

117

Eq. (3.37) thermal resistance of 129.4 K/W for the LD_1 and 80.6 K/W for the SOA_1. Main contributions to the thermal resistance are from the InP device and the interfaces between copper mount and diodes. The difference between the two ther-mal resistances is mainly due to differences in the interfaces. The copper itself does not really contribute to the thermal resistance. Below threshold the pulsed curve has a higher output power than the cw curve. This unexpected behaviour is due to the considerable higher temperature of the pulsed curve at such low currents. In order to perform experiments at constant cavity temperatures, the corresponding sensor temperature for a given current has be determined. Fig. 3.24 gives the sensor temperature that has to be adjusted in order to maintain a given cavity temperature for the different currents.

2. Current <-> Carrier-Density RelationsThe relationship between the carrier density and the injection current is needed in order to express the material properties with respect to the carrier densities. A convenient approach to determine the carrier density versus the injection current

below threshold is based on the experimental determination of the differential carrier lifetime [3.41-3.43]

, (3.38)

with the internal differential quantum efficiency and the volume of the ac-

0 50 100 150Current [mA]

0

20

40

60Se

nsor

-Tem

pera

ture

Tsens

[o C]

Tcav=20oC

Tcav=30oC

Tcav=40oC

Tcav=50oC

Tcav=60oC

Fig. 3.24 Sensor temperatures that have to be used in order to maintain a constantcavity temperature of 20, 30, 40, 50 or 60oC for all currents of SOA_1.

I

N I( )τd

N I( ) 1qVact------------- ηiτd I'( ) I'd

0

I

∫=

ηi Vact

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

118

tive section. The differential-carrier lifetime (as defined in Eq. (2.41)) can be de-termined from the sub-threshold modulation response [3.44]

, (3.39)

where is the small signal SOA current modulation frequency and and hold for the modulated and the CW optical power, respectively.Measured values of the differential carrier lifetimes of our SOA_1 derived from the sub-threshold small-signal modulation responses are shown in Fig. 3.25. decreas-es with increasing current as it is expected from Eq. (2.41). The differential carrier lifetimes did not change within the measurement tolerances, when the cavity tem-perature was varied.The internal quantum efficiency is determined from the slope efficiencies of the pulsed current versus output power curves and the experimentally determined mirror and internal losses

, (3.40)

where we have used the definition and relations from (2.64)-(2.65). With the output-

τd

s ω( )δS

-------------- 1

1 ωτd( )2+------------------------------=

ω sδ S

0 20 40 60 80 100Current [mA]

Diff

.Car

rierL

ifetim

e[n

s]

0

2

4

6

10

Fig. 3.25 Variation of the below threshold differential carrier lifetimes with the in-jection current of the SOA_1. The inset shows the measured small-signalmodulation response at I=60 mA. The differential-carrier lifetime was ob-tained by fitting Eq. (3.39) at 1 MHz and at the 3 dB modulation frequencyinto the measured curve.

8-80

-85

-906 7 8 9Modulation-Frequency [10x Hz]

Res

pons

eP m

od[d

Bm

]

ν3dB=150 MHzτd = 1.86 ns

FitExp.

τd

ηi

dPoutdI

------------- hν2q------ ηi

αmαm αint+-----------------------⋅ ⋅=

3.5 Appendix

119

power measured above the threshold (100 mA) of the SOA_1 at oC we find an external quantum efficiency of 0.21. With the mirror and total losses meas-ured at 1.54 μm we derive an internal quantum efficiency of 0.29.The carrier density obtained by evaluating the measured differential lifetimes and the measured quantum efficiency in the integral Eq. (3.40) is depicted in Fig. 3.26. The active layer volume in Eq. (3.40) is 258 μm x 3.6 μm x 0.15 μm. The curve of Fig. 3.26 will allow us to express the gain and refractive-index changes as a function of the carrier densities.

3. Output-PowerA typical output power versus current curve from a measurement on the SOA_1 with a cavity temperature at oC is given in Fig. 3.27. At ~100 mA the device starts lasing. The curve was obtained by adjusting the sensor temperature at 20oC and operating the device with a duty cycle of 2%. This duty cycle is poor enough so that the device does not heat up. For comparison we have also displayed the P-I curve of a 16oC cw-operated device. Since the temperature of the cw device increases with increasing current we observe a significant output-power degradation.

Tcav 20=

0 20 40 60 80 100Current [mA]

0.0

1.0

2.0

3.0C

arrie

rDen

sity

[1·1

018cm

-3]

Fig. 3.26 Carrier density versus current injection of the SOA_1 diode from Fig. 3.1as found by measurement of the differential lifetimes and internal quantum efficiencies

V

Tcav 20≅

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

120

Fig. 3.27 Power versus current curve of the SOA_1 operated at ~20oC cavity tem-perature and operated with 16oC sensor temperature. The curve shows how heat effects strongly perturb the P-I curve. The cavity temperature raises in the experiment while the sensor temperature is kept at 16oC and whereby the output power significantly decreases.

0 50 100 150Current [mA]

200

Out

put-P

ower

mW

]

0

2

4

6

8

Tcav=~20oC

Tsens=16oC

3.6 References

121

3.6 References

[3.1] J.L. Pleumeekers; “POSEIDON: A simulator for optoelectronic semiconduc-tor devices”; Ph.D. thesis, Delft University, 1997

[3.2] E. Anderheggen, J.G. Korvink, M. Roos, G. Sartoris, H.U. Schwarzenbach; “NM-SESES: Semiconductor Sensor and Actuator Simulation”; NM Nu-merical Modelling GmbH, Alte Landstrasse 88, CH-8800 Thalwil, Switzer-land, 1996

[3.3] Ch. Holtmann; “Polarization insensitive semiconductor optical amplifiers in InGaAsP/InP for 1.3 μm wavelengths exploiting bulk ridge-waveguide structure”; Ph.D. thesis ETH-Zürich, Switzerland, Hartung-Gore Verlag, Konstanz 1997

[3.4] B.W. Hakki, T.L. Paoli; “Gain spectra in GaAs double-heterostructure injec-tion lasers”; J. of Appl. Physics, vol. 46, no. 3, pp.1299-1306, March 1975

[3.5] D.T. Cassidy; “Technique for measurement of the gain spectra of semicon-ductor diode lasers”; J. of Appl. Phys., vol. 56, no. 11, pp. 3096-3099, Dec. 1984

[3.6] C.H. Henry, R.A. Logan, F.R. Meritt; “Measurement of gain and absorption spectra in AlGaAs buried heterostructure lasers”; J. of Appl. Phys., vol. 51, no. 6, pp. 3042-3050, 1980

[3.7] M.P. Kesler, C. Harder; “Spontaneous emission and gain in GaAlAs quantum well lasers”; J. Quantum Electron., vol. 27, no. 6, pp. 1812-1818, June 1991

[3.8] R.O. Miles, M.A. Dupertuis, F.K. Reinhard, P.M. Brosson; “Gain measure-ments in InGaAs/InGaAsP multiquantum-well broad-area lasers”; IEE Pro-ceedings-J., vol. 139, no. 1, pp. 33-38, Feb. 1992

[3.9] G.P. Agrawal, N.K. Dutta; “Long-wavelength semiconductor lasers”; Van Nostrand Reinhold, 1986

[3.10] K.A. Andrekson, N.A. Olsson, R.A. Logan, D.L. Coblenty, H. Temkin; “Novel technique for determining internal loss of individual semiconductor lasers”; Electron Lett., vol. 28, no. 2, pp. 171-172, Jan. 1992

[3.11] G.E. Shtengel, D.A. Ackerman; “Internal optical loss measurements in 1.3 μm InGaAsP lasers”; Electron Lett., vol. 31, no. 14, pp. 1157-1159, July 1995

[3.12] P.A. Besse; “Modal reflectivities and new derivation of the basic equations for semiconductor optical amplifiers”; ETH Diss. No. 9608, 1992

[3.13] J.S. Gu; “Femsc”; Institute of Quantum Electronics, ETH-Zuerich; 1991[3.14] S.E.H. Turley, G.H.B. Thompson, D.F. Lovelace; Electron. Lett. vol. 15, pp.

256, 1979

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

122

[3.15] B. Zhao, T.R. Chen, S. Wu, Y.H. Zhuang, Y. Yamada. A. Yariv; “Direct measurement of the linewidth-enhancement factors in quantum lasers of dif-ferent quantum-well barrier heights”; Appl. Phys. Lett. vol. 62, no. 14, pp. 1591-1593, April 1993

[3.16] Landolt-Börnstein; “Numerical data and functional relationships in science and technology”; Vol. 22, Semiconductors, Subvolume a; Springer Verlag Berlin

[3.17] L.D. Westbrook, B. Eng; “Measurement of dg/dN and dn/dN and their de-pendence on photon energy in l=1.5 μm InGaAsP laser diodes”; IEE Pro-ceedings, vol. 133, Pt. J. No. 2, pp. 135-142, April 1986

[3.18] J. Bouley, J. Charil, Y. Sorel, G. Chaminant; “Injected carrier effects on mo-dal properties of 1.55 mm InGaAsP lasers”; J. Quantum Electron., vol. 19, pp. 696-9973, 1983

[3.19] J. Manning, R. Olshansky, C.B. Su; “The carrier induced change in AlGaAs and 1.3 mm InGaAsP diode lasers”; J. Quantum Electron., vol. 19, pp. 1525-1530, 1983

[3.20] Landolt-Börnstein; “Numerical data and functional relationships in science and technology”; Vol. 17a, Semiconductors; Springer Verlag Berlin

[3.21] C.H. Henry; “Theory of linewidth of semiconductor lasers”; J. of Quantum Electron., vol. 18, no. 2, pp. 259-264, Feb. 1982

[3.22] K. Vahala, L.C. Chiu, S. Margalit, A. Yariv; “On the linewidth enhancement factor in semiconductor injection lasers’; Appl. Phys. Lett., vol. 42, no. 8, pp. 631-633, April 1983

[3.23] K. Furuya; “Dependence of linewidth enhancement factor a on waveguide structure in semiconductor lasers”; Electron. Letters; vol. 21, no. 5, pp. 200-201, Feb. 1985

[3.24] B. Zhao, T.R. Chen, A. Yariv; “A comparison of amplitude-phase coupling and linewidth enhancement in semiconductor quantum-well and bulk la-sers”; J. of Quantum Electron., vol. 29, no. 4, p. 1027-1030, April 1993

[3.25] Y. Huang, S. Arai, K. Komori; “Theoretical linewidth-enhancement factor of Ga1-xInxAs/GaInAsP/InP strained-quantum-well structures”; Photon. Technol. Lett., vol. 5, no. 2, pp. 142-145, Feb. 1993

[3.26] T. Yamanaka, Y. Yoshikuni, W. Lui, K. Yokoyama, S. Seki; “Theoretical analysis of extremely small linewidth-enhancement factor and enhanced dif-ferential gain in modulation-doped strained quantum-well lasers”; Appl. Phys. Lett, vol. 62, no. 11, pp. 1191-1193, March 1993

α

α

3.6 References

123

[3.27] W. Rideout, B. Yu, J. LaCourse, P.K. York, K.J. Beernink, J.J. Coleman; “Measurement of the carrier dependence of differential gain, refractive in-dex, and linewidth-enhancement factor in strained-layer quantum well la-sers”; Appl. Phys. Lett, vol. 56, no. 8, pp. 706-708, Feb. 1990

[3.28] H.D. Summers, I.H. White; “Measurement of the static and dynamic linew-idth enhancement factor in strained 1.55 μm InGaAsP lasers”; Electron. Lett., vol. 30, no. 14, pp. 1140-1141, July 1994

[3.29] D.J. Bossert, D. Gallant; “Gain, refractive index and α-parameter in InGaAs-GaAs SQW broad-area lasers”; Photon. Technol. Lett., vol. 8, no. 3, pp. 322-324, March 1996

[3.30] T. Keating, J. Minch, C.S. Chang, P. Enders, W. Fang, S.L. Fang, S.L. Chuang, T. Tanbun-Ek, Y. K Chen, M. Sergent; “Optical gain and refractive index of a laser amplifier in the presence of pump light for cross-gain and cross-phase modulation”; Photon. Technol. Lett., vol. 9, no. 10, pp. 1358-1360, Oct. 1997

[3.31] R.M. Jopson, K.L. Hall, G. Eisenstein, G. Raybon, M.S. Whalen; “Observa-tion of two-colour gain saturation in an optical amplifier’; Electron. Lett., pp. 510-512, May 1987

[3.32] F. Girardin, St. Pajarola, G. Guekos; “Nonlinear parameters of bulk InGaAsP photonic devices”; Proc. Materials for Nonlinear Optics, EOS topical meet-ing, 1997

[3.33] Ch. Zellweger; “Characterization of LDs and SOAs”; ETH diploma thesis, Institute for Quantum Electronics, March 1998

[3.34] K. Kikuchi, M. Kakui, Ch.-En Zah, T.-P. Lee; “Observation of highly nonde-generate four-wave mixing in 1.5 mm travelling wave semiconductor optical amplifiers and estimation of nonlinear gain coefficient”; J. of Quantum Elec-tron., vol. 28, no. 1, pp. 151-156, Jan. 1992

[3.35] D. Gershoni, C.H. Henry, G.A. Baraff; “Calculating the optical properties of multidimensional heterostructures: Application to the modelling of quater-nary quantum well lasers”; J. of Quantum Electron., vol. 29, no. 9, pp. 2433-2449, Sept. 1993

[3.36] G.P. Agrawal; “Fiber-Optic Communication Systems”; New York: Wiley, 1992, ch. 8

[3.37] I.D. Henning, M.J. Adams, J.V. Collins, “Performance prediction from a new optical amplifier model”; J. Quantum Electron., vol. 2, pp. 609-613, 1985

[3.38] A.W. Willner, W. Shieh; “Optimal spectral and power parameters for all-op-tical wavelength shifting: Single stage, fanout, and cascadibility”; J. of Lightwave Technol., vol. 13, no. 5, pp. 771-781, May 1995

3. Material Parameters of 1.55 μm bulk InGaAsP Lasers and Amplifiers

124

[3.39] R. Dall’Ara; Institute for Quantum Electronics, ETH Zuerich Switzerland, 1993

[3.40] D. Kuksenkov, S.Feld, C. Wilmsen, H. Temkin, S. Swirhun, R. Leibenguth; “Linewidth and α-factor in AlGaAs/GaAs vertical cavity surface emitting lasers”; Appl. Phys. Lett, vol. 66, no. 3, pp. 277-279, Jan. 1995

[3.41] W. Rideout, B. Yu, J. LaCourse, P.K. York, K.J. Beernink, J.J. Coleman; “Measurement of the carrier dependence of differential gain refractive index, and linewidth-enhancement factor in strained-layer quantum-well lasers”; Appl. Phys. Lett., vol. 56, no. 8, pp. 706-708, 1990

[3.42] St. Pajarola; “Dual-polarization external cavity diode laser for the optical generation of millimeter-waves”; Ph.D. Thesis Swiss Federal Institute of Technology; Diss. ETH No. 12633, 1998

[3.43] P. Granestrand, K. Fröjdh, O. Sahlén, B. Stoltz, J. Wallin; “Gain characteris-tics of QW Lasers”; European Conference on Optical Communication (ECOC’98), pp. 431-432, Sept. 1998

[3.44] H. Kressel, J.K. Butler; “Semiconductor lasers and heterojunction LEDs”; Academic Press, 1977

125

4 Modified Multimode-Interference Couplers

Following a short theoretical summary of the actual multimode-interference (MMI) theory, we introduce two new types of MMIs. The first type is a wavelength-multi-plexer MMI while the second works as a mode-converter-combiner, which is used to convert fundamental-order modes into higher-order modes and to combine the first-order with another zero-order mode.

4. Modified Multimode-Interference Couplers

126

4.1 Summary of Multimode-Interference Theory

J. Leuthold, J. Eckner, E. Gamper, P.A. Besse, H. MelchiorPublished in Introduction and Appendix of Journal of Lightwave Technology

vol. 16, no. 7, pp. 1228-1239, July 1998

(The following summary was with some modifications published in the introduction and the appendix as part of a larger paper on multimode interference. Here we put this summary at the begin of the chapter since it lays the fundament for the theoret-ical understanding of the MMIs that are subsequently introduced.)Integrated optics makes extensively use of mode couplers in order to split and com-bine light in it’s way through an optical waveguide structure. Standard techniques in-clude digital optical switches like X- and Y-junctions or directional couplers. But digitial-optical switches have considerable lengths and put high technological re-quirements due to the sharp intersection angles and directional couplers need a tight control of the waveguide structures since they make use of the overlap between two modes in waveguides that are close to each other.New couplers based on multimode interference [4.1-4.3] have attracted considerable interest in the last few years. They are of interest since they are compact [4.4], have high design tolerances, large optical bandwidths and show polarization insensitivity when strongly guided structures are used [4.5, 4.6].Meanwhile they have found applications as splitters and combiners [4.7-4.8], as mode converters [4.9, 4.10, Chapter 4.4 on page 147] and as multiplexers [4.11, Chapter 4.3 on page 136].Here we sum up the presently known standard-MMI theory, since we are going to use it extensively throughout the whole chapter. The summary is based on the works published in Ref. [4.14, 4.7, 4.8, 4.15, 4.16].

1. Self Images and the Term Multimode InterferenceMultimode interference (MMI) couplers base on self-imaging [4.1-4.3], which goes back to the Talbot effect, discovered 1836 [4.17]. In multimode waveguides, self-im-aging is the property to reproduce an input field profile in single or multiple images at periodic intervals along the propagation direction of the guide.In integrated optics an MMI is determined by it’s geometry. It typically consists of one or several narrow input ports, that are guided into a wider waveguide. After a certain interval the wider waveguide is stopped and one or several output waveguides are appended to extract the light.

4.1 Summary of MMI Theory

127

Physically, at the transition from the narrow to the wider waveguide, the optical mode of the input guide is adiabatically decomposed into the eigenmodes of the wid-er waveguide. The different new eigenmodes of the wider waveguide then propagate each with it’s own propagation velocity along the wider waveguide. Depending on the position along the wider waveguide, their superpositions form new interference patterns. At some periodic intervals the input profile is reproduced and can be ex-tracted into new output waveguides.Interval lengths, where one or several input profile images are reproduced are given by [4.14, 4.15]

, with , (4.1)

where N denotes the number of input and output arms of the MMI, M is a number giving other possible MMI lengths with N input and output guides, the effective refractive index, the wavelength in vacuum and the equivalent MMI width, which is the geometric width of the MMI including the penetration into the neigh-bour material of the waveguide. The position of the in- and output waveguides is de-termined by the unique parameter a (see Fig. 4.1).

2. Classification of MMIs We classify a MMI as general MMI if the light intensity from either of the N possible input guides is equally distributed onto the N possible output guides and we classify MMIs as overlap MMI if the input-guide position is chosen such that the output-im-ages overlap and unequally distributed output ratios arise.

General NxN-MMIsIf the input guide position, described by the parameter a, lies between: we obtain a general MMI. In a general MMI, the N-fold images have the same shape and intensity as the input mode and only differ in the phase relations between each other.The phase relations at the output arms j in relation to the input arms i are given for the general MMI of Fig. 4.1 for a wave propagating as by [4.15]

i+j even: (4.2)

i+j odd: . (4.3)

is a constant phase shift depending on the MMI-length.

LNM M

N----- 3Lc⋅= Lc

4n'effWeq2

3λ-----------------------=

n'effλ Weq

0 a W N⁄< <

ikz iωt–( )exp

Δφij φ0 π– π4N-------– j i–( ) 2N j– i+( )=

Δφij φ0π

4N-------– j i 1–+( ) 2N j– i– 1+( )=

φ0

4. Modified Multimode-Interference Couplers

128

(a)

a

aa

aa

aa

aa

a

a

a

LN

Weq/N

Weq/N

Weq/N

Weq/N

Weq/N

Weq

inputs i

N

N-1

N-2

1

2

3

1

2

3

N-2

N-1

N

outputs jN odd

Fig. 4.1 General NxN-MMI coupler with (a) N odd input and output guides and (b) N even input and output guides. The positions of the N in- and output-waveguides are defined by the parameter a and the MMI width Weq. [4.15]

a

aa

a

aa

aa

a

a

LN

Weq/N

Weq/N

Weq/N

Weq/N

Weq

inputs i

N

N-1

1

2

3

1

2

N-2

N-1

N

outputs jN even(b)

4.1 Summary of MMI Theory

129

The positions of the in- and output guides is shown in Fig. 4.1(a) for N odd. The case of N even can be treated analogously. Overlap MMIsOn the other hand we obtain overlap MMIs if a takes the value: or

. They are derived by letting or in the MMI in Fig. 4.1. By doing that, one observes that usually two input and output guides of Fig. 4.1 over-lap. The constructive and destructive interference of the two overlapping output arms lead to new intensity distributions at the MMI output. The resulting new MMI con-sists of possible in- and output guides in the middle and two half guides at the MMI edges. We will use the notation given in Fig. 4.2 for overlap MMIs.As these MMIs can be considered as a special case of two overlapping input arms of a general NxN-MMI the resulting phase relations at the corresponding output arms are given as a superposition of the phase relations given in (4.2) and (4.3). If one ad-ditionally studies the mirror characteristics, the following phase relations can be written down [4.16]:For a symmetrical input image at input arm and output port , with even, one gets

for (4.4)

and for an antisymmetrical input image

for (4.5)

is a number greater or equal 1. We only use MMIs with .Now that we have an overlapping image the signals at the output guides do not have any more equal intensities. For a symmetrical input image at input arm the ampli-tudes at the output arms are given according to

. (4.6)

The amplitudes at the output of an antisymmetrical input image from input arm are given as

. (4.7)

a 0=a W N⁄= a 0→ a W N⁄→

N 1–

i′ j′ i′ j′+

Δφi′j′ i′2 j′2+( )MN-----

π4--- j– ′Mπ

2---π2---– 0

π–= Ai′j′

S 0>0<

Δφi′j′ i′2 j′2+( )MN-----π

4--- j– ′Mπ

2--- 0

π–= Ai′j′

A 0>0.<

M M 1=

i′Ai′j′

S j′

Ai′j′S 2

N-------- N j′–( )i′M

N----- π

2---sin=

Ai'j'A j′

i′

Ai′j′A 2

N-------- N j′–( )i′MN-----

π2---cos=

4. Modified Multimode-Interference Couplers

130

Special Overlap MMIs: 1xK and 2xK MMIsImportant special cases of the NxN-overlap MMIs are the “symmetric” and “restrict-ed interference MMIs”. They have found applications as 1xK and 2xK splitters, re-spectively. 1xK MMIs [4.14, 4.7] are a special case of overlap MMIs, that have one input guide at the centre. They are derived of MMIs with N=4K and an input guide at the centre, which splits the input mode onto K uniform images. This result follows directly when calculating the output intensities of an MMI in the overlap picture with

and . These 1xK MMIs are often called “symmetric interference MMIs”. Analogously, 2xK MMIs [4.8] are another special case of overlap MMIs. They have the input guide at or and are often referred as “restricted in-terference MMIs. With , a length after Eq. (4.1) and an input guide at

, the input mode is split onto K images. This result can be verified with the theory of the overlap MMIs, when using an input guide at with . For even the input at and lead to identical output distribu-tions.

inputs i’ outputs j’N odd or even

N-1 1

2

Weq/N

N-1

2

1

LN

Weq/N

Weq/N

Weq/N Weq

0 or N

N0 or N

N M

Fig. 4.2 NxN-overlap-MMI with N-1 access-waveguides within the MMI and twohalf access-waveguides at the MMI-edge. All waveguide positions aresolely defined through the MMI-width Weq and N [4.16].

i' 2K= N 4K=

2 3 Weq⋅⁄ 1 3 Weq⋅⁄N 3K=

2 3⁄ Weqi′ 2K= N 3K=

K 1 3⁄ Weq 2 3⁄ Weq

4.1 Summary of MMI Theory

131

Weq/2

Weq

Weq/2K

Weq/K

Weq/2K

k=1

k=2

k=K

. . .

(a) N=4K

(b) N=3K

Weq/3

Weq

Weq/K

2Weq/3K

k=1

k=2

k=K. .

.

LN = 3Lc/4KM

LN = 3Lc/3KM

2Weq/K

Fig. 4.3 Geometries of special overlap MMIs. (a) 1xK-MMI and (b) 2xK-MMI for K even, as given in [4.7, 4.8, 4.35].

4. Modified Multimode-Interference Couplers

132

4.2 Design Guidelines for MMIs

1. MMI Widths and MMI LengthsOn the one hand, MMI widths should be designed as narrow and as short as possible to attain wide optical bandwidths [4.5, Ref. 4.6 and Chapter 4.6 on page 177]. On the other hand MMI, widths have to be sufficient wide to allow propagation of higher-order modes.In this section we discuss the contributions of the higher-order modes into which a fundamental mode is decomposed if it enters an MMI. The percentage contributions of each of the higher-order modes depends on the input-guide position and the rela-tive width of the MMI in comparison with the input-guide-rib width. The result of a mode-analysis calculation is depicted in Fig. 4.4 for various input-guide positions. The figure shows that an optical mode introduced at the centre ( ) is decomposed into even higher-order modes because of symmetry rela-tions. A fundamental mode introduced at is decomposed into 0th, 1st and 3rd, 4th and 6th, 7th order modes. In these MMIs every third mode is suppressed, while in MMIs with an input at every fourth higher-order mode is suppressed. Gen-erally, we state that the MMI width has to be chosen such that at least the first 10 higher-order modes still propagate. However, if the input guide forms one line with the MMI-edge ( ), fewer higher-order modes are needed and the width can be chosen considerably smaller.

0th orde

1st orde

2nd ord

3rd orde

4th orde

5th orde

6th orde

7th orde

8th orde

0th order1st order2nd order3rd order4th order5th order6th order7th order8th order

Weq/2Weq/3Weq/4WRib/2

0th

2nd

4th

6th

8th

0th

1st

3rd

4th

6th7th

1st

4th5th

6th

0th

2nd

Fig. 4.4 Percentage of decomposition of a fundamental mode into higher-order modes when it is introduced into a MMI as a function of the input-guide positions. Relevant cases are input-guide positions at , ,

and when input-guide forms a line with the MMI edge ( ).Weq 2⁄ Weq 3⁄

Weq 4⁄ WRib 2⁄

Dec

ompo

sitio

n[%

]

0

10

20

30

40

0th4th1st3rd2nd

5th

Input-Guide Positions

Weq 2⁄Weq 3⁄

Weq 4⁄

WRib 2⁄

4.2 Design Guidelines for MMIs

133

The decomposition remains similarly independent of the waveguide structure. Nev-ertheless, we had to perform calculations for a concrete optical waveguide. We picked out a standard MMI-waveguide structure as used in the subsequent all-optical switches. The structure is shown in Fig. 4.5. The data of Fig. 4.4 were calculated with

this structure. The MMI has a geometric width of μm. This corresponds to an equivalent width of μm. The input-guide rib width was

μm. Next we investigated how the mode decomposition is modified when the input-guide width varies. We have performed our calculation for two configurations which are of practical interest. On the one hand a general 2x2 MMI of length and on the other hand a restricted-interference 2x2 MMI of length , where the input-guide position is at (Fig. 4.6). The width of the two MMIs is chosen as small as possible. However, they must be sufficient wide so that the two in- and output waveguides do not touch each other. Since the in- and output guides of the MMI in

InP

InP

1.28-Q

1.10-Q

300 nm570 nm50 nm

Width w

Fig. 4.5 MMI-waveguide structure.

n’=3.403n’=3.296n’=3.168

Wg 16,3=Weq 16,8=

WRib 3,0=

3Lc 2⁄3Lc 6⁄

Weq 3⁄

(a) General 2x2-MMI (b) Restricted Interference 2x2-MMI

Wg

WRib

Wg

WRib

Fig. 4.6 Two different types of 2x2-MMIs. (a) General 2x2-MMI with input guides at the MMI edges. (b) Restricted-interference 2x2-MMI with input guides at .Weq 3⁄

Weq/3

L = 3Lc/2L = 3Lc/6

4. Modified Multimode-Interference Couplers

134

Fig. 4.6(a) are quite far apart, it is possible to minimize this MMI type, although the length formula suggests a long MMI. Indeed, it is possible to minimize the general 2x2-MMI length to the length of the restricted interference 2x2-MMI. Finally we have chosen the geometry of our structures for the simulations as follows: (a) the general 2x2-MMI has a width =9.5 μm and a length of 470 μm, (b) the restricted-interference 2x2-MMI exhibits a width =16.8 μm and a length of 425 μm. The mode decomposition for a varying input-guide width of the general 2x2-MMI type with the input guide aligned to the MMI edge is depicted in Fig. 4.7(a) while the mode decomposition of a restricted 2x2-MMI is depicted in Fig. 4.7(b) for var-

WgWg

0th

order

1st

order

a

3rd

order

4th

order

Column

6th

order

7th

order

0th order

1st order

2nd order

3rd order

4th order

5th order

6th order

7th order

8th order

0th order1st order2nd order3rd order4th order5th order6th order7th order8th order

1.0 2.0 3.0 4.0Rib-Width [μm]

0

10

20

30

40

50

Dec

ompo

sitio

n[%

]

(a)

Fig. 4.7 Mode-decomposition for a fixed MMI width but varying input-guide widthsof (a) a general 2x2-MMI with input guides at the MMI edge (b) a restrict-ed 2x2-MMI with input guide at . The increase of the input-ribwidth reduces the number of guided higher-order modes.

Weq 3⁄

Rib-Width [μm]1.0 2.0 3.00

10

20

30

Dec

ompo

sitio

n[%

]

(b)0th order1st order

3rd order4th order

6th order7th order

4.2 Design Guidelines for MMIs

135

ying input-rib width. In conclusion, increase of the input-rib width reduces the number of higher-order modes propagating within the structure.

2. Trap WaveguidesBecause of nonidealities in input waveguide structures and in MMI waveguides or due to spontaneous emission from amplifiers, disturbing reflections, etc,... evanes-cent fields and higher-order modes may be generated. In order to avoid perturbations from these unwanted fields, we have added trap waveguides which extract part of these disturbances. We have depicted this kind of a trap waveguide for a restricted 2x2-MMI in Fig. 4.8. The depicted 2x2-MMI with trap waveguide has been tested within a device that contained active and passive components. Experimentally we found that approximately 10% of the undesired total spontaneous emission can be extracted with trap waveguides.

Trap Waveguides

2x2-MMI

Fig. 4.8 Trap waveguide concept shown for a restricted 2x2-MMI.

InputGuides

OutputGuides

4. Modified Multimode-Interference Couplers

136

4.3 Guided Wave 1.30/1.55 μm Wavelength Division Multiplexers based on Multimode Interference

J. Leuthold, E. Gini, H. Melchiorintended for submission

AbstractThree possible concepts for very compact 1.30/1.55 μm Wavelength Division Mul-tiplexers (WDM) based on multimode interference (MMI) are discussed and first implementations are presented. The best crosstalks and polarisation insensitivity are attained with a new concept based on 1x2-MMIs. The WDM principle is general enough to be used for multiplexing two signals of other wavelengths.

1. IntroductionSimple low cost wavelength multiplexers (WDM) for the 1.30 and 1.55 μm commu-nication bands are key components for bi-directional transmitter-receiver devices in local area optical networks. To keep the costs low monolithic integration of the WDMs with the transceivers is prerequisite and therefore duplexers on InP materi-als, which are compatible with lasers and photodetectors, are needed. Several differ-ent waveguide-type demultiplexing schemes based on directional couplers [4.18], Mach-Zehnder Interferometers [4.19, 4.20], asymmetrical Y-junctions [4.21], Phased Array [4.22, 4.23] and MMIs [4.11] have been presented. While good per-formances were attained with cascaded directional couplers [4.24], cascaded MZIs [4.25] and Phased Array multiplexers [4.23], the former were found to have large di-mensions and the latter are intricate to design.Due to the compact size, the ease in design and fabrication and the polarisation in-sensitivity MMI-couplers [4.1] gained an increasing popularity in recent years [4.3]. Meanwhile they have found applications as splitters and combiners [4.7, 4.8], as mode converters [Chapter 4.4 on page 147] and as multiplexers [4.11].In this work we compare three different concepts of 1.30/1.55 μm MMI-WDMs and demonstrate first implementations. Experiments with the three device concept show that a new MMI-WDM approach based on 1x2-MMIs delivers the best performance. The new devices are compact, insensitive for light of different polarisation, deliver high signal transmissions over a large spectral range and are simple to design. They have the potential for applications in transceivers [4.12] and in absorption-type all-optical devices [4.13] where high transmissions are required and crosstalk plays a minor role.

4.3 WDM based on MMI

137

2. MMI-WDMs2.1 StructureThe structures of the different MMI type WDMs, which were realized and compared are shown in Fig. 4.9.A first 1.30/1.55 μm MMI-WDM is depicted in Fig. 4.9(a). The WDM is derived of general 1x1 MMIs [4.3]. A general 1x1 MMI mirrors an input image unchanged to an output image, whereas the output guide lies at a cross position relative to the input

1.30 μm

1.55 μm

(a) General 1x1-MMI Type

1.30 μm

1.55 μm6*restricted 1x1-MMI’s at 1.55

Weq/3

Weq/3

Weq/3

(b)

Weq=12.3μm

LWDM=2530 μmRestricted 1x1-MMI Type

Weq/4

Weq/4

Weq5*1x2-MMI’s at 1.30 6*1x2 MMI’s at 1.55

1.30 μm

1.55 μm

Wda1

a2

(c) Restricted 1x2-MMI Type

1x2-MMI

Weq=18.0μm

L=2060μm L1x2=415μm

Fig. 4.9 Three different types of multimode interference couplers, that can be used to multiplex respectively demultiplex 1.30 μm and 1.55 μm signals. The MMI WDMs are built up, by cascading 5 MMIs optimized for wavelength 1.30 μm. The resulting MMI has precisely the length of a cascade of 6 MMIs of the same type at wavelength 1.55 μm. However the in- and outputs of the even numbered cascade of MMIs at 1.55 are at the same positions. Those for the odd numbered 1.30 MMIs are displaced.

5*restricted 1x1-MMI’s at 1.30

5*1x1-MMI’s at 1.30 6*1x1-MMI’s at 1.55 Weq=6.9 μm

LWDM=2515 μm

4. Modified Multimode-Interference Couplers

138

guide. Optimizing these 1x1 MMIs for the wavelength at 1.30 μm and merging the same MMI five times leads to a new long MMI. Since five is an odd number, the 1.30 light is mapped onto the cross output port. But the length of an MMI depends as

on the wavelength, so that the same long MMI matches also into six 1x1-MMIs optimized for light at 1.55 μm. Since six is an even number, the 1.55 μm light is coupled out into the bar output port. By using the long MMI in either of the two ways, multiplexing of two wavelengths is attained. The device of Fig. 4.9(a) is anal-ogous to the 0.98/1.55 μm MMI-WDM of Ref. [4.11].The WDM of Fig. 4.9(b) has a larger width and almost the same length as the WDM of Fig. 4.9(a). It is derived of restricted 1x1 MMIs. In restricted 1x1-MMIs [4.3] the input port is chosen at 1/3 of the MMI width, so that interference effects can be ex-ploited to reduce the required length for an 1x1 MMI by a factor three. However in order to avoid mode overlap between the output waveguides, the MMIs width has to be increased, which in turn demands for longer cavity lengths, so that the MMI length remains alike. To couple out the wavelength at 1.55 μm into the cross output, six 1x1 MMIs are merged and to couple out the wavelength at 1.3 μm into the bar output, five 1x1 MMI are merged. Again, the two merged MMIs have the same length at the same width and can be used for either operation. The WDM of Fig. 4.9(b) corresponds to the MMI WDM principally proposed in Ref. [4.26].A new and shorter MMI WDM is given in Fig. 4.9(c). The MMI is built up of sym-metric-interference 1x2 MMIs. A symmetric-interference 1x2 MMI [4.3] maps a mode from the central input guide onto two modes of equal intensity at Weq/4 and 3Weq/4 from the MMI edge. When adding a second 1x2 MMI the initial mode profile is regenerated in the middle of the MMI output. The choice of an input guide position at the centre helps to exploit interference effects, which reduce the MMI length by a factor of eight in comparison with a general 1x1 MMI. At a wavelength of 1.55 μm six of these 1x2-MMI are merged. Since six is an even number, the mode at 1.55 μm is mapped from the central input-position to the central output. At wavelength 1.30 μm the same MMI corresponds to a symmetric interference 1x2-MMI merged five times in the demultiplexer region. At the output, this signal is mapped onto the two modes at and 3Weq/4. We append a separate 1x2-MMI to map the two waveguides with the 1.30 μm signals onto a single output-guide and append a central output-waveguide to the main MMI-cavity for extracting the 1.55 μm signal.Generally, an MMI of length may be used as wavelength multiplexer if it obeys the equation

, (4.8)

where and are odd and even integers and and are the respective lengths of 1x1-MMI or corresponding MMIs optimized for the wavelengths and

L 1 λ⁄∼

Weq 4⁄

LWDM

LWDM M1L1 M2L2= =

M1 M2 L1 L2λ1

4.3 WDM based on MMI

139

. Because two wavelengths can principally be multiplexed if

(4.9)

2.2 MMI-Length ConsiderationsThe design tolerances relax with decreasing MMI length [4.5]. To estimate the lengths of the different MMI type WDMs we use the MMI length expressions given in Ref. [4.14][4.16]

. (4.10)

The length is given trough the number Ni determining the MMI type (for example: general NxN-MMI type, restricted interference -MMI with or sym-metric interference -MMI with ). It is further determined by , which gives possible MMI device lengths of a given MMI-type, the equivalent width Weq, which essentially corresponds to the geometric width but takes into account the penetration into the neighbouring material, the effective refractive index and the wavelength .The length of an MMI can be reduced by minimizing the MMIs width . Howev-er the width is limited by the smallest allowed distance between two neighbouring waveguides that should neither touch each other nor allow a mode overlap. When is the minimum distance between two waveguides and is the minimum rib width of waveguides, we find for the smallest allowed MMI-width in the case of the gen-eral 1x1-MMI-WDM that , in the case of the restricted 1x1-MMI-WDM and in the case of the symmetric interference 1x2-MMI-WDM . On the other hand the MMI-width may not be too small, since a certain width is necessary for the propagation of higher modes within the MMI-cavity. Additionally the design tolerances relax the wider the MMI-widths. A deviation of e.q. 0.5μm from the design width is more severe for small ’s than for large ones.We define as a figure of merit for the MMI length

. (4.11)

It gives a measure for the MMI-length independent of the waveguide structure. Note that this definition of the figure of merit for the length of MMIs delivers the length ratio in comparison to the well known 2x2-MMIs. Both the general 2x2-MMI and the restricted 2x2-MMI have of 1 at 1.50μm.

λ2 L 1 λ⁄∼

λ1 λ2⁄ M1 M2⁄≅

LMMIMiNi------

4neff λi( )Weq2 λi( )

λi------------------------------------------=

2xK Ni 3K=1xK Ni 4K= Mi

Wgneff

λiWeq

dw

Weq 2w d+=Weq 3 w d+( )=

Weq 4 w d+( )=

Weq

LF.o.M.LMMI

4neff w d+( )2⋅-------------------------------------=

LF.o.M.

4. Modified Multimode-Interference Couplers

140

In Table 4.1 we give the figure of merit lengths for the different MMI-WDMs. The

symmetric 1x2-MMI-WDM delivers the shortest MMIs and has at the same time the largest MMI-width. It will surely be the preferred demultiplexer. The figure of merit length of the general 1x1-MMI-WDM delivers a relatively small length too. How-ever it may not be used that small, since the minimum width of this type of MMI is critically small and considers amounts of higher order modes propagating in the MMI would be cut off.

2.3 Design ConsiderationsFinally we want to find the precise geometries and structures for polarisation inde-pendent MMI-WDMs. Eq. (4.8) gives a rough estimate of the number of MMIs that have to be cascaded to multiplex two wavelengths. However Eq. (4.8) neglects the material dispersion.To determine the geometry and the structure to multiplex two specific wavelengths, the following search procedure is used. First a MMI, that can be used for demulti-plexing is selected (see for example Table 1) and it’s geometric MMI-width is fixed. Depending on the structure, the geometric width will determine the equivalent MMI-width which additionally includes the penetration of the mode into the neighbouring material. Second, Eq. (4.8) is used to determine the number and

of MMIs which have to be cascaded for each of the two given wavelengths and . Last we can detune the two wavelengths that are to be multiplexed by the proper choice of the waveguide structure. This detuning is due to the material dis-persion, which strongly influences the effective refractive index and the equivalent MMI-width. To find the two multiplexing wavelengths for a given MMI-WDM structure, we solve for a given wavelength the following equation selfconsist-ently for the second wavelength

, (4.12)

Table 4.1: Length Figure of Merits for Different MMI-WDM Types

Type WeqCritic due tocut-off Weq

LF.o.M.

General 1x1-MMI-WDMNxN-type with N=1

yes ~8.7

Restricted 1x1-MMI-WDM2xK type with N=3K and K=1

no 11.6

Symmetric 1x2-MMI-WDM1xK-type with N=4K and K=2

no 7.7

Weq 2w d+=

Weq 3 w d+( )=

Weq 4 w d+( )=

Weq

Wg

WeqM1

M2 λ1λ2

λ1λ2

LMMI λ1( ) LMMI λ2( )=

4.3 WDM based on MMI

141

or written out with Eq. (4.10)

. (4.13)

The effective indices and equivalent widths are calculated with the effec-tive index method. Mandatorily the material dispersions have to be considered.Subsequently we consider the waveguide structure given in Fig. 4.10. We investigate the effect on the two multiplexing wavelengths, when the ridge waveguide structure is modified.

Depending on the choice of , and the waveguide structure, two different wavelengths are multiplexed. Fig. 4.11 shows how by variation of the quaternary-layer thickness the optimum wavelength at output 2 varies between 1.45 and 1.70 μm, while the wavelength at output 1 is fixed at 1.30 μm.

M1N1-------

4neff λ1( )Weq2 λ1( )

λ1--------------------------------------------

M2N2-------

4neff λ2( )Weq2 λ2( )

λ2--------------------------------------------=

neff Weq

Q-1.07

InP

InP

e=0.100 μmd=0.520 μm

Fig. 4.10 Polarisation insensitive 1.30/1.55 MMI-WDM structure, as derived from Fig. 2 & 3.

M1 M2

Thickness of Quaternary-Layer d [μm]0.3 0.4 0.5 0.6 0.7

Wav

elen

gth

[μm

]

1.30

1.40

1.50

1.60

1.30 μm Output

1.55 μm Output

5 & 6 MMI

6 & 7 MMI7 & 8 MMI

Fig. 4.11 Two signals at specific wavelengths can be multiplexed by an appropriate choice of the quaternary layer width and the number of cascaded MMIs (of the general 1x1, restricted 1x1, restricted 1x2-type,...) . Here we fixed the MMI length for optimum output at 1.30 μm. When varying the quater-nary layer thickness, the optimum output wavelength of the 1.55-output varies in a wide range. Using a cascade of 6 and 7 or 7 and 8 MMIs for the 1.3/1.55 multiplexer other wavelength ranges can be mastered.

d

4. Modified Multimode-Interference Couplers

142

Beyond, to make the structure polarisation insensitive we made use of the freedom to vary the bandgap wavelength of the quaternary material and the freedom to etch into the guiding quaternary layer. Fig. 4.12 demonstrates that for a quaternary layer thickness around d=0.520 μm the MMI-WDM becomes polarisation insensitive for both, the wavelength at 1.30 and 1.55 μm. This optimum was found for a quaternary layer with λgap=1.07 μm and by etching 100 nm into the quaternary layer.

2.4 Simulation of Transmission and Crosstalk Characteristics for a 1.3/1.55 MMI-WDM with Optimized Waveguide Structure

A mode-analysis calculation for the three structures of Fig. 4.9 reveals, that the structure (c) delivers the best design tolerances together with the best crosstalks. The results are shown in Fig. 4.13. The MMI-geometries for the simulation are listed in Table 4.2.

Table 4.2: Geometries of MMI-WDM’s

Type Length Width Wg Rib-Widths Distance d

General 1x1 2515 μm 6.0 μm 2.1 μm 1.8 μm

Restricted 1x1 2530 μm 11.4 μm 2.3 μm 1.8 μm

Restricted 1x2 2060 + 415 μm 17.1 μm 2.7 μm 1.8 μm

Thickness of Quaternary-Layer d [μm]0.3 0.4 0.5 0.6 0.7

Wav

elen

gth

[μm

]

1.30

1.40

1.50

1.60

1.30 μm Output

1.55 μm OutputTE

TM

TETM

Fig. 4.12 For a 1.30/1.55 multiplexer the 1.07-Q quaternary layer thickness has to be chosen around d=0.520 μm. The figure additionally shows that the po-larisation sensitivity is for both wavelengths minimal. For optimizing the polarisation dependence we used the freedom, to choose the bandgap of the quaternary layer and the etch-depth into the guiding layer. The opti-mum was found for λgap=1.07 μm and a 100 nm etch depth into the qua-ternary layer.

4.3 WDM based on MMI

143

3. ExperimentsThe three MMI-WDMs presented in Fig. 4.9 were realized to demonstrate and com-pare the multiplexing characteristics of the different types. We measured the cou-pling efficiencies and the crosstalk of the three structures around 1.30 μm and around 1.55 μm.The MMI-WDM of Fig. 4.9 were produced as ridge waveguide structures on a dou-ble-heterostructure InP/InGaAsP/InP wafer. The ridge structure was optimized with the procedure proposed above to obtain polarisation insensitive MMIs at both mul-tiplexing wavelengths (Fig. 4.10). The epitaxial layers were grown by Low-Pressure Metal Organic Chemical Vapour Deposition (LP-MOVPE) on (100) Indium Phos-phide. The structure of the wafer has an InP buffer layer, a 520 nm waveguiding qua-ternary layer (1.07 μm InGaAsP) and a 1500 nm InP upper cladding layer. Dry etching 1600 nm into the InP cladding layer formed the ridge. After cleaving, a broad-band anti-reflection coating around 1.45 μm was applied. Measurements of the optical properties were then performed on an optical bench. The dimensions of the MMI-WDM are listed in Table 4.2. The angle between the two crossing waveguides of the MMI-WDM (c) is 19o. Fig. 4.14 shows the measured transmission and crosstalks for each of the devices of Fig. 4.9 around the wavelength at 1.30 and 1.55 μm. The best transmission and cross-talk characteristics are obtained with the ’restricted 1x2-MMI-WDM’. The transmis-sion losses are 2 and 1 dB at 1.30 μm and 1.55 μm respectively. Crosstalk exceeding

Type A, TE

Type D, T

Type N, TE

Type A, TE

Type D, TE

Type N, TETran

smis

sion

[dB

]

-15

-10

-5

0

-20

1.275 1.300 1.325Wavelength [μm]

1.30 Output

1.55 Output

-25

Tran

smis

sion

[dB

]

-15

-10

-5

0

-20

1.500 1.550 1.60Wavelength [μm]

-25

1.30 Output

1.55 Output

Fig. 4.13 Simulation of the transmission and crosstalk characteristics, that can be obtained with each of the three MMI-WDMs. The simulations show, that with the restricted 1x2-MMI-WDM the best transmission characteristics and the best crosstalks are achievable. To obtain a high crosstalk at 1.55 μm, the peak wavelength at the 1.55 output has to be shifted towards the 1.56 μm range.

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TE

TM

TE

TM

-15

-10

-5

0

1.30 Output

(c) Restricted 1x2-MMI-WDM Type:

TE

TMTE

TMTran

smis

sion

[dB

]

-15

-10

-5

0

1.280 1.300 1.320Wavelength [μm]

1.30 Output

1.55 Crosstalk Output

(a) General 1x1-MMI-WDM Type:

Tran

smis

sion

[dB

]

-15

-10

-5

0

-201.500 1.550 1.60

Wavelength [μm]

1.30 Crosstalk Output

1.55 Output

TE

TM

TE

TMTran

smis

sion

[dB

]

-15

-10

-5

0

1.280 1.300 1.320Wavelength [μm]

1.30 Output

1.55 Crosstalk Output Tran

smis

sion

[dB

]

-15

-10

-5

0

-201.500 1.550 1.60

Wavelength [μm]

1.30 Crosstalk Output

1.55 Output

(b) Restricted 1x1-MMI-WDM Type:

Tran

smis

sion

[dB

]

1.280 1.300 1.320Wavelength [μm]

1.55 Crosstalk Output Tran

smis

sion

[dB

]

-15

-10

-5

0

-201.500 1.550 1.60

Wavelength [μm]

1.30 Crosstalk Output

1.55 Output

1.30 μm1.55 μm

5*1x1-MMI at 1.306*1x1 MMI at 1.55

1.30 μm1.55 μm

5*restricted 1x1-MMI’s at 1.306*restricted 1x1-MMI’s at 1.55

5*1x2-MMI’s at 1.306*1x2-MMI’s at 1.55

1.30 μm

1.55 μm

(images scaled with 68.5%)

Fig. 4.14 Transmission characteristics of the MMI-WDMs from Fig. 4.9(a), (b) and (c) for the output at 1.30 μm and the output at 1.55 μm. As predicted, the restricted 1x2-MMI-WDM shows the best transmission characteristics. Polarisation insensitive demultiplexing with 1 and 2 dB transmission loss-es at 1.55 and 1.30 is demonstrated for this device. Crosstalks of 18 and 10 dB at the respective wavelengths are found.

TETM

TETM

TETM

TETM

TETM

TETM

4.3 WDM based on MMI

145

12 and 20 dB at the respective wavelengths are attained. Polarisation insensitive multiplexing is found. In comparison to the MMI-WDM from Fig. 4.14(a) and (b) the cross-talk is improved by over 5 dB. We attribute this improvement to the short length and large width of the MMI-cavity.Improvement of the transmission and crosstalk characteristics are obtained by using strongly instead of weakly guided structures, by adding trapwaveguides at the MMI-edges to extract evanescent light, by optimizing the angle between the crossing waveguides and by reducing MMI-cavity lengths through use of parabolically ta-pered MMIs [4.28].

4. Future Applications as MMI-WDM Add-Drop SwitchesMinor changes are necessary to turn MMI-WDM’s into wavelength selective add-drop switches.Fig. 4.15 shows a MMI-WDM add-drop switch to add or drop a signal at 1.30 with one at 1.55 μm. When the phase-shifters are not used, the switch is in the cross state for the signal at 1.30 and it is in the bar state for the 1.55 μm wavelength. This means that a signal at 1.30 μm can be added or dropped. When a phase-shift of π is applied, the switch is in the bar state for both the signals at 1.30 and for the signals at 1.55 μm.The add-drop WDM of Fig. 4.15 is derived from the restricted 1x1-MMI-WDM of Fig. 4.9(b). It is obtained by splitting up the MMI-WDM at the centre into two parts and adding two waveguides with phase-shifters. The mapping characteristics of these MMIs are such that in the middle of these MMI-WDMs, the 1.55 signal is di-rected to the cross port, but the 1.30 signal is split up into two signals of equal inten-sities. When guiding the signals onto waveguides, the phase-relations between these

1.30 μm

1.55 μm

Weq/3

Weq/3

Weq/3

Weq

Phase-Shifters(1.30 μm)

Weq/3

1.30 μm

1.55 μm1.30 μm

Fig. 4.15 Add-drop MMI-WDM to add, drop or leave a 1.30 μm signal in the signal path of a 1.55 μm signal. The switch is obtained by dividing the MMI-WDM of Fig. 4.9(b) into two parts of equal length and by connecting the two segments with waveguide phase-shifters. The field distribution in the central region of a MMI-WDM allows to alter the phase-relations of the signal that is split up into two waveguide branches and to direct this sig-nal to the bar or cross output.

LWDM/2 LWDM/2

( )1.55 μm

1.30 μm

4. Modified Multimode-Interference Couplers

146

two branches can be changed such that the 1.30 signal is directed towards the cross or the bar output port depending on the phase-shift that is induced. The intensity of the phase-shift has no effect onto the 1.55 signal, since all the light is guided in a sin-gle waveguide. Right behind the switch, the 1.55 signal will be in the bar state.The add-drop switch of Fig. 4.15 switches the 1.30 μm wavelength. A similar switch that allows to switch the 1.55 μm wavelength can be designed analogously with MMI-multiplexers that are built up of 6 and 7 1x1-MMIs respectively (refer to Fig. 4.11).Similar add-drop principles exist for other wavelengths and other MMI-type WDMs (including those of Fig. 4.9(a) and (c)).

ConclusionsThree concepts for monolithically integrateable WDMs based on MMI are com-pared. Although further fabrication optimization is needed, experiments with first implementations show that a new MMI-WDM approach delivers the best perform-ance. The device shows polarisation insensitive demultiplexing with 2 and 1 dB ex-cess loss in comparison to straight waveguides and crosstalks exceeding 12 and 20 dB at 1.30 μm and 1.55 μm, respectively. The new MMI-WDM might find applica-tions where compactness, polarisation insensitivity, high transmissions over a large spectral range are needed.

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4.4 Multimode Interference Couplers for the Conversion and Combining of Zero and First Order Modes

J. Leuthold, J. Eckner, E. Gamper, P.A. Besse, H. Melchior

Journal of Lightwave Technologyvol. 16, no. 7, pp. 1228-1239, July 1998

Abstract Optical waveguide-mode combiners for fundamental and first order modes, based on multimode-interference (MMI) couplers are presented. These devices convert a fun-damental mode into a transversal first order mode and combine it in lossless fashion with a second fundamental mode. They can separate zero and first order modes in a common waveguide and allow the splitting and combining of zero and first order modes with non-uniform power splitting ratios. Realizations in InGaAsP/InP are demonstrated. These new components have successfully been integrated into all-op-tical switches and were found to have advantageous characteristics in all-optical de-vices.

1. IntroductionIntegrated waveguide couplers, that generate first order modes and that allow to combine them with zero order modes have attractive applications in photonic inte-grated circuits. They are already used as adiabatic asymmetrical Y-junctions [4.29]in 2x2 Digital Optical (DOS)-Switches [4.30] and in Mach-Zehnder interferometer (MZI) switches [4.29]. A TE/TM-mode splitter has been reported, which transforms the TM-mode selectively into a first order mode and then extracts it using an asym-metrical Y-junction [4.31]. Although these Y-junctions work well, they have consid-erable lengths and sharp intersection angels, which poses a technological challenge. Pedersen et al. [4.32] recently described Y-junctions of good quality in InGaAsP/InP. Their couplers’ Y junctions can be used as mode converters and mode combiners but they are longer than 1.5 mm.Multimode interference (MMI) couplers [4.1-4.3] on the other hand, have found a considerable interest in the last few years. They feature compactness, high design tolerances, a large optical bandwidth and polarisation insensitivity when strongly guided structures are used [4.4, 4.5]. Currently MMIs are used as 3 dB power split-ters and combiners [4.33], as power splitters with free selection of power splitting

4. Modified Multimode-Interference Couplers

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ratio (so called butterfly MMIs) [4.34], as multileg-splitters, that can act as 1xN, 2xN or NxN [4.7, 4.8, 4.35] splitters and they have recently been used for wavelength-multiplexing [4.11].In this paper we introduce new types of short MMI-converter-combiners, that con-vert a fundamental mode into a transversal first order mode and that allow to com-bine the first order mode with a second fundamental mode. The ease with which the couplers can be integrated and the high coupling efficiencies make these couplers at-tractive for applications in all-optical devices, that use optical signals to control an optical data-signal [4.36-4.40]. The new couplers have successfully been used to in-troduce the control-signal as a first order mode into all-optical switches and to mod-ulate the zero order mode data-signal [4.41, 4.42] (see Chapter 6.1 on page 226). Due to the difference in the mode’s symmetry, the control-signal can easily be separated from the data-signal after signal-processing even if the two signals copropagate and have the same wavelength.After a short introduction on multimode interference in paragraph II, we explain the principle for mode-conversion in a first section. We propose three different kinds of converters, which should be chosen depending on the applications. In a second sec-tion we show, how these MMIs can be used as mode combiners for a zero order mode in combination with a second input guide. The opposite process - the separation of modes of different orders into different waveguides - is discussed in a last section. The manifold possibilities of zero order mode MMIs, like splitting and combining, are guaranteed for a first order mode also. They are discussed in paragraph III. Ex-perimental results are given in paragraph IV. The experiments are in good agreement with the theoretical predictions.

2. Principles of MMI-Converter-CombinersMMI-converter-combiners are waveguide couplers, that convert a fundamental or-der mode into a first order mode and that allow to couple a second fundamental mode into the same output-waveguide. A demonstration of the converter-combiner princi-ple for a simple structure is given in Fig. 4.16. The mode conversion principle can be derived from the presently known MMIs. A classification, a summary of the presently known theory and the notation used throughout this paper has been given in Chapter 4.1 on page 126. The theory section was part of the appendix of the original work submitted to J. of Lightwave Technol-ogy, but has been moved at the begin of this chapter since it is valid for all-MMI types presented in the whole chapter.

4.4 MMI-Converter-Combiner

149

A B

Fig. 4.16 Beam propagation simulation (BPM) of a ridge MMI waveguide structure demonstrating the MMI-converter-combiner principle. The left hand side shows how a fundamental mode introduced at the wider input-guide is completely mapped onto the wider output-guide. The right hand side shows how a second fundamental mode is converted with a 66% efficiency into a first order mode and mapped into the same wider output-waveguide. The remaining 33% of the light are decoupled through the thinner output on the left side of the MMI.

4. Modified Multimode-Interference Couplers

150

2.1 Mode Conversion Based on Multimode InterferenceThe principle for the conversion of a fundamental mode into a first order mode by use of MMIs is discussed first. We use general MMIs to explain the basic mode con-version mechanism and then improve the conversion efficiency of these MMIs.The method of multi-mode interference allows to convert a fundamental mode into a first order mode. A first order TE01 or TM01-mode is characterized by its intensity profile and by the phase relation within the mode. Two steps lead to the desired mode characteristics:1.) The intensity profile of the first order mode is realized by positioning an input arm of the MMI such, that the output guides of the MMI touch each other to form a wider waveguide that transmits two beams in the form of a first order mode. Refer-ring to Fig. 4.1(a) this implies the parameter to take the value of approximately one half of the rib width.

In a first approach the rib width of the fundamental mode input guide and the rib width of the first order mode output guide are for strongly guided struc-tures with sufficient accuracy related by

(4.14)

A careful analysis reveals that the Gaussian beam waist of the zero order mode guided in the waveguide with rib-width and the beam waist of the first or-der mode guided in a rib of width are related by [4.6]

(4.15)

(The beam waist is the width of the mode, where the amplitude has decayed on of the maximum value. Due to the definition of first order Gaussian modes, is the span of the mode, where it’s amplitude has decayed on a value of 0.86 of the peak maximums.)

2.) The phase distribution within a first order mode varies by π from one intensity peak to the other. This phase-profile is automatically obtained when using an MMI-input guide at the MMI edge. With the phase relations given in Eq. (4.2)and Eq. (4.3) we derive for adjacent output guides of a general NxN-MMI that uses the inputport i=1

for , with . (4.16)

Thus the phases at the adjacent output guides and for are always automatically phase-shifted with π between each other,

if input is used. The same applies for the opposite input guide i=N for rea-sons of symmetry. :

a

w0w1

w1 2w0≅

d0w0 d1

w1

d1 3d0=

d0e 1–

d1

Δφ1 j Δφ1j– π= j 2m= j 2m 1+= 1 m N 1–≤ ≤

j 2m= j 2m 1+=1 m N 1–≤ ≤

i 1=

4.4 MMI-Converter-Combiner

151

a

2a

w1

L3=3Lc/3

a

j=1

j=2j=3

Input

i=1

Weq/3

Weq

w0

a

a

2aw1

Outputs(b) 66% Mode Converter MMI

(a) 50% Mode Converter MMI

Input

i=1

L4=3Lc/4

a

Outputsj=1

j=2j=3

j=4

Weq/4

Weq

w0

(c) 100% Mode Converter MMIInput

i=1

L4=3Lc/4 Outputsj=1

j=2j=3

j=4

Weq/4

Weq

i=4

Phase-Shifter50%-MMI-Converter1x2-MMI

Fig. 4.17 Multimode-interference (MMI) couplers to convert a zero order mode into a first order mode. (a) MMI-converter with a 50% conversion effi-ciency of a fundamental mode into a first order mode. Each of the two out-er waveguides contain 25% of the intensity. (b) 66%-MMI-converter, that converts a fundamental mode with a 66% efficiency into a first order mode. The remaining 33% of the light intensity is mapped into a zero or-der mode. (c) MMI-converter realized with a 1x2-MMI, a phase-shifter and the 50%-MMI-converter of Fig. 4.17(a) with two inputs. A zero order mode is completely converted into a first order mode.

4. Modified Multimode-Interference Couplers

152

Mode-combiners that use general-MMIs with N=4 and N=3 are shown in Fig. 4.17(a) and (b). The first one couples 25% of the power of the fundamental input mode into the fundamental modes of the outer outputs and 50% into the first order mode of the central output. In the following lines we refer to this MMI as 50%-MMI-converter. The N=3 mode converter MMI, or 66%-MMI-converter, couples 33% of the power of the fundamental input mode into the fundamental mode of the output-waveguide j=1 and a fraction of 66% to the first order mode of the wider j=2,3 output waveguide.The conversion efficiency can be improved to 100% by modifying the 50%-MMI-converter. Consider the phase relations of the 50%-MMI-converter respectively it’s primal version, the general 4x4-MMI in Table 4.3. If we introduce two coherent sig-nals at input-arms i=1 and 4 with a phase retardation of π between each other, then the phases of the two signals in the output j=2 and the phases of the two signals in the output j=3 interfere constructively and they interfere destructively in the output-arms j=1 and j=4. Therefore the signal in the output j=2,3 of the first order mode su-perpose and the zero order mode outputs j=1,4 are deleted. That’s exactly what we are looking for. A possible realisation of such a device is given in Fig. 4.17(c). It con-sists of an 1x2-MMI for splitting the input mode into two equal parts, followed by a phase-shifter region to adapt the phases to the required phase shift of π. The two sig-nals in the two branches are then guided into the 4x4-MMI, whose input-arms are positioned to form a 50%-MMI-converter.

2.2 MMI-converter-combinersThe above MMI-converters that couple light from one input to a first order output are of interest since, with a second input they allow a zero order mode to be exited into the same output waveguide. We call these new couplers MMI-converter-com-biners.The MMI-converter-combiners of Fig. 4.18(a)-(d) and Fig. 4.24 combine three op-erations in a single device: 1.) The complete projection of a fundamental mode A into a common output, 2.) the conversion of a zero order mode B into a first order mode, which is mapped into the common output guide and 3.) when going to the

Table 4.3: Phase-Relations of the general 4x4 MMI

N=4 j=1 2 3 4

i=1 -4/4∗π -3/4∗π -7/4∗π -4/4∗π

i=2 -3/4∗π -4/4∗π -4/4∗π -7/4∗π

i=3 -7/4∗π -4/4∗π -4/4∗π -3/4∗π

i=4 -4/4∗π -7/4∗π -3/4∗π -4/4∗π

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153

generalisation, the splitting of the zero order as well as the first order mode onto Koutput-waveguides. The generalisation is discussed in the Appendix. The different structures are compared in Tab. 2.

50%-MMI-converter-combiner A MMI-converter-combiner with high technological tolerances is obtained by mod-ifying the 50%-MMI-converter. The new MMI has a 100% coupling efficiency for the fundamental mode A into the wider output guide and a 50% first order mode con-version efficiency of a fundamental mode B into the same output waveguide.Fig. 4.18(a) illustrates how the mapping characteristics of two different MMIs have been combined to form the new 50%-MMI-converter combiner. The new MMI in-cludes the 1xK-MMI-splitter with K=1 mentioned in Appendix A.1, and the 50%-MMI-converter of Fig. 4.17(a). For K=1 these two MMIs have the same geometry (the same length for the same width ). The resulting 50%-MMI-converter-combiner possess the in- and outputs of both MMIs. We state that the inputs of the two MMIs are at different positions, however the outputs of the MMI-splitter and the output of the MMI-converter for the first order mode merge. A zero order mode A introduced at the centre is completely mapped into the waveguide output at the centre. The zero order mode B launched into the MMI at the edge is converted into two zero order modes and a single first order mode. The first order mode is imaged into the output port, that already guides the image of the zero order mode A.The 50%-MMI-converter-combiner of Fig. 4.18 is of special interest, since it has got in- and output guides that are positioned either at the centre or the edge of the MMI, so that a technological deviation in the rib widths of the MMIs can not shift the waveguide arms out of their relative correct position. For this reason that sort of MMI should be preferred when high technological tolerances are desired.

66%-MMI-converter-combinerThe 66%-MMI-converter allows to introduce a MMI-converter-combiner with good broadband-characteristics, a complete coupling of the fundamental mode A into the wider output waveguide, a 66% first order mode conversion efficiency of the funda-mental mode B and the superposition of the former with the latter into the wider out-put-waveguide.The 66%-MMI-converter-combiner is derived in analogy to the previous coupler by combining the 66%-MMI-converter given in Fig. 4.17(b) and the 2xK-MMI-splitter with , which has the input at 2/3Weq, as described in Appendix A.1. These MMIs have got the same length for the same MMI-width Weq. In Fig. 4.18(b) we have laid the former MMI into the latter. Again we can state, that the inputs are at different positions, but the outputs of the waveguides guiding the first order mode

L4 1 4⁄ 3Lc⋅= Weq

K 1=

4. Modified Multimode-Interference Couplers

154

50%-MMI-Converter-Combiner(a)

66%-MMI-Converter-Combiner

(c) 100%-MMI-Converter-Combiner, Type 1

L4=3Lc/4

Weq

1x2-MMI50%-MMI-Converter

A

BWeq/2

Weq/2

L8=3Lc/8

W’eq/2

W’eq/2

W’eq/4

Phase-Shift: Δφ=π

50%-MMI-Converter

Phase-Shift Δφ=π/2

A

B

L4=3Lc/4(d) 100%-MMI-Converter-Combiner, Type 2

2x2-MMI

Weq/2

Weq/2

L2=3Lc/2

(b)L3=3Lc/3

outputs

L4=3Lc/4

Weq/3

Weq

Weq/4WeqA

B

A

B

4.4 MMI-Converter-Combiner

155

and the output guiding the zero order mode of the MMI-splitter join. A beam propa-gation (BPM) simulation is given in Fig. 4.16. It shows the intensity evolution, when introducing a fundamental mode once at the MMI-splitter input port and once at the MMI-converter input port.The 66%-MMI-converter-combiners have small device sizes in comparison to the access waveguide widths and therefore have a large optical bandwidth [4.6]. They are preferentially used, when good broadband-characteristics are required.• 100%-MMI-converter-combiner

Two versions of MMI-converter-combiners delivering a superposition of the funda-mental mode A with the fundamental mode B that is completely converted into a first order mode are demonstrated subsequently. These MMIs base on the MMI-converter with 100% conversion efficiency presented in Fig. 4.17(c).A realization of such a 100%-MMI-converter-combiner is given in Fig. 4.18(c). It shows how a zero order mode A is guided to the second-MMI, that acts as a 1x1-MMI for the fundamental mode when the central input of the MMI-converter is used. (For an explanation refer to the description of the 50%-MMI-converter-combiner.) The zero order mode B is only effected by the structure of the 100%-converter as out-lined in Fig. 4.17(d), so that it is completely converted into a first order mode. By using the symmetric interference 1x2-MMI-splitter from Appendix A.1, we use a short MMI with the corresponding good broadband-characteristics. The length of the 1x2-MMI is . Thus, this variant of a 100% MMI-converter-combiner is recommended when the spectral bandwidth is of relevance. However this structure contains waveguides that cross each other, which might require increased technolog-ical efforts.The crossing waveguides can be circumvented by using an other MMI-splitter, which introduces the two signals A and B at different input guides and couples them out such, that they are correctly positioned for the 100%-MMI-converter. The gen-eral 2x2-MMI offers these possibilities. Fig. 4.18(d) illustrates this MMI in combi-nation with the MMI-converter. When using the central input port of the 2x2-MMI, the input-mode is mapped onto itself at the centre. We refer to the simulation depict-ed in Fig. 4.19(a), which shows how the zero order mode introduced at the central

Fig. 4.18 MMI converter-combiner structures coupling a zero order mode A com-pletely into the wider output-arm and mapping mode B as a first order mode into the same output-waveguide. The different structures have a (a) 50%, (b) 66% and (c) and (d) 100% conversion efficiency for the mode B to be converted from a zero into a first order mode. The MMI-converter-combiners include the 1x1 overlap-MMI characteristics for mode A with the conversion capabilities proposed in Fig. 4.17 for mode B.

L8 3Lc 8⁄=

4. Modified Multimode-Interference Couplers

156

input of the MMI is mapped onto the central output after the distance , which is exactly the length of the general 2x2-MMI. However when coupling the mode B at the edge into the 2x2-MMI, the mode B coupled in through this input-arm is split into two branches. The phase-shifters are used again to adapt the phases of the two modes in the outer branches. In contrast to the MMI-converter-combiner, that is built with the 1x2-MMI, only a phase-shift adaptation of is needed be-tween the splitter and converter-MMI, since the 2x2-MMI already delivers a phase-shift difference of between the two output-arms, as one can derive from the phase relations given in Appendix A.1. As there are no crossing waveguides, this structure should show improved technological tolerances.

2.3 Decoupling of modes of different orderA decoupler, that allows to separate modes of different order is useful, for applica-tions in all-optical devices, where first order mode control-signals are used for signal processing but have to be extracted afterwards [4.41, 4.42]. In comparison with the wavelength demultiplexing in integrated optics, the demultiplexing of modes of dif-ferent order or more precise of different symmetry is simple.The 100%-MMI-converter-combiners of Fig. 4.18(c) and (d) can be used as decou-plers when operated reversely. This is a direct consequence of the reciprocity theo-rem in optics. These decouplers guide the light depending on the symmetry of the mode onto separate waveguides.Noteworthy are the short and highly efficient MMI-filters, that have been discussed elsewhere [4.10]. They are useful, if the first order modes are not any more used af-terwards. In contrast to the MMI-converter-combiners with 100%-conversion effi-ciency the first order mode is not mapped into a single waveguide but split off and mapped onto several different waveguides. The devices rely on the different map-ping characteristics of symmetric and antisymmetric modes in 1xK-MMI splitters. Fig. 4.19 displays the mode evolution of a symmetric zero order and an antisymmet-ric first order mode introduced at the centre of the MMI. At MMI lengths

with K=1,2,3,... the zero order mode is clearly mapped to out-put-positions different from those of the first order mode. MMI-filters with are useful if the fundamental mode has to be split into K branches and has to be freed from antisymmetrical modes. In principle the bent waveguide connecting the 2x2-MMI and the 50%-MMI-con-verter in the MMI-converter-combiner of Fig. 4.18(d) can be omitted. The phase-shift of π/2 which is necessary for a complete conversion can as well be obtained by twisting the two MMIs against each other, see Fig. 4.24(b). However, structure (c) and (d) were superior - both in simulation as well as in experiment.

L 3Lc 2⁄=

π 2⁄

π 2⁄

L4K 1 4K( )⁄ 3Lc⋅=K 1>

4.4 MMI-Converter-Combiner

157

3. Power-Splitter for Zero and First Order ModesIn this paragraph we discuss MMI splitters, that can handle zero and first order modes. Such devices are useful in integrated optics to allow for the simultaneous sig-nal processing of zero and first order modes. So far, the behaviour of first order modes in zero order mode devices has not yet been studied.

3Lc12

--------- 3Lc8--------- 3Lc

6--------- 3Lc

4---------

3Lc2

---------

3Lc10

---------

Fig. 4.19 MMI-filter and short MMI-splitter can be identified with this simulation for the symmetric (zero order) mode and the antisymmetric (first order) mode introduced at the central input guide. MMI-filters, that project the symmetric and the antisymmetric modes onto different output-guides, have lengths , , . Short MMI-splitters have MMI-lengths , , . The formation of 1, 3, and 5 images of a zero and a first order mode, respectively, at identical output-positions are clearly visible.

L 3Lc 4⁄= 3Lc 8⁄ 3Lc 12⁄L 3LC 2⁄= 3Lc 6⁄ 3Lc 10⁄

4. Modified Multimode-Interference Couplers

158

3.1 MMI-couplers with uniform power-splitting ratiosThe theory of the general NxN-MMIs of Appendix A.2.1. is valid for light with an arbitrary mode profile. Therefore it remains valid for zero order modes as well as for first order modes. Thus, the structure given in Fig. 4.1 allows to split optical modes of any order onto N output guides and it allows to combine modes of any order into a common output guide.Besides these known NxN-MMIs, there exist short devices, that split the fundamen-tal and the first order modes into K common outputs. They exist as 1x1, 1x3, 1x5,... etc. splitters. These devices correspond to overlap-MMIs with lengths ,

, ,... With other words, they have lengths with =2, 6, 10,.... ac-cording to Eq. (4.1). They have an input guide at the centre and the output guides at equidistant positions of . Fig. 4.19 gives the light distribution in an MMI for a fundamental and a first order mode respectively. For MMI-lengths ,

, the formation of 1, 3 and 5 images of a zero and a first order mode respectively at identical output-positions is clearly visible.

3.2 MMI-couplers with non-uniform power-splitting ratiosSubsequently we discuss the existence of non-uniform first order mode power-split-ting ratios for MMIs of the general and MMIs of the overlap type. Recently MMIs with free selection of splitting and combining ratios for zero order modes have found attention [4.34]. We pick up this idea to vary the first order mode output-intensity ratios. As an interesting result we find, that the symmetric and antisymmetric modes split up nonuniformly jointly when general-MMIs are used and they split up non-uniformly in an opposite way if overlap-MMIs are used.We start discussing non-uniform but fix power splitting ratios of first order modes in overlap-MMIs and then turn to the effect of the butterfly modification in the general- and overlap-MMIs.

MMIs with fix, nonuniform power-splitting and combining ratiosWe derive MMIs with fix, nonuniform power-splitting and combining ratios from the overlap-MMI (see Fig. 4.1(b)) with , ,.. . Their length is given with Eq. (4.1)

. (4.17)

For waveguide-input positions with =1, 2,..., 2K-1 and waveguide-output positions at with j’=1, 2,..., 2K-1 and

even numbers and x measured from the bottom MMI-edge, one obtains with Eq. (4.4) and Eq. (4.5) in the overlap-notation the following output-intensity distri-

L 3Lc 2⁄=3Lc 6⁄ 3Lc 10⁄ LN N

Weq K⁄L 3Lc 2⁄=

3Lc 6⁄ 3Lc 10⁄

N 2K= K 2 3 4, ,=

L 12K------- 3Lc⋅=

xi'in i′W 2K( )⁄= i′

xj'out W j′W 2K( )⁄–=

i′ j′+

4.4 MMI-Converter-Combiner

159

butions

for symmetric modes and (4.18)

for antisymmetric modes. (4.19)

The following conclusions can be stated from Eq. (4.18) and Eq. (4.19):If one chooses the same input-guide for a normalized symmetric and a normalized antisymmetric mode, the sum of the intensity of the symmetric and antisymmetric mode at the output guides remains constant. In Fig. 4.20(a) we have exemplarily

Ai'j'2 2

K---- π

2---i′ j′i′ π

4K-------–⎝ ⎠

⎛ ⎞2sin=

Ai'j'2 2

K---- π

2---i′ j′i′ π

4K-------–⎝ ⎠

⎛ ⎞2cos=

L2K=3Lc/(2K)

Weq/2K

Weq/2K

Weq

Weq/2K

Weq/2K

Weq/K

Weq/K

. . .

A

B . . .

. . .

1

3

2K-1

(b)

L2K=3Lc/(2K)

Weq/(2K)

Weq/2K

Weq

Weq/2K

Weq/2K

Weq/K

Weq/K

. . .

. . .

. . .

1

3

2K-1

(a)

BAi’=1

i’=1

Outputs j’

Outputs j’

Input i’

Input i’

i’=2K-1

Fig. 4.20 MMI-splitters with fix, nonuniform power-splitting ratios for two modes of different symmetry. (a) Case where the two modes use the same input-guide. The intensity distribution in the output-guides depends on the sym-metry of the input-mode. (b) Case where the antisymmetric mode uses in-put =1 and the symmetric mode uses input-guide =2K-1. i′ i′

j′

4. Modified Multimode-Interference Couplers

160

chosen . Because has to be even, we only find output guides at with q=1, 2, ..., K.

Likewise one obtains a constant intensity distribution at the output guides j’ (for this case the outputs and have to be considered also, and give together a full output), if one chooses for the normalized symmetric mode an input guide at

and for the normalized antisymmetric mode an input guide at , with p a number between .

However if one chooses for the symmetric mode the input guide and for the antisymmetric mode the input guide at , then both modes show at the output j’ the same nonuniform intensity distribution Fig. 4.20(b)

MMIs with free selection of power splitting and combining ratiosThe output-intensity distribution of the different output guides can be changed by ap-plying the so called butterfly MMI concept. Butterfly MMIs are obtained by dividing a rectangular MMI-box into two sections and down- and uptapering the two sections. With the extent of the taper the output intensity ratio varies. This effect has been dis-cussed in ref. [4.34] for zero order modes.Butterfly-MMIs can be derived from general-MMIs. Since the general-MMI theory remains valid independent of the mode form, the non-uniform power-splitting ratios are for a zero as well as a first order mode identical. The deviation from the uniform power-splitting ratio only depends on the taper.The butterfly MMI concept can similarly be applied to the overlap-MMIs. We dis-cuss the just mentioned N=2K overlap-type for the case K=2 (Fig. 4.21a). This MMI is of interest since it allows with a free choice of the splitting ratios the nonuniform distribution of a zero or/and a first order mode into the two output waveguides. To give the splitting ratio as a function of the taper, we have identified the taper by the ratio Weq1/Weq0, where Weq0 is the MMI-width at the begin and end and Weq1 the MMI-width in the middle of the MMI. The total length of the butterfly-MMI has then to be adapted depending on the widths Weq1 and Weq0 to

, with . (4.20)

In Fig. 4.21(b) we have depicted the splitting ratio for a symmetric and antisymmet-ric input mode using input port =1 as a function of Weq1/Weq0. The variation in the splitting ratio at the output and are clearly visible.

i′ 1= i′ j′+j′ 2q 1–=

j′ 0= j′ 2K=

i′ 2p=i′ 2K 2p–= 1 p K 1–≤ ≤

i′ 2p 1–=i′ 2K 2p– 1+=

LN1N---- 3Lc⋅= Lc

4nWeq0Weq13λ

--------------------------------=

i′j' 1= j' 3=

4.4 MMI-Converter-Combiner

161

4. ExperimentFor experimental demonstration of the mode conversion and mode combining effi-ciency we have realized the 50%, the 66% and both types of the 100%-MMI-con-verter-combiners.We demonstrate how the 50%-MMI-converter-combiner of Fig. 4.18(a) couples a zero order mode A introduced from the wider input waveguide with a 100% coupling efficiency into the central output guide and how the zero order mode B from the low-er left input guide is coupled with a 50% and 25% efficiency as a first order mode

0.50 0.75 1.00 1.25 1.500

10

20

30

40

50

60

70

80

90

100x10-2

Symmetric Modes

Antisymmetric Modes

Taper Ratio W1/W0

Ligh

t in

Out

put-G

uide

in %

j’=3

j’=1

100

50

00.50 0.75 1.00 1.25 1.50

W0

W1

(a) L4 Outputs j’

j’=1

j’=3

(b)

Fig. 4.21 MMI splitter of Fig. 4.20(a) with K=2 and butterfly geometry. The butter-fly geometry allows to deviate from the fix splitting ratios given by the rec-tangular MMI-geometry. Depending on whether one down- or uptapers the middle MMI-width W1 in relation to the MMI-width W0 at the front and back of the MMI, the splitting ratio varies. The Graph shows the var-iation of the intensities in the output-guides j’=1 and j’=3 as a function of the taper-ratio for the symmetric (solid line) and the antisymmetric mode (dotted line).

Symmetric ModesAntisymmetric Modes

4. Modified Multimode-Interference Couplers

162

into the central output guide and as zero order mode into the two outer output guides, respectively. Analogously we demonstrate, that the 66%-converter-combiner of Fig. 4.18(b) couples the fundamental mode A completely into the wider output guide and it converts the zero order mode B with a 66% efficiency into a first order mode, which is mapped into the wider output waveguide also. The remaining 33% of the light are coupled as a zero order mode into the thinner output guide. The 100%-MMI-converter-combiners of Fig. 4.18(c) and (d) couple the mode A and B with a 100% efficiency as a zero and a first order mode into the same output waveguide.

4.1 Structure The 50%, the 66% and the 100% -MMI-converter-combiner of Fig. 4.18(a)-(d) were produced as ridge waveguide structures on a double-heterostructure InP/InGaAsP/InP wafer, optimized for light at a wavelength around 1.5 μm (inset of Fig. 4.22).The epitaxial layers were grown by Low-Pressure Metal Organic Chemical Vapour Deposition (LP-MOVPE) on (100) Indium Phosphide. The structure of the wafer has an InP buffer layer, a 250 nm waveguiding quaternary layer (InGaAsP with corre-sponding material gap wavelength of 1.19 μm) and a 1590 nm InP upper cladding layer. Dry and wet etching 1500 nm into the InP cladding layer formed the ridge. Af-ter cleaving, a broad-band anti-reflection coating was applied. Measurements of the optical properties were then performed on an optical bench.The 50%-MMI-converter-combiner has a dimension of 12.0 μm x 340 μm and the 66%-MMI-converter-combiner has a dimension of 9.0 μm x 270 μm. The mask lay-out of the MMI-converter-combiner with 100% efficiency of Fig. 4.18(c) is depict in Fig. 4.22. It consists of a 1x2-MMI which is designed as small as 12.0 μm x 150 μm and a 100%-MMI-converter, which has a geometry of 12.0 μm x 340 μm. The

Phase-Shift-Region: Δφ=π

50%-MMI-

Inputs

B

A

Symmetrical 1x2 MMIconverter-combiner

InP

InP

90 nm1500 nm

w or W

1.19-InGaAsP 250 nm

Fig. 4.22 Mask layout of the 100%-MMI-converter-combiner from Fig. 4.18(c). The 1x2-MMI is as small as 12 x 150 μm and the 50%-MMI-converter is 12 x 340 μm. Curve radii of the waveguides are 1000 μm. The inset shows the cross section through the InP/InGaAsP/InP waveguide-structure.

4.4 MMI-Converter-Combiner

163

1x2-MMI splits the signal B into two identical branches of light that have equal phas-es behind the 1x2-MMI. The signals are then guided into two slightly different bent curves with radiuses of R=997 μm and R=1003 μm to induce the required phase shift of π as mentioned in paragraph II. Finally the two signals are completely converted into a first order mode signal in the 100%-MMI-converter. To guide the signal A to the centre of the 100%-MMI-converter we have to cross waveguides. We have bent the input waveguides to obtain an intersection angle of 18 degrees. The widths of the in- and output guides before and behind the MMI-converter-combiners are 1.8 μm and 3.8 μm for the zero and first order mode waveguides, respectively [4.6]. Behind the MMIs we uptapered the thinner waveguides to 2.5 μm and the wider waveguide to 4.5 μm to avoid for unnecessary high waveguide losses. The MMI-converter-com-biner with 100% efficiency of Fig. 4.18(d) is built up of a 2x2-MMI of 13.0 μm x 780 μm and a 100%-MMI-converter of 13.0 μm x 390 μm. The total angle of the bent curves between the two MMIs is 0.6 degrees for a curve radius of 995, 1000 and 1005 μm. This induces a phase-shift between the two outer bent waveguides. The in- and output waveguides are designed with rib-widths of 1.7 μm and 4.3 μm respectively. Again the waveguides were uptapered behind the MMIs.

4.2 Experimental ResultsThe functioning of the 50%-MMI-converter-combiner, Fig. 4.18(a), the 66%-MMI-converter-combiner, Fig. 4.18(b), and the 100%-MMI-converter of Fig. 4.18(c) are demonstrated in Fig. 4.23(a)-(c). In Table 4.4 we have summarized the excess loss of all four devices under investigation in comparison to straight 3 μm waveguides.

Table 4.4: Comparison between the different MMI-Converter-Combiner structures

MMI-Converter-Combiner

Advantage of Structure

Theoret. Coupl. Efficiencies0th 1st

order

Measured Excess Losses0th 1st

order

50%Fig. 4.18(a)

Simple structure;high design tolerances

100% 50% 0.1 dB 0.3 dB

66%Fig. 4.18(b)

simple structure,large optical bandwidth

100% 66% 0.0 dB 0.3 dB

100%Type 1Fig. 4.18(c)

100% Conversion;large optical bandwidth

100% 100% -0.2 dB 0.7 dB

100%Type 2Fig. 4.18(d)

100% Conversion;no intersection angles

100% 100% -0.1 dB 0.6 dB

π 2⁄

4. Modified Multimode-Interference Couplers

164

(a) 50% MMI-Converter-Combiner

(b) 66% MMI-Converter-Combiner

(c) 100% MMI-Converter-Combiner

Fig. 4.23 Video camera images of the output intensity distribution of the MMI-con-verter-combiner shown in Fig. 4.18. (a) 50%-MMI-converter-combiner, (b) 66%-MMI-converter-combiner and (c) 100%-MMI-converter-combin-er Type 1. The left sides show the image generated by the zero order mode A, which is completely mapped onto itself. In (a) and (b) small cross-talks into the adjacent outputs are visible on the left photos. The right sides show the first order modes, which are generated with 50%, 66% and 100% when the second input-guide is used. The predicted output-distributions of 25:50:25 and 33:66 in (a) and (b) respectively for the zero and first order modes are found with good accuracy.

4.4 MMI-Converter-Combiner

165

The output intensity distribution of the 50%-MMI-converter-combiner is depicted in Fig. 4.23(a). The left photo shows how TE-light of wavelength 1.5 μm from the input mode A is completely mapped as a fundamental mode onto the wider waveguide. This output shows a 0.1 dB excess loss in comparison to a straight 3 μm rib waveguide of the same length. At the position of the two outer output ports, only a slight crosstalk is visible. The cross talk is asymmetric due to technological inhomo-geneities. The right picture of Fig. 4.23(a) shows the output guides of the MMI of the mode B, which is mapped onto two fundamental modes, which are visible at the outer sides and onto a first order mode in the centre. 54% of the light is coupled into the first order mode. The fundamental modes on the left and right side of the wider waveguide contains 46% of the light. The outer outputs guide less light than the ex-pected 50% because of the weak guiding in the sharp S bends of the outer waveguides behind the MMI, which were designed to avoid cross talk with the cen-tral waveguide. The total excess loss of the first order mode in comparison to a 3 μm straight waveguide of equal length is 0.3 dB. The losses were measured by projecting the output-beams through a lens onto a photo-diode. Although an estimate of the conversion efficiencies can be obtained from the figures, by taking the integral over the light shape, we used the values measured with the photo-diodes because the sig-nal shape from the camera does only approximately linearly correspond with the sig-nal intensity. The TM-mode showed an analogous behaviour, tough slightly worse, because the structures were not designed to be polarisation insensitive.For the 66%-MMI-converter-combiner we measured a 0.0 dB excess loss for the mode A that was projected into the wider waveguide and for the fundamental mode B, which is converted into a first order mode we found a 0.3 dB excess loss in com-parison to the 3 μm waveguide. The output intensity distributions of mode A and B behind the MMI are given on the photos of Fig. 4.23(b). The left photo shows how 99% of the light is coupled into the wider output. The photo on the right displays the zero and the first order modes containing 32% and 68% of the light. The peak of the zero order mode is smaller in comparison to the first order mode peak. On the one hand this is due to the higher propagation losses in the S-bend of the zero order mode waveguide behind the MMI and on the other hand it is an effect of the unequal rib-widths of the output waveguides at the cleave position. The area in the integral over the zero order mode shape corresponds quite exactly to half of the area of the integral over the first order mode shape as one expects from theory. To proof the predicted good broadband-characteristics of this device we measured the 1 dB optical band-width. It was found to be as large as 160 nm for the first order mode. The zero order mode bandwidth is even better.The measured excess losses of the MMI-converter-combiner with 100% conversion efficiency of Fig. 4.18(c) were similar to those of the above devices. The mode B, that was converted into a first order mode showed a 0.7 dB excess loss and the mode

4. Modified Multimode-Interference Couplers

166

A that kept its zero order mode geometry gave a 0.2 dB excess gain in comparison with the 3 μm waveguide. A 0.2 dB excess gain lies within the design and measure-ment tolerances but might also be due to the lower waveguide losses in the MMIs, which have a considerable wider rib-width in comparison to the 3 μm rib-widths of the straight waveguides. The output-intensity distributions of the device are present-ed on Fig. 4.23(c). To avoid high excess losses with this structure three design pa-rameters have to be optimized carefully. First of all, the phase shifter introduced between the two MMIs has to be adjusted to π. A deviation of Δφ from the required phase shift leads to an excess loss of dB. Second the intersec-tion angle of the waveguides has to be maximized and third, a possible cross-talk be-tween the three waveguides that are leaded to the 50%-MMI-converter-combiner has to be suppressed by choosing strongly guided waveguide structures with small cur-vature radii. The last two problems can be circumvented by using the structure of Fig. 4.18(d) instead. However the MMI-converter-combiner of Fig. 4.18(d) has a worse spectral bandwidth. For this structure we found a 0.1 dB excess gain for the zero order mode and a 0.6 dB excess loss for the mode that was converted into a first order mode.

5. ConclusionsWe have proposed and realized new types of short and low-loss MMI-couplers that allow to combine a fundamental mode with a second mode, that is converted into a first order mode. The performances of the 50% and 66%-MMI-converter-combiners realized in InGaAsP/InP have excess losses below 0.3 dB. The 100%-MMI-convert-er-combiners with conversion efficiencies of theoretically 100% show excess losses below 0.7 dB. The 100%-MMI-converter-combiners can also be used to decouple modes of different symmetry. Additionally we have proposed new possibilities to split zero and first order modes uniformly and nonuniformly. The new MMIs have found applications in all-optical switches and wavelength-converters, where they can be used to introduce a first order mode optical signal to control the data-signal processing. They are of interest because of the simple and efficient coupling scheme. Moreover first order mode optically controlled devices allow copropagating data- and control-signals even at the same wavelength, since the different symmetry of the signals always guarantees an easy separation of the two signals, so that external wavelength-filters for the extraction of control-signals are unnecessary with these devices.

10lg Δφ 2⁄( )2cos( )–

4.4 MMI-Converter-Combiner

167

6. Appendix: Generalisation of the Concept to Converter-Combiner MMIs with K outputs

In the appendix we outline the generalisation of the conversion principle. We dem-onstrate, how the converter-combiners presented in paragraph II can be generalized to allow for the splitting up into K common output guides for both the zero order and the first order modes.

Generalized Mode ConversionWe first mention, that for the generation of first order modes, any NxN-MMIs can be used in full analogy to the N=3 and N=4 MMIs with the 66% and 50% conversion efficiencies presented in Fig. 4.17. Because of Eq. (4.16), which guarantees the first order mode phase-relations for any general NxN-MMI, first order modes are gener-ated whenever the input guide is positioned at or near the MMI-edge. Such MMIs allow for N odd to generate one zero order mode and first order modes and for N even, they generate 2 zero order modes and first order modes.A complete conversion into K first order modes can be obtained by modifying the 100%-converter of Fig. 4.17(c). We only have to replace the 50%-MMI-converter of Fig. 4.17(c), which is derived of a general 4x4-MMI by a general 4Kx4K-MMI. The new MMI is shown in Fig. 4.24(a). As in the 100%-MMI-converter a signal is first launched into the 1x2-MMI and split into two parts. The phases of the two signals have to be properly adjusted before introduced through the inputs i=1 and i=4. The required phase shift amounts to for K odd and to 0 degrees for K even. Due to con-structive and destructive interferences of the two signals from input i=1 and i=4K in the general 4Kx4K-MMI, only K output-ports guide light. We have given the output-guides j, that guide light, in the general-MMI notation. As we have more than one signal at the output, it is of interest to know the phase relations of the first order modes. For simplicity we introduce the k-notation as given on Fig. 4.24(a). The phases of the first order modes at the output of the converter can then be calculated with Eq. (4.2) to Eq. (4.5) to be

. (4.21)

Generalized MMI-Converter-CombinerWe now derive two versions of MMI-converter-combiners with a 100% coupling ef-ficiency for both modes that additionally split the modes completely into K outputs. All we have to do is to append a central input guide on the second MMI of the 100%-MMI-converter given in Fig. 4.24(a). Fig. 4.24(b) shows two realizations of gener-alized 100%-MMI-converter-combiners with K outputs. They have the geometry of

N 1–( ) 2⁄N 2–( ) 2⁄

π

ΔΦk 2k 1–( )2 πK----⋅ 2k 1–( )π–=

4. Modified Multimode-Interference Couplers

168

. . .

Outputs

j=4K-2,

Genralized 50%-MMI-Converter

(a) Generalized 100%-MMI-Converter with K outputs Inputs

i=1

L4=3Lc/(4K)

Weq/Ki=4K

Phase-Shifter

. . .

j=2,3

j=6,7

. . .

1x2-MMI

Weq/K

Weq/K

Weq

Outputs

k=K

k=1

k=2

. . .

4K-1

Input

. . . B

A. .

.

. . .

Phase-Shifter

1x2-

(b)

12

K

. . .

Generalized 100%-MMI-converter-combiners with K outputs

Generalized 50%-MMI-Converte

2x2-MMI

Phase-Shifter Section

A

B

. . . . .

. . .

.

1

2

K

. . .

Generalized 100%-MMI-Converter

Generalized 50%-MMI-Converter

A

B

. . . . .

. . .

.

1

2

K

. . .

2x2-MMI-sectionAngled Phase-Shifter Section

MMI

4.4 MMI-Converter-Combiner

169

the generalized 100%-MMI-converter with K first order mode output guides. More-over the second MMI has the same geometry as the symmetric interference 1xK-MMI. With the central-input guide the MMI can be used as 1xK-splitter for the zero order mode A. Fortunately the outputs of the generalized MMI-converter and the 1xK-MMI splitter have the same positions. The intensities of the zero order mode image A and first order mode image B are uniformly split onto the K output-waveguides. Last we mention, that both the MMI-converter-combiner with 50% and 66%-effi-ciencies can be generalized analogously, so that they combine the conversion, com-bining and splitting of zero and first order modes into K outputs.

AcknowledgmentsThis work was in part supported by the Swiss research programme in optics.

Fig. 4.24 (a) Generalized MMI-converter that converts a fundamental mode with 100% efficiency into K first order modes. (b) MMI-converter-combiner with K outputs and a 100% coupling efficiency for the mode A onto K zero order modes and for the mode B onto K first order modes. The structures are obtained by adding the central waveguide to the generalized 100%-MMI-converter of (a)

4. Modified Multimode-Interference Couplers

170

4.5 Spatial Mode Filters realized with Multi-Mode In-terference Couplers

J. Leuthold, R. Hess, J. Eckner, P.A. Besse, H. Melchior

Optics Lettersvol. 21, no. 11, pp. 836-838, June 1996

AbstractSpatial mode filters based on multi-mode interference (MMIs) couplers that offer the possibility of splitting off the antisymmetric from the symmetric modes are present-ed, and realizations in InGaAsP/InP are demonstrated. Measured suppression of the antisymmetric first-order modes at the output for the symmetric mode is better than 18dB. Such MMIs are useful to monolitically integrating mode-filters with all-opti-cal devices, which are controlled through an antisymmetric first order mode. The fil-tering out of optical control-signal is necessary for cascading all-optical devices. Another application is the improvement of on-off ratios in optical switches.

1. IntroductionSpatial mode filters are of interest for single and low order transverse mode photonic integrated circuits, for example, for the separation of symmetric zero-order and an-tisymmetric first-order modes in all-optical switches or for the elimination of spuri-ous unwanted modes. When coupling fibres are coupled to waveguide components, small deviations from the optimal alignement often lead to the excitation of disturb-ing higher order modes. In practical photonic integrated circuits there is thus a need to eliminate such unwanted first-order and loosely guided radiation modes.To suppress higher-order modes, narrow waveguides, designed to guide only the fundamental modes [4.43] are used in combination with bends. However, the first order mode in particular is lost only after an appreciable distance. Additionally, when narrow waveguides and sharp bends are used, technology related inhomoge-neities cause excitation of radiating and higher order modes. Burns and Fenner [4.29] propose adiabatic asymmetrical Y-junctions as mode-combiner and mode-splitter for symmetric and antisymmetric transverse modes. Although these Y-junc-tions work well [4.29, 4.30], they are relatively long and require sharp intersection angles. Pedersen et al. recently described 3-dB couplers with integrated adiabatic asymmetrical Y-junctions of good quality in InGaAsP/InP. The couplers’ Y-junc-

4.5 MMI-Filters

171

tions are used as mode-filters and they are longer than 1.5 mm [4.32].In this Letter we report on spatial mode filters that are based on multi-mode interfer-ence (MMI) couplers [4.1, 4.3], offering the possibility of splitting off antisymmetric from symmetric modes. By way of reciprocity such MMIs can be derived from the so-called converter-combiner MMIs with a 100% conversion efficiency [4.9], which permits the conversion of a fundamental mode into a first-order mode and combining its lossless combination with a second fundamental mode. However, if there is no need for the antisymmetric modes to be used later, a shorter and simpler solution ex-ists. In contrast to the converter-combiner MMIs 100% conversion efficiency, which map the symmetric and the antisymmetric modes onto different waveguides, the MMI proposed here splits off the antisymmetric modes and maps them onto several different waveguides, so that this MMI will have an advantage in such applications as spatial mode filtering for antisymmetric modes. As with all MMIs the one present-ed here is characterised by compactness, tolerance in fabrication, good broadband-characteristics [4.5] and polarization insensitivity when strongly guided structures are used.

2. Principle of MMI-FiltersThe principle of the spatial MMI mode filters proposed here relies on the different mapping characteristics of symmetric and antisymmetric modes in MMI couplers. Consider a MMI-structure of equivalent width W, which essentially corresponds to the geometric width but takes into account penetration into the neighbouring mate-rial, effective refractive index n, whose coupling length Lc at the operational wave-length λ is defined as [4.14]

. (4.22)

According to theory [4.14, 4.16] a MMI images a symmetric mode that is introduced at an input at the centre of a MMI, to K output waveguides of equal intensity, provid-ed the MMIs length is

, (4.23)

with N=4K. Choosing K=1, this MMI maps the mode onto a single output guide at the centre, Fig. 4.25(a). When we consider an antisymmetric mode at the central input of the coupler, the general MMI theory [4.15] has to be applied. According to this theory, a MMI imag-es a single-mode waveguide input, onto N single-mode output waveguides of equal

Lc4nW2

3λ--------------=

LN1N---- 3Lc⋅=

4. Modified Multimode-Interference Couplers

172

intensity, provided its length is and the input guide position does not lie at with . In Fig. 4.25(b) we show a MMI with a single input-

guide of width 2w. This wide input guide of width 2w can be considered as the com-position of 2 input guides that touch each other such that neither lies at the centre of the MMI. If we choose N=4, each of these inputs is imaged onto 4 output guides. The MMI of Fig. 4.25(b) has 2 output guides (j=1 and j=4) of width w and one single in-put guide of width 2w, which again can be considered the composition of 2 output guides (j=2 and j=3) that touch each other. For N even the output-image is always symmetric, so that also the 4 output-images of the 2 input guides overlap for reasons of symmetry. In agreement with [4.16], constructive interferences appears at the out-er waveguides, while the images in the central outputs interfere destructively. The same is valid for every antisymmetric mode.Combining the two MMIs for the symmetric and the antisymmetric modes, we ob-tain the spatial MMI-filter in Fig. 4.25(b), which separates the symmetric, zero order (black coloured) and the antisymmetric, first order modes (shaped white).In fact every MMI of length LN (Eq. (4.23)) with N=4K and K an arbitrary number and the input guide at the centre can serve as mode-filter. The output-distributions of

L4=1/4 3Lc

W/2W

2w

(a) N=4

A

Inputs Outputsj=1

j=2j=3

j=4

L4=1/4 3Lc

W/2

W

2w

B

(b)N=4

k=1

A

Fig. 4.25 Mapping characteristic of two MMI’ss of the same length for (a) a symmet-ric, zero order mode (filled shapes) and (b) a zero order mode and an an-tisymmetric, first order mode (open shapes). Using an input-guide at the centre of the MMI, (a) the symmetric mode is mapped for lengths L=3/N Lc with N=4K and K=1 onto one central output-guide. (b) using the cen-tral input-guide for an antisymmetric mode, it is mapped onto the two out-er output-guides.

LNm W N⁄( ) m 1 … N, ,=

4.5 MMI-Filters

173

this MMI-filters can be calculated analogously to the case of the K=1 or N=4 MMI-filter. We give the output-intensity-distribution of these MMI-filters in Fig. 4.26 for an arbitrary K. Such structures are of use, because they combine the filtering and splitting characteristic.

3. ExperimentFor experimental testing of the filtering concept we have realized a device consisting of two MMIs (Fig. 4.27). The first MMI is a converter-combiner-MMI with 50% conversion efficiency [4.9]. This MMI couples a zero order mode A introduced in the central waveguide onto itself at the central output of the MMI, but converts a zero order mode B from the lower left hand input onto a first order mode B’ at the central output containing 50% of the light and two zero order modes b1 and b4, each con-taining 25% of the light each. The second MMI is the MMI-filter of Fig. 4.25(b) whose efficiency in separating the two modes of different orders is measured below.We have grown a double-heterostructure InP/InGaAsP/InP wafer to fabricate the de-vice. Etching 1500 nm into the 1590 nm InP cladding formed a ridge. The quaternary film layer thickness is 250 nm and has a gap wavelength of λg=1.19 μm. The dimen-sions of the converter-combiner MMI and the MMI-filter are 12.0 μm x 340 μm. The rib-widths measured just before and behind the MMI of the wider central

W/2

W/2K

W/2K

LK=1/(4K) 3Lc

. . .

. . .

. . .

Outputs

W

A B

Input

k=1

k=2

k=K

w

2w

2w

Fig. 4.26 Spatial mode filters with MMI couplers, separating symmetric (zero-order) and antisymmetric (first-order) modes. The symmetric mode A is mapped onto K symmetric output modes. The antisymmetric mode B is mapped onto 2 symmetric and K-1 antisymmetric modes.

4. Modified Multimode-Interference Couplers

174

waveguides are 4.0 μm, and those of the thinner waveguides are 2.0 μm.To demonstrate the fundamental and first order mode-separation capabilities of the MMI-filter, we first introduce the zero order transverse electric (TE) mode A at the central input of the structure from Fig. 4.27. As the converter-combiner MMI in the structure of Fig. 4.27 guides the mode completely onto the central input of the MMI-filter and the filter guides the mode completely onto the central input of the MMI-filter and the filter should map this symmetric, zero-order mode onto the central ouput, we expect the zero-order mode to be fully mapped onto output mode a. Fig. 4.28(a) is a photograph of the output-intensity distributions after the MMI-filter for the output-modes a, b1 and b2, corresponding to the left hand side of the outputs in Fig. 4.27. The right hand side with outputs b3 and b4, was verified to have a similar intensity distribution. We measured at a wavelength of λ=1.5 μm. The photograph shows that the zero order mode A is transmitted onto output mode a. Spurious cross-talk at the positions of the output modes b1 and b2 can be observed. Compared with that of a straight 3 μm waveguide of equal overall length, the excess loss of mode Amapped onto output mode a is 0.2 dB. Second, we use the input guide designed for the mode B, which converts the TE00-mode in the first MMI into an antisymmetric first order TE01-mode and then guides it into the MMI-filter. According to the map-ping characteristics of the MMI-filters, this mode should be filtered out at the posi-tion of the output mode a and instead be guided to the output modes b2 and b3. Fig. 4.28(b) demonstrates this filtering. As the conversion efficiency in the converter-

Converter-Combiner-MMI MMI-Filter

A

B

L=340 μm

W=1

2 μm

L=340 μm

W=1

2 μm

Radius of Curves =400 μmw=4.0 μm

w=2.0 μmb1

b2

a

b3

b4

B’

b4

b1

A

Outputs

Fig. 4.27 Device consisting of two MMIs in sequence to couple a zero-order and a first-order mode into the same waveguide and then to separate them. The first MMI generates the first order mode B’ and lets the zero order mode A pass depending on the input-guide used. The second MMI is the proposed filter-MMI of Fig. 4.26a), which maps the first order mode B’ onto the out-put modes b2 and b3 and the zero order mode A onto the output mode a.

4.5 MMI-Filters

175

combiner MMI for the first-order mode is 50%, half the light is converted and the other half is guided onto output modes b1 and b4. The outer output-modes b1 and b4guide less light in comparison to the modes b2 and b3 due to the weakly guiding in the sharp S-bends of the outer waveguides, which were designed to avoid any cross-talk with the central waveguide. Quantitatively, we measured a 18 dB suppression for the first-order mode at the central output guide a in comparison to the zero order mode of Fig. 4.28(a). This 18 dB already take into consideration the experimentally measured 3.2 dB penalty of the mode conversion in the converter-combiner MMI. An estimate of the suppression factor for the first-order mode can also be obtained from Fig. 4.28(b). The intensity of one mode is the integral over the light shape shown in the photograph. (The light shape at the overexposed weak intensities is de-fined by the middle of the curve’s line.) The intensity at output a divided by the in-

b1 b2 ab1 b2 a

b2b2b1 aa

(a)

(b)

Fig. 4.28 Fundamental and first order mode-separation capabilities of MMI-filter measured with the structure of Fig. 4.27. (a) Transmission of the zero-or-der mode from input mode A to output a with spurious crosstalk to outputs b1 and b4 and b2 and b3. (b) Input mode B converted to first order mode B’ in converter-combiner MMI, which demonstrates the filtering efficien-cy for the first order mode at output a. The first order mode is instead mapped onto the outputs b2 and b3. A part of the light from the generation of the first order mode can be found at the outputs b1 and b4.

4. Modified Multimode-Interference Couplers

176

tensity b2, a and b3 together gives the suppression ratio. Performing the integration we obtain a suppression factor of 15-17 dB, which is similar to the measured 18 dB. The deviation probably comes from the fact that our camera supports weak signals but suppresses strong signals. Simulations demonstrate that suppression ratios for antisymmetric modes of >25 dB for weakly guiding (for example our structure) to >60 dB for strongly guiding (deeply etched) structures are possible. To attain these suppression ratios, technology related waveguide inhomogeneities have to be avoid-ed. For the transverse-magnetic (TM) mode at the same wavelength we measured a 10 dB suppression of the TM01-mode.

4. ConclusionsMode filters for extraction of antisymmetric modes based on MMIs have been real-ized and experimentally tested. A suppression ratio of 18 dB for undesired first-order TE modes has been measured. These MMI-filters can be used to make all-optical de-vices cascadable. Before an optical data-signal is guided into the next cascade of an all-optical devices the control-signal has to be extracted, or else the data-processing in the next cascade will be disturbed. When a first order mode control-signal is used instead of a zero-order mode [4.41], the short and simple MMI-filter of Fig. 4.25(b) can efficiently extract the control-signal without disturbing the zero order mode data-signal. Other applications might include the improvement of on/off ratios in op-tical switches so that they can be used to extract small fractions of light from unde-sired first-order modes. Such small light fractions in an undesired antisymmetric first order mode, excited in the off state of an optical switch, can limit the on/off ratio considerably.

4.6 Optical Bandwidths of MMIs

177

4.6 Optical Bandwidths and Design Tolerances of Multimode-Interference Converter-Combiners

J. Leuthold, P.A. Besse, R. Hess, H. MelchiorProc. of European Conference on Integrated Optics, ECIO’97,

Stockholm, Apr. 2-4 1997, pp 154-157(The subsequent article is a slight modification of the conference proceedings in or-der to submit it to a journal)

AbstractAnalytical expressions and experimental measurements of optical bandwidths for both the fundamental and the first order modes of MMI-converter-combiners are giv-en.It is shown, that this new MMI type offers wide optical 1 dB bandwidths exceed-ing 140 nm at 1.5μm. The MMI-converter-combiners are advantageous for introducing optical control-signals into all-optical devices and for applications in wavelength-converters.

1. IntroductionIntegrated optical couplers are becoming key components for signal routing in mod-ern optical telecommunication networks. Couplers based on the multimode interfer-ence (MMI) effect [4.1, 4.8] have found a growing interest in recent years, since they fulfil the requirements of polarisation insensitivity, large optical bandwidths, small dimensions and high fabrication tolerances.Recently MMI couplers have been proposed as MMI-converter-combiners and MMI-filters [4.9, 4.10]. MMI-converter-combiners convert a fundamental mode into a first order mode and allow to combine this signal with a second fundamental mode. They are of interest for applications in all-optical switches and wavelength-convert-ers, where they can be used to introduce an optical signal as a first order mode into a nonlinear medium to control the data-signal processing. First order mode control-signals were found to be advantageous for several reasons [4.41]: First, the nearly lossless combining of the first order mode control-signal with the zero order mode data-signal into a single waveguide is feasible with new and short MMI-converter-combiners. This would be impossible with equally short 2x2-couplers. Second, in combination with MMI-filters a simple and nearly complete separation of the data- and control-signal after data-processing is possible [4.10]. This allows bidirectional operation of the all-optical switches without the use of external wavelength filters. Realisations of such dual order mode (DOMO) all-optical devices using these MMIs have recently been published [4.41]. However these converter-combiner and filter

4. Modified Multimode-Interference Couplers

178

MMIs will only find acceptance if they have broad optical bandwidths and large de-sign tolerances not only for zero order modes [4.5] but also for the first order modes.Here we present design criteria and results for broadband MMI-converter-combin-ers, that are used to introduce first order mode control-signals in optically controlled devices. We find large optical 1 dB bandwidths exceeding 140 nm for the new MMIs. Although their bandwidth is reduced by a factor 3 in comparison to conven-tional zero order mode MMIs, they have wide bandwidths and high design toleranc-es because of their small dimensions.

2. Theory Several converter-combiner MMI structures have already been presented [4.9]. Without loss of generality we start our discussion by considering the 50%-MMI-converter-combiner of Fig. 4.29. This MMI-converter-combiner converts 50% of the light of input guide i=1 onto a first order mode at output j=2,3 and couples 100% of the light from input guide i=2,3 onto the same output. The conversion efficiency of this structure amounts only to 50%, but in combination with a second MMI, this cou-pler turns out to be the key component for MMI-converter-combiners with 100% ef-ficiency, Fig. 4.30(a).

aw0

w1Weq

Weq/2Weq/2

LN=3Lc/4

i=1

i=2,3

j=1

j=2,3

j=4

Inputs Outputs

Q-1.2InP

InP

w

100 nm250 nm

(b)

(a)

n=3.17n=3.39

Fig. 4.29 (a) 50%-MMI-converter-combiner with the relevant design parametersLN, Weq, w0, w1, and a. (b) The heterostructure used for the devices.

4.6 Optical Bandwidths of MMIs

179

The optical bandwidth and the design tolerances are mainly determined by three con-tributions.

. (4.24)

The first term arises due to variations in the wavelength or the width of the MMI. If these parameters vary, the focus of the first order TEM01 output mode shifts away from the MMI output guide position. This results in a loss . The second term in Eq. (4.24) arises if the geometry-parameters , , and are not properly opti-mized. is the model dependent term. It is used to compensate deviations due to approximations. E.g, in our model we use one dimensional Hermite-Gauss modes. The error introduced with this approximate modes in comparison with a more refined

1.40 1.50 1.60 1.70Wavelength [μm]

0.0

1.0

2.0

3.0

Exce

ssLo

ss[d

B]

TEM00-Mode

TEM01-Mode

(b)

Phase-Shifter: Δϕ=π

1x2-MMI

(a)

50%-MMI-converter-combiner

Fig. 4.30 (a) 100% efficiency MMI-converter-combiner: 100% of the (black) zero order mode is mapped into the output-guide and 100% of the (white) zero order mode is converted into a first order mode that is coupled into the same output-waveguide. (b) The optical bandwidth of this device for both modes calculated with the refined model proposed in this work.

αtot α01 αG αM+ +=

α01w0 w1 a

αM

4. Modified Multimode-Interference Couplers

180

model based on a Mode-Analysis calculation is smaller than = -0.1 dB.Consider an optimized MMI of length [4.8]

, (4.25)

where is the effective refractive index, is the equivalent width, which essen-tially corresponds to the geometric width but takes into account penetration into the neighbouring material, is the wavelength and is the number of output-images of a MMI, whose input-positions do not lie at , with =1,...,N. If the wavelength or the equivalent width , of the structure is varied, the focus of the first order mode image displaces by a distance from the ideal output-position at

. The first order mode image of the MMI then deviates at the output from the ideal form of a TEM01-mode. This excess loss contribution can be calculated by overlap-ping the Hermite-Gauss Mode TEM01 having the beam waist at with a TEM01-mode focused at . The overlap gives

(4.26)

with

. (4.27)

is the waist of a TEM01-Hermite-Gauss mode. With the δλ−term in Eq. (4.27) we can calculate the optical-bandwidth and with the -term in Eq. (4.27) the fabri-cation tolerances in respectively. The overlap is related to the excess loss

for TEM01-modes by . It is of interest to compare the expression for first order mode MMI-devices with the excess losses of the well known zero order TEM00-mode MMI devices. With Eq. (4.26) and the expression given in Ref. [4.5] one can find the simple relation

. (4.28)

This expression is very useful, because in MMI-converter-combiners a zero- and a first order mode are combined into the same waveguide and Eq. (4.28) relates the bandwidths of these two modes.Next we derive a formula to express the losses due to unoptimized geometry-parameters , and . The input-rib width determines the Gaussian beam

αMLN

LN1N----

4nWeq2

λ----------------⋅=

n Weq

λ Nm Weq N⁄( ) m

λ WeqδL

LN

LNLN δL+

T011 4 δZ2⋅+

1 5 δZ2⋅ 4 δZ4⋅+ +------------------------------------------------

⎝ ⎠⎜ ⎟⎛ ⎞ 3 2⁄

=

δZ 2λ

πnd12------------ δL⋅ 2L

πnd12------------ δλ⋅– 4Lλ

πnd12Weq

---------------------- δWeq⋅–= = =

d1δWeq

Weq T01α01 α01 10– T01( )10log⋅=

α00

α01 3 α00⋅=

αGw0 w1 a w0

4.6 Optical Bandwidths of MMIs

181

waist of the zero order mode at the MMI-input. corresponds to the -ampli-tude decay waist of the Gaussian beam. Analogously the choice of the output-rib width determines the Gaussian beam waist of the first order mode at the MMI-output waveguide. is defined in the standard expression for first order Her-mite-Gauss modes. The quantities , and respectively , and have to be chosen such that there is a maximum overlap between the first order Hermite-Gauss eigenmode of the output waveguide and the first order mode formed by the zero order mode. The overlap which gives the excess loss is

. (4.29)

With Eq. (4.29) we find the minimum excess loss of 0.2 dB for the mode which is converted into a first order mode when

and . (4.30)

In a more refined model the contributions and should not be treated sepa-rately. Instead one should directly overlap the two zero order Gauss-Hermite modes with the first order mode at the MMI output. We have done this calculation and ob-tain

(4.31)

with

(4.32)

and

and . (4.33)

When setting we obtain again Eq. (4.29).

3. Comparison with Mode Analysis and ExperimentWe start by comparing the analytical expressions with simulations from a Mode-Analysis calculation and study the design tolerances of the converter-combiner MMIs. In a second step the analytical expressions are validated with experiments.The discussion is held for a 50%-MMI-converter-combiner (Fig. 4.29) of Length

d0 d0 e 1–

w1 d1d1

w0 w1 a d0 d1 a

TG αG

TG 64a2 d0d13

d02 d1

2+( )3-------------------------= 8 a2

d02 d1

2+-----------------–

⎝ ⎠⎜ ⎟⎛ ⎞

exp

αG

d0 2 a⋅= d1 6 a⋅=

α01 αG

αtot αG01 αM+=

TG01 64a2d0d13 1 A2+

B2 A2d14+

------------------------

32---

= 8a2 A2d12 B+

A2d14 B2+

------------------------–⎝ ⎠⎜ ⎟⎛ ⎞

exp

A 2= δZ B A2 1+( )d02 d1

2+=

δZ 0=

4. Modified Multimode-Interference Couplers

182

=340 μm, width =13.0 μm, which corresponds to a geometric width of =12.0 μm, =1.8 μm, =1.6 μm and =5.15 μm. To relate the beam waists

and of the zero and first order modes used in the analytical expressions with the waveguide width and we applied a mode analysis routine to calculated the mode profiles of the zero and first order modes in the waveguides with the widths

and . The beam waist is then the width of the mode, where the amplitude has decayed on of the maximum value. Similarly the beam waist of the first order mode gives, due to the definition, the span of the mode, where it’s amplitude has decayed on a value of 0.858 of the peak maximums (the outer mode slopes of the first order mode are relevant). For TE-modes in the structure of Fig. 4.29(b), we found, in the range of interest, the following linear fit between d and w

and . (4.34)

We mentioned above, that the relations of Eq. (4.30) must be fulfilled for an opti-mized design of an MMI-converter-combiner. Fig. 4.31(a) shows the losses of the 50%-MMI-converter-combiner versus the first order output-rib width w1 once cal-culated with the simple analytical model that obeys Eq. (4.30) and once calculated with a lengthy mode analysis simulation. The mode analysis simulation confirms the result of the analytical model excellently. For a given rib-width =1.80 μm, we predict with Eq. (4.30) and Eq. (4.34), that the optimum conversion efficiency will be found for a rib-width =5.20 μm (minimum of dashed line). When varying in a Mode-Analysis simulation we find the lowest losses for a rib width =5.25 μm, which is very close to the result from the analytical formula (minimum of solid line).The width of MMIs is sensitive to deviations from the appropriate value. In Fig. 4.31(b) we have calculated the design tolerances for the MMI-width of the 50% MMI converter-combiner under investigation. Again we obtain similar results for the curves calculated with the mode-analysis, the analytical model based on Eq. (4.24) and the refined analytical model based on Eq. (4.31). The design tolerances of the MMI-converter-combiners are comparable to those of commonly used 2x2-MMIs for zero order modes [4.5]. The optical bandwidth obtained with a Mode-Analysis simulation and the optical bandwidth obtained with the analytical model based on Eq. (4.24) and the refined model based on Eq. (4.31) are compared in Fig. 4.31(c). All three curves were cal-culated for the above device. The models deliver similar results and reveal an im-pressive 1 dB optical bandwidth of 200 nm around a wavelength of 1.55 μm for an optimized device.For testing the model we have realized a 4x4-MMI-converter combiner of length

=340 μm, a geometric width of =11.8 μm, =1.8 μm, =1.4 μm and =3.8

LN WeqWG w0 a w1d0 d1

w0 w1

w0 w1 d0e 1– d1

d0 0,69w0 1,16+= d1 0,63w1 0,92+=

w0

w1 w1w1

LN WG w0 a w1

4.6 Optical Bandwidths of MMIs

183

4.0 5.0 6.0Rib-Width w1 [μm]

3.0

3.2

3.4Lo

ss[d

B]

Mode Analysis Simulation

Simple Analytical Model

Mode Analysis Simulation

Refined Analytical ModelSimple Analytical Model

3.0

4.0

5.0

Loss

[dB

]

3.0

4.0

5.0

Loss

[dB

]

1.40 1.50 1.60 1.70Wavelength [μm]

Mode Analysis Simulation

Refined Analytical ModelSimple Analytical Model

5.20 5.25

Fig. 4.31 50% MMI-converter-combiner design tolerances for (a) first order mode rib width w1, (b) MMI-width Wg and optical bandwidth (c).

(a)

(b)

(c)

12.5 13.0 13.5MMI-Width Wg [μm]

4. Modified Multimode-Interference Couplers

184

μm. For our measurements we have chosen a device with a smaller MMI-width in comparison to the above device. This will shift the optimal wavelength to the lower range of the wavelength spectrum of our apparatus. We then can measure the optical bandwidth from the optimal wavelength up to the upper limit of our external cavity. This restriction onto one half of the spectrum makes sense, since the optical band-width is fairly symmetric around the optimal wavelength and because the devices have a quite broad bandwidth. The geometry of the realized structure deviates some-what from the optimized structure as used for the simulations in Fig. 4.31. Besides a smaller MMI-width, we have a smaller first order mode waveguide rib width in within the realized structure. Since deviates in the experiment from the rib-width recommended by Eq. (4.30), we expect higher losses and a worse wavelength-spec-trum. The experimental measurement are given in Fig. 4.32(a), where the excess losses in comparison to 3.0 μm waveguides are plotted. The higher first order mode losses in the output waveguides are already considered. They amount to 0.9 dB per mm, which is quite high - but reasonable if one considers that we have small ribs (as found out above: too small ribs) at the MMI-output. The 1 dB bandwidth for this de-vice leads to an optical bandwidth of 140 nm. The calculated curves are in excellent agreement with the measured points. Extrapolating from the experimentally verified theory shows, that under consideration of the design criteria from Eq. (4.30) optical bandwidth of 200 nm are feasible. The optical bandwidth of a 66%-MMI-converter-combiner [4.9] is shown in Fig. 4.32(a). We find a 1 dB optical bandwidth of 160 nm. This converter can be designed as small as =280 μm with a geometric width of =8.8 μm. The other parame-ters were chosen as in the previous combiner. The device is of interest because of its simplicity and due to the 66% conversion efficiency of the first order mode.Additionally we have measured the bandwidth of a 3x3 MMI-converter-combiner [4.9]. This converter can be designed as small as =280 μm with a geometric width of =8.8 μm. The other parameters were chosen as in the previous combiner. The 1 dB optical bandwidth is over 160 nm, Fig. 4.32(b). The device is of interest be-cause of its simplicity and due to the high 66% conversion efficiency for the first or-der mode.Last but not least we apply Eq. (4.24) to the MMI-converter-combiner of Fig. 4.30(a). As this device has conversion efficiencies of 100% for both modes, it might be of relevance in the near future. In this device the first order mode passes the 1x2-MMI before it enters the 4x4-MMI-converter-combiner. The total losses for the first order mode therefore amount to

. (4.35)

The losses in the 1x2-MMI are small. The calculated optical bandwidths for the

w1w1

LN WG

LNWG

αtot α00 αG01 αM αφ+ + +=

α00

4.6 Optical Bandwidths of MMIs

185

zero and the first order modes are given in Fig. 4.30(b), for the case of appropriately adjusted phase-shifters: .

4. ConclusionsSimple expressions that relate the optical bandwidths of zero with first order modes and relevant design geometry relations have been found for converter-combiner MMIs. A refined model gives a general expression for optical bandwidths and design tolerances of these MMIs. Experimental measurements confirm the relations and show large optical 1 dB bandwidths for the MMI-converter-combiner exceeding 140 nm.

BW_mc3x3_comp.mif.data

0.0

1.0

2.0

3.0

4.0

0.0

1.0

2.0

3.0

1.45 1.50 1.55Wavelength [μm]

1.45 1.50 1.55Wavelength [μm]

Fig. 4.32 Measured and calculated optical bandwidths of (a) the 50% MMI-convert-er-combiner and (b) 66% MMI-converter-combiner. Large optical 1 dB bandwidths of (a) 140 nm and (b) 160 nm are found for the first order modes.

Exce

ssLo

ss[d

B]

Exce

ssLo

ss[d

B]

1st Order Mode

0th Order Mode

1st Order Mode

0th Order Mode

(b)

(a)

αΦ 0=

4. Modified Multimode-Interference Couplers

186

5. AppendixIn the paper we calculate with TEM0q-Hermite-Gauss modes. Here we give the def-inition of a TEM0q-mode and explain some details of the calculations. The amplitude of a Hermite-Gauss mode propagating in z-direction and a focus at z=0 can be written as

(4.36)

with

, (4.37)

(4.38)

and

. (4.39)

The index q assigns the lateral order of the mode (q=0 for a zero order mode and q=1 for a first order mode), which diverges in the x-direction. The transversal mode is always assumed to be a zero order mode, whose y-profile is maintained by the struc-ture. describes the evolution of the phase in z-direction. It is not of interest for our considerations. is the beam waist in the focal point for a mode of order q. It evolves with z according to Eq. (4.37). gives the curvature of the phase front as a function of z. And is the abbreviation for Gauss-Hermite polynomials of or-der q. The mode intensity is normalized to unity.The excess losses for a displaced first order mode MMI-output focus is obtained by overlapping Eq. (4.36) at with the amplitude function at

. (4.40)

Analogously we find the losses due to unoptimized geometry parameters. We over-lap two zero order modes that form a first order mode profile and have a first order

A0q x z,( ) 12p p!⋅--------------- 8

π dq2 z( )⋅

--------------------⎝ ⎠⎜ ⎟⎛ ⎞ 1 4⁄

Hq2 2xdq z( )-------------⎝ ⎠

⎛ ⎞ eiψ z( )eiπnx2

λR z( )--------------

e

4x2–dq

2 z( )------------

⋅=

dq z( ) dq 1 2δZ( )2+( )⋅= q 0 1,=

R z( ) z 1 12δZ----------⎝ ⎠

⎛ ⎞ 2+⎝ ⎠

⎛ ⎞=

δZ 2λzπndq

2------------=

ψ z( )dq

R z( )Hq

z 0= z δL=

T01 A01 x δL,( ) A01 x 0,( )⟨ | ⟩ 2 A01 x δL,( )A01 x 0,( ) xd∞–

∞∫

2= =

4.6 Optical Bandwidths of MMIs

187

mode phase distribution with a real first order Gauss-Hermite mode

. (4.41)

The refined model includes both effects. We therefore have to calculate

. (4.42)

AcknowledgmentsPart of the work was carried out within the Swiss "optique" project.

TG A00 x a– 0,( ) A00– x a+ 0,( ) A01 x 0,( )⟨ | ⟩ 2=

TG01 A00 x a– δL,( ) A00– x a+ δL,( ) A01 x 0,( )⟨ | ⟩ 2=

4. Modified Multimode-Interference Couplers

188

4.7 References

[4.1] O. Bryngdahl, “Image formation using self-imaging techniques”, J. Optical Society of America, vol. 63, pp. 416-419, April 1973

[4.2] R. Ulrich and T. Kamiya, “Resolution of self-images in planar optical waveguides”, J. Optical Society of America, vol. 68, no.5, pp. 583-592, 1978

[4.3] L.B. Soldano, E.C.M. Pennings, “Optical multimode interference devices based on self-imaging: principles and applications”, J. of Lightwave Tech-nol., vol. 13, pp. 615-627, April 1995

[4.4] R.J. Deri, E. C. M. Pennings, A. Scherer, A. S. Gozdz, C. Caneau, N.C. An-dreadakis, V. Shah, L. Curtis, R. J. Hawkins, J.B.D. Soole, J. I. Song, “Ultra-compact monolithic integration of balanced, polarization diversity photo-detectors”, ECOC’92, Berlin, Germany, Sept. 1992, pp. 457-460

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[4.6] J. Leuthold, P.A. Besse, R. Hess, H. Melchior, “Wide Optical Bandwidths and High Design Tolerances of Multimode-Interference Converter-Combin-ers”, Proc. of European Conference on Integrated Optics, ECIO'97, Stock-holm, Sweden, Apr. 1997, pp. 154-157

[4.7] J.M. Heaton, R.M. Jenkins, D.R. Wight, J.T. Parker, J.C.H. Birbeck, K.P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides”, Appl. Phys. Lett. vol. 61, pp. 1754-1756, Oct. 1992

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[4.42] J. Leuthold, E. Gamper, M. Dülk, P.A. Besse, R. Hess, H. Melchior; “Cas-cadable Optically Controlled Space Switch with High and Balanced Extinc-tion Ratios”; 2nd Optoelectronic and Communications Conference (OECC'97), Seoul, Korea, July 97, paper 9C3-4

[4.43] H. Kogelnik, V. Ramaswamy, Appl. Opt., vol. 13, pp. 1857, 1974

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192

193

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

In this chapter we describe advanced designs to eliminate the principal extinction-ratio limitations in MZI-SOA based all-optical switches. In a first section we discuss the various sources that limit the extinction ratios of all-optical devices and show how these can be overcome. Among the various all-optical devices, the MZI-SOA switch is special, since for an ideal switching performance, both the phase and the signal intensity have to be controlled in order to obtain com-plete extinctions. However, this is difficult, since the strong optical signal that is in-troduced to the phase shift also modulates the intensity of the signal to be controlled. This difficulty is discussed in a second section, where we show how one can take these effects into account by an advanced design.

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

194

5.1 Extinction-Ratio Limitations in All-Optical Devices

High extinction ratios and low crosstalk are prerequisite for the application of the all-optical components in optical telecommunication systems.We distinguish mainly three different causes limiting the extinction ratios and rais-ing the crosstalk of all-optical devices. These are noise, non-ideality of the strong pulsed source needed to control the device and switching-related shortcomings.Noise is introduced in the nonlinear medium of actively driven all-optical devices. Both the spontaneous emission and the signals themselves, i.e. data and control sig-nal, contribute to the noise. The individual contributions are spontaneous-emission shot noise and spontaneous-spontaneous beat noise from mixing of the signal with the ASE itself, signal-shot noise and signal-spontaneous-beat noise from mixing of the signal with the ASE [5.1, 5.2]. Additional noise is generated if the incident opti-cal-signal powers or polarizations are fluctuating. These fluctuations may have their origin in the signal sources and the transmitting network but may also stem from set-up instabilities when coupling the fibre to the device. Last but not least it has to be emphasized that especially cross-phase modulated (XPM) all-optical devices are sensitive to thermal fluctuations, since small changes in the phase of the interferom-eter severely influence the output power. To suppress the spontaneous-emission noise wavelength filters are applied at the output of the processing units. Polarization noise can be reduced by the use of polarization-insensitive devices. Monolithic inte-gration makes the devices compact and stable and thus eliminates efficiently the thermal fluctuation problems.The non-ideality of the control- respectively pump signal may severely degrade the device performance. Control-signal jiiter and oscillations between the expected con-trol-signal peaks cause non-intended energy transfer to the nonlinear medium, which in turn results in an unexpected switching. However, impressive progress has been attained in recent years towards stable ultrashort high-repetition-rate laser sources [5.3-5.5].Beyond, physical limitations related to the different all-optical switching principles, reduce the extinction ratios. All-optical devices based on the fourwave-mixing prin-ciple are limited by the unsatisfactory conversion efficiency. Cross-gain modulated devices suffer from the impossibility to saturate an optical amplifier so strongly that a second signal completely disappears [5.6]. In this respect, only cross-phase mod-ulated devices have the potential to achieve ideal switching. By phase adaptations within the interferometric configuration it is relatively easy to produce devices with one ideal output-guide. However, when considering the two output guides of e.g. a MZI-SOA all-optical device, only unbalanced extinction ratios are attained. This is caused by disturbing gain saturations related to the phase changes due to carrier de-

5.1 Extinction-Ratio Limitations in All-Optical Devices

195

pletion.Subsequently we demonstrate, how switching with almost ideal extinction ratios can be attained with MZI-SOA all-optical switches by the introduction of specific asym-metries.

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

196

5.2 All-Optical Space Switches with Gain and Principally Ideal Extinction-Ratios

J. Leuthold, P.A. Besse, J. Eckner, E. Gamper, M. Dülk, H. Melchior

IEEE J. of Quantum Electronicsvol. 34, no. 4, pp. 622-633, April 1998

AbstractAsymmetric Mach-Zehnder interferometer (MZI) configurations are proposed to build all-optical space switches with gain and principally ideal extinction ratios. Ac-tually three asymmetries in MZI configurations with semiconductor optical amplifi-ers (SOAs) on their arms are discussed. The asymmetries in the all-optical switches are necessary to overcome the extinction-ratio limitations that are due to the disturb-ing gain changes that arise when control signals are introduced into the SOAs to in-duce the necessary refractive-index change for switching. Starting from a generic MZI configuration with SOAs on the arms, a description in terms of transmission matrices is used and applied to identify 1x2 and 2x2 all-optical switch configurations with high on-state transmissions and close to ideally large extinction ratios. The the-oretical predictions are verified and found to be in excellent agreement with experi-ments for a switch with symmetric MZI-splitters in a monolithically integrated InP waveguide version, that allows operation with equally or unequally biased SOAs.

1. IntroductionFuture high capacity fibre-optical communication networks based on optical time di-vision multiplexed and wavelength division multiplexed systems require high-speed add-drop multiplexers, demultiplexers and switches with high extinction ratios and gain. As electronics encounter difficulties at high speeds, optically controlled com-ponents are of interest. Indeed, high-speed switching, multiplexing and demultiplex-ing of a data signal with a control signal have been demonstrated with optically controlled semiconductor optical amplifiers (SOAs) providing not only the neces-sary nonlinearity for switching but also gain. SOAs have been used in fibre loops [5.7] and in Mach-Zehnder Interferometer (MZI) configurations. MZIs with optical nonlinearities, both passive [5.8] and active (MZI-SOA) [5.9] have demonstrated pi-cosecond speeds. An advantage of the MZI-SOA version is that it allows for mono-lithic integration rendering the switches stable and compact [5.10, 5.11]. In these

5.2 All-Optical Space Switches with Gain and Principally Ideal Extinction-Ratios

197

MZI-SOA all-optical switches the optical control signals are introduced onto one arm of the MZI to deplete the carriers in the corresponding SOA. This causes a gain saturation and also a refractive index change which is used for switching. When the control signal is turned on the data signal is switched from the cross into the bar state. The high-speed capabilities are based on the fast carrier depletion times, whereas the carrier relaxation times are a limiting parameter. Ways to overcome this limit were found by operating with two control pulses [5.8, 5.12] or by spatially displacing the SOAs [5.9, 5.11]. However, in symmetric configurations the on-off ratios in the non-switched state and in the switched state are not equal. In the switched state, the “off” is not optimal due to unequal gain in the two SOAs. To be practically useful, all-op-tical switches will need improved and balanced extinction ratios. Two different ver-sions of all-optical switches, that overcome these extinction-ratio limitations, have recently been proposed or demonstrated. The improvements were obtained by opti-mizing both, the bias current of the SOAs and the phases in the two arms of the MZI [5.13] or, alternatively, by using two asymmetric splitters with reciprocal splitting ra-tios [5.14, 5.15].This paper, after presenting MZI-SOA all-optical multiplexer, demultiplexers switches and their capabilities in more detail, will analyse modifications, which im-prove the extinction ratios. Three MZI-SOA configurations are described, discussed and compared. Configurations with unequally biased SOAs (A) asymmetric splitters (B1) and (B2) and two pairs of asymmetrically arranged SOAs having different al-pha factors each (C). The analysis allows us to introduce new 1x2 and 2x2-all-optical switches (type B1 and C). We perform experiments to confirm the model and give design criteria for the different configurations.The inequality of the output extinction ratios for a basic symmetric MZI-SOA switch is shown in paragraph 2. The formalism which describes the physics of all-optical switches is presented in paragraph 3. It will be used in paragraph IV to find the new structures and the operation conditions with the best extinction ratios. In the end we compare the theory with experiments. The design tolerances imposed by the struc-tures are discussed and compared in the appendix.

2. Basic MZI-SOA All-Optical SwitchAt first a symmetric MZI-SOA all-optical switch is considered and the inequality of the output extinction ratios is discussed. In Fig. 5.1 we have depicted a symmetric MZI switch in the non-switched and switched state. The MZI-switch comprises two 50:50 splitters, two SOAs (SOA1 and SOA2) and two couplers C to introduce the control signal into the SOA section on the MZI-arms. The SOAs are equally biased to provide an identical gain each. In the non-switched state, when the control signal is absent, the input signal is directed toward it’s cross port , supposing that Pin PX

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

198

the phase relations are correctly adjusted. The extinction ratio (the power in the switched-off output guide divided by the power in the switched-on output guide) for this state would be ideally large, were it not for imperfections. The extinction ratio of 29 dB in Fig. 5.1(b) refers to the value attained in the experiment of paragraph 5.In the switched state, an optical control signal saturates SOA1 and, as a conse-quence, induces a gain and refractive-index change. A data signal passing through the MZI experiences the phase shift and is redirected from the cross output port to the bar output port . However, since the gain in SOA1 changes while the gain of SOA2 remains unchanged, the “off” at the -port is not optimal. In the

Pin

PX2

SOA2

SOA11

50:5050:50

P=PX

= 13dB PC=on

Nonswitched State

SwitchedState

PC

P=

PX

t

t

t

PXP=

= 29dB PC=off

P=

(b)

PC

C

Attained Extinctions:

(a)C

Fig. 5.1 (a) All-optical switch based on a Mach-Zehnder Interferometer (MZI) con-figuration comprising two 50:50 splitters for dividing and combining the data signal and two equally biased Semiconductor Optical Amplifiers (SOAs). (b) In the non-switched state, without applied control signal , the data signal is directed to the cross port. Experimentally an almost ideal extinction of 29 dB is found (see also Fig. 5.8). The phase shift of ~π, which is necessary to switch the signal into the bar-output is provided by the con-trol signal coupled into SOA1 through coupler C. However, since the gain in SOA1 changes, while the gain of SOA2 remains, the ’off’ in Px is not optimal and the extinction ratio is reduced to 13 dB.

PinPC

PCPin

PX P=PX

5.2 All-Optical Space Switches with Gain and Principally Ideal Extinction-Ratios

199

experiment we find a reduced extinction ratio of 13 dB. To obtain an all-optical MZI-SOA switch with equal extinction ratios in its different states with and without con-trol signal, we have to symmetrize the extinction ratios of the switch. This can be achieved by reducing the current supply of SOA2 which is not influenced by the con-trol signal . With the new current settings we achieve, that the gain difference of the non-switched and switched state before the second 50:50 splitter becomes iden-tical. Subsequently, balanced extinction ratios for both states attaining 20 dB are reached. We mention that this unequally biasing of an MZI-SOA all-optical switch requires additional phase shifting to compensate the undesired phase shifts, that oc-cur when differently biasing the SOAs. The basic MZI-SOA demonstrated the need of introducing asymmetries (unequally biasing) to improve the performance of the switch. In analogy we can introduce other asymmetries to improve the switch. An 1x2 all-optical switch with principally ideal extinction ratios in both outputs is obtained by allowing for unequally biased SOAs and asymmetric splitters in the MZI configuration (paragraph IV. B). A 2x2 switch with ideal extinction ratios for data signals from both inputs into both outputs is ob-tained with an additional pair of amplifiers asymmetrically arranged in the MZI-con-figuration (paragraph IV. C).

3. Analysis 3.1 ConfigurationA generalized MZI-SOA all-optical switch, encompassing all types of all-optical switches that are discussed, is given in Fig. 5.2. The MZI is formed by two splitters

and for dividing and combining the data signals or , two couplers C to introduce the control signals and , and SOAs providing the necessary nonlinearity for switching. Two phase shifters for controlling the phase offsets in the MZI are added. Without control signal the data signals from the input ports and

are directed to the respective cross port and . Suitable optical con-trol signals and , by means of carrier density related refractive-index changes, induce phase shifts of π in the MZI such as to switch the signals to their respective bar output ports and . The power-splitting ratios of the split-ters and can deviate from the 50:50 splitting ratio. The all-optical switches of Fig. 5.2 can be used with a zero order mode control signal, but also in the dual order mode configuration with a first order mode control signal [5.15, 5.16].

3.2 ModelWe develop the analysis for the generalized MZI-SOA all-optical switch of Fig. 5.2.To describe the switch of Fig. 5.2 we use a notation with 2x2-matrices [5.17]. The total transfer matrix for the field amplitude of a data signal in the non-switched

PC

SA SB Pin 1, Pin 2,PC1 PC2

Pin 1,Pin 2, PX 1, PX 2,

PC1 PC2

P 1,= P 2,=SA SB

Pin

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

200

state, when no control signal is applied, and in the switched state, when or is applied, is given by

(5.1)

where the first and last matrices give the 2x2-multimode interference (MMI) split-ters and with varying bar-transmission intensity probabilities and . They are related to waves at the optimum wavelength of the splitters propagating in the positive direction: . The definition of the bar power-transmission probabilities is illustrated in Fig. 5.3. The phase relations of MMI-splitters are dis-cussed in Ref. [5.18-5.20]. The second matrix describes the phase shifts induced from the phase shifters. and are the static phase offset on MZI-arm 1 and 2, respectively. The third matrix gives the single-pass gain , and the induced

SA

SOA2

PC2

Pin,1

Pin,2

P=,1

PX,1SB

C

SOA1Inputs

1

2

Outputs

1

2

PC1

PX,2

P=,2

PhaseShifter 2

PhaseShifter 1

I1

I2

Δφ1

Δφ2

C

Fig. 5.2 Generalized MZI-SOA all-optical switch with phase shifters to adapt phase-shift offsets and splitters and that allow asymmetric split-ting ratios. Depending on the phase relations in the MZI-arms, the data signal is mapped onto the cross and bar-output guide and

is mapped onto its respective cross and bar-output guides . The phase relation within the MZI-arms is changed when a control signal

and/or is introduced through the couplers into SOA1 and/or SOA2, respectively.

SA SB

Pin1 PX,1 P=,1Pin2 PX,2 P=,2

PC1 PC2 C

PC1 PC2

t c i– sB 1 sB–

1 sB– i– sB

eiΔφ1 0

0 eiΔφ2

⋅ ⋅=

G1eiΔϕ1 0

0 G2eiΔϕ2

i– sA 1 sA–

1 sA– i– sA

,⋅

SA SB sA sB

i+ kz ωt–( )

Δφ1 Δφ2G1 G2

5.2 All-Optical Space Switches with Gain and Principally Ideal Extinction-Ratios

201

phase shifts , that the data signal experiences when it passes SOA1 and SOA2. with are the induced phase shifts in the SOAs due to unequally biasing the offset currents (contribution ) and due to the carrier depletion effect from a control signal (contribution ), respectively. is a cou-pling constant, taking into account the coupling efficiencies of couplers . Without restriction to generality, we set which is true for the device presented in Ref. [5.15-5.16], for instance.With any gain change, a phase change is associated according to the Kramers-Kronig relation. The alpha factor relates these two quantities in linear approximation, so that we can rewrite the gain with in terms of the unperturbed single pass gain and the phase change, respectively

. (5.2)

The first equal sign in (5.2) relates the gain change averaged over the SOA of length to the amplifier gain . It holds for travelling wave amplifiers, which ide-ally should be used in the all-optical switches. The second equal sign goes back to the definition of the alpha factor which is the ratio of the refractive-index change and the gain change

, (5.3)

where is the wavelength of the data signal and the effective refractive-index change which determines the total phase change through . The al-pha factor is a material constant depending on the wavelength, the current-density and the material used for the SOAs. We assume that the alpha factor remains con-

Pin P=

PX

Bar- and cross-power transmission probability:

s =Pin

P= 1-s =Pin

PX

Splitting ratios:

r =PX

P=2 x 2

Fig. 5.3 Definitions of the cross- and bar-power transmission probabilities and the definition of the splitting ratio for a 2x2-splitter used throughout this text.

Δϕ1 Δϕ2Δϕj Δϕj

I ΔϕjC+≡ j 1 2,=

ΔϕjI

ΔϕjC c

Cc 1=

αGj j 1 2,=

Go

Gj Go eΔgjL⋅ Go e

2Δϕj–α

---------------⋅= =

ΔgjL Gj

α

α 4π–λ

--------- ΔnΔg-------⋅≡

λ ΔnΔϕj ΔnL2π λ⁄=

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

202

stant, for a given operation point. This is reasonable since the devices are operated at a fixed wavelength at gain maximum and the carrier densities are only moderately modulated by the control signals, because we are working with the cross-phase mod-ulation (XPM) effect rather than the cross-gain modulation (XGM) effect [5.21]. Re-fined amplifier gain calculations, considering higher order perturbation of strong optical signals, should be performed with a multisection SOA model. However also in a multisection model the total phase change and the total gain change obey Eq. (5.2) as long as the alpha factor remains constant. To calculate with this model we additionally assume that our control signals are long enough (longer than ~1ps in 1.55-InGaAsP) so that intraband relaxation effects, that would modify the value of the alpha factor, are not involved.The bar and cross output-powers of the all-optical switch can now be calculated by evaluating Eq. (5.1) under consideration of Eq. (5.2). It is important to note that the matrix t is a transfer matrix for field amplitudes. To obtain relations for the output-powers, we have to square the matrix elements of the matrix . Using the defini-tions for the bar and cross output-powers given in Fig. 5.2, we find for a data signal from input guide 1

, (5.4a)

and for a data signal from input guide 2

, (5.4b)

with

(5.5a)

(5.5b)

(5.5c)

, (5.5d)

where the splitting ratios of the couplers are defined as

, (5.6)

tij t

P=,1 t112Pin 1,= PX 1, t21

2Pin 1,=

PX 2, t122Pin 2,= P=,2 t22

2Pin 2,=

t112 C 1 2– rArB Δφ Δϕ12+( )e

Δϕ12– α⁄cos rArBe

2Δϕ12– α⁄+⎝ ⎠

⎛ ⎞=

t212 C rB 2 rArB Δφ Δϕ12+( )e

Δϕ12– α⁄cos rAe

2Δϕ12– α⁄+ +( )=

t122 C rA 2 rArB Δφ Δϕ12+( )e

Δϕ12– α⁄cos rBe

2Δϕ12– α⁄+ +( )=

t222 C rArB 2 rArB Δφ Δϕ12+( )e

Δϕ12– α⁄cos– e

2Δϕ12– α⁄+( )=

rAsA

1 sA–--------------≡ rB

sB1 sB–--------------≡

5.2 All-Optical Space Switches with Gain and Principally Ideal Extinction-Ratios

203

and the coupling variable is defined as

. (5.7)

For convenience we have used the notation with the relative phase shift of the phase in SOA1 with respect to the phase in SOA2

, (5.8)

with analogous definitions for and and with the definition for the phase-offset

. (5.9)

3.3 Expressions for the Extinction Ratios To estimate the quality of the switching process extinction or on-off ratios are de-fined. For some applications it is useful to determine the extinction of the signal in the switched-off output port in comparison to the output port guiding the signal. That is the extinction ratio. For other applications it might be more useful to discriminate between the “1” in comparison to the “0” in the output from one port. That is the on-off ratio. Although these two definitions are different, they lead to similar results. Es-pecially in the case of an ideally high extinction ratio the on-off is ideally high too. For that reason we restrict the discussion on the extinction ratios. The expressions of the on-off ratios as well as the differences when working with on-off ratios are given in appendix A.In accordance with the upper nomenclature, the extinction ratios in the non-switched state , in the absence of a control signal (superscript ), and in the switched state

, in presence of a control signal (superscript ) are defined as power ratios

(5.10)

with being the power in the switched-on output guide and being the power in the switched-off output guide for signals coupled in through input guides j=1,2.In the non-switched state the terms in (5.10) have to be used with . Simi-larly, one has to use (5.10) with in the switched state. is positive when the control signal is coupled into SOA1 and is negative when the control signal is coupled into SOA2.With Eq. (5.4a)-(5.9) the expressions for the extinction ratios of Eq. (5.10) can now be expressed as a function of the five variables , , , and . How-

C

C c2 G2 1 sA–( ) 1 sB–( )⋅⋅=

Δϕ12

Δϕ12 Δϕ1 Δ– ϕ2≡

Δϕ12I Δϕ12

C

Δφ Δφ1 Δφ2–=

XN NXC C

XjN POn

POff----------

PC off

PX,jP= j,----------

PC off

= = XjC POn

POff----------

PC on

P= j,PX,j----------

PC on

= =

POn POff

Δϕ12C 0=

Δϕ12C 0≠ Δϕ12

C

Δϕ12C

Δ φ, Δϕ12I Δϕ12

C sA sB

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

204

ever some of the variables depend on each other. We can constrain the space of so-lutions to such cases, that have not only high extinction ratios, but also an ideal on-state. This implies

(5.11)

and

(5.12)

Eq. (5.11) shows, that the phase shift induced from the unequally biased SOAs has to be compensated with the phase shifters. Eq. (5.12) expresses the fact that, for switching a MZI, a phase shift difference of π has to be introduced between the two MZI arms. The equations are obtained when maximizing the output-powers of the switched on output guides for signals from input 1 or 2, respectively with regard to the variable and .With the definition of the extinction ratios of Eq. (5.10) under consideration of the restrictions imposed by Eq. (5.11)-(5.12) we obtain for a data signal at input guide 1 (upper sign) and those for a data signal at input guide 2 (lower sign)

and (5.13a)

. (5.13b)

4. Specific MZI-SOA ImplementationsWe now discuss three asymmetries that lead to three MZI-SOA all-optical switch configurations with improved or almost ideal extinction ratios (Table 5.1). We chose the operation conditions such that they are appropriate for applications as add- or drop-multiplexers or add-drop multiplexers.

Δφ Δ– ϕ12I=

Δϕ12C

0 PC off ,

π PC on . ,⎝⎜⎛

±≅

Δφ Δϕ12C

X1/ 2

N cosh 2Δϕ12I α⁄ rAln rBln±+−( ) 4⁄[ ]2

2Δϕ12I α⁄ rAln rBln+−+−( ) 4⁄[ ]2sinh

----------------------------------------------------------------------------------------=

X1/ 2

C 2Δϕ12I α⁄ 2Δϕ12

C α⁄+ rAln+− rBln+−( ) 4⁄[ ]2cosh

2Δϕ12I α⁄ 2Δϕ12

C α⁄+ rAln rBln±+−( ) 4⁄[ ]2sinh----------------------------------------------------------------------------------------------------------------------=

5.2 All-Optical Space Switches with Gain and Principally Ideal Extinction-Ratios

205

High speed add-drop multiplexing with switching speeds as fast as 1 ps has been demonstrated with two optical control-pulses consecutively coupled into the two SOAs of the MZI-SOA switch [5.22]. The method uses a first control signal to switch a data signal from one output guide into the other and a second to reset the switch. When the first control signal is introduced, for example into SOA1, the de-vice switches due to refractive-index changes caused by ultrafast carrier depletion in SOA1 from the cross into the bar state. After a time interval corresponding to a bit length or alternatively a package length, the second control signal is introduced in the opposite SOA. This switches the device back to the initial switching state - using again the subpicosecond carrier depletion effect. As the carrier regeneration time is much slower than the time delay between the two control-pulses, SOA1 and SOA2 regenerate together up to the initial carrier-injection level until the next switching cy-cle excites them again.Subsequently we discuss the effect of the first control signal. The effect of the second control pulse is briefly discussed here. It mainly resets the phase differences in the MZI so that the all-optical device switches back to the initial state. In addition the second control pulse saturates the second SOA, so that the over-all-gain of the data signal is suppressed right after the switching window. The gain saturation depends on the value of the alpha factor (Fig. 5.4). The higher the alpha factor, the smaller the gain saturation. Fig. 5.4 was obtained by evaluating of Eq. (4) for the worst case, where both control pulses are turned on: . Normally the two control pulses are introduced consecutively, so that the carriers re-generate during the time interval and the gain saturation is slightly less severe.

Table 5.1: Design parameters of all-optical MZI-SOA switches with high extinctions

Device Fig Characteristics

Number of inputs with high

extinctions;Best extinctions

Unequally BiasedSwitch

(A) Fig. 5.5

1 1 90o -90o 1 type of SOAs,uncritical design tolerances

2 ;~20 dB each

(depends on α)

One AsymmetricSplitter

(B1) Fig. 5.6

1 90o -90o 1 type of SOAs,good wavelength bandwidth, especially with 1x2-splitter

1 ;ideal

Two AsymmetricSplitters

(B2)

0o 0o 1 type of SOAs 1 ;ideal

Two Alpha FactorSwitch

(C) Fig. 5.7

1 1 0o 0o,0o

2 types of SOAs with differ-ent alpha factors

2 ;ideal

rA rB Δφ Δϕ12I

e π– α⁄

eπ α⁄ e π– α⁄

PXΔϕ1

C Δϕ2C π= =

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

206

4.1 All-optical switches with symmetric splittersWhen the splitters and are symmetric - that means when they have a 50:50 splitting ratio, which corresponds in our notation to

, (5.14)

we can state, that the switching characteristic is independent of the input guide that is used. The cross and bar states of a signal from input 1 and the cross and bar states from a signal from input 2 behave alike. This statement holds with and without ap-plied control signal. It even remains valid when the SOAs are unequally biased. To verify the statement it has to be shown, that and , when us-ing Eq. (5.14) in Eq. (5.5a)-(5.5d). Consequently we will find that 2x2-MZI-SOA add-drop multiplexers are advantageously built with symmetric splitters and . In contrast, 1x2-MZI-SOA add or drop multiplexers can more advantageously be de-signed with asymmetric splitters.

1) Symmetric splitters, equally biased all-optical switch: , Equally biased all-optical 2x2 MZI-SOA switches with symmetric splitters lead to unbalanced extinction ratios (Fig. 5.1). It is illustrative to use Eq. (5.13b) to plot the extinction ratios of the switched -state as a function of the alpha factor. The dot-ted line in Fig. 5.5(b) shows how the attainable extinction ratio of the -state im-

Alpha Factor

Gai

n Sa

tura

tion

[dB

]

0 5 10 12.5

0

-5

-10

-15 2.5 7.5

Fig. 5.4 Saturation of the overall-gain right after resetting an all-optical switch to the initial state with a second control pulse (Fig. 5.2). The first control-pulse was used to switch from the cross into the bar state and the sec-ond control signal is introduced after a time interval to reset the switch. The saturation decreases with increasing alpha factor.

PC1PC2

SA SB

rA rB 1= =

t11 t22= t12 t21=

SA SB

rA rB 1= = Δϕ12I 0=

XC

XC

5.2 All-Optical Space Switches with Gain and Principally Ideal Extinction-Ratios

207

proves with an increasing alpha factor. For an alpha factor of 7.4, the extinction ratio of the -state is limited to moderate high 13 dB. An alpha factor of 7.4 corresponds to the experimentally determined static value at the gain-maximum of our device presented in paragraph V.

XC

1

2 50:50

I1

I2

1

2

Δφ=Δφ1-Δφ2=0I1=I2

Δφ1

Δφ2

Equally Biased Switch: Unequally Biased Switch: I1>I2 Δφ=Δφ1-Δφ2=π/2

50:50

PC1

All-Optical MZI-SOA Switch Type A

Alpha Factor

Extin

ctio

n-R

atio

[dB

]

0 5 10 12.5

30

20

10

0

Equally Biased

2.5 7.5

Unequally Biased

(b)

(a)

Fig. 5.5 (a) All-optical 2x2 switch with equally and unequally biased SOAs and symmetric (50:50) splitters. (b) The attainable extinc-tion ratios improve - the higher the value of the material dependent alpha factor. The dotted line shows the attainable extinction for the ’equally bi-ased switch’ in the switched-state (PC1 on) and the solid line shows the improved and balanced extinctions for the ’unequally biased switch’ in both the non-switched and switched state.

I1 I2=( )I1 I2≠( )

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

208

2) Symmetric splitters, unequally biased all-optical switch: , The equally biased all-optical switch with symmetric splitters clearly demonstrated the unsatisfactory situation, of a switch with only one state (the -state) having a good extinction ratio. However, high and balanced extinction ratios can be realized by unequally biasing the SOAs [5.13]. With Eq. (5.11)-(5.13b) we are able to deter-mine the operation parameters of the unequally biased switch with balanced extinc-tions. By requiring we find

and . (5.15)

which means, that an additional current prebias on SOA1 - the amplifier guiding the first control signal - has to be applied. The optimum for the additional gain is reached, when the associated phase shift is -π/2. To reach high extinction ratios, this induced phase shift has to be compensated with the active phase shifters in accord-ance with the second equation of (5.15).Attainable extinction ratios as function of the alpha factor, with unequally biased SOA currents under operation conditions as required by Eq. (5.15), are given by the solid line of Fig. 5.5(b). For an alpha factor of 7.4 the extinction ratios reach 20 dB for both states. To attain higher extinction ratios with these devices, research will have to focus on materials with larger alpha factors. The device operates with signals from either 1 or 2 inputs.

4.2 All-optical switch with asymmetric splittersPure add- or drop-multiplexer applications require only 1x2 switches. For such ap-plications condition (5.14) is too restrictive. When allowing for asymmetric splitters in combination with unequally biased currents, the extinction ratios can principally be ideal for signals from one of the two inputs.The operational parameters for this switch are found by requiring that the upper

-term in the expressions of Eq. (5.13a)-(5.13b), for example for a signal from input 1, be zero. Two of the three variables , and are used to solve the two equations of the non-switched and switched extinction states.In subsection B1.) we give a device with only one asymmetric splitter and unequally biased currents. This device is simpler in design than the device of subsection B2.) with two different asymmetric splitters.

rA rB 1= = Δϕ12I 0≠

XN

XN XC=

Δϕ12I π 2⁄–= Δφ π 2⁄=

2sinhrA rB Δϕ12

I

5.2 All-Optical Space Switches with Gain and Principally Ideal Extinction-Ratios

209

1) All-optical switch with one asymmetric splitterThe conditions for ideal extinction ratios of a data signal from input 1 with a control-signal introduced into SOA1 are due to Eq. (5.13a)-(5.13b)

For an ideal -state: (5.16a)

For an ideal -state: (5.16b)

Since we only need two variables to solve Eq. (5.16a)-(5.16b) we can require, that splitter is symmetric. This leads us to a device with

(5.17a)

(5.17b)

=> (5.17c)

corresponds to a simple 50:50 splitter. The splitting ratio leads us to an asymmetric splitter. As asymmetric splitters the so called 2x2-butterfly MMIs as pro-posed in Ref. [5.20] can be used. They possess the splitting characteristics described by the matrix for the MMI splitter given in Eq. (5.2). The current bias has to be cho-sen asymmetrically such that the associated phase shift amounts to and due to Eq. (5.11) an offset phase shift of is needed.In Fig. 5.6 we show an asymmetrically biased 1x2-all-optical switch with a 50:50 splitter and a 40:60 splitter. A 40:60 splitter has to be chosen due to Eq. (5.17b) for SOAs with an alpha factor of 7.4. On the right side of Fig. 5.6 we give the design tolerances of the extinction ratios as a function of the alpha factor. For signals from input 1 we find ideal extinction ratios in the non-switched and switched state around an alpha factor of 7.4. The extinction ratio degrades when the alpha factor deviates too much from this value, for which the switch was designed. Using the switch for signals from input 2, the resulting extinction ratios are poor, since the switch can only be optimized for signals from one of both inputs.We can replace the 2x2-MMI splitter SA with a 1x2-splitter since input guide 2 is not used. This will allow to drop the offset of the phase shifters. The phase shifter is not anymore needed, because the coupling matrix of the 2x2-MMI, given in Eq. (5.2), contains a phase delay between the two splitted signals, which drops with a 1x2-splitter for reasons of symmetry. Therefore we can rewrite Eq. (5.17c) for the

X1N 2Δϕ12

I α⁄ rA( )ln– rB( )ln– 0=

X1C 2Δϕ12

I α⁄ rA( )ln– rB( )ln+ 2– π α⁄=

SA

rA 1=

rB e π α⁄–=

Δϕ21I π 2⁄–= Δφ π 2⁄=

rA 1= rB

Δϕ21I π 2⁄–=

π 2⁄

π 2⁄

π 2⁄

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

210

version with a 1x2-splitter SA

=> (5.18)

A second advantage is worth being mentioned. When using for example the 1x2-MMI proposed in [5.18] instead of a 2x2-MMI splitter, the wavelength bandwidths and fabrication tolerances are much more relaxed, since 1x2-MMIs are shorter than 2x2-MMIs [5.24].

(b)

(a)

Fig. 5.6 Unequally biased 1x2 all-optical switch with one asymmetric splitter. For a signal from input 1, ideal extinction ratios are achieved in the non-switched (PC1 off) and switched (PC1 on) state with a 40:60 splitter for SOAs with an alpha factor around 7.4. The extinction ratios for a signal from input 2 are poor. This 1x2-switch is optimized for signals from input

Alpha Factor

Extin

ctio

n-R

atio

[dB

]

0 5 10 12.5

40

30

20

02.5 7.5

10

rB=40:60

Signals from

Signals from Input 1

Input 2

1

240:60

Δφ=Δφ1-Δφ2=π/2I1>I2

Δφ1

Δφ2

I1

I2

PC1

1

2 50:50

All-Optical MZI-SOA Switch Type B1

Δϕ12I π 2⁄–= Δφ 0=

5.2 All-Optical Space Switches with Gain and Principally Ideal Extinction-Ratios

211

2) All-optical switch with two asymmetric splittersA 1x2-multiplexer with ideal extinction ratios and symmetrically biased currents can be derived of Eq. (5.16a)-(5.16b) by requiring that . However it is found, that for obtaining ideal extinctions, two asymmetric splitters with unequal splitting ratios are needed

(5.19a)

. (5.19b)

For an alpha factor of 7.4, the former splitting ratio corresponds to a 60:40 splitter and the later corresponds to a 40:60 splitter. It is worth mentioning, that the set of variables in Eq. (5.19a)-(5.19b) not only leads to ideal extinction ratios of , , but also of . This version of an all-optical switch with two asymmetric splitters corresponds to the device proposed in [5.14]. An experimental demonstration was given in Ref. [5.15].

4.3 All-optical switch with two pairs of asymmetrically operated SOAsA 2x2-MZI-SOA all-optical switch with ideal extinction ratios, Fig. 5.7, can be ob-tained with two sets of amplifiers: SOA and SOA , both having different alpha fac-tors and . For changing from the non-switched into the switched state, the two control signals and are simultaneously introduced from the left hand side into the amplifiers SOA and SOA . The power of the control signals is chosen such that the gain saturation in the two SOAs is identical, but the phase shift induced be-tween the two MZI-arms amounts to π. This choice is always possible when the two SOAs have different alpha factors. As a result the gain in the two MZI-arms is sym-metric in the non-switched state, but also symmetric in the switched state. To reset the phase difference of π in the MZI, similar control signals can be introduced from the right side.The phase shifts of the two control signals, which are necessary for switching follow from Eq. (5.13a) and (5.13b). To find explicit expressions, we have to extend the the-ory developed for the all-optical switches of Fig. 5.2 to the case of Fig. 5.7. As the new switch has two gain-mediums we have to substitute

, and (5.20)

. (5.21)

The terms with a containing an upper index “I” and “C” are replaced due to their definitions in full analogy. The restrictions (5.11) and (5.12) are still valid and must

Δϕ21I 0=

rA eπ α⁄=

rB e π α⁄–=

X1N X1

C

X2N

′ ″α′ α′′

P′C1 P″C1′ ″

Δϕ12 Δϕ′12 Δϕ″12+→

Δϕ12 α⁄ Δϕ′12 α′⁄ Δϕ″12 α″⁄+→

ϕ

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

212

be adapted according to the substitution (5.20).Finally we can rewrite the four conditions for ideal extinction ratios for a signal from input guide 1 and a signal from input guide 2 in the switched and non-switched state in analogy to those of Eq. (5.16a)-(5.16b). Solving for this new set of equations leads

SOA2’P’’C1

50:50C

C

SOA1’’

1

2

1

2

P’C1SOA1’

SOA2’’

50:50

Δφ1

Δφ2

All-Optical MZI-SOA Switch Type C

Idea

l Idea

lPX,1,P=,1,

Induced Phase Δφ’c12 from Control Signal P’C1

Nor

mal

ized

Pow

er [d

B]

α‘ =8.0α‘‘=2.0

0

-20

-10

0

-15

-5

5

PX,2P=,2

Induced Phase Δφ’’c12 from Control Signal P’’C1

8/6π4/6π

0 2/6π1/6π

Extin

ctio

n

Extin

ctio

n

-25

(a)

(b)

Fig. 5.7 (a) 2x2 all-optical switch with symmetric splitters and two pairs of ampli-fiers (SOA and SOA ) with different alpha factors each. This allows to build a 2x2-all-optical switch with principally ideal extinction ratios. (b) A perfect switching ’on’ of the bar and a perfect switching ’off’ of the cross state is found, when simultaneously introducing the two control signals

and into the switch. This ideal switching behaviour is attained for signals from input 1 and 2.

′ ″

P′C1 P″C1

5.2 All-Optical Space Switches with Gain and Principally Ideal Extinction-Ratios

213

us to

(5.22)

(5.23)

and . (5.24)

In other words, the all-optical switch with ideal extinction ratios for signals from ei-ther input guide has symmetric splitters SA and SB (see the discussion of Eq. (5.14)), both pair of amplifiers may be symmetrically biased: and con-sequently active phase shifters are due to Eq. (5.11) not needed. With the (+)-sign in Eq. (5.12) - an arbitrary choice for introducing the control signal from the left side we find that the control-signal powers must be chosen such that they induce the phase shifts given in Eq. (5.24).A possible realisation contains symmetric splitters, equally biased amplifiers SOA having for example alpha factors of 8 and amplifiers SOA with alpha factors of 2, such that the control signal introduced through provide a phase shifts of

and . Although the alpha factor is a material constant, it’s value varies considerably within the wavelength spectrum and for dif-ferent carrier injection levels. Values of 8 and 2 for alpha factors are reasonable, since this range is covered when shifting the bandgap and/or varying the carrier in-jection level in amplifiers [5.23, 5.25]. When simultaneously turning on the control signals and the power in the bar states and switches on and the power in the and switches off, as depicted in Fig. 5.7(b). We find ideal extinction ratios for signals from input 1 and input 2. In comparison to the all-optical switch with asymmetric splitters we have now a device with ideal off-states for all in- and output guides. However, the price is a considerably reduced ’on’ of the bar state.For obtaining four ideal extinction ratios, it is in principle only necessary to have two unequal SOAs with a different alpha factor each. The most simple version would therefore work with totally two amplifiers. One with an alpha factor on MZI-arm 1 and one with an alpha factor on MZI-arm 2. It would however be more difficult to adjust offset phases and unequal carrier relaxation times could degrade the per-formance.

rA rB 1= =

Δϕ′12I

α′--------------

Δϕ″12I

α″---------------+ 0=

Δϕ′12C 1

1 α″ α′⁄–-------------------------π= Δϕ″12C α″– α′⁄

1 α″ α′⁄–-------------------------π=

Δϕ′12I Δϕ″12

I 0= =

′″

CΔϕ′12

C 8 6⁄ π⋅= Δϕ″12C 2 6⁄ π⋅=

Δϕ′12C Δϕ″12

C P 1,= P 2,=PX 1, PX 2,

α′α″

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

214

5. Experiments To demonstrate all-optical switches with optimized extinction ratios, we have real-ized the MZI-SOA all-optical switch of Fig. 5.5. Static and dynamic experiments have been performed. We first verify the improvement of the extinction ratio when going from an all-optical switch with symmetric splitters and equally biased SOAs to an all-optical switch with symmetric splitters and unequally biased SOAs. Sec-ond, we show that the ’unequally biased’ version delivers the predicted high and bal-anced extinction ratios under dynamic conditions for switching windows of 25 ps. Our device uses 2x2-MMI-splitters and with 50:50 splitting ratios. The same MMIs are used to couple in a zero order mode optical control signal through the cou-plers .The InGaAsP/InP MZI-SOA-switches were grown by two-step Metal Organic Va-pour Phase Epitaxy on (001) InP. The SOA layers with a 0.22 μm thick 1.58μm-In-GaAsP active layer were grown first. Subsequently the region outside the 500 μm long SOA areas are etched, followed by a regrowth of the passive waveguide layer (0.6 μm thick 1.28 μm-InGaAsP followed by a 1.6 μm thick InP cladding). The two different layers are butt coupled [5.26]. The ridge waveguides were formed by etch-ing 1.7 μm into the SOA and waveguide heterostructure. The 2x2-MMIs were etched during the same process step as the waveguides. They have dimension of 450 μm by 17.5 μm. A heavily doped InGaAs layer was placed on top of two waveguide sections to provide good contacts with gold pads. These two sections formed the electrooptic phase shifters. The total size of the switch chip is 9 x 1.3mm. The waveguide facets were antireflection coated and the chip thinned to allow high heat dissipation and ease of cleaving.The improvement towards two high and balanced extinction ratios of an all-optical switch, when gradually moving from the symmetrically biased to the unequally bi-ased operation mode is given in Fig. 5.8. The improvement is due to intentionally reducing the current bias in the SOA that is not influenced by the optical control sig-nal and an adaptation of the phase-shifter voltages. Since there exists a relation be-tween asymmetrically biasing the SOAs and the induced phase in the phase shifters, see Eq. (5.9), we express the asymmetry by the reverse voltage applied on the phase shifters. The experiment was carried out under static conditions since that allows for higher measurement precision. The experiment was performed with a cw signal

coupled through input guide two at λPin2=1.58 μm with a signal power of -19 dBm. A high power DFB-laser provided a control signal at a wavelength of 1.554 μm. It was coupled into SOA1 through the 2x2-MMI . A power of +3.6 dBm, measured in the fibre, was needed to induce a phase shift of π. In Fig. 5.8 we show the measured fibre-to-fibre gains of the bar ( ) and cross-state

SA SB

C

Pin 2,

C

G= P 2,= Pin 2,⁄=

5.2 All-Optical Space Switches with Gain and Principally Ideal Extinction-Ratios

215

( ), in the non-switched state in the absence of a control signal (PC off), as well as in the switched state with an applied control signal (PC on). The op-timal ’equally biased’ switch state at the right of Fig. 5.8 is obtained for SOAs with almost equal bias currents of ISOA1=190 and ISOA2=182 mA. The difference in the bias currents is due to fabricational inhomogeneities. To reach this optimal state with the smallest possible (PC off), an offset-phase-correction voltage of 3V was ap-plied to phase shifter one. A negative reverse voltage on the x-axis indicates that phase shifter one was in use and a positive reverse voltage shows that phase shifter two was used.We observe how the cross state GX(PC on) improves from an unsatisfactory off at

dB to an off-state at -14 dB, when decreasing the voltage applied to phase shifter

10

0

-10

-20

Fibr

e-to

-Fib

re G

ain

[dB

]

XC=13dB

XN=29dB

XN=XC=20dB

R

XNR

G=C

GXC

G=N

Asymmetrically Biased Switch

SymmetricallyBiased Switch

-Δϕ12IR

0 30 60 90Phase-Shifter Phase ΔΦ [degrees]

-3 0 3 6 9Applied Voltage on Phase Shifters [V]

GXN

Fig. 5.8 Measured and calculated fibre-to-fibre gain for the all-optical switch giv-en in Fig. 5.5 demonstrating the improvement towards high and balanced extinction ratios when going from the symmetrically biased operation mode with highly dissimilar extinction ratios (13 and 29 dB) to the une-qually biased operation mode with two similar extinction ratios of 20 dB. At an offset voltage of -3V, the symmetrically biased mode is found for nearly equal SOA1 and SOA2 currents of 192 mA and 182 mA respectively. Unequally biasing the SOAs by decreasing the SOA2 current from 182 to 162 mA while compensating the induced phase shift with the active phase shifters leads to the unequally biased operation mode at an offset voltage of V. is the extinction ratio used to calculate the alpha fac-tor, the only fit parameter of the calculated curves.

V 7,6= XNR

GX PX 2, Pin 2,⁄=

G=

7–

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

216

one and increasing the voltage applied on phase shifter two, while asymmetrically decreasing the current through SOA2 such that a minimal G=(PC off) is maintained. When the reverse voltage applied to phase shifter two reaches 6V, the extinction ra-tios of the switched off G=(PC off) state and the switched off GX(PC on) state are similar and the extinction ratio for both states is 20 dB. In this best unequally biased switch state, the SOA2 current has to be set to 160 mA.We have calculated the gain as a function of induced phase in the phase shifters by using Eq. (5.4b)ff. and (5.11)-(5.12) and found an excellent agreement with the measured points (see the drawn lines in Fig. 5.8). The only disagreement between calculated and measured values can be seen in the off state below -20dB of the bar output . For output signals as weak as -41 dBm (the input-signal power is -19 dBm and the fibre-to-fibre gain is -22 dB) the amplified spontaneous emission leads to contributions, that can not be neglected. To compare the experiments with the the-oretical curves we had to relate the applied voltage with the phase shift induced in the phase shifters. Since the voltages applied to the phase shifters are small, we could linearly interpolate the induced phase shifts with the voltage scale [5.27]. It is worth noting that for fitting all four curves we used only one fitting parameter, name-ly the alpha factor, and only one reference point R (see Fig. 5.8). This reference point is necessary because coupling losses and induced phase-shift-to-applied voltage characteristics are different for every device and have to be determined experimen-tally.The alpha factor was extracted from the measured curves as follows: Experimental-ly, the extinction ratio (see Fig. 5.8) measured from the point R, where the curves of the two off-states are crossing, up to the state can be determined with good accuracy. The extinction ratio , or with other words at the point R, is given by the formula (5.13a). The only unknowns in expression (5.13a) are the alpha factor, which we are looking for, and the phase shift induced by the current asymmetry. But is determined since it is the induced current phase shift, where the two curves of the switched off states cross. It is found by equating Eq. (5.5a) with (5.5b). It’s value is given by

= . (5.25)

Eq. (5.13a) and (5.25) then allow to calculate the alpha factor from the measured ex-tinction ratio It becomes

. (5.26)

Δφ

G =N

Δφ

XNR

GXN

XNR XN

Δϕ12I

Δϕ12I

Δϕ12IR α 2

1 e π– α⁄+-----------------------⎝ ⎠

⎛ ⎞ln–

XNR

α π–2

e2 1 X

NR⁄⎝ ⎠⎛ ⎞atanh

------------------------------------- 1–

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

ln

⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞ 1–

⋅=

5.2 All-Optical Space Switches with Gain and Principally Ideal Extinction-Ratios

217

With an extinction ratio of =20.5 dB we obtain an alpha factor of . The experiments were performed statically. For very short control-

pulses, where intraband relaxation times may become important, the alpha factor must be adapted.The result of a switching experiment with short optical control-pulses is shown in Fig. 5.9. The unequally biased all-optical switch was operated with two optical con-trol-pulses consecutively coupled into the two SOAs of the MZI-SOA switch. The first control signal was inserted into SOA1, to switch the cw-data signal from the cross into the bar output. The second control-pulse was inserted with a delay of ~25 ps into SOA2 to reset the switch. Balanced opening windows for the bar ( ) and the cross ( ) state with a duration of 25ps (FWHM) were obtained. The extinction ratio of the opening window is about 16 dB for both states ( and ). The static value of 20 dB is difficult to attain, due to high noise. Part of the noise is due to me-chanical instabilities of the measurement bench, another part rises from the amplifi-ers that were used to amplify the output signals in order to provide enough power for the photodiode. For these dynamic experiments we used control-pulses with 3-4 ps width (FWHM), 114 MHz repetition rate and a peak power of +22dBm, measured in the fibre. The data-signal power was increased to -2 dBm to compensate the losses in the wavelength-filters and isolators that were needed for the dynamic ex-periment at the outputs. Clearly visible is the reduction of the overall gain of the -signal right after the switching window due to saturation in the SOAs. The saturation is 3.5 dB, which is in good agreement with the value predicted in Fig. 5.4 for an alpha

XNR

α 7,4 0,5±=

0 100 200 400

1.0

0.0

0.5

Out

put-P

ower

[mW

]

Time [ps]

PX

P=0 100 200P C

-Inte

nsity

Time [ps]

300

0.024

0.87

XN=1

6dB XC

=16d

B

Fig. 5.9 Balanced 25 ps (FWHM) opening and closing windows of the unequally biased MZI-SOA all-optical switch. Extinction ratios exceeding 16 dB at both the bar and cross output are obtained by unequally biasing.P= PX

P=PX

XN XC

Pin

PX

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

218

factor of 7.4. The inset of Fig. 5.9 shows the control-pulse. The oscillations behind the pulse are the photodiode’s ringing. All signals shown in Fig. 5.9 were taken with a fast photodiode (10 ps rise time) and a 50 GHz sampling oscilloscope.

6. ConclusionsAll-optical space switches in Mach-Zehnder Interferometer configurations with semiconductor optical amplifiers and principally ideal extinction ratios are pro-posed. It is shown that asymmetries or high alpha factor media are necessary for at-taining high extinction ratios. New 1x2 and 2x2-switch configurations are presented. The theoretical predictions are verified and found to be in excellent agreement with experiments for a switch with symmetric MZI-splitters in a monolithically integrat-ed InP waveguide version, that allows operation with symmetrically or unequally bi-ased SOAs. While symmetrically biased operation of a symmetric all-optical switch achieves unbalanced extinction ratios of 13 dB and 29 dB, asymmetric operation al-lows balanced extinction ratios of 20 dB. Extrapolating from experimentally verified theory shows that, today’s extinction ratio limitations can be extended below 29 dB for both states, when at the same time disturbing noise sources are suppressed.

AcknowledgementsE. Gini is acknowledged for growth of the wafer and Ch. Holtmann for help in the characterisation. Part of the work was carried out within the ACTS-HIGHWAY project of the European Union and the Swiss optics project. Funding was from the Swiss Confederation.

5.2 All-Optical Space Switches with Gain and Principally Ideal Extinction-Ratios

219

AppendixApp.1 Formulation in Terms of On-Off RatiosThe on-off ratios are defined as the power ratios between the switched-on ( ) and switched-off ( ) signals in the same output guide. In the case of a 2x2-switch we distinguish between the bar and the cross on-off ratios. For a signal intro-duced at input guide j, with we define

. (5.27)

Using the definition of the on-off ratios and the restrictions imposed by Eq. (5.11)-(5.12) we find for a data signal at input 1 (upper sign) and a signal at input 2 (lower sign)

(5.28)

. (5.29)

There is no difference in the choice of operation parameters, independent of whether we optimize for on-off or extinction ratios as long as we are looking for ideally high states. For example the condition for an ideally high bar on-off state and the condition for an ideally high extinction ratio require both that the bar output sig-nal vanishes when no control signal is applied: . Analogously the condition of an ideally high cross on-off state and the condition of an ideally high extinction ratio require both that: .However in the case of an all-optical switch with symmetric splitters it makes a dif-ference whether we require that the two extinction ratios of the two states are iden-tical, that means or if we require that . The former condition leads us to set the operation parameters according to Eq. (5.15) but the latter condi-tion requires

and . (5.30)

POnPOff

O= OXj 1 2,=

O j,=POnPOff----------

P=

P= j, PC on

P= j, PC off

------------------------= = OX j,POnPOff----------

PX

PX j, PC off

PX j, PC on

-------------------------= =

O= 1/ 2, e

Δϕ12C

α---------------– 2Δϕ12

I α⁄ 2Δϕ12C α⁄ lnrA+− lnrB+−+( ) 4⁄[ ]

2cosh

2Δϕ12I α⁄ lnrA+− lnrB+−( ) 4⁄[ ]

2sinh

----------------------------------------------------------------------------------------------------------------------⋅=

OX 1/ 2, e

Δϕ12C

α--------------- 2Δϕ12

I α⁄ lnrA+− lnrB±( ) 4⁄[ ]2

cosh

2Δϕ12I α⁄ 2Δϕ12

C α⁄ lnrA+− lnrB±+( ) 4⁄[ ]2

sinh----------------------------------------------------------------------------------------------------------------⋅=

O=,jXj

N

P= j, PC off 0=OX,j

XjC PX j, PC on

0=

XN XC= O= OX=

Δϕ12I α

2--- 1

2--- 1

2---e

2π–α

---------+

⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

ln= Δφ Δϕ12I–=

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

220

In the experiment from Fig. 5.8 the operation point of identical on-offs can be found for a phase-shift offset , which is in agreement with Eq. (5.30). This has to be compared with the operation point of identical extinction ratios which is found at a phase-shift offset of .Alpha factors can be extracted from (5.29) once is measured. For example, at the symmetrically biased operation point of the symmetric MZI-SOA switch (see Fig. 5.8), (5.29) simplifies to

, (5.31)

which delivers for an on-off ratio of 15.0 dB (found in the experiment of Fig. 5.8) an α-factor of 7.2. This is in good agreement with the value of 7.4 found above.

App.2 Design Tolerances of the Asymmetric ComponentsFor finding the ideal operation points of the unequally biased MZI-SOA all-optical switch (Type A), it may be useful to know the relation between unequally biasing the SOAs and the extinction ratio. This dependency can easily be calculated with Eq. (5.13a)-(5.13b). All-optical switches with one or two asymmetric splitters (Type B1 and B2) have re-stricted design tolerances. Depending on the value of the alpha factor of the SOAs, the asymmetries in the couplers have to be chosen. So, as the design accuracy of the asymmetry influences the device performance for the obtainable extinction ratios, the design tolerances are of interest.Fig. 5.10 gives the design tolerances for the asymmetric coupler in the all-optical switch (Type B1) with one asymmetric splitter (see Fig. 5.6). We have plotted the attainable extinction ratios as a function of the bar intensity transmission of the asymmetric splitter with Eq. (5.13a)-(5.13b). The alpha factor was set to 7.4. The corresponding appropriate bar transmission intensity with ideal extinctions lies at 0.40. Deviations from these values lead to a degradation of the extinction ratio per-formance.Considering the version with two asymmetric splitters, we expect at first glance more restricted design tolerances for the choice of the splitting ratios, since a wrong choice in the asymmetries gives twice rise to losses. However this point of view is wrong. When the correct splitting ratios are not hit, it is still possible to unequally bias the SOA and balance out the asymmetries in the extinction ratios of the two states. Under consideration of this additional freedom, we find, that the 1x2-all-op-tical switch with one or two asymmetric splitters are absolutely identically critic to deviations from the correct splitting ratio. For this reason Fig. 5.10 also shows the attainable extinction ratios for the all-optical switch with two asymmetric splitters as

Δφ 71,4o=

Δφ 90,0o=OX

α π–1 4 OX⁄–( )ln

--------------------------------------=

OX

sB

5.2 All-Optical Space Switches with Gain and Principally Ideal Extinction-Ratios

221

a function of the bar-transmission s. In the case of the all-optical switch with two asymmetric splitters one has to set as is suggested by the condition of Eq. (5.19a)-(5.19b), which requires reciprocity of the splitting ratios. This result can be understood with the analytical expressions that were derived. To get a 1x2 all-optical switch with two good extinction ratios, we need to integrate two of the three available asymmetries ( ). Therefore the choice of at least one asymmet-ric splitter is mandatory. For the next asymmetry we are free to choose any combi-nation between asymmetric splitting ratio and unequally biased SOAs. For the extinction ratios it makes no difference whether one manipulates the asymmetries in the splitter or unequally biases the SOA currents, since mathematically they are treated alike in the expression of the extinction ratio Eq. (5.13a)-(5.13b). The wrong choice of the asymmetry in one of both splitters can always be compensated by the freedom to unequally bias the SOA currents.The design tolerances for the all-optical switch with two pairs of SOAs with different alpha factors (Type C) are relaxed. From Eq. (5.24) one can derive, that two input guides with ideal extinction ratios are feasible, as soon as one has two amplifies with a different alpha factor. It is not necessary to hit two determined materials with pre-cise values of alpha factors. Nevertheless it is useful to have a large contrast in the alpha factors, because this reduces the necessary phase shift of the control signal for a complete switching.

Δφ = π/2α=7.4

Bar-Transmission s in Splitter0.0 0.5 1.0

Extin

ctio

n-R

atio

[dB

]

30

20

10

0

ΔϕI12=-π/2

Fig. 5.10 Extinction ratios for different asymmetric splitting ratios of the unequally biased all-optical switch with one asymmetric splitter (Fig. 5.6). The x-axis gives the bar-transmission of the asymmetric splitter. When the alpha fac-tor is set to 7.4, optimal extinctions are obtained for a 40:60 splitter. Devi-ations from the 40%, or s=0.4 bar power transmission value leads to performance losses as shown by the figure.

s 1 sA– sB= =

rA rB Δϕ12I, ,

5 All-Optical MZI-Based Devices with Enhanced Extinction Ratios

222

5.3 References

[5.1] T. Saitoh, T. Mukai; “1.5 mm GaInAsP Travelling-Wave Semiconductor La-ser Amplifier”, IEEE J. Quantum Electron., vol. 23, no. 6, pp. 1010-1020, June 1987

[5.2] Ch. Holtmann, P.A. Besse, H. Melchior; “Polarization resolved, complete noise characterization of bulk ridge-waveguide semiconductor optical am-plifiers”; Proc. Conf. on Optical Amplifiers and Their Applications OAA’95, OSA Techn. Digest Series, vol. 18, pp. 115-118

[5.3] G. Raybon, P.B. Hansen, U. Koren, B.I. Miller, M.G. Young, M. Newkirk, P.P. Iannouen, C.A. Buurus, J.C. Centanni, M. Zirngibl; “2 Contact, 1 cm long, monolithic extended cavity laser actively mode-locked at 4.4 GHz”;, Electron. Lett, vol. 28, no. 24, pp. 2220, Nov. 1992

[5.4] M. Suzuki, H.Tanaka, N. Edagawa, K. Utaka, Y. Matsushima; “Transform-limited optical pulse generation up to 20-GHz repetition rate by a sinusoidal-ly driven InGaAsP electroabsorption modulator”; J. Lightwave-Technology. vol. 11, no. 3, pp. 468-473, March 1993

[5.5] A. Takada, H. Miyazawa; “30 GHz picosecond pulse generation from active-ly mode-locked erbium-doped fiber laser”; Electron. Lett, vol. 26, no.3, pp. 216-217, Feb. 1990

[5.6] A.E. Willner, W. Shieh; “Optimal spectral power parameters for all-optical wavelength shifting: single sate, fanout, and cascadability”; J. of Lightwave Tenchnol., vol. 13, no. 5, pp. 771-781, May 1995

[5.7] M. Eiselt; “Optical loop mirror with semiconductor laser amplifier”, El. Lett., vol. 28, no. 16, pp 1505-1507, July 1992

[5.8] S. Nakamura, K. Tajima, Y. Sugimoto; “Experimental investigation on high-speed switching characteristics of a novel symmetric Mach-Zehnder all-op-tical switch”, Appl. Phys. Lett., vol. 65, pp. 283-285, July 1994

[5.9] K. I. Kang, T.G. Chang, I. Glesk, P.R. Prucnal, R.K. Boncek; “Demonstration of ultrafast, all-optical, low control energy, single wavelength, polarization independent, cascadable, and integrateable switch”, Appl. Phys. Lett., vol. 67, pp. 605-607, July 1995

[5.10] R. Hess, J. Leuthold, J. Eckner, Ch. Holtmann, H. Melchior; “All-optical space switch featuring monolithic InP-waveguide semiconductor optical amplifier interferometer”, Proc. Optical Amplifiers and their Applications, June 1995, vol. 18, OSA Tech. Digest Series, PD2

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[5.11] E. Jahn, N. Agrawal, M. Arbert, H.-J. Ehrke, D. Franke, R. Ludwig, W. Pieper, H.G. Weber, C.M. Weinert; “40 Gbit/s all-optical demultiplexing us-ing a monolithically integrated Mach-Zehnder interferometer with semicon-ductor laser amplifiers”, Electron. Lett., vol. 31, pp. 1857-1858, Oct. 1995

[5.12] M. Vaa, B. Mikkelsen, K.S. Jepsen, K.E. Stubkjaer, M. Schilling, K. Daub, E. Lach, G. Laube, W. Idler, K. Wünstel, S. Bouchoule, C. Kazmierksi, D. Mathoorasing; “A bit-rate flexible and power efficient all-optical demulti-plexer realised by monolithically integrated Michelson Interferometer”, Proc. European Conference on Optical Communication - ECOC’96, Oslo, Sept. 96, ThB.3.3

[5.13] J. Leuthold, J. Eckner, Ch. Holtmann, R. Hess, H. Melchior, “All-optical 2x2 switches with 20 dB extinction ratios”, Electronics Letters, Vol. 32, No. 24, pp.2235-2236, Nov. 1996

[5.14] P.A. Besse, H. Melchior; “Improved extinction ratios in all-optical switches based on Mach-Zehnder configuration”, IEEE Photon. Technol. Lett., vol. 9, no. 1, pp. 55-57, Jan. 1997

[5.15] J. Leuthold, E. Gamper, M. Dülk, P.A. Besse, J. Eckner, R. Hess, H. Mel-chior; “Cascadable all-optical space switch with high and balanced extinc-tion ratios”; 2nd Optoelectronic and Communications Conference (OECC'97), July 97, Seoul, Korea, paper 9C3-4

[5.16] J. Leuthold, J. Eckner, P.A. Besse, G. Guekos, H. Melchior; “Dual Order Mode (DOMO) All Optical space switch for bidirectional operation”, Proc. Optical Fibre Communication Conference (OFC) 96, San Jose, Feb. 1996, ThQ1

[5.17] S. Wang; “Semiconductors and Semimetals“; Vol. 22, Lightwave Communi-cations Technology Part E, Vol. Ed. W. T. Tsang, Academic Press, 1985, pp. 80 ff.

[5.18] R.M. Jenkins, R.W.J. Devereux, J.M. Heaton; “Waveguide beam splitters and recombiners based on multimode propagation phenomena”, Opt. Lett., vol. 17, pp. 991-993, July 1992

[5.19] L.B. Soldano, F.B. Veerman, M.K. Smit, B.H. Verbeek, A.H. Dubost, E.C.M. Pennings; “Planar monomode optical couplers based on Multimode Interfer-ence effects”, J. Lightw. Technol., vol. 10, p.1843-1849, Dec. 1992

[5.20] P.A. Besse, E. Gini, M. Bachmann, H. Melchior; “New 1x2 Multi-Mode In-terference couplers with free selection of power splitting ratios”, Proc. ECOC’94, 669-672, Sept. 1994

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[5.21] K.E. Stubkjaer, T. Durhuus, B. Mikkelsen, C. Joergensen, R.J. Pedersen, C. Braagaard, M. Vaa, S.L. Danielsen, P. Doussiere, G. Garabedian, C. Braver, A. Jourdan, J. Jacquet, D. Leclerc, M. Erman, M. Klenk; “Optical Wave-length Converters”, Proc. ECOC’94, pp. 635-642, Sept. 1994

[5.22] S. Nakamura, K. Tajima; “Analysis of subpicosecond full-switching with a symmetric Mach-Zehnder all-optical switch”, Jpn. J. Appl. Phys., vol. 35, no. 11A, pp. L1426-1429, Nov. 1996

[5.23] M. Osinsik, J. Buus; “Linewidth broadening factor in semiconductor lasers - an overview”, J. of Quantum Electron., vol. 23, pp. 9-29, Jan. 1987

[5.24] P.A. Besse, M. Bachmann, H. Melchior, L.B. Soldano, M.K. Smit; “Optical bandwidth and fabrication tolerances of multimode interference couplers”, J. of Lightwave Technology, vol. 12, no. 6, pp. 1004-1009, June 1994

[5.25] D.J. Bossert, D. Gallant; “Gain, refractive-index, and α-Parameter in In-GaAs-GaAs SQW broad area lasers”, IEEE Photon. Technol. Lett., vol. 8, pp. 322-324, March 1996

[5.26] T. Brenner, E. Gini, H. Melchior; “Low coupling losses between InP/In-GaAsP Optical Amplifiers and monolithically integrated waveguides”, Photon. Technol. Lett., vol. 5, pp. 212-214, Feb. 1993

[5.27] J. Faist, F.K. Reinhart; “Phase modulation in GaAs/AlGaAs double heter-ostructure. II. Experiment”, J. Appl. Phys., vol. 67, pp. 7006-7012, June 1990

225

6 All-Optical Devices with Integrated Data- and Control-Signal Separation Schemes

This chapter is devoted to new devices that allow all-optical operation with simulta-neous separation of the control and data signals at the output after signal processing.All-optical operation usually requires two signals, i.e. a strong control or a pump sig-nal to modulate a data signal. Usually high-performance external wavelength filters are used for this purpose. Such additional components are expensive, lead to addi-tional losses, may degrade the pulse shape of ultrashort pulses and inhibit monolithic integration of several components. Therefore, monolithically integrated data- and control-signal separation schemes are highly desirable.In a first section we present two control- and data-signal separation schemes, that al-low monolithic integration and which can be used in cross-phase modulated (XPM) all-optical devices as well as in cross-gain modulated (XGM) devices. The second section gives information how data and control signals can be separated within a po-larisation-insensitive fourwave-mixing all-optical device. The presented mode sep-aration schemes are based on interferometric principles and therefore do not degrade the pulse shape of short pulses.

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226

6.1 All-Optical Mach-Zehnder Interferometer Wavelength Converters and Switches with Integrated Data- and Control-Signal Separation Scheme

J. Leuthold, P.A. Besse, E. Gamper, M. Dülk, St. Fischer, G. Guekos, H. Melchior

IEEE J. of Lightwave Technologyaccepted for publication in June 1999 issue

AbstractAll-optical Mach-Zehnder Interferometer (MZI) wavelength converters and switch-es with monolithically integrated data- and control-signal separation schemes are re-ported. Two schemes to separate the data from the strong control signals are discussed. A first dual-order mode configuration uses modes of different symmetry for the data and control signals. A second configuration uses additional MZIs to sep-arate the two signals. The control-signal separation permits to operate the devices with co-propagating data and control signal without distortion of the control signal in the data-signal output. Co-propagative operation allows for shorter switching win-dows compared to the counter-propagative operation. Further, this concept enables cascading of several devices since the control signal is filtered out and does not dis-turb the signal processing in a next cascade of devices. The all-optical switches are characterized under static and dynamic 10 GHz conditions.

1. IntroductionFuture optical high-capacity transmission systems are expected to rely on wave-length division multiplexing (WDM) and time division multiplexing (OTDM) [6.1]. In WDM and OTDM systems wavelength converters, demultiplexers and cascada-ble fast switches with gain and high extinction ratios are needed.All-optical devices [6.2-6.5] have the potential to meet the needs of ultrafast WDM [6.4] and OTDM [6.6] systems. They have already demonstrated impressive switch-ing speeds down into the sub-picosecond range [6.7, 6.8], exceeding by far the op-erational range of electronically controlled devices. A most promising all-optical operation scheme is based on a Mach-Zehnder Interferometer (MZI) configuration with semiconductor optical amplifiers (SOAs) on the MZI arms [6.9]. For switching a strong optical control signal is directed into one of the two SOAs positioned in the

6.1 Control-Signal Separation in All-Optical Wavelength Converters and Switches

227

arms of the MZI. Thereby the control signal depletes the carriers in the SOA, which through the carrier-related refractive-index change leads to a phase shift. If the phase shift amounts to π, a signal of another wavelength introduced into the MZI is switched from one output port to the other. An advantage of this MZI-SOA config-uration is that it allows for monolithic integration, making the device stable and com-pact [6.10, 6.11]. Another advantage of the configuration is its versatility as switch [6.11], multiplexer [6.12] and wavelength converter [6.5]. Recently 80:10 GBit/s de-multiplexing has been reported with such MZI-SOA configurations [6.13] and im-provements of on-off ratios were demonstrated with asymmetric MZI configurations [6.14, 6.15].At the output of these MZIs, external wavelength filters are needed to separate the control and data signals. However, these filters introduce losses, modify the pulse form of short signals, lack of integration and are expensive. To circumvent external wavelength filters, counter-propagating control and data signals can be used. But this prevents bidirectional operation, whereby reducing the functionality of the network. In addition, we will show that co-propagating control and data signals allow for shorter switching windows. These disadvantages can be overcome with all-optical devices that have a monolithically integrated control- and data-signal separation scheme. Furthermore, an integrated mode-separation scheme makes the devices compact and allows for cascadation of several all-optical devices on a chip without distortion of the control signal in the subsequent switches.In this paper we discuss and compare all-optical device concepts with a monolithi-cally integrated mode-separation mechanism for the control and data signals [6.16]-[6.20]. These devices feature:

• Bidirectional operation without external wavelength filters for separation of thecontrol signal at the output.

• Switching and wavelength conversion in co-propagative direction at any wave-lengths within the gain spectrum (even when control and data signal have the same wavelength)

• Cascadability of several devices on a single chip• Control-signal extraction for reuse

Section II shows that efficient control-signal separation schemes are necessary, es-pecially since co-propagating control and data signals provide shorter switching win-dows in comparison with counter-propagating control and data signals. In Section III we present two different concepts for control- and data-signal separation. The first is based on a dual-order mode configuration, that is operated with control- and data-signal modes of different symmetry. It exploits the symmetry-related characteristics for control-signal separation. A second concept uses interleaved MZIs to split off the control signal from the data signal. Static and dynamic experiments at 10 GHz were performed to characterize and compare the devices. In addition, we have experimen-

6 All-Optical Devices with Integrated Data- and Control-Signal Separation Schemes

228

tally determined the relations between the data- and control-signal wavelengths and powers. All these experimental results are given in section IV.

2. Co- and Counter-Propagating Operation with Versatile All-Optical Devices

In this section we will briefly discuss the use of counter-propagating control and data signals to circumvent control-signal filtering. We will show how this filtering scheme lacks switching speed.We consider all-optical devices as sketched in Fig. 6.1. This waveguide MZI-SOA all-optical devices comprise two SOAs, two phase-shifter sections and four 50:50 multimode interference (MMI) splitters [6.14]. The first two 50:50 MMI splitters on the left and on the right side form the MZI and are used to split up and combine the input signal Pin. The other two 50:50 MMI splitters are used to introduce the control signals PC1 and PC2 into the SOAs. The phase shifters are adjusted such that the cw input signal Pin is mapped into the cross output PX when none of the control signals PC1 and PC2 is applied. For switching we launch two consecutive optical control-pulses PC1 and PC2 with a time delay of ~25 ps into the respective SOAs. The first control signal induces a carrier-related refractive index change such that the cw input signal switches from the cross into the bar state. The consecutive second control sig-

PC2

Pin P=

PX

PC1

50:50

Co-Propagating Data and Control Signals Counter-Propagating Data and Control Signals

SOA2 Phase-Shifter 2

SOA1Phase-Shifter 1

Pin

50:50

50:50

50:50

Fig. 6.1 MZI-SOA wavelength converter, multiplexer or switch with phase-shifters and 50:50 MMI splitters. Depending on the phase-relations in the MZI arms, the data signal (wavelength λin) is mapped onto the cross and bar-output guide . The phase-relation within the MZI arms is changed when a control signals and/or (wavelength λC) are in-troduced through the outer 50:50-MMI couplers into SOA1 and/or SOA2, respectively.

Pin PXP=

PC1 PC2

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nal resets the switch [6.21].The device of Fig. 6.1 is versatile in the sense that it can be used for switching [6.11], add-drop multiplexing [6.12] and wavelength conversion [6.5]. For switching or multiplexing an input signal Pin , control signals PC1,2 are applied to direct Pin into the PX or the P= channels. The remaining input channel can be used to add informa-tion if the device is operated as add-drop multiplexer. For wavelength conversion, the signal to be converted is introduced as a control signal PC1,2. It thereby modu-lates a cw signal, introduced through the Pin input guide, which carries the new wavelength. The converted signal is directed into the P= output port. The experiment with co- and counter-propagating input and control signals is depict-ed in Fig. 6.2. The transients of the co-propagating switching windows (dashed lines) are clearly steeper. In the counter-propagating experiment (dotted lines), the speed is limited by the time a control signal needs to spread from one end to the other in the nonlinear SOA medium. This propagation time amounts to 6 ps for our SOAs with lengths of 0.5 mm. Using co-propagating control and data signals, both signals prop-agate with a similar speed and the transients of the switching window should only be limited by the carrier depletion time and the response time of the photodiode (here: ~10 ps photodiode rise time). For the dynamic experiments we used control pulses with 3-4 ps width (FWHM), 114 MHz repetition rate and a peak power of +22 dBm, measured in the fibre. A strong control signal was needed since the wavelength of the pulsed source was not really at the gain maximum of the amplifier. The data-sig-nal power was -2 dBm. The wavelengths of the control signal and the input sig-

0 50 100 150 200Time [ps]

Out

put-P

ower

[a.u

.]

Counter-Propagating

Bar-Output P=

Co-Propagating

Fig. 6.2 Comparison of the switching transients for co- and for counter-propagat-ing data and control signals. The transients are steeper in the co-propa-gating case. Experiments were performed with the switch from Fig. 6.1with identical operation parameters for the two cases.

Pin

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nal were 1.530 μm and 1.554 μm. The amplifier gain maximum lies around 1.560 μm. Further details of the experiment have been given in Ref. [6.14].High performance external wavelength filters can be used to extract the co-propa-gating control signals. But external wavelength filters modify the pulse form of the signal that is transmitted and in addition become partially transparent for ultrashort signals, which need to be filtered out. They become transparent because most filter-ing principles (Fabry-Perot filters, Bragg gratings, arrayed waveguide gratings) re-quire superposition of reflected parts of the wave with itself. However it takes some time to build up the superpositions. To circumvent such signal-rejection problems in the co-propagating operation mode, new configurations exploiting other filtering principles will be needed. Subsequent-ly we discuss versatile all-optical devices with monolithically integratable mode-separation schemes.

3. All-Optical Control- And Data-Signal Separation Schemes3.1 The Dual-Order Mode (DOMO) ConfigurationA first all-optical concept to separate and reuse control and data signals is based on a dual-order mode (DOMO) configuration, in which the control signal propagates as a first-order mode and the data signal propagates as a zero-order mode (Fig. 6.3). Due to the different symmetries of control and data signal, the first-order mode con-trol signal can be easily separated after signal processing and reused. Control-signal separation is even guaranteed for ultrashort pulses in co-propagative direction. An-other advantage of this concept is the absence of any fourwave mixing of the first with the zero-order modes. Since the two signals are distinguished rather by mode symmetry instead of the wavelengths and since there is no fourwave mixing, signal processing is possible for any incoming data signal - even at the wavelength of the control signal. This might be of advantage for the construction of wavelength flexi-

Fig. 6.3 MZI dual-order-mode all-optical configuration, where data (Pin) and control signal (PC1 and PC2) propagate as zero- and first-order modes, respectively. (a) The first configuration is for reuse of the control signal after signal processing. So-called 100%-MMI-converter-combiner (100%-cc) completely introduce and convert the zero-order-mode control signal as a first-order mode into the SOAs and reconvert and extract it behind the SOAs (inset). (b) The second type allows a strong control-sig-nal filtering. The control signals are converted with a 66% conversion ef-ficiency (in 66%-cc) into first-order-mode signals (inset) and they are filtered out in MMI-filters (inset). Both configurations use asymmetric MMI-splitters with a 60:40 and a 40:60 splitting ratio for attaining high on-off ratios.

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SOA2

PinPC

PC Pin

Output

2x2-MMI

390μm780 μm

13 μm

MMI

Phase-Shift -π/2

Input

PC2

Pin

PC1

SOA1 P=

PX

100%-cc 100%-cc

100%-cc100%-cc

PC1

PC2

40:6060:40

(a) Dual-Order-Mode Configuration with Control-Signal Extraction: Type A

1st

320 μm

10 μm390 μm

(b) Dual-Order-Mode Configuration with Control-Signal Filtering: Type B

13 μm

66%

33%

100%

PC2

Pin

PC1

P=66%-cc

40:60

SOA1MMI-Filter

Absorber

66%-cc SOA2FilterMMI-

PX60:40

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ble networks. Another aspect is worth being mentioned. When introducing the con-trol signal through 50:50 splitters onto the MZI arms as shown in Fig. 6.1, 50% of the input as well as 50% of the control signal is lost. However, with a special coupler concept for zero and first-order modes it is possible to introduce 100% of the control signal onto the MZI arms without losses for the data signal.Subsequently we discuss two different DOMO configurations. The first configura-tion features complete control-signal recycling and -10 dB crosstalk of the control signal into the data-signal channels. The second configuration attains -23 dB cross-talk but it radiates out the control signal (Table 1).

DOMO Configuration with Control-Signal Extraction: Type AAn all-optical device with separate in- and outputs for the data and control signals is shown in Fig. 6.3(a) [6.17-6.19]. While introducing the control signals into the de-vice, they are completely converted into first-order modes. After signal processing they are converted back into fundamental zero-order modes and extracted through separate output ports.The configuration comprises two asymmetric multimode interferometer (MMI) cou-plers with a 60:40 and a 40:60 splitting ratio for data-signal splitting. The asymmetry in the splitters guarantees for high extinction ratios [6.14]. Two two-section SOAs on the MZI arms provide the necessary nonlinearities for switching. The SOAs are divided into two sections. This enhances the switching speed [6.22] and allows for gain and small offset-phase adaptations. The total length of the SOAs is 1 mm. The two 100%-MMI-converter-combiners (100%-cc) on the left are used to introduce the control signal (PC1,2) as a first-order mode into the SOAs and the two 100%-cc’s on the right are used to extract the first-order mode control signal from the signal path of the data signal. These 100%-cc’s completely convert the zero-order-mode control signals (PC1,2) into first-order modes while the input signal Pin passes the coupler unaltered and without loss. The inset of Fig. 6.3(a) shows the conversion principle. Details of the 100%-MMI-converter-combiners are outlined in Ref. [6.23]and their optical bandwidths are discussed in Ref. [6.24].

DOMO Configuration with Strong Control-Signal Filtering: Type BA DOMO all-optical MZI-SOA device with strong control-signal filtering at the data-signal outputs is shown in Fig. 6.3(b). The high control-signal filtering becomes possible with smaller and simpler converter MMIs (66%-cc), which convert the con-trol signal with less efficiency but higher quality into a first-order mode. Since we do not reuse the control signal we can filter it out after signal processing in a simple manner and with high efficiency. The configuration is shown in Fig. 6.3(b). It is built analogously to the preceding configuration of Fig. 6.3(a). Instead of the 100%-MMI-converter-combiners we have integrated the simple 66%-MMI-converter-combiners (66%-cc’s). These 66%-

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MMI-converter-combiners convert the zero order control signal with a 66% conver-sion efficiency into a first-order mode and guide the remaining 33% of the control signal into an absorbing layer, whereas the input signal passes the MMI unchanged [6.23], first inset of Fig. 6.3(b). The main advantage of these MMI-converter-com-biners is the large optical bandwidth. The 1 dB optical bandwidth exceeds 160 nm [6.24]. A strong suppression of the control signal can be attained by radiating the first-order mode out of the waveguide in the S-bends behind the SOAs. First-order mode rejections up to -38 dB were found with such all-optical switches [6.16]. Al-ternatively MMI-filters can be used [6.25]. These MMI-filters split off antisymmet-ric from symmetric modes, second inset of Fig. 6.3(b). They map the symmetric zero-order mode from the central input guide onto the central output guide and the antisymmetric first-order mode onto the two outer output guides, where the signal is either absorbed in an active layer or guided out of the device. The 1 dB optical band-width of these MMI-filters exceeds 140 nm [6.24].

Discussion of the Switching Efficiency in a DOMO DeviceAn efficient carrier depletion in the SOAs is a requirement for fast and efficient switching. In order to obtain a large carrier related refractive index change, the over-lap between the control and the data signal should be large.To find the optimum overlap between zero-order mode input signals and first-order mode control signals, we have calculated the zero- and first-order modes with a finite element solver as a function of the SOA rib widths. Fig. 6.4 shows a large overlap for zero and first order TM modes, displaying the strong guiding of the first order

SOA-Rib-Width wSOA [μm]

TM

TE

Ove

rlap

0.3

0.4

0.5

0.6

0.7

2.0 3.0 4.0 5.0 6.0

p-InPQ-1.28Q-1.55n-InP

160 nm230 nm

wSOA

Fig. 6.4 Mode overlap of the zero-order input signal with the first-order mode con-trol-signal in the ridge-waveguide SOA (inset). A large overlap is required to exploit the SOAs nonlinearity for switching. The strong guiding of the TM mode results in a large overlap for the TM mode.

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TM mode in our rib-waveguide SOA structure (inset Fig. 6.4). The simulation led us to choose SOA rib widths of 3.5 μm, since for these waveguide widths we reach overlaps of 70%. Polarisation insensitive operation for the control signal requires other SOA waveguide geometries.To further characterize the SOAs we compared the gains of the zero and the first-order modes in the amplifiers. In order to do that, we have fabricated a 66%-MMI-converter-combiner together with an amplifier, Fig. 6.5(a). This test device converts a zero-order mode B in the MMI-converter-combiner with a 66% conversion effi-ciency into a first-order mode and then guides it into the amplifier. A zero-order mode A passes the MMI-converter-combiner lossless and is then guided into the am-

0 50 100 150-40

-30

-20

-10

0

10

Current [mA]

Gai

n [d

B]

1st Order Mode

0th Order Mode

SOA

(100) InP

66%

33%

100%SMF

SMF

B

A

Lens

66%-cc

(a)

(b)

Fig. 6.5 (a) Monolithically integrated 66%-MMI converter combiner (66%-cc) with an SOA to test the amplification of the zero and the first-order-mode signal in the SOA. (b) Zero- and first-order-mode gains measured with the above test device. Under consideration of the 2 dB conversion loss for the first order mode, we find for the first-order mode a 3 dB weaker amplifica-tion. The zero-order-mode gain was once measured with the more stable fi-bre-to-fibre setup (dashed line) and once with a fibre-lens setup (dotted line). The first-order mode was measured with the fibre-lens setup.

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plifier. Fibre-to-lens gains were measured for weak input signals A and B. Addition-ally fibre-to-fibre gains were measured for the zero-order mode A. The fibre-to-fibre measurement of mode A was then used to get a reference gain for the two fibre-lens measurements. From Fig. 6.5(b) we learn that the first-order mode gain is reduced by 5 dB in comparison to a zero-order mode. Two of these five dBs are due to the 33% of the control signal which are lost in the 66%-MMI-converter-combiner so that there remains for the first order mode a 3 dB weaker amplification.

In total, we expect similar or only slightly higher switching powers for the DOMO configuration in comparison with the zero-order mode all-optical devices of the Fig. 6.1 type. This is due to the fact, that in the DOMO configuration the coupling effi-ciencies of the control signal into the SOA reaches 66% or even 100% and the over-lap between the data and control signal is ~70%. Whereas in the Fig. 6.1 type configuration the coupling efficiency reaches at most 50% due to the 3 dB loss of both, the data and control signal in the 50:50 splitters, but the overlap of the data and control signal attains 100% and the signal amplification in the SOA is higher.Another issue to be addressed is the group delay difference between the data and control signal. The group velocities for the two signals are slightly dissimilar since they are propagating in different modes (zero- and first-order modes). If the group delay difference were too large, it would affect the timing relationship of data- and control-signal pulses. Fortunately the differences are negligible for structures as giv-en in Fig. 6.4. Such structures deliver a group delay difference around 60 fs between the zero- and the first-order mode for a propagation length of 1 mm and effective group refractive indices of 3.306 for the zero and 3.288 for the first-order TE modes. The group delay difference for the TM mode is alike. Such small delays are much shorter than our typical signal FWHM in the ps range.

3.2 The Interleaved MZI-ConfigurationA second concept that allows to separate and to reuse the control signal after signal processing makes use of additional MZIs which are placed on the MZI arms of an exterior MZI (Fig. 6.6) [6.20]. These additional interferometers are used to direct the control signals by interference onto separate output ports. Since the control and the data signal propagate as fundamental modes within the SOA, a perfect overlapping between both signals is assured. In contrast to the DOMO configuration, efficient fourwave mixing takes place and can be exploited for polarisation-independent op-tical-phase conjugation with an integrated filtering mechanism of the pump and sig-nal waves [6.26].The configuration allowing control- and data-signal separation with purely funda-mental modes (Fig. 6.6) uses 4 SOAs (SOA1 to 4). These form three MZIs: an exte-rior-MZI (1:2 - 3:4) with two additional MZIs (1 - 2) and (3 - 4) on each arm of the exterior MZI. The additional MZIs comprise two 50:50 MMIs and a two-section

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SOA on each MZI arm. The SOAs are divided into two sections of 500 and 250 μm length. A π phase shift, inserted either in the upper or the lower branch of the addi-tional MZIs on the arm (1 - 2) and (3 - 4), respectively, guarantees that both the con-trol (PC1 and PC2) and the input signal (Pin) are mapped into their respective bar output ports. This provides a separation of the control and the data signal since the control signals are mapped from the outer input guides to their respective outer out-put guides and the data signals are mapped from their inner input guides to their re-spective inner output guides. The exterior MZI is formed by the 50:50 MMI splitter, a π/2-phase shifter, the two additional MZIs and an asymmetric 40:60 MMI splitter. The asymmetry in the splitting ratio of the MMI, the phase shifter and an asymmetry in the current bias is needed to provide high extinction ratios both in the state without and with applied control signals [6.14]. The phases of the whole configuration are adopted in such a way that a signal Pin is mapped into the cross output PX when none of the control signals PC1 and PC2 are applied. When control signals are applied, the input-signal Pin is directed from the cross into the bar output P=. The data- and con-trol signals propagate together through the SOAs, but before and behind the MMI couplers they are completely separated.

4. Static and Dynamic CharacterizationsTo compare the mode-separation efficiency for co-propagating data and control sig-nals with the DOMO and interleaved all-optical configuration, we fabricated the MZI-SOA devices and performed static and dynamic experiments.

PC2

Pin P=

PX

PC1

40:60

SOA1

SOA2

Δφ=π

Δφ=π/2 50:50

SOA4

SOA3

Δφ=π

Interleaved MZI Configuration

50:50

50:50

50:50

50:50

Fig. 6.6 Interleaved MZI all-optical device, with a mode-separation scheme be-tween input and control signal based on the integration of additional MZIs. These additional MZIs (formed by SOA 1-2 and SOA 3-4) are placed on the arms of the exterior MZI. The phases are adapted to map the control signal and the data signal to their respective bar outputs.

6.1 Control-Signal Separation in All-Optical Wavelength Converters and Switches

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The all-optical device of Fig. 6.3 and Fig. 6.6 were fabricated as InP/InGaAsP/InP-waveguide ridge structures in a two step LP-MOVPE process on (001) InP. The SOA sections with a 230 nm thick 1.55 μm-InGaAsP active layer was grown first (inset Fig. 6.4). The ridge amplifier structures were defined lithographically and unwanted material was removed, so that in a second step butt coupled waveguide and MMI lay-ers could be grown. These passive waveguides consist of a 570 nm thick guiding 1.28 μm-InGaAsP layer sandwiched between InP cladding layers. The waveguides and the MMIs were formed by etching through the InP cladding layer, 100 nm re-spectively 300 nm into the guiding quaternary layer. The total size of the chip is 1.5 x 9 mm. The output waveguides are tilted and a two-layer anti-reflection coating was applied on the facets in order to minimize reflections.

4.1 Control- and Data-Signal Separation EfficienciesTo make statements on the efficiencies of the mode separation between the control and the data signal at the different outputs we first measured output spectra under static conditions.The output spectra of an experiment with co-propagating control and input signals at the bar P=, the cross PX and the control-signal output PC1 are shown in Fig. 6.7. To each of the devices from Fig. 6.3 (a), (b) and Fig. 6.6 belongs a complete record of spectra, whereas the solid lines show the spectra of the non-switched state when no control signal is applied and the dashed lines show the spectra of the switched state when the control signal PC1 is turned on.High and symmetric on-off ratios between 16 and 20 dB were measured at the bar P= and cross PX outputs independent of the device type. Such on-off ratios are higher than usual [6.16] since asymmetric MZI configurations were used [6.14-6.15].

Extraction of the control signal after signal processing is possible with the DOMO device type A (Fig. 6.3 (a)) and the interleaved MZI-device (Fig. 6.6). Over 90% of the control signal coupled into the PC1 output can be reused. In this first demonstra-tion of the DOMO type A device, 10% of PC1 were still coupled into the P= and PXoutputs. The loss can be reduced by optimization of fabrication technology.

The crosstalk of the control signal into the data-signal channels P= and PX was dB in the DOMO device type A and the crosstalk was reduced to a value below to -27 dB for the two devices of the DOMO type B and the interleaved MZI de-

vice. Obviously, these two configurations are favourably for applications where a strong control-signal separation is required, e.g. if data and control signal may have the same wavelength. For operation of the DOMO device with the control-signal extraction scheme (Type A) an input signal Pin of 0.4 dBm (measured in the fibre before the device) at 1.554 μm is used. Lossless fibre-to-fibre transmission is found. For switching a -

10–22–

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0

-20

-40Optic

al-S

pect

ral D

ensi

ty [d

Bm/A

]

PX-Output

On-Off17 dB

PC1

Pin

Pin

PC1Pin

On-Off17 dB

1.550 1.570

PC1-Output

PC1

Pin

Pin

1.550 1.570

0

-20

-40

1.550 1.570

PC1Pin

On-Off18.5 dBPin

Pin

On-Off18 dB

0

-20

-40

0

-20

-40

Optic

al-S

pect

ral D

ensi

ty [d

Bm/A

]

1.550 1.5601.550 1.560

Pin

Pin

On-Off16 dB

0

-20

-40

0

-20

-40

Optic

al-S

pect

ral D

ensi

ty [d

Bm/A

]

1.54 1.55Wavelength [μm]

PC1

Dual-Order-ModeAOSType A

Dual-Order-ModeAOSType B

AOSwithInter-leavedMZIs

P=-Output

Pin

Pin

Pin

On-Off20 dB

PC1

1.54 1.55Wavelength [μm]

Pin

Pin

PC1

1.54 1.55Wavelength [μm]

Pin

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1 dBm control signal PC1 (at 1.569 μm) is needed. The two segments of SOA1 and SOA2 were operated with 188+194 mA and 192+188 mA, respectively.The DOMO device with the strong control-signal filtering (Type B) was operated with input signals Pin at a wavelength of 1.554 μm and signal powers of 1.1 dBm. The power was measured in the fibre before the light entered the chip. For switching a -2 dBm strong control signal PC1 was introduced. Although the conversion effi-ciency of the 66%-MMI-converter combiner is reduced by 33% in comparison with the 100%-MMI-converter-combiner of the above all-optical device, we needed less power for switching, because the control-signal wavelength was chosen at 1.560 μm, where this device has its gain maximum. We were free to choose the wavelength around the gain maximum, since the large optical bandwidths of the 66%-MMI-con-verter combiner allowed us to maintain a small crosstalk of the PC into the P= an PXstate within a range of 40 nm around 1.55 μm. And indeed, the crosstalks of the con-trol signal in the bar P= and cross PX outputs are considerably suppressed in com-parison with the spectra of the type A device. The fraction of the control signal that coupled into the bar P= and cross PX outputs was smaller than -24 and -22 dB, re-spectively. To operate the device, currents of 197+166 mA and 181+186 mA were applied on the two-sections of SOA1 and SOA2, respectively.For operation of the interleaved MZI device a cw input signal Pin at -3.5 dBm and a wavelength at λin=1.541 μm was used. The power once again was measured in the fibre before the device. For switching we used a 2.4 dBm strong control signal PC1at λC=1.554 μm. A -22 dBm and -27 dBm small fraction of the control signal cou-pled into the cross and bar output ports of the switch, whereas +2.1 dBm were recy-cled into the PC1-output. The currents applied on the two segments from SOA1 to SOA4 were 279+28, 253+12, 179+36 and 249+28 mA.

Fig. 6.7 Output spectra at the PC1, the P= and the PX output-guides demonstrating the on-offs of the data-signal channels and the control-and data-separa-tion efficiencies of the different switches from Fig. 6.3(a), (b) and Fig. 6.6. Solid lines represent the nonswitched state and dashed lines the switched state, when the control signal is applied. At the P= and PX outputs on-offs between 16 and 20 dB are attained independent of the device-type. With the dual-order-mode device type A and the interleaved MZI device 90% of the control signal and more is regained at the PC1-output for reuse. Only a weak crosstalk of the control signal PC1 couples into the data-signal channels P= and PX of the dual-order-mode device type B and the inter-leaved MZI. The crosstalk lies between -22 and -27 dB.

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4.2 Dynamic PerformanceThe dynamic performances are demonstrated with experiments at 10 GHz.An experiment with the DOMO configuration type A and 25 ps windows on both the bar P= and cross PX outputs as it would be required in a 40:10 GBit/s demultiplexing experiment is depicted in Fig. 6.8. Two consecutive optical control pulses were used to control the switching window length [6.21]. A first control signal PC1 was intro-duced into SOA1 to induce the necessary π-phase shift for switching. It determined the rising and falling edge of the P= and PX opening window respectively. The sec-ond control signal was introduced with a time delay into SOA2. It was used to reset the phase relations within the switch. The control pulses had a FWHM of 8 ps and a wavelength around 1.557 μm (available wavelength of pulsed source). The peak powers of PC1 and PC2 were 14 dBm and 12 dBm respectively. Opening and closure times of the switching windows are limited by the length of the control-pulses and the response time of the photo diode and the oscilloscope. For the dynamic experi-ments we used the same current settings and the same input signal as in the static experiment, except that the input-signal power was raised onto 3 dBm [6.27]. The amplified spontaneous emission noise limited the obtainable extinction ratios. Nev-ertheless we attained 10 dB extinction ratios with external wavelength filters at the outputs.A comparison between the control signal PC1 at the input of the DOMO all-optical device type A and the extracted control signal at the PC1 output is shown in Fig. 6.9. We ascertain that the pulse shape was maintained although spontaneous emission (ASE) has been added and the stronger parts of the signals were amplified less due to gain saturation. The measurements were performed without wavelength filters at the outputs. The pulse heights of the two curves are resized onto the same level to allow the comparison of the pulse shapes. The pulse sequence shows some unexpect-ed resonances and power that goes below zero. These effects are due to the photodi-odes ringing.The DOMO configuration type B and the interleaved MZI all-optical device can be operated without additional external wavelength filters for control-signal filtering at the outputs, since they have included an efficient control- and data-signal separation scheme. A 10 GHz experiment with the interleaved MZI device and co-propagating input and control signals is depicted in Fig. 6.10(a). Two consecutive optical control pulses PC1 and PC2 were launched with a time delay of ~20 ps into the respective input ports. The data signal are directed into a photodiode without any additional wavelength filters between output port and photodiode. The input signal of 1.7 dBm was transmitted lossless. To check the separation efficiency of the control signal, the input signal was switched off. Only slight modulation from the control signals are visible in the ASE background noise (see Fig. 6.10a). The overall amplified sponta-

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0

1.0

0.2

0.4

0.6

0.8

Time [ps]

Out

put-P

ower

[mW

]

0 50 100 150 200

PX-Output

P=-Output25 ps

Fig. 6.8 Ten GHz experiment showing 25 ps windows at the bar (P=) and cross (PX) output of the dual-order-mode type A all-optical device.

Fig. 6.9 Pulse shape of the control signal before signal processing in the all-opti-cal device (solid lines) and extracted pulse shape at output PC1 (doted line). The pulse shape is maintained, except of the spontaneous emission which has been added. The curves are from an experiment with the dual-order-mode type A all-optical device.

0 100 200 300 400Time [ps]

Out

put-P

ower

[a.u

.]

0

1.0

0.2

0.4

0.6

0.8

Extracted PC1 after DevicePC1 in Fibre before Device

6 All-Optical Devices with Integrated Data- and Control-Signal Separation Schemes

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2.0

1.5

1.0

0.5

0

Out

put-P

ower

[mW

]

PX-Output

P=-Output

ASE-Noise

0 200Time [ps]

100

Out

put-P

ower

[a.u

.]

PX

P=

0 50 100 150 200Time [ps]

2.0

1.5

1.0

0.5

0

Out

put-P

ower

[mW

]

PX-Output

P=-Output

0 200Time [ps]

100O

utpu

t-Pow

er[a

.u.]

PX

P=

25 ps

25 ps

Fig. 6.10 Bar and cross output ports opening windows in a 10 GHz experiment with the interleaved MZI all-optical device: (a) when no external wavelength filters are used (b) with wavelength-filters to extract the ASE from the de-vice and from the EDFA’s which were inserted to compensate for filter-losses. For comparison we show in the insets the output signals of a co-propagating experiment with the all-optical device of Fig. 6.1. Top inset: Severely distorted PX and P= output signals when no wavelength filters are used. Bottom inset: The output signals can only be used in combina-tion with high performance external-wavelength filters.

6.1 Control-Signal Separation in All-Optical Wavelength Converters and Switches

243

neous emission (ASE) noise is fairly high which is not unusual for ridge amplifier structures. To enhance the recovery times of the carriers within the SOA, higher cur-rents were applied in comparison with the static experiment [6.4]. The currents ap-plied on the two segments from SOA1 to SOA4 were 280+127, 254+106, 212+142 and 250+149 mA. The need for increasing the currents in the second segment re-stricted the possibility to use them for phase adaptations, so that on-off ratios were reduced. In future designs real phase shifters should be used. Further improvement of the on-off ratios should be possible with faster photodiodes that have no ringing and improved pulsed control-signal source. Unfortunately there is considerable noise in the switched-off state of our pulsed source as can be seen in Fig. 6.9. For this dynamic experiment, control signals at λC=1.557 μm, a FWHM of 8 ps and a peak power of 6.5 dBm were used.In Fig. 6.10(b) we show the opening windows of the same experiment but with ex-ternal wavelength filters at the outputs. In practice only the ASE has to be filtered out at the end of the switching matrix (for example with a wavelength multiplexer). The control signal itself is already filtered by the inner MZI and can be reused or in-terchanged within the matrix of cascaded devices. The filtering losses of the external wavelength filters had to be compensated with Erbium Doped Amplifiers (EDFAs). For comparison we have depicted in the insets of Fig. 6.10 a copropagating experi-ment with a MZI-SOA device of the Fig. 6.1 type, when no external filters are used (top inset) and when filters are used (bottom inset). Without filters the signals are se-verely distorted by the control signals. The figure in the top-inset shows, that the PX-output is dominated by the two control-signal powers, whose peaks are clearly visi-ble. In the P=-output the signal peak and the two control-signal peaks merge to form one sole large peak. The output signal of the Fig. 6.1 type configuration can only be used when external wavelength filters that are added at the output (bottom inset).The DOMO configuration type B delivers similar dynamic curves as depicted in Fig. 6.10 for the interleaved MZI device. Since they resemble each other we do not repeat them here.

4.3 General Operation Characteristics of MZI All-Optical DevicesIn order to convert or to switch the wavelength with minimum power and large on-offs we have to answer the following questions:

• How does the wavelength λin and the power of the input signal Pin influence the on-off ratios?

• How does λin and Pin influence the power of the control signal needed for switching?

• Which control-signal wavelength λC is most efficient for switching?• Is there a difference between the diverse types of all-optical devices?

Switching with minimum power is not only desirable to save device costs or to ex-

6 All-Optical Devices with Integrated Data- and Control-Signal Separation Schemes

244

tend device lifetime. It is also desirable because it results in smaller coupling of the control signal into the PX and P= outputs, thereby reducing the control-signal cross-talk into these outputs. We have characterized each of the devices of Fig. 6.1, Fig. 6.3(a), (b) and Fig. 6.6. They differentiate in the way how they introduce and separate the control-signal mode. This leads to advantages and disadvantages for each of the devices expressing itself in the control-signal coupling efficiencies, the crosstalk of the control signal into the data-signal channels, the control-signal extraction efficiency and whether fourwave mixing can be observed or not (Table 6.1).But the principal switching be-haviour as a function of the input-signal wavelength, input-signal power and control-signal wavelength was found to be alike for all types of switches under investigation. Therefore we can restrict the subsequent discussion onto one device. We will depict the operation characteristics of the DOMO type A all-optical device with the con-trol-signal extraction scheme, Fig. 6.3(a).

The dynamic experiments were achieved at the available wavelength of the pulsed source. To extend our measurements onto the whole spectrum of interest, we need an external cavity. Since this cavity is unpulsed, the following measurements are performed statically. The static measurements show principle tendencies for the

Table 6.1: Characteristic of All-Optical Devices with Mode Separation Schemes

Device&

Characteristic

100%-Dual-Order Mode

(Fig. 6.3a)

66%-Dual-Order Mode (Fig. 6.3b)

Interleaved-MZI

(Fig. 6.6)

Coupler-Type andLength

100%-MMI-cc~ 1 mm

66%-MMI-cc280 μm

50:50 splitters~500 μm

Crosstalk ofPC into P=/X channel < -10 dB < -23 dB < -24 dB

PC Recycling yes no yes

FWM no no yes

Operation at Wave-length of Control Sig

possible possible not possible

Co-propagative Operation

possible,weak external fil-

ters necessary

possible, without external filters

possible, with-out external fil-

ters

Polarization Inde-pendent Operation

possible for either the PC or Pin

possible for either the PC or Pin

possible for both optical sig.

6.1 Control-Signal Separation in All-Optical Wavelength Converters and Switches

245

characteristics of the all-optical devices, which should remain valid for the dynamic performance as long as we don’t operate with subpicosecond pulses, where new ef-fects become important [6.28]. Additionally, in order to transfer the static results onto dynamic experiments, one should keep in mind that the powers required for switching increase the shorter the pulses become, since the amount of depleted car-riers within the SOAs relies on the totally transferred energy of the control pulses [6.29].The on-off ratios and the control-signal power PC needed to induce a π phase shift are depicted in Fig. 6.11 as a function of the input-signal wavelength. Due to the asymmetries implemented in the switch [6.14] and due to the large optical band-widths of the MMI-couplers [6.24], we find high on-off ratios exceeding 20 dB over the whole spectrum of 80 nm. The control-signal power of PC, which was needed for

Input-Signal Wavelength [μm]1.530 1.550 1.570 1.590

λC=1.540 μm

Fibr

e-to

-Fib

reG

ain

[dB

]

-30

-20

-10

0

Non-SwitchedState

(PC off)

SwitchedState

(PC on)

0.2

0.4

0.6

0.8

1.0

1.2

Pin=-1.5 dBm

PX

PX

P=

P=

Pow

erP C

forS

witc

hing

[mW

]

Fig. 6.11 (a) Static extinction ratios of the dual-order-mode switch, exceeding 20 dB in the non-switched (solid lines) and switched state (dashed lines) over a large wavelength range. (b) For inducing a phase-shift of π in the SOAs, the control-signal power has to be increased with increasing wavelength.

6 All-Optical Devices with Integrated Data- and Control-Signal Separation Schemes

246

switching, had to be increased from 0.24 to 1.20 mW when shifting the data-signal wavelength from 1.520 to 1.600 μm. The power needed for switching increases since the refractive index change decreases as we move to longer wavelengths. The largest refractive index change for a given carrier density change is obtained some 10 nm below the band edge at 1.550 μm [6.30]. In comparison with section III.A, we decreased the currents applied on the SOAs so that gain and phase adaptations could be achieved more easily and higher on-offs were attained. The two segments from SOA1 and SOA2 were operated with 101+105 and 181+99 mA and the experiments were performed with input signals Pin of dBm power.The influence of the input-signal power Pin onto the on-off ratios and the power of the control signal that is needed for switching is shown in Fig. 6.12. An increase of the input-signal power leads to a gain saturation but does not degrade the on-off ra-tios. However the gain saturation causes a dramatic increase of the control-signal

1,5–

Fibr

e-to

-Fib

reG

ain

[dB

]

-30

-20

-10

0

Pow

erP C

forS

witc

hing

[mW

]

0.2

0.4

0.6

0.8

1.0

-25 -20 -15 -10 -5 0 5Data-Signal Power [dBm]

λin=1.540 μmλC=1.554 μm

Non-SwitchedState

(PC off)Switched

State(PC on)

PX

P=

PX

P=

Fig. 6.12 (a) An increase of the data signal power leads to gain saturation but does not degrade the on-off of the input signals. (b) However, an increase of the input-signal power requires stronger control signals for switching.

6.1 Control-Signal Separation in All-Optical Wavelength Converters and Switches

247

power which is needed for inducing the π phase shift.The switching characteristic of the input signal as a function of the control-signal power is depicted in the top Fig. 6.13. The power required for switching as a function of the control-signal wavelength is shown in the bottom Fig. 6.13. The minimum power for switching is found at 1.560 μm, which is the gain maximum of the ampli-fier (see Fig. 6.11). In agreement with Fig. 6.11 less power is needed for switching an input signal Pin with smaller wavelength λin.

Control-Signal Power [mW]0.0 0.1 0.2 0.3 0.4 0.5 0.6

Fibr

e-to

-Fib

reG

ain

[dB

]

-30

-20

-10

0

1.530 1.550 1.570 1.590Control-Signal Wavelength [μm]

-4

-2

0

2

4

6

λin=1.540 μmλin=1.550 μm

Pin=-1.5 dBm

PX-Output

P=-Output

Pow

erP C

forS

witc

hing

[dB

m]

Fig. 6.13 Switching curve of the data-signal output channels as a function of the control-signal power. The modulation of the signal in the bar and cross outputs is large at small control signal powers and the switching curves gets flatter the higher the control signal. (b) The smallest control-signal power for switching is needed for control-signal wavelength at the ampli-fier gain maximum (see Fig. 6.11).

6 All-Optical Devices with Integrated Data- and Control-Signal Separation Schemes

248

Resuming, we can state that switching with smallest control-signal powers is found for input-signal wavelengths below the gain maximum and for control signals at the gain peak. However, switching with input-signal wavelengths below the gain peak is at the expense of gain. For this reason switching is preferentially performed with input and control signals near the gain peak. Our new configurations with separate ports for the data and control signal allow high signal-separation efficiencies even when the two signals have the same wavelength. With asymmetric MZI configura-tions, high on-offs are maintained for all operation conditions.

5. ConclusionsWe have realized and compared new all-optical MZI-SOA devices with a monolith-ically integrated scheme for separation of the data and control signals at the in- and outputs.

Two different concepts are presented. The first is based on a DOMO configuration, where data and control signal propagate as modes with different symmetry and where the even and odd symmetry of the data and control signals is exploited for mode separation. We have realized two versions. One version features complete con-trol-signal extraction and -10 dB crosstalk of the control signal into the data-signal channels. The second version attains approximately -23 dB crosstalk but it radiates out the control signal. The second concept uses interleaved MZIs to separate the sig-nal paths of the data and control signals. With this configuration the crosstalk be-tween data- and control-signal channels is smaller than -24 dB. Dynamic 10 GHz experiments with co-propagating data and control signals demonstrated lossless switching with almost negligible distortions from the control signals in the data sig-nal output even when no external wavelength filters are used to split off the control signals.All optical switches with a separation scheme for data and control signals have ap-plications in future cascaded and bidirectionally operated monolithically integrated high speed switching arrays. The freedom to separate the data and control signals al-lows to freely interchange data and control signal and to reuse the control signal, which leads to interesting system design possibilities.

AcknowledgmentsW. Vogt is acknowledged for elaboration of the wafer processing, E. Gini for growth of the wafer and W. Hunziker for contributing the fibre array. We thank Prof. Dr. U. Keller and G. Spühler for allowing access to 3 ps pulses for device characterization. Prof. F. Kneubühl is thanked for inspiring discussions and helpful comments. The work was in part supported by the Swiss research programme in optics.

6.2 Pump-Signal Separation in Optical Phase Conjugaton Devices

249

6.2 Polarization Independent Optical Phase Conjugation with Pump-Signal Filtering in a Monolithically Integrated Mach-Zehnder Interferometer Semiconductor Optical Amplifier Configuration

J. Leuthold, F. Girardin, P.A. Besse, E. Gamper, G. Guekos, H. MelchiorPhoton. Technol. Letters,

vol. 10, no. 11, pp. 1569-1571, Nov. 1998

AbstractDemonstration of polarization-independent optical phase conjugation by fourwave mixing in a monolithic MZI-SOA configuration. An integrated filtering mechanism for the pump signal enhances the conjugate-to-pump ratio by 15 dB compared to the single SOA case.

1. IntroductionThe chromatic dispersion in optical fibres causes a degradation of the signal in high bit-rate optical networks. This reduces the transmission lengths considerably, in par-ticular since the presently installed optical links mainly consist of non dispersion-shifted fibres. The transmission length can be drastically increased by using disper-sion compensation methods [6.31]. One method relies on using dispersion shifted fi-bres. But replacement of the existing standard single-mode fibres is expensive. Another solution is the use of optical phase-conjugation (OPC) by fourwave mixing (FWM) in semiconductor optical amplifiers (SOAs) or in dispersion shifted fibres. This method allows high bit-rate transmission over hundreds of kilometres of stand-ard single-mode fibres [6.32-6.33]. Two drawbacks of this method are its polariza-tion dependence and the tight filtering required at the output of the converter which degrades the spectral shape of the signal. To reduce the polarization dependence, a polarization diversity scheme has been proposed with two orthogonally polarized pumps [6.34-6.38]. Filtering out of the control signal at the output has been realized in all-optical switches by means of Mach-Zehnder Interferometer (MZI) based de-vices [6.39]. In this all-optical switch interleaved MZIs were used to introduce and split off the optical control signal.In this paper, we propose and demonstrate a polarization independent OPC with a fil-tering mechanism for the pump and the input signal in a new integrated device based

6 All-Optical Devices with Integrated Data- and Control-Signal Separation Schemes

250

on an MZI-SOA configuration (see section 6.1, [6.40]). This is the first demonstra-tion of polarization independent OPC in an integrated configuration. The rise of the noise figure due to the increased number of active devices can be overcome by the use of long SOAs [6.41-6.42].

2. ConfigurationThe principle of polarization independent phase conjugation with the integrated MZI-SOA configuration is shown in Fig. 6.14(a). The MZI device consists of 50:50 multi-mode interference (MMI) splitters and SOAs. The SOAs are split into two sec-tions of 500 and 250 μm to allow for phase adaptations. The inputs I1 and I4 are used to send two orthogonally polarized pump signals (PTE and PTM) into SOA1 and SOA2 respectively, which are biased in order to provide the same FWM efficiency. The randomly polarized input signal, Sin, is introduced into one of the remaining in-puts (I3). The TM (respectively TE) component of the signal interferes in SOA1 (re-spectively SOA2) with the pump and generates a phase conjugated signal. The two parts are recombined in the output coupler of the device. As the conjugated fields have the polarization state of the corresponding pump fields, they do not interfere and are equally separated into the two outputs. The intensity of the conjugated output field (CX and C=) is then polarization independent. On the other hand the two parts of the amplified input signal interfere in the output coupler because they keep the po-larization when passing the SOAs. This allows to direct the input signal into the cross output (O2) when properly adjusting the phase-relations within the MZI con-figuration. The bar output (O3) then guides only a small fraction of the input signal.The second configuration, presented in Fig. 6.14(b), allows besides a polarization in-dependent phase conjugation a filtering of the pump PTE, PTM and of the signal field Sin. It uses 4 SOAs (SOA1 to 4) which form three MZIs: (1-2), (3-4) and (1:2-3:4). A π phase shift, inserted either in the upper or the lower branch of the MZIs on the arm (1-2) and (3-4), respectively, guarantees that both the pump (PTE and PTM) and the signal (Sin) are mapped into their respective bar output ports. This provides a fil-tering of the pump since the pump signals are mapped from the outer input guides to their respective outer output guides. On the other hand the signal Sin is mapped from the I3 input guide into O2 but not into O3. This filtering mechanism does not depend on the pump and signal wavelengths. Polarization independent phase conjugation follows the same scheme as described in Fig. 6.14(a). It is attained by FWM of the TE and TM parts of the signal Sin in the MZIs (1-2) and (3-4) guiding the TE and TM pump signal respectively. The conjugated field is equally distributed onto the two middle output guides. For that reason the bar output (O2) mainly guides the con-jugated field with severely suppressed amounts of the pump and signal fields. We subsequently call this configuration: Interleaved MZI configuration.

6.2 Pump-Signal Separation in Optical Phase Conjugaton Devices

251

3. ExperimentsThe integrated device of Fig. 6.14(b) was realized to demonstrate polarization insen-sitive phase conjugation and pump-signal filtering. The MZI device of Fig. 6.14(a) is obtained by operating the interleaved MZI device of Fig. 6.14(b) solely with SOA2 and SOA3. The remaining SOAs are used as absorbers. Implementation of the configuration was carried out in analogy to the MZI-SOA all-optical switches re-

SOA1

SOA2

50:50

50:50SOA3

SOA4

Δφ=π

Δφ=π

50:50

50:50

50:50

I1

I2

I3

I4

O1

O2

O3

O4

λ

50:50

λ

λ

λ

λ

λ

λ

PTM

PTE

Sin

SX

S=

PTE

PTM

C=

CX

SOA1

SOA2

50:50

50:50

50:50

I1

I2

I3

I4

O2

O3

λ

50:50

λ

λ λ

λ

PTM

PTE

Sin

SX

S=

PTE

PTM

C=

CX

PTM

PTE

(a)

(b)

InP

InP ‘

‘

‘ ‘

‘

‘

Fig. 6.14 (a) Monolithically integrated MZI configuration with SOAs for polariza-tion insensitive FWM and signal filtering. Two pump signals (PTE and PTM) are launched into SOA1 and SOA2 respectively and transform the input signal Sin into its phase conjugated Cout. (b) MZI configuration as given in (a) but with additional MZIs on each arm. This allows to filter out the pump signals (PTE and PTM) from the generated conjugate field Cout.

6 All-Optical Devices with Integrated Data- and Control-Signal Separation Schemes

252

ported earlier [6.40]. The two orthogonal pumps are generated with a DFB laser whose signal is amplified in an erbium doped amplifier. The pump signal is then split with a coupler into two signals and their polarizations are adjusted with two polari-zation controllers. The pump powers injected into the chip are +5 dBm each. They are decorrelated one from the other by using a 6 km long fibre delay. The decorrela-tion of the two pump signals is not mandatory. It only eases the adjustment of the currents in the MZI, since two decorrelated pump signals do not produce disturbing interferences at the outputs O2 and O3. The signal Sin is provided by an external cav-ity laser followed by a polarization controller. The injected power is maintained at -8 dBm. The spectrum at the output O2 and O3 is observed. An output spectrum of an optical phase conjugate experiment with a single SOA is shown in Fig. 6.15. The experiment was carried out with the interleaved device of Fig. 6.14(b) but with only one of the four SOAs in operation. We use it as a reference to compare the performance of the single SOA case with the proposed more ad-vanced OPC devices. The solid line shows the best and the dashed line the worst po-larization cases. The different polarization cases were obtained by rotating the polarization of the input signal. There is no phase conjugated signal (Cout) in the worst polarization case and there is no filtering of the pump or signal field either.

-50

-40

-30

-20

-10

0

Opt

ical

Spe

ctra

l Den

sity

[dB

m/A

]

Fig. 6.15 (a) Output-spectrum of a FWM experiment with a single SOA. In the worstpolarization case (dashed) the output conjugate (Cout) vanishes. The singleSOA output spectrum has to be compared with the phase conjugate exper-iments of the more advance configurations as shown below.

1.553 1.554 1.555 1.556Wavelength [μm]

Cout

SoutPTE+PTM

6.2 Pump-Signal Separation in Optical Phase Conjugaton Devices

253

With the operation scheme of Fig. 6.14(a) we find a strong reduction of the polariza-tion dependency of the conjugate Cout signal. The spectral output-density at the bar output (O3) of the OPC experiment is displayed in Fig. 6.16. It shows how the po-larization sensitivity of the conjugated signal (Cout) due to the polarization state of the input signal Sin is reduced below 1 dB. In addition, the figure reveals a reduction of the signal field S=. Sin is directed as SX signal into the cross output O2 so that the corresponding field S= at the bar output O3 is suppressed. SOA2 and SOA3 were operated with 350 and 325 mA respectively. With the polarization insensitive scheme, the ASE level increases by 3 dB.An experiment with a polarization dependence below 1 dB together with a pump to conjugate power ratio, which is improved by 15 dB is depicted in Fig. 6.17. The fig-ure displays the spectral output-densities at both the cross, Fig. 6.17(a), and bar, Fig. 6.17(b), outputs of the interleaved device from Fig. 6.14(b). The polarization sensi-tivity of the conjugate signal at the bar output is smaller than 1 dB. In addition we see, how the signal field Sin is mapped into the cross output port (SX peak in the spectra of Fig. 6.17(a)) but is strongly suppressed in the bar-output spectra (S= peak of Fig. 6.17(b)). The SOA currents were 356, 387, 365 and 418 mA in SOA1 to SOA4 respectively. The use of an adjustable phase-shifting section in all four arms of the switch can bring a drastic improvement of the filtering performance.

1.553 1.554 1.555 1.556

-50

-40

-30

-20

-10

0

Wavelength [μm]

C=

S=

PTE+PTMO

ptic

al S

pect

ral D

ensi

ty [d

Bm

/A]

Fig. 6.16 Polarization insensitive conjugate signal (Cout) obtained by FWM with the MZI configuration from Fig. Fig. 6.14(a). The output spectrum re-veals a residual polarization dependence below 1 dB. Depicted is the bar-output spectrum with the S= signal, which is reduced relative to Fig. 6.15. The solid line gives the best polarization case and the dashed line the worst.

6 All-Optical Devices with Integrated Data- and Control-Signal Separation Schemes

254

The interleaved configuration with four SOAs has an increased ASE noise level. However the OPC efficiency increases by 3 dB too. Previously very good ASE-to-signal noise performances were obtained with 1.5 mm long SOAs [6.42]. This shows a way to overcome the ASE background noise in our and other configurations. Moreover we have to note that the high ASE noise in the bar-output of Fig. 6.17(b) can only in part be due to the increased number of SOAs. We attribute it to uninten-

1.553 1.554 1.555 1.556

-50

-40

-30

-20

-10

0

Wavelength [μm]

PTE+PTM

S=C=

CX

SXPTE+PTM

-50

-40

-30

-20

-10

0Cross-Output O2

Bar-Output O3

Opt

ical

Spe

ctra

l Den

sity

[dBm

/A]

Opt

ical

Spe

ctra

l Den

sity

[dBm

/A]

(b)

(a)

1.553 1.554 1.555 1.556

Fig. 6.17 Output-spectra at the cross (a) and bar (b) outputs obtained with the con-figuration of Fig. 6.14(b). The spectra demonstrate polarization insensi-tive phase conjugation (Cout) with pre-filtering of the pump (PTE and PTM). Compared with the single-SOA case a 15 dB enhancement of the conjugate-to-pump ratio is obtained. In addition the spectra show, how the signal field Sin is directed towards the cross output (a) but is filtered out in the bar output (b).

Wavelength [μm]

6.2 Pump-Signal Separation in Optical Phase Conjugaton Devices

255

tional reflections that occurred at the interface between the optical spectrum analyzer and the fibre. The reflections are particularly disturbing when the pump and signal fields are strong.Within the new configurations, part of the conjugate power is lost at the output. Ap-proaches based on discrete components with polarisation beam splitters [6.37] per-mit the achievement of polarisation independent operation with lower conjugate-signal losses. However these concepts allow no pump and signal filtering and do not have the compactness of the integrated devices.In conclusion, we have proposed and demonstrated a new scheme for polarization independent FWM with filtering of the pump and signal fields at the output. The very good noise performance obtained previously in 1.5 mm long SOAs [6.42] shows that the increase of the ASE background in the proposed configuration is not an obstacle. Thus, this method presents a compact and stable solution for the optical phase con-jugation needed for dispersion compensation.

AcknowledgementsW. Vogt is acknowledged for technology development and wafer processing, E. Gini for growth of the wafer and W. Hunziker for contributing the fibre array. Part of the work was supported by the Swiss Office for Science and Education.

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6.3 References

[6.1] O. Wada; “Femtosecond optoelectronic devices for Terabit/sec communica-tion and signal processing systems”; Proc. 2nd Optoelectr. Commun. Conf. (OECC'97), July 97, Seoul, Korea, pp. 124-125

[6.2] H. Kawaguchi; “Proposal for a new all-optical waveguide functional de-vice”; Optics Letters, vol. 10, no. 8, pp. 411-413, Aug. 1985

[6.3] M.C. Farries, D.N. Payne; “Optical fibre switch employing a Sagnac inter-ferometer”; Appl. Phys. Lett., vol. 55, no. 1, pp. 25-26, July 1989

[6.4] T. Durhuus, B. Mikkelsen, C. Joergensen, S.L. Danielsen, K.E. Stubkjaer, “All-optical wavelength conversion by semiconductor optical amplifiers”; J. of Lightwave Technol., vol. 14, no. 6, pp. 942-954, June 1996, invited paper

[6.5] J.M. Wiesenfeld, “Wavelength conversion for optical networks”; Proc. 2nd Optoelectr. Commun. Conf. (OECC'97), July 97, Seoul, Korea, pp. 426-427, invited paper

[6.6] S. Kawanishi: “100 Gbit/s optical communication systems”, Proc. OFC’96, 1996, pp. 136-137, invited paper

[6.7] S.R. Friberg, A.M. Weiner, Y. Silberberg, B.G. Sfez, P.S. Smith, “Femtosec-ond switching in a dual-core-fiber nonlinear coupler”; Optics Letters, vol. 13, no. 10, pp. 904-906, Oct. 1988

[6.8] S. Nakamura, Y. Ueno, K. Tajima; “Ultrafast (200 fs) and high-repetition (10 GHz) operations of symmetric Mach-Zehnder all-optical switch”; Proc. 5th International Workshop on Femtosecond Technology, Tsukuba, Japan, Feb. 1998, p. 64

[6.9] C. Joergensen, T. Durhuus, B. Mikkelsen, K.E. Stubkjaer, “Wavelength con-version at 2.5 Gbit/s using a Mach-Zehnder Interferometer with SOAs”; Proc. Optical Amplifiers and their Applications, Yokohama, Japan, July 1993, pp. 154-157

[6.10] M. Schilling, T. Durhuus, C. Joergensen, E. Lach, D. Baums, K. Daub, W.Idler, G. Laube, K. Stubkjaer, K. Wünstel; “Monolithic Mach-Zehnder Interferometer based optical wavelength converter operated at 2.5 Gb/s with extinction ratio improvement and low penalty”; Proc. European Conference on Optical Communication (ECOC’94), Firenze, Italy, Sept. 1994, pp. 647-650

[6.11] R. Hess, J. Leuthold, J. Eckner, C. Holtmann, H. Melchior, “All-optical space switch featuring monolithic InP-waveguide semiconductor optical amplifier interferometer”, in Optical Amplifiers and their Applications, June 1995, vol. 18, OSA Tech. Digest Series, pd. PD2

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257

[6.12] E. Jahn, N. Agrawal, H.-J. Ehrke, R. Ludwig, W. Pieper, H.G. Weber; “Mon-olithically integrated asymmetric Mach-Zehnder interferometer as a 20 Gbit/s all-optical add/drop multiplexer for OTDM systems”; Electron. Lett., vol. 32, no. 3, pp. 216-217, Feb. 1996

[6.13] R. Hess, M. Caraccia-Gross, W. Vogt, E. Gamper, P.A. Besse, M. Dülk, E. Gini, H. Melchior, B. Mikkelsen, M. Vaa, K.S. Stubkjaer, S. Bouchoule; “All-optical demultiplexing of 80 to 10 Gb/2 signals with monolithic inte-grated high-performance Mach-Zehnder Interferometer”, Photon. Technol. Lett., vol. 10, no. 1, pp. 165-167, Jan. 1998

[6.14] J. Leuthold, P.A. Besse, J. Eckner, E. Gamper, M. Dülk, H. Melchior; “All-optical space switches with gain and principally ideal extinction ratios”; IEEE J. of Quantum Electronics, vol. 34, no. 4, pp. 622-633, April 1998

[6.15] K. Morito, J. Leuthold, H. Melchior; “Dynamic Analysis of MZI-SOA All-Optical Switches for Balanced Switching”, Proc. European Conference on Optical Communications (ECOC’97), Edingburgh, Great Britain, Sept. 97, pp. 81-84

[6.16] J. Leuthold, J. Eckner, P.A. Besse, G. Guekos, H. Melchior, “Dual-order mode (DOMO) all-optical space switch for bidirectional operation”, in Pro-ceedings of the Conference on Optical Fiber Communication OFC’96, San Jose, Feb. 1996, pp. 271-272

[6.17] J. Leuthold, E. Gamper, M. Dülk, P.A. Besse, J. Eckner, R. Hess, H. Mel-chior; “Cascadable all-optical space switch with high and balanced extinc-tion ratios”; 2nd Optoelectronic and Communications Conference (OECC’97), July 97, Seoul, Korea, pp. 184-185

[6.18] J. Leuthold, P.A. Besse, E. Gamper, M. Dülk, St. Fischer, H. Melchior; “Cas-cadable dual-order mode all-optical switch with integrated data- and control-signal separators”; Electron. Lett., vol. 34, no. 16, Aug. 1998, pp. 1598-1600

[6.19] C. Janz, F. Poingt, F. Pommereau, F. Gaborit, D. Ottenwälder, I. Guillemont, B. Dagens, M. Renaud; “All-active dual-order mode Mach-Zehnder wave-length converter for co-propagative operation”; El. Lett., vol. 34, no. 19, pp. 1848-1849, Sept. 1998

[6.20] J. Leuthold, P.A. Besse, E. Gamper, M. Dülk, W. Vogt, H. Melchior; “Cas-cadable MZI-all-optical switch with separate ports for data and control sig-nals”; in Proced. ECOC’98, Spain, Madrid, pp. 463-464, Sept. 1998

[6.21] M. Vaa, B. Mikkelsen, K.S. Jepsen, K.E. Stubkjaer, M. Schilling, K. Daub, E. Lach, G. Laube, W. Idler, K. Wünstel, S. Bouchoule, C. Kazmierksi, D. Mathoorasing, “A bit-rate flexible and power efficient all-optical demulti-plexer realised by monolithically integrated Michelson Interferometer”, Proc. European Conf. Opt. Comm. - ECOC’96, Oslo, Sept. 96, ThB.3.3

6 All-Optical Devices with Integrated Data- and Control-Signal Separation Schemes

258

[6.22] J.H. Chen, F.S. Choa, P.S. Cho, J.S. Wey, J. Goldhar, D.L. Butler, G.L. Burdge; “The gain decompression effect and its application to very fast wavelength conversions”; Photon. Technol. Lett, vol. 9, no.6, pp. 755-757, June 1997

[6.23] J. Leuthold, J. Eckner, E. Gamper, P.A. Besse, H. Melchior: “Multimode in-terference couplers for the conversion and combining of zero- and first-order modes”, J. of Lightwave Technol., vol. 16, no. 7, pp. 1228-1239, July 1998

[6.24] J. Leuthold, P.A. Besse, R. Hess, H. Melchior, “Wide optical bandwidths and high design tolerances of multimode-interference converter-combiners”, Proc. of European Conference on Integrated Optics, ECIO’97, Stockholm, Apr. 2-4 1997, pp 154-157

[6.25] J. Leuthold, R. Hess, J. Eckner, P.A. Besse, H. Melchior, “Spatial-mode fil-ters realized with multimode-interference couplers”, Optics Letters, Vol. 21, pp. 836-839, June 1996

[6.26] J. Leuthold, F. Girardin, P.A. Besse, E. Gamper, G. Guekos, H. Melchior; “Polarization independent optical phase conjugation with pump-signal filter-ing in a monolithically integrated Mach-Zehnder interferometer semicon-ductor optical amplifier configuration”; Photon. Technol. Letters, vol. 10, no. 11, pp. 1569-1571, Nov. 1998

[6.27] F. Girardin, G. Guekos, A. Houbavlis; “Gain recovery of bulk semiconductor optical amplifiers”; Photon. Technol. Lett., vol. 10, no. 6, pp. 784-786, June 1998

[6.28] K.L. Hall, G. Lenz, A.M. Darwish, E.P. Ippen; “Subpicosecond gain and in-dex nonlinearities in InGaAsP diode lasers”; Optics-Communications., vol.111, no.5-6, pp.589-612, Oct. 1994

[6.29] J.M. Wiesenfeld; “Wavelength Conversion Techniques”; Tutorial Sessions of Conference on Optical Fiber Communication OFC’96, San Jose, Feb. 1996, Tutorial Session TuP

[6.30] C.H. Henry, R.A. Logan, K.A. Bertness, “Spectral dependence of the change in refractive index due to carrier injection in GaAs lasers”; J. Appl. Phys, vol. 52, no. 7, pp. 4457-4461, July 1981

[6.31] A. Yariv, D. Fekete, D.M. Pepper, “Compensation for channel dispersion by nonlinear optical phase conjugation”, Optics Letters, vol. 4, no. 2, pp. 53-54, Feb. 1979

[6.32] D.D. Marcenac, D. Nesset, A.E. Kelly, M. Brierley, A.D. Ellis, D.G. Moodie, C.W. Ford , “40 Gbit/s transmission over 406 km of NDSF using mid-span spectral inversion by four-wave-mixing in a 2 mm long semiconductor opti-cal amplifier”; Electron. Lett., vol. 33, no. 10, pp. 879-880, May 1997

6.3 References

259

[6.33] S. Watanabe, S. Takeda, G. Ishikawa, H. Ooi, J.G. Nielsen, C. Sonne, “Simul-taneous wavelength conversion and optical phase conjugation of 200 Gb/s (5*40 Gb/s) WDM signal using a highly nonlinear fiber four-wave mixer”; in Proc. ECOC’97, Sept. 1997, paper TH3A-1

[6.34] T. Hasegawa, K. Inoue, K. Oda; “Polarization independent frequency con-version by fiber four-wave mixing with a polarization diversity technique”; Photon. Technol. Letters, vol. 5, no. 8, pp. 947-949, Aug. 1993

[6.35] R.M. Jopson and R.E. Tench, “Polarization-independent phase conjugation of lightwave signals”, Electron. Lett., vol. 29, no. 25, pp. 2216--2217, Dec. 1993.

[6.36] G. Hunziker, R. Paiella, D.F. Geraghty, K.J. Vahala, and U.Koren, “Polariza-tion-independent wavelength conversion at 2.5 Gbit/s by dual-pump four-wave mixing in a strained semiconductor optical amplifier”, Photon. Tech-nol. Lett., vol. 8, no. 12, pp. 1633--1635, Dec. 1996

[6.37] J.P.R. Lacey, S.J. Madden, M.A. Summerfield, “Four-Channel Polarization-Insensitive Optically Transparent Wavelength Converter”, Photon. Technol. Lett., vol. 9, no. 10, pp. 1355--1357, Dec. 1997

[6.38] I. Zacharopoulos, I. Tomkos, D. Syvridis, T. Sphicopoulos, C. Caroubalos, E. Roditi, “Study of polarization insensitive wave mixing in bulk semiconduc-tor optical amplifiers”, Photon. Technol. Lett., vol. 10, pp. 352-354, March 1998

[6.39] J. Leuthold, P.A. Besse, E. Gamper, M. Dülk, W. Vogt, H. Melchior; “Cas-cadable MZI-All-Optical switch with separate ports for data and control sig-nals”; Presentation at ECOC’98, pp. 463-464, Sept. 1998

[6.40] R. Hess, J. Leuthold, J. Eckner, C. Holtmann, H. Melchior, “All-optical space switch featuring monolithic InP-waveguide semiconductor optical amplifier interferometer”, in Proc. Optical Amplifiers and their Applications 95, vol. 18, OSA Tech. Digest Series, June 1995, pd. PD2

[6.41] A. D’Ottavi, A. Mecozzi, S. Scotti, F. Cara Romeo, F. Martelli, P. Spano, R. Dall’Ara, J. Eckner, G. Guekos; “Four-wave mixing efficiency in travelling wave semiconductor optical amplifiers at high saturation”; Appl. Phys. Lett., vol. 67, no. 19, pp. 2753-2755, Nov. 1995

[6.42] F. Girardin, J. Eckner, G. Guekos, R. Dall’Ara, A. Mecozzi, A. D’Ottavi, F. Martelli, S. Scotti, P. Spano, “Low noise and very high efficiency four-wave mixing in 1.5 mm long semiconductor optical amplifiers”; Photon. Technol. Letters, vol. 9, no. 6, pp. 746-748, June 1997

6 All-Optical Devices with Integrated Data- and Control-Signal Separation Schemes

260

261

7 Conclusions and Outlook

7 Conclusions and Outlook

262

7.1 Conclusions

We have characterized and introduced new MZI-SOA all-optical switches.The semiconductor material equations have been applied to calculate and quantify the different contributions from the nonlinear effects of our devices. Dominant ef-fects are the bandfilling, the bandgap shrinkage and the plasma effect.To optimize the devices we have determined the material gain, refractive-index changes and alpha factors of the bulk 1.55 μm InGaAsP material system. The refrac-tive-index changes due to carrier injection were determined in the 1.480 to 1.600 μm wavelength range. At the gain peak we found . For the refractive-index change due to temperature variations we found

. The alpha factors were determined as a function of the wavelength, the carrier density and the temperature. At gain peak, the alpha factor does not vary with carrier density. But it rises with increasing wavelength and it de-creases with increasing temperature.In order to allow new functionalities in all-optical devices we have introduced new types of MMIs. The first type is a wavelength-multiplexing MMI. This MMI shows polarization-insensitive demultiplexing with 2 and 1 dB excess loss in comparison to straight waveguides and crosstalks exceeding 12 and 20 dB at 1.30 μm and 1.55 μm, respectively. The second type of a new MMI is a mode-converter-combiner MMI, which is used to convert fundamental order modes into higher order modes and which allows to overlap the first order with another zero order mode. We have realized MMI-converter-combiner with a conversion efficiency of 50%, 66% and 100%. The 50% and 66%-MMI-converter-combiners realized in InGaAsP/InP have excess losses below 0.3 dB. The 100%-MMI-converter-combiners have excess loss-es below 0.7 dB. We have developed simple relations to determine the optical band-widths of the new first-order mode MMIs. Experimental measurements confirm the relations and show large optical 1 dB bandwidths for the MMI-converter-combiner exceeding 140 nm.We have shown that asymmetries or high alpha-factor media are necessary for attain-ing high extinction ratios with all-optical devices. New asymmetric 1x2 and 2x2-switch configurations have been presented. With the new devices, balanced extinc-tion ratios exceeding 20 dB on both output ports of a 2x2 all-optical switch were demonstrated. Finally, we have introduced new devices that allow to perform all-optical operation, while at the same time the control and the data signal are separated at the output. Two different concepts were presented. The first is based on a dual-order-mode configu-ration, where data and control signal propagate as modes with different symmetry and where the even and odd symmetry of the data and control signals is exploited for

dn' dN⁄ 1,8– 10 20– cm 3–⋅=

dn' dT⁄ 1,0 10 3– K 1–⋅=

7.1 Conclusions

263

mode separation. We have realized two versions. One version features complete con-trol-signal extraction and -10 dB crosstalk of the control signal into the data-signal channels. The second version attains approximately -23 dB crosstalk. The second concept uses interleaved MZIs to separate the signal paths of the data and control sig-nals. With this configuration the crosstalk between data- and control-signal channels is smaller than -24 dB. Dynamic 10 GHz experiments with co-propagating data and control signals demonstrated lossless switching with almost negligible distortions from the control signals in the data-signal output even when no external wavelength filters are used to split off the control signals.

7 Conclusions and Outlook

264

7.2 Outlook

All-optical devices are promising tools for multiplexing, demultiplexing and switch-ing in ultrafast telecommunication networks. They most probably will soon be ap-plied in commercial systems.Different all-optical techniques and configurations are competing with each other. But the MZI-SOA based devices belong to the most versatile all-optical devices, that can be used as multiplexers, demultiplexers, switches and wavelength converters. In addition, they offer switching with smallest control-signal powers, noise reduction and high extinction ratios.The presented new MZI-SOA all-optical devices with an asymmetry and a control-signal separation scheme have several advantages in comparison with symmetric MZI-SOA all-optical devices. They allow all-optical signal processing while at the same time the control signal is separated from the data-signal path, even when the input signal and the control signal co-propagate. Operation is possible even for input and control signals at the same wavelength. Besides, the good performance of the conventional switches is maintained. These unique features make them attractive as candidates for use in future all-optical telecommunication systems.

265

Appendix

In this appendix we outline the derivation of the material gain and the nonlinear ma-terial-gain compression factor. We have collected the most important InGaAsP ma-terial parameter formulas and give a list of the material parameters that have been used in this work. The appendix is concluded with a list of symbols and acronyms.

Appendix

266

App. A Material Gain

The material gain is relevant for the XPM and XGM all-optical devices. Therefore we outline its calculation.The material gain is equivalent to the fractional increase of photons per unit length

, (A.1)

where is the photon flux (in units of s-1m-2) and and the total tran-sition rate of photons per unit volume and time from the conduction to the valence and from the valence to the conduction band. Calculations of Eq. (A.1) have been given in many textbooks [A.1, A.2], publica-tions [A.3] and simulation tools [A.4]. Hence we do not need to repeat all the details. The transition rates used in Eq. (A.1) are usually calculated with Fermi’s golden rule under the assumption that the wave interacts with the electrons via dipole transitions. After some calculations one finds [A.1, A.2]

,(A.2)

where the index “h” stands for either heavy hole (hh) or light hole (lh) transitions, indicates the effective group index of the waveguide, the photon transi-

tion energy from a conduction band state at energy to a valence energy state at energy and is the energy gap of the semiconductor, as illustrated in Fig. A.1.The lower index of the material gain denotes explicitly, that the formula applies for single-particle transitions. Thus many particle scattering effects are excluded. The notation is used rather than to emphasize, that we consider carrier den-sities in thermal equilibrium.The total material gain is the sum over all transitions

. (A.3)

In Eq. (A.2) represents the squared dipole matrix element

, (A.4)

gm

gm1Φ----dΦ

dz-------≡Wc v→ Wv c→–

Φ---------------------------------------=

Φ Wc v→ Wv c→

gm s,h Ecv( ) 1

Ech-------- e2πh

ε0cm0---------------

n'g,eff

n'2------------- Mave

h 2ρred

h Ecv Eg–( ) fc Ec( ) fv Ev( ) 1–+[ ]⋅=

n'g,eff EcvEc

Ev Egs

gm gm

gm s, gm s,hh gm s,

lh+=

Mh

Maveh 2

waveh m0

m*------ 1–⎝ ⎠⎜ ⎟⎛ ⎞ Eg Δs.o.+( )Eg

Eg 2Δs.o. 3⁄+( )--------------------------------------

m02------⋅ wave

h Epm02-------------⋅= =

App. A Material Gain

267

where is the elementary mass, the true electron mass and the spin-orbit splitting energy. The factor is a probability factor for the occurrence of a dipole transition between an electric field having the polarisation and an electron with a momentum in the direction. The transition probabilities are summarized in text-books such as [A.1][A.5]. They are:

(A.5)

where . However, there are no restrictions imposed to the direction into which the vectors in bulk material are pointing, such that the photons of a gi-ven polarization interact with an equal probability with all of them. Sweeping the

-vector over all directions yields an average factor for light of any po-larisation (TE or TM) and independent of the transition (c->lh, c-> hh). Occasionally

Fig. A.1 Emission of a photon with energy Ech. Thereby an electron is transferred from the conduction into the valence band. is the semiconductor ener-gy gap. and are the electron and hole energies taken from the con-duction and valence band edge. indicates the split-off energy while

and are the fermi energies in the conduction and valence band. It should be noticed that two coordinate systems are used for the conduction- and one for the valence-band states. They have their origins at the respec-tive band edges.

EgEc Ev

Δs.o.Efc Efv

Ec

E

Efc

Efv

Eg

Δs.o.

k

k

Conduction Band

Heavy Hole BandLight Hole BandSplit-Off Band

Ech=hω

Ev

E

m0 m* Δs.o.wave

h

ek

whh 1 k e⋅2 , for hh-transitions–=

wlh 13--- k e⋅

2 , for lh-transitions,+=

e k 1,= =k

k waveh 1 3⁄=

Appendix

268

the dipole-matrix elements are given in terms of the Kane matrix element as de-fined on the right-hand side of Eq. (A.4).

in Eq. (A.2) is the reduced density of states

. (A.6)

Here indicates the electron’s wavenumber. Since nowadays practically all SOAs are nonintentionally doped one can use the k-selection rule, which states that the conduction and hole wavenumbers must be identical: . is the number of states that can be occupied in a volume . The density of states of Eq. (A.6) depends on the energy bands chosen. Since the va-lence bands are flat, the higher energy states are weakly occupied. Therefore they can be approximated with parabolic bands

. (A.7)

However, the conduction-band effective mass is small. Consequently nonlinear dis-persion should be considered [A.6]

. (A.8)

An approximation for the nonparabolicity factor has been given in [A.7]

. (A.9)

Solving Eq. (A.8) for one obtains

. (A.10)

Eq. (A.7) and (A.10) are then used to evaluate on the basis of

. (A.11)

Ep

ρredh

ρredh Ech( ) 1

V---

dNstatesdktr

------------------- 1dEdktr------------------

E Ech=

=

ktr

ktrc ktr

h= NstatesV

Evh ktr( )

h2ktr

2

2mvh------------=

Ec 1 αnpEc+( )⋅h

2ktr

2

2mc------------=

αnp

αnp1

Eg------ 1

mcm0------–⎝ ⎠

⎛ ⎞ 1EgΔs.o

3 Eg Δs·o·+( ) Eg 2Δs.o. 3⁄+( )-------------------------------------------------------------------–⎝ ⎠

⎛ ⎞=

Ec

Ec ktr( )1– 1 4αnp

h2ktr

2

2mc------------++

2αnp----------------------------------------------------=

Ecv

Ecv Ec Eg Ev+ +=

App. A Material Gain

269

The last term in Eq. (A.2) is the population-inversion factor. It is the difference be-tween the probabilities for electron downward transitions (light emission) and elec-tron upward transitions (light absorption)

. (A.12)

and are the electron and hole occupation probabilities for the conduction and valence bands

(A.13)

where the quasi-fermi levels for the conduction and valence band are deter-mined via the electron and hole density and by

(A.14)

The density of states depends on the dispersion relations of the respective bands. For the conduction and valence bands (including spin degeneracy), one obtains with an analog definition of Eq. (A.6)

(A.15)

When deriving the expression of Eq. (A.2), we assumed that an electron in a conduc-tion band state would remain in that state forever if there were no interactions with photons. In reality, interactions with phonons and other electrons occasionally scatter the electron into another conduction band state. For high carrier injections, carrier relaxation times are believed to be 100 fs or less in 1.55 InGaAsP semiconductors [A.8]. An exponential decay implies a Lorentzian lineshape. Hence the energy of

fcfv 1 fc–( ) 1 fv–( )– fc fv 1–+=

fc fv

fc Ec( ) 1

1Ec Efc–

kBT-------------------⎝ ⎠

⎛ ⎞exp+-------------------------------------------- =

fv Ev( ) 1

1Ev Efv–

kBT-------------------⎝ ⎠

⎛ ⎞exp+--------------------------------------------,=

Efc EfvN P

N ρc Ec( )fc Ec( ) Ec ,d0

∞

∫=

P ρvlh Ev( ) ρv

hh Ev( )+[ ]fv Ev( ) Ev .d0

∞

∫=

ρc Ec( ) 2π2-------

mc

h2------

⎝ ⎠⎜ ⎟⎛ ⎞ 3 2/

1 2αnEc+( ) Ec αnEc2+=

ρvh Ev( ) 2

π2-------mv

h

h2------

⎝ ⎠⎜ ⎟⎛ ⎞

3 2/

Ev= with h=lh, hh.

Appendix

270

each state and each transition is no longer discrete but is spread over a range of on each side of the transition.

To include the spectral broadening of each transition, we convolve the expression for gain with a spectral lineshape function over all transition energies to thus obtain (e.g. [A.1, A.2])

, (A.16)

where is taken from (A.3) and represents a normalized lineshape func-tion. Different functions can be employed. The most common are the• Lorentzian lineshape function

, (A.17)

which corresponds to an exponential decay of the excited carriers. is the in-traband relaxation time. Corresponding to the dipole-dephasing time introduced in many textbooks for the two-level approximation [A.9].

• Exponential broadening

, (A.18)

as introduced in [A.3] to understand the exponential tail (Urbach tail) on the low-energy sides of the absorption and emission edges in undoped semiconduc-tors due to interactions of the carriers with vibrating lattice.

• Secant hyperbolic broadening function

, (A.19)

which has been employed in the material gain model since it matched the experi-mental results [A.10].

For nonparabolic bands the integration of Eq. (A.16) becomes quite intricate. There-fore it is more appropriate to evaluate the integral by performing the integration in the -space. This is simpler since the -states are always uniformly distributed.

ΔE h τ2⁄≈ 7 meV=

Ecv

gm Eϖ( ) gm s, Ecv( )L Eϖ E– cv( ) Ecvd∫=

gm s, L E( )

L E( ) 1πh------

1 τ2⁄

E h⁄( )2

1 τ2⁄( )2+--------------------------------------------=

T2

L E( ) 12 h τ2⁄( )-------------------- E

h τ2⁄------------–

⎝ ⎠⎜ ⎟⎛ ⎞

exp=

L E( )τ2

πh------ E

h τ2⁄------------⎝ ⎠

⎛ ⎞sech=

ktr ktr

App. A Material Gain

271

This yields the material gain

(A.20)

where now the 3-dimensional density of states is simply

, (A.21)

with the factor two taking into account the spin degeneracy.

gm Eϖ( ) 1Eϖ------- e2πh

ε0cm0---------------

n'g,eff

n'2-------------=

Mave0

∞

∫×2ρ ktr( ) fc Ev ktr( )( ) fv Ec ktr( )( ) 1–+[ ]L Eϖ Ecv ktr( )–( )dktr

ρ ktr( ) 1V---

dNstatesdktr

------------------- 2ktr

2

2π2---------⋅= =

Appendix

272

[A.1] P.S. Zory; “Quantum Well Lasers”; Academic Press, San Diego, California 1993

[A.2] A. Yariv, “Quantum Electronics” third Ed.; John Wiley & Sons, New York, 1989

[A.3] D. Gershoni, C.H. Henry, G.A. Baraff; “Calculating the optical properties of multidimensional heterostructures: Applications to the modelling of quater-nary quantum well lasers”; J. of Quantum Electron., vol. 29, no. 9, pp. 2433-2450, Sept. 1993

[A.4] J.L. Pleumeekers; “POSEIDON: A simulator for optoelectronic semicon-ductor devices”; Ph.D. thesis, Delft University, 1997

[A.5] G. Bastard; “Wave mechanics applied to semiconductor heterostructures”; Les Editions de Physique, Halsted Press, Les Ulis CEDEX A France, 1988

[A.6] B. Sermage, D.S. Chemla, D. Sivco, A.Y. Cho; “Comparison of Auger re-combination in GAInAs-AlInAs multiple quantum well structure and in GaInAs”; J. Quantum Electron., Vol. QE-22, PP. 774-780, 1986

[A.7] S.L. Chuang, J. O’Gorman, A.F.J. Levi; “Amplified spontaneous emission and carrier pinning in laser diodes”; J. Quantum Electron., vol. 29, no. 6, pp. 1631-1639, June 1993

[A.8] K.L. Hall, G. Lenz, A.M. Darwish, E.P. Ippen; “Subpicosecond gain and in-dex nonlinearities in InGaAsP diode lasers”; Optics Communications, vol. 111, pp. 589-612, 1994

[A.9] R. W. Boyd; “Nonlinear Optics”; Academic Press Inc., San Diego, Califor-nia, 1992

[A.10] W.W. Chow, S.W. Koch, M. Sargent III; “Semiconductor-Laser Physics”; Springer Verlag, 1994

App. B Nonlinear Gain Compression

273

App. B Nonlinear Gain Compression

In this appendix we briefly outline the approach of Mecozzi and Mørk to describe SHB and CH [B.1]. To describe SHB and CH, we have to represent the local carrier densities and local temperature at the respective conduction and valence band by the rate equations

and (B.1)

. (B.2)

An equation similar to (B.1) and (B.2) can be used for the local hole density . In addition, an expression of the material gain in terms of the local carrier densities is required. For that reason is linearized in and

. (B.3)

The parameters in Eq. (B.1)-(B.3) are the carrier-carrier scattering relaxation time , the group velocity of light in the semiconductor, the photon density and

the fermi density of the electrons in the conduction band at the electron tempera-ture . Due to carrier heating, will be higher than the lattice temperature . The first term at the right side of Eq. (B.1) determines the recovery times of the local carriers and the second term describes the carrier depletion by stimulated emission. Other processes like carrier injection and spontaneous emission can be neglected versus these two effects and within the timescale. The first term in Eq. (B.2) indicates the temperature recovery contribution due to car-rier-phonon scattering with the temperature relaxation time . The second term concerns temperature increase due to free-carrier absorption (FCA), where is the FCA coefficient. The next term accounts for the temperature increase due to carrier injection and the last term accounts for stimulated emission. is the energy densi-ty.In Eq. (B.1) is the gain cross section and the density of available states in the optically coupled region and is obtained by integration over the density of states. It should be noticed that the local carrier densities have to be summed up in order to obtain the total carrier density (e.g. Eq. (A.12) of Appendix A).In the case where the pulse lasts much longer than the scattering times and ,

n˜

p˜

n˜

∂t∂

----- n˜

n˜

–τ2 n

˜,

------------– vggmS–=

dTn˜dt---------

Tn˜

TL–τh n

˜,

------------------Tn

˜∂

U∂ n˜

---------+–N

σbNhωgm

------------------Un

˜∂

dN---------Tn

˜

En˜

–+⎩ ⎭⎨ ⎬⎧ ⎫

vggmS=

p˜

gm n˜

p˜

gmaNvg------ n

˜p˜

N0–+( )=

τ2 n˜

, vg Sn˜Tn

˜Tn

˜TL

τh n˜

,σn

˜

Un˜

aN N0

τ2 n˜

, τ2 p˜

,

Appendix

274

i.e. several picoseconds, one can solve the rate equation (B.1) for the local carrier densities in the adiabatic limit and write

. (B.4)

Moreover, one can linearize and instead express as the sum of an equi-librium term at the lattice temperature and a contribution due to the local temper-ature change

. (B.5)

An expression for is obtained from Eq. (B.2) in analogy to (B.4) by neglecting the derivative. The relevant thing is that the temperature changes are proportional to

. (B.6)

A new expression for the non-equilibrium material gain in terms of the material gain at equilibrium is now gained by insertion of Eqs. (B.4)-(B.6) and the same expressions for the local hole densities in (B.3)

, (B.7)

where is

, (B.8)

with the differential gain, the total carrier density and the transparency carrier density. This corresponds to the linearized equilibrium ma-terial-gain from Eq. (A.20).The parameter is the nonlinear gain-suppression parameter

, (B.9)

n˜

n˜

τ2 n˜

, vggmS–=

n˜

n˜

Tn˜

( )= n˜TL

ΔTn˜

Tn˜

TL–=

n˜

n˜

TL( )n˜

∂Tn

˜∂--------

N

ΔTn˜

+=

ΔTn˜

vggmS

ΔTn˜

Kn˜vggmS=

gmgm

p˜

gmgm N TL,( )1 ∈totS+-------------------------=

gm

gmaNvg------ n

˜p˜

N0–+( ) a N Ntr–( )= =

a gmd /dNλ

= N Ntrgm

∈tot

∈tot ∈SHB ∈CH+=

App. B Nonlinear Gain Compression

275

where due to the derivation

and (B.10)

. (B.11)

The photon density is related to the power of a signal measured at the facet by

, (B.12)

with the effective area of the waveguide, the waveguide width and the thickness of the waveguide.Eq. (B.7) is the new material gain expression and replaces the equilibrium material gain when gain compression becomes relevant and when the pulse widths are at least in the picosecond range.

[B.1] A. Mecozzi, J. Mørk; “Saturation induced by picosecond pulses in semicon-ductor optical amplifiers”; J. Opt. Soc. Am. B; vol. 14, no. 4, pp. 761-770, April 1997

∈SHB aN τ2 c, τ2 v,+( )=

∈CH ∈CHn˜

∈CHp+ vg Kn

˜τh n,

˜

gm∂Tn

˜∂--------- Kp

˜τh p,

˜

gm∂Tp

˜∂---------+

⎝ ⎠⎜ ⎟⎛ ⎞

–= =

S

Pout hωAeffvgS=

Aeff 1 Γ⁄ wd⋅= wd

gm

Appendix

276

App. C InGaAsP Material Parameters

Active components at the 1.3 μm and 1.55 μm absorption windows of optical fibres require materials with a bandgap at the respective wavelength. The In1-xGaxAsyP1-y material system covers the whole spectrum and beyond. It is therefore of special in-terest. Below we list the material parameters, used for calculations and simulations in the thesis.The InP, In1-xGaxAs and In1-xGaxAsyP1-y compounds form crystals with the zinc-blende arrangement. The zinc-blende structure is based on the cubic space group in which the lattice atoms are tetrahedrally bound in network ar-rangements related to those of the group IV (diamond-type) semiconductors. They lack a symmetry centre and, hence, exhibit second-order nonlinear susceptibilities [C.1].Unfortunately, not all material parameters for quaternary InGaAsP have been meas-ured. For this reason it has become common practice to use Vegard’s law, in which a particular property Q of a given quaternary composition (x,y) is calculated from the accurately known values of the respective property BMN of the constituent binary compounds MN [C.1]

(C.1)

In order to match the two lattice grids of In1-xGaxAsyP1-y and InP materials, the two indexes x and y have to fulfil the “lattice match” condition [C.3]

. (C.2)

The bandgap energy of undoped In1-xGaxAsyP1-y lattice-matched to InP at 300 K is [C.4]

[eV]. (C.3)

The carrier-concentration dependence of the bandgap [C.5] is

, (7.4)

with the free electron and hole carrier concentrations and . The theory discussed in chapter 2 yields = . The results of the experiments de-scribed in chapter 3 do not confirm such a high bandgap shrinkage. On the contrary,

F43m Tdd( )

Q x y,( ) 1 x–( )y BInAs⋅ 1 x–( ) 1 y–( )BInP xy+ + BGaAs⋅=

x+ 1 y–( )BGaP.

x 0,4562y1 0,031y–-------------------------=

Eg 1,35 0,72y– 0,12y2+=

ΔEgdEgdN--------- N1 3⁄ P1 3⁄+( )=

N PdEg dN⁄ 2,35– 10 10– eVm⋅

App. C InGaAsP Material Parameters

277

the bandgap shrinkage estimated in the textbook of Agrawal [C.4] is confirmed by our experiments. Agrawal assumes for InGaAsP materials

. (C.5)

The temperature dependence of the bandgap is usually described by the Varshni equation [C.1]

(C.6)

where is the bandgap energy at 0 K, while α and β are coefficients. For In1-xGaxAsyP1-y/InP the coefficients have been determined to be α=0.49 meV/K and β=327 K independent of the composition [C.1]. Near and above room temperature

varies approximately linearly with temperature. In In1-xGaxAsyP1-y materi-als one finds with the aid of Vegard’s law[C.1]

. (C.7)

Other textbooks quote experiments for InGaAsP lattice-matched to InP which result-ed in -0.348 meV/K [C.2] and meV/K [C.3] for room temperature.The effective masses of In1-xGaxAsyP1-y are listed in [C.3]:• the electron mass in the conduction band

, (C.8)

• the heavy hole mass in the valence band

, (C.9)

• the light hole mass in the valence band

, (C.10)

• the split-off energy [C.6]

[eV]. (C.11)

The refractive index of III-V semiconductors is best fitted with the Aframowitz mod-

dEg dN⁄ 1,6– 10 10– eV m⋅[ ]⋅=

Eg TL( ) Eg 0( )αTL

2

β TL+---------------–=

Eg 0( )

Eg TL( )

dEg dTL⁄ 0,40– 0,03y+( ) meV/K=

0,325–

mc m0⁄ 0,080 0,039y–=

mhh m0⁄ 1 y–( ) 0,79x 0,45 1 x–( )+[ ] y 0,45x 0,4 1 x–( )+[ ]+=

mlh m0⁄ 1 y–( ) 0,14x 0,12 1 x–( )+[ ] y 0,08x 0,026 1 x–( )+[ ]+=

Δs.o. 0,119 0,300y 0,107y2+ +=

Appendix

278

el [C.7]. In the range where one can write

(C.12)

where the parameters and as functions of the y concentration for InP lat-tice-matched In1-xGaxAsyP1-y were fitted to experimental data by Sartoris [C.8] with the result

, and (C.13)

. (C.14)

Carrier and temperature related refractive index changes of 1.55 μm InGaAsP have been determined in this work (see chapter 3.2 and 3.4). For a change from carrier density to carrier density we found

(C.15)

with the peak wavelength, and .For the variation of the refractive index with temperature we found in chapter 3.2

(C.16)

The static dielectric constant is given as [C.3]

(C.17)

Bandgap discontinuities of InGaAsP lattice-matched to InP are related by [C.9]

, (C.18)

This relation has been tested for samples spanning the alloy range from In0.53Ga0.47As to InP in [C.9]. indicates the conduction-band discontinuity ver-sus InP.Alpha factors and for InGaAsP are given in chapter 3.3 and 3.4.For the material gain we have given in chapter 3.4 a polynomial fit func-tion, that can be used in order to get values based on experiments.

E Eg<

n2 E( ) 1EdE0------

Ed

E03------E2 Ed

E03------ 1

2 E02 Eg

2–( )--------------------------

2E02 Eg

2 E2––

Eg2 E2–

----------------------------------⎝ ⎠⎜ ⎟⎛ ⎞

E4⋅ln⋅ ⋅+ + +=

Ed E0

E0 3,34977 1,23842y 0,24236y2+–=

Ed 28,48475 4,34776y–=

N1 N2

Δn' λ ΔN,( ) 1,8 10 26– m 3–⋅– 5,0 10 26– m 3–

μm--------- λ λp N( )–( )⋅ ⋅+ ΔN⋅=

λp N( ) ΔN N2 N1–= N N1 N2+( ) 2⁄=

dn'

dTcav-------------- 1,0 10 3– K 1– 1,5 10 4– K 1–⋅±⋅=

ε 1 y–( ) 8,4x 9,6 1 x–( )+[ ] y 13,1x 12,2 1 x–( )+[ ]+=

ΔEc 0,39 ΔEg⋅=

ΔEc

αN λ N Tcav S, , ,( ) αT λ N Tcav, ,( )

gm λ N,( )

App. C InGaAsP Material Parameters

279

[C.1] S. Adachi; “Physical Properties of III-V semiconductor compounds”; John Wiley & Sons, New York, 1992

[C.2] A. Katz; “Indium Phosphide and related materials: Processing, technology, and devices”; Artech House, Norwood, USA, 1992

[C.3] G.P. Agrawal, N.K. Dutta; “Long-wavelength semiconductor lasers”; Van Nostrand Reinhold, 1986

[C.4] R.E. Nahory, M.A. Pollack, W. Johnston, K.L. Barns, Appl. Phys. Lett., vol. 33, pp. 660, 1978

[C.5] P.A. Wolff; “Theory of the band structure of very degenerate semiconduc-tors”; Phys. Rev., vol. 126, pp. 405-412, 1962

[C.6] Landolt-Börstein; “Numerical Data and Functional Relationships in Science and Technology”, Vol. 22 Semiconductor, Subvolume a (Extensions to vol. III/17), Springer Verlag Berlin

[C.7] M.A. Afromowitz; “Refractive index of Ga1-xAlxAs”; Solid State Commu-nications, vol. 15, pp. 59-63, 1974

[C.8] G. Sartoris, ETH-Zürich postdiploma Thesis, Micro- and Optoelectronics In-stitute, 1989

[C.9] S.R. Forrest, P.H. Schmidt, R.B. Wilson, M.L. Kaplan; “Relationship be-tween the conduction-band discontinuities and band-gap differences of In-GaAsP/InP heterojunctions”; Appl. Phys. Lett., vol. 45, no. 11, pp. 1199-1201, Dec. 1984

Appendix

280

App. D Material Parameters Used in the Calculations

In the following we list the values of the material parameters, that were used in our calculations. The refractive-index changes caused by the optical nonlinearities of chapter 2.1 are given for a bulk 1.55 μm-In1-xGaxAsyP1-y amplifier lattice-matched to InP.The qua-ternary material compositions (x and y) and the differential carrier-relaxation times were determined by comparison of the calculated and measured material gains of the device investigated in chapter 3. Comparison of calculated and measured material gains, refractive-index changes etc. has also been made in chapter 3. All other pa-rameters have been derived from the formulas given in App. A.

In Table D.1 is the electron mass =9.110·10-31 kg.

Table D.1: Material Parameters Used for 1.55-μm In1-xGaxAsyP1-y Amplifiers

Parameter Symbol Remark Value Ref.

As Concentration y from exp. §3 85% Ch. 3

Ga Concentration x 40% (C.2)

Intraband-Relaxation Time τ2 from exp. §3 60 fs Ch. 3

Electron Effective Mass mc 0.0469·m0 (C.8)

Heavy Hole Eff. Mass mhh 0.4447·m0 (C.9)

Light Hole Eff. Mass mlh 0.0596·m0 (C.10)

Energy Bandgap Eg no Carriers 0.8247 eV (C.3)

Spin-Orbit Splitt-off energy 0.2967 eV (C.11)

Kane Matrix Element Ep 24.46 eV (A.4)

Refr. Index n‘ 3.4703 (C.12)

Effective Group Refr. Index n‘g from exp. §3 Ch. 3

Dielectricity Constant ε 12.043 (C.17)

Bandgap-Shrinkage dEg/dN N=P -1.6·10-10eVm3 (C.5)

Gain Compression SHB 8.4·10-23 m-3 [2.29]

Gain Compression CH 5.6·10-23 m-3 [2.29]

Δs.o.

∈SHB

∈CH

m0 m0

App. E Useful Formulas

281

App. E Useful Formulas

Sometimes it is useful to represent expressions in terms of the intensity , the optical power and the photon density rather than the electric field . For this reason we discuss here their relations.If the electric field of a signal in a particular segment of an amplifier or laser is de-fined as

, (E.1)

the corresponding intensity is

(E.2)

and in terms of the power and the photon density

, (E.3)

where one uses the effective area

(E.4)

with , and having the usual meaning. For a signal intensity in a segment of a laser or amplifier one uses rather than , since part of the energy is in the clad-ding layers.In an amplifier the signal power changes while propagating along the cavity. To re-late the power at a particular segment with the input power one uses

, (E.5)

where is the amplifier gain. In the case of an ideal amplifier (small facet reflectiv-ities) one can express the gain by the single-pass gain

. (E.6)

The single pass gain is limited by the facet reflectivities. It reaches it’s maximum at

IP S E

E t( ) 12--- Eoe iωt– c.c.+( )=

I 12---

ε0εμ0μ--------- Eo

2 12---cε0n' Eo

2 PAeff---------= = =

P hω vg Aeff S⋅ ⋅ ⋅=

Aeff

Aeffw d⋅

Γ-----------=

w d ΓAeff A

PSig G Pin,Sig⋅=

GG GSP

GSP z( ) eΓgmz

=

Appendix

282

threshold, where it becomes

. (E.7)

Finally, the relation between input power and photon density in an amplifier segment at position z along the propagation direction is given by combining Eq. (E.3)-(E.6)

. (E.8)

In a laser one usually measures the signal powers at the input or output of the device. The relation between the spontaneous or amplified spontaneous (ASE) photon den-sity inside a laser cavity with the power coupled out through the laser cavity is after [E.1]

, (E.9)

where and denote the powers emitted from each individual facet. The ratio between these powers can be written as [E.2]

(E.10)

and the average losses through the facets are

, (E.11)

where is the length of the laser and and are the mirror reflections.

[E.1] G.P. Agrawal, N.K. Dutta; “Long-wavelength semiconductor lasers”; Van Nostrand Reinhold, 1986

[E.2] L.F. Tiemeijer; “Optical properties of semiconductor lasers and laser ampli-fiers for fibre optical communication”; Technical University of Denmark, Ph.D thesis, 1992

GSPmax 1

R1R2-----------------≅

Pin,Sig eΓ– gmz

hω vg Aeff SSig⋅ ⋅ ⋅⋅=

SSp Pout,Sp

Pout,Sp hω Lαm Aeff v⋅ g S⋅ Sp⋅ ⋅ Pout,Sp1 Pout,Sp2+= =

Pout,Sp1 Pout,Sp2

Pout,Sp1Pout,Sp2-------------------

R2R1------

1 R1–( )1 R2–( )

--------------------⋅=

αm1

2L------ 1R1R2------------ln=

L R1 R2

App. F List of Symbols and Acronyms

283

App. F List of Symbols and Acronyms

ConstantsConstant Description Value and SI-unitsc speed of light in vacuum 2.99792·108 m/s

reduced Planck constant 1.05459·10-34 JsPlanck constant 6.62618·10-34 Js

kB Boltzman constant 1.38041·10-23 J/Km0 elementary rest mass 9.10953·10-31 kg

pi 3.141592q elementary charge 1.60219·10-19 C

permittivity of vacuum 8.854188·10-12 As/V/mmagnetic field constant

SymbolsSymbol Description Value and SI-units

effective area of the waveguide m2

differential peak-gain coefficient m2

magnetic flux T, peak wavelength parameterization parameters m3μm, m6μm

active layer thickness m, drift diffusion coefficient m2/s

, , electric fields V/m=N/Celectric field, where the time dependent term is split off V/mphoton energy Jbandgap energy J

, electron and hole energy measured from the bottom of the bands Jtransition energy from a conduction to a valence band state JKane dipole matrix element Jelectric field distribution perpendicular to propagation direction -

, electron and hole occupation probability -material gain m-1

hh

π

ε0μ0 4π 10 7– Vs/Am⋅

Aeffa0Bb0 b1dDN DPE Eω EΩ

Eo

EϖEgEc EvEcvEpF x y,( )fv fcgm

Appendix

284

material gain at thermal equilibrium m-1

material gain at peak wavelength m-1

linear gain m-1

modal gain m-1

net modal gain m-1

amplifier gain -, single pass gain and maximum single pass gain -

modal gain integrated over the laser cavity length -magnetic field strength A/mlight intensity J·s-1·m-2

current density A/m2

vacuum wavenumber 1/mwavenumber in medium of complex refr. index 1/melectron-hole transition wavevector 1/mdiode cavity length mconduction band mass kghole effective mass kglight hole band mass kgheavy hole band mass kgdipole matrix element -

, complex scalar and tensor refractive index -real refractive index -unperturbed real refractive index -real effective refractive index -real effective group refractive index -electric field second-order refractive index -intensity dependent second-order index of refraction -local electron density m-3

electron carrier density m-3

transparency carrier density m-3

donor density m-3

acceptor density m-3

local hole density m-3

gmgpgLggGGSP GSP

max

h τ( )HIjkvac

kn nktrLmcmhmlhmhhMaven nijn′n′0n′effn'g,effn'2n'2nNN0ND

+

NA-

p˜

App. F List of Symbols and Acronyms

285

hole carrier density m-3

electric polarisation As/m2

, power coupled in and out through one output facet W, bar and cross output power W, splitting ratios of coupler A and B -

scalar and tensor coefficients of linear electrooptic effect -spontaneous emission per volume, time and wavelength m-4s-1

, refractive-index change parameters m-3, m-3μm-1

Schockley-Read-Hall photon rate m-3s-1

total spontaneously emitted photon rate m-3s-1

spontaneously emitted photon rate coupling into one mode m-3s-1

auger recombination rate m-3s-1

amplified spontaneous emission m-3s-1

, facet reflectivities -, bar transmission probability of coupler A and B -

photon density m-3

amplified spontaneous emission photon density m-3

signal photon density m-3

time slattice temperature K

, local electron and hole temperature Kdiode cavity temperature Ksensor temperature Kgroup velocity of light m/sapplied voltage Vwidth of active layer mspace vector mwavelength parametrization parameter m3μm

x,y,z space coordinates m

intensity absorption coefficient m-1

, unperturbed and second order intensity absorption coefficient m-1

internal losses m-1

PPPin PoutP= PXrA rBrijkrspr0 r1RSRHRSpTotRSpRAugerRASER1 R2sA sBSSASESSigtTLTn

˜Tp

˜Tcav

Tsens

vgVwxz0

αα0 α2αi

Appendix

286

mirror losses average over a device m-1

total losses m-1

, modal and active-layer alpha or linewidth-enhancement factor -, , , alpha factor parametrization parameters -

, modal and active-layer temperature factor -coupling probability of spont. emission into the waveguide -coupling factor of tot. spont. emission into one lasing mode -confinement factor -permittivity -nonlinear gain suppression factor due to carrier heating m3

nonlinear gain suppression factor due to spectral hole burning m3

nonlinear gain suppression factor m3

internal quantum efficiency -external quantum efficiency -vacuum wavelength m

, , wavelength at peak, at begin of the zero-gain region, at bandgap mcarrier mobility m2V-1s-1

frequency chirp Hzcarrier density m-3

, Hakki-Paoli and Cassidi terms - effective carrier lifetime s

differential carrier lifetime s, carrier-carrier scattering relaxation time of electrons and holes s, carrier-phonon scattering relaxation time of electrons and holes s

phase of an optical field -, phase-shift due to current effects, due to control-signal nonlinearities

phase-shift induced from a phase-shifter -, , linear, second and third-order susceptibilities -

complex effective optical susceptibility -, real and imaginary part of effective optical susceptibility -

electrical potential V, frequencies of the optical fields in radian 1/s

αmαtotαN αnαN0 αN1 αN2 αN3αT αtββ∗

Γε∈CH∈SHB∈totηiηextλλp λz λ0μΔνρρhp ρcasτeτdτ2 n

˜, τ2 p

˜,

τh n˜

, τh p˜

,ϕΔϕI ΔϕC

Δφχ 1( ) χ 2( ) χ 3( )

χeffχ′eff χ″effψω ω′

App. F List of Symbols and Acronyms

287

Acronymsa.c. alternating currentcw continuous waveAOS All-Optical SwitchASE Amplified Spontaneous EmissionBGR Bandgap Renormalization (Bandgap Shrinkage)CH Carrier HeatingDFG Difference Frequency GenerationDOMO Dual Order ModeEDFA Erbium Doped Fibre AmplifierFCA Free Carrier AbsorptionFP Fabry-PerotFSR Free Spectral RangeFWHM Full Width at Half MaximumFWM Four-Wave MixingKK Kramers-Krönig LD Laser DiodeMI Michelson InterferometerMMI Multimode InterferenceMMI-cc Multimode Interference converter-combinerMZI Mach-Zehnder InterferometerNOLM Nonlinear Optical Loop MirrorOR Optical RectificationOSDM Optical Space Division MultiplexingOTDM Optical Time Division MultiplexingPDFFA Praseodymium-Doped Fluoride Fibre AmplifierPLL Phase Locked LoopRZ Return-to-zero SHB Spectral Hole BurningSHG Second Harmonic GenerationSOA Semiconductor Optical AmplifierSPM Self Phase ModulationTHG Third Harmonic Generation

Appendix

288

TOAD Terahertz Optical DemultiplexerTPA Two-Photon AbsorptionWDM Wavelength Division MultiplexingXGM Cross Gain ModulationXPM Cross Phase Modulation

289

Acknowledgements

The present work was performed in the group of Micro- and Optoelectronics of Prof. Dr. H. Melchior, at the Institute of Quantum Electronics of the Swiss Federal Insti-tute of Technology. First of all I wish to thank Prof. Dr. H. Melchior for the unrestricted academic free-dom he offered me. His scientific advice and his profound knowledge invaluably contributed to this work.Furthermore I would like to thank Prof. Dr. G. Guekos for many encouraging and fruitful discussions, Prof. Dr. F.K. Kneubühl for his readiness to co-examine this the-sis, his efforts to improve my English, for the interesting discussions and for sharing his wisdom beyond the scientific horizon. I am very thankful to Prof. Dr. H. Jäckel for reading the thesis in the shortest possible time. It was always a pleasure to discuss with him at the “prioritaire optiques” meetings.I thank Prof. Dr. U. Keller for allowing access to 3 ps pulses for device characteriza-tion and G. Spühler for providing the pulses.I am much indepted to Dr. P.A. Besse for sharing many of his ideas that contributed to the success of this work. His clear and straightforward thinking opened new doors for a novel methodical approach in my own understanding. Special thanks I owe to all the colleagues providing the technology. E. Gamper and Dr. J. Eckner for processing the all-optical devices and lasers and R. Hess for MMI processing, Dr. E. Gini for wafer growth, device processing and his social activities and Dr. W. Vogt for technological developments.Dr. W. Hunziker is acknowledged for providing fibre arrays. Thanks to his skills we attained highest coupling efficiencies.It is a pleasure to thank M. Dülk for his perseverance in providing short pulses. As an office room mate, I very much appreciated his cheerful nature as well as the pro-found conversations at night times.Throughout the work I benefited from the technical experience of Dr. Ch. Holtmann, Dr. St. Pajarola and R. Dall’Ara. Many measurements could not have been realized without their support. Furthermore, I thank M. Blaser and M. Bitter for many useful hints and for providing the ultra-high speed photodiodes.It was a privilege to meet M. Mayer, Ch. Zellweger and M. Caraccia-Gross and act as adviser of their diploma thesis and to guide X. Comby in his semester work. Many results, that have been elaborated in the frame of these studies, have substantially contributed to the chapter on material characterisations in this thesis.Our Unix administrator E. Wildermuth deserves a special medal for the competent

290

support and his colleague from the PC side, G. Hagn, deserves the second medal for solving the most impossible problems on an impossible operating system.I would like to express my appreciation to A. Müller, Ch. Graf and M. Ebnöter for quick technical support at all times. P. Pfammatter and V. Bürgisser for the arrange-ments of hotels and other administrative things.I thank St. Fischer, Ch. Velez, L. Alimkulo, M. Lanker and R. Annen for many in-teresting discussions.The Swiss research programme in optics is acknowledged for supporting part of the work.Beyond I wish to thank all my friends with whom I spent the spare time within the last years. A. Jung for his constant efforts to urge me to spend more time on the pleasant things of life and his cooking lessons, R. Dangel, M. Von Arx and V. Ziebart and many others for all those happy moments that make life valuable. I should also mention P. Huber, S. Miesler and M. Rutz of the Toggenburg valley with whom I spent many saturday evenings.Finally, I wish to express my deep gratitude towards my parents and my grand-moth-er for their support in all aspects of life. My fiancee, Barbara, merits a special thank for encouraging me to pursue my scientific interests.

Zürich, 25th November, 1998Jürg Leuthold

291

List of Publications

Patent:[1] J. Leuthold, P.A. Besse, M. Bachmann, PCT-Patent:

PCT/CH96/00035

Related to Multimode Interference (MMI):[2] J. Leuthold, R. Hess, J. Eckner, P.A. Besse, H. Melchior;

“Spatial mode filters realized with multimode interference couplers”, Optics Letters, Vol. 21, pp. 836-839, June 1996

[3] J. Leuthold, P.A. Besse, R. Hess, H. Melchior, “Wide Optical Bandwidths of MMI Converter-Combiners”, Proc. of Europ. Conf. on Integr. Optics, ECIO'97, Apr. 1997, pp 154-157

[4] J. Leuthold, J. Eckner, E. Gamper, P.A. Besse, H. Melchior; “Multimode interference couplers for the conversion and combining of zero- and first-order modes”; J. of Lightwave Technol., vol. 16, no. 7, pp. 1228-1239, July 1998

Related to MZI-SOA All-Optical Switches:[5] R. Hess, J. Leuthold, J. Eckner, C. Holtmann, H. Melchior;

“All-optical space switch featuring monolithic InP-waveguide SOA-MZI”, Optical Amplifiers and their Applications, June 1995, vol. 18, pd. PD2

[6] J. Leuthold, P.A. Besse, E. Gamper, M. Dülk, St. Fischer, G. Guekos, H. Mel-chior, "All-Optical Mach-Zehnder Interferometer Wavelength Converters and Switches with Integrated Data- and Control-Signal Separation Scheme", accepted for publication in June 1999 issue of J. Lightwave Technol.

Related to All-Optical Switching with Ideal Extinction Ratios:[7] J. Leuthold, J. Eckner, Ch. Holtmann, R. Hess, H. Melchior;

“All-optical 2x2 switches with 20 dB extinction ratios”, Electronics Letters, Vol. 32, No. 24, pp.2235-2236, Nov. 1996

[8] J. Leuthold, P.A. Besse, H. Melchior; “Optically Controlled Space Switches with Gain and Principally Ideal Ex-tinction Ratios”, Proc. of Europ. Conf. on Integr. Optics, ECIO'97, Apr. 1997, pp 555-558

292

[9] K. Morito, J. Leuthold, H. Melchior; “Dynamic Analysis of MZI-SOA All-Optical Switches for Balanced Switch-ing” Proc. European Conference on Optical Communications ECOC’97, Eding-burgh, pp. 2.81-2.84, Sept. 97

[10] J. Leuthold, P.A. Besse, J. Eckner, E. Gamper, M. Dülk, H. Melchior; “All-optical space switches with gain and principally ideal extinction ratios”; IEEE J. of Quantum Electronics, vol. 34, no. 4, pp. 622-633, April 1998

Related to All-Optical Dual Order Mode Configuration Switch-es:[11] J. Leuthold, J. Eckner, P.A. Besse, G. Guekos, H. Melchior,

“Dual-order (DOMO) all-optical space switch for bidirectional operation”, Proc. of the Conf. on Optical Fiber Comm. OFC'96, Feb. 1996, pp. 271-272

[12] J. Leuthold, E. Gamper, M. Dülk, P.A. Besse, J. Eckner, R. Hess, H. Mel-chior; “Cascadable All-Optical Space Switch with High and Balanced Ext. Ratios”; 2nd Optoel. and Comm. Conf. (OECC'97), July 97, Korea, pp. 184-185

[13] J. Leuthold, P.A. Besse, E. Gamper, M. Dülk, St. Fischer, H. Melchior; “Cascadable dual-order mode all-optical switch with integrated data- and control-signal separators”; Electron. Lett., vol. 34, no. 16, Aug. 1998, pp. 1598-1600

Related to All-Optical Interleaved MZI Configuration Switches:[14] J. Leuthold, P.A. Besse, E. Gamper, M. Dülk, W. Vogt, H. Melchior;

“Cascadable MZI-all-optical switch with separate ports for data- and con-trol-signals”; ECOC’98, Sept. 1998, Madrid, Spain, pp. 463-464

Related to Four-Wave Mixing:[15] J. Leuthold, F. Girardin, P.A. Besse, E. Gamper, G. Guekos, H. Melchior;

“Polarization independent optical ahase conjugation with pump-signal filter-ing in a monolithically integrated Mach-Zehnder interferometer semicon-ductor optical amplifier configuration”, Photon. Technol. Letters, vol. 10, no. 11, pp. 1569-1571, Nov. 1998

Related to Material Research:[16] J. Leuthold, M. Mayer, Ch. Zellweger, J. Eckner, G. Guekos, H. Melchior;

“Optical Material gain of bulk 1.55 μm InGaAsP/InP approximated by pol-

293

ynomial model”, Submitted for publication

293

Curriculum Vitae

1973-85 Elementary and high school

1985-86 Service in the Swiss Army

1986-91 Physics undergraduate studies at the Swiss Federal Institute of Tech-nology

1991 Licence as an instructor in Physics

1992 Laboratory work on Nuclear Magnetic Resonance in the group of Noble price winner Prof. Dr. R.R. Ernst (6 months)

1992-98 Postgraduate studies at the Institute of Quantum Electronics of Swiss Federal Institute of Technology (ETH) Zürich, Switzerland, which is headed by Prof. Dr. H. Melchior. Work on the presented thesis and on related topics.

Jürg LeutholdBorn on July 11th 1966Citizen of Nesslau SG, Switzerland,Married

294

295

Index

Numerics3R regeneration................................. 12

Aa.c. quadratic stark effect .................. 54a.c. stark effect .................................. 52absorption

bleaching...................................... 41free-carrier ............................. 46, 48intervalence-band ........................ 43saturation ..................................... 41two-photon................................... 48

absorption coefficient ....................... 38second-order ................................ 54

add-drop multiplexer .............. 208, 229advantages of

all-optical processing................... 12different separation schemes ..... 244

Aframowitz model .......................... 277all-optical .......................................... 36

device with control sig. separ. ... 225multiplexer................................. 229switch

asymmetric splitters.............. 208generalized............................ 199symmetric splitters

equally biased ...................... 206unequally biased................... 208

two pairs asym. biased SOAs 211versatile...................................... 228wavelength converter................. 229

alpha factor ....................... 62, 103, 201change with

current density ...................... 104photon density ........................ 62temperature........................... 104wavelength............................ 104

parametrization .......................... 114

relation with extinction ratio...... 205amplification

of zero and first-order-mode...... 234amplified

signal recombination term ........... 69spont. emission ............................ 16spont. emission term .............. 69, 70

amplifiererbium-doped fiber (EDFA) ........ 10gain-shifted EDFAs ..................... 10Praseodym-Doped Fluor Fibre .... 10Raman.......................................... 10SOA ............................................. 10

amplifier-gain change ............... 72, 201analysis

See "theory on"asymmetric

SOA current bias ................. 73, 208splitter .................................. 71, 208

attenuation losses ................................ 8Auger recombination ........................ 69

Bbandfilling effect............................... 41bandgap

carrier concentr. dependence ..... 276discontinuities of InGaAsP........ 278energy of InGaAsP .................... 276related refr.-index changes........... 43renormalization (BGR) ................ 44shrinkage................................ 43, 44shrinkage of InGaAsP................ 276temperature dependence ............ 277

bandwidthavailable transmission.................... 8optical of MMI........................... 177

296

Barbara......................................... i, 290See also "the sun shines effect"

beam propagation ........................... 149beam waist ...................................... 150bidirectional operation .............. 17, 227breakthroughs ................................... 10broadening, spectral........................ 270Burstein-Moss effect......................... 41butterfly MMI ................................. 160

Ccalculation

extinction ratio........................... 204material gain .............................. 266MMI bandwidth......................... 178MMI mode decomposition ........ 132MMI-WDM ............................... 141nonlinear effects .......................... 40nonlinear gain compression....... 273

carrier densityfunction for local - ..................... 273versus current............................. 119

carrier heating (CH)............ 14, 47, 273carrier mobility ................................. 68carrier-related refr.-index change ..... 97cascadability of several devices...... 227Cassidy method................................. 85causality ............................................ 39characterization of

all-optical devices...................... 236MMI bandwidth......................... 181MMI-converter-combiner.......... 161MMI-filter.................................. 173MMI-WDM ............................... 143

chirp ...................................... 16, 17, 63classification of

MMI........................................... 127nonlinearities ............................... 13

clock recovery................................... 12coefficient

free carrier absorption................ 273

comparison of all-optical devices ... 244confinement factor

definition...................................... 61determination ............................... 89

continuity equation ........................... 68control signal

extraction ................................... 227influence of

control-signal wavelength .... 243input-signal power ................ 243input-signal wavelength........ 243

jiter............................................. 194power ......................... 245, 246, 247

copper coaxial cable ........................... 8co-propagating signals .... 227, 228, 244counter-propagating signals............ 228coupling

efficiency ................................... 235factor of sp. emis. into wg ........... 70length of MMIs.......................... 127

cross-gain modulation........... 13, 16, 40See also "XGM"

cross-phase modulation .. 13, 15, 40, 53extinction ratio enhancement..... 196See also "XPM"

crosstalk ofcontrol-signal in all-optical........ 244MMI-WDM ............................... 144

current density vector ....................... 68

Dd.c. stark effect.................................. 52definition

alpha factor .................................. 62confinement factor....................... 61effective group refr. index ........... 61extinction ratio........................... 203group velocity .............................. 61on-off ratio ................................. 219

degenerate fourwave mixing............. 52density of states......................... 42, 268

297

design tolerancesasymmetric MZI-SOA............... 220MMI........................................... 177

determination ofSee experiments on

dielectric constant of InGaAsP ....... 278difference-frequency generation. 13, 51differential carrier lifetime........ 57, 118differential equation

See "equation"digital optical switches ................... 147dimension

all-optical devices .............. 214, 237MMI........................... 133, 142, 182MMI-converter-combiner.......... 162MMI-filter.................................. 173MMI-WDM ............................... 142

dipole matrix element ............... 42, 266dipole-dephasing time..................... 270directional coupler .......................... 136disadvantage

of dispersion shifted fibres ........ 249dispersion.......................................... 16

chromatic ................................... 249compensation methods .............. 249conduction band......................... 268group velocity .............................. 61tolerance acceptable..................... 11

drift diffusion coefficient.................. 68Drude model ..................................... 46dual-order mode configuration . 22, 230

Eeffect

a.c. quadratic Stark effect ............ 54a.c. stark....................................... 52bandfilling.................................... 41bandgap and refr. index ............... 43Burstein-Moss.............................. 41d.c. stark....................................... 52electroabsorption ......................... 52

electrooptic Kerr .......................... 52electrorefractive ........................... 52Franz-Keldysh ............................. 52linear electrooptic ........................ 50optical-Kerr.................................. 52plasma.......................................... 46Pockels......................................... 50pressure and bandgap................... 43quadratic-electrooptic .................. 52Talbot......................................... 126temperature and bandgap............. 43

effectivearea............................................. 281carrier lifetime ............................. 57dielectric function ........................ 65group refractive index.......... 61, 102masses in InGaAsP .................... 277refractive index ............................ 61susceptibility ................................ 38

Einstein relation ................................ 70electric field .................................... 281

inside the amplifier ...................... 65electroabsorption

effect ............................................ 52modulator..................................... 11

electrooptic Kerr effect ..................... 52electrorefractive effect ...................... 52energy

bandgap...................................... 266of InGaAsP ........................... 276

conduction ................................. 266photon transition ........................ 266split-off ...................................... 267

of InGaAsP ........................... 277valence ....................................... 266

equationall-optical

extinction ratio...................... 203on-off ratios .......................... 219

differentialcontinuity................................ 68

298

continuity (local carrier)....... 273Helmholtz ............................... 36mode-profile ........................... 69phases of signals............... 66, 69photon density .................. 66, 69temperature (local) ............... 273wave equation......................... 37

material gain .............................. 271in terms of local carriers....... 273

MMIbandwidth ............................. 178length .................................... 127phases ................................... 127

Varshni ...................................... 277excess loss

MMI................... 144, 163, 174, 181experiments on

all-optical extinction ratios ........ 214all-optical separation schemes... 236alpha factor ................ 104, 106, 216losses in LD and SOAs................ 87material-gain................................ 90MMI optical bandwidth............. 181MMI-converter-combiner.. 161, 163MMI-filter.................................. 173MMI-WDM ............................... 143refractive-index change ............... 95

external modulation .......................... 11external quantum efficiency ..... 64, 119extinction ratio ............ 16, 17, 237, 245

definition.................................... 203enhancement .............................. 196limitations .................................. 194

extractioncontrol-signal ............................. 232

Ffactor

alpha .................................... 62, 103band nonparabolicity ................. 268coupling of sp. emission .............. 70

linewidth-enhancement........ 62, 103nonlinear gain suppression .. 49, 274population-inversion............ 70, 269

Fermi function .......................... 42, 269Fermi’s golden rule......................... 266fibre

nondispersion-shifted................... 14fibre-to-fibre gain...................... 10, 234filter

control-signal ............................. 232external wavelength................... 230monolithically integrateable ...... 230

filteringpump signal................................ 249

fit functionSee also "parametrization"......... 108

formulaSee "equation"

fourwave mixing ,13, 16, 40, 230, 244, 249degenerate.................................... 52nondegenerate.............................. 52

Franz-Keldysh effect ........................ 52free spectral range............................. 84free-carrier absorption................. 46, 48

Ggain

integrated ..................................... 67material ................................ 90, 266modal ........................................... 84net-modal ............................... 85, 86single pass.................... 67, 200, 281

gain compression ................ 14, 47, 273CH................................................ 47factors .......................................... 49phenomenol. expression .............. 48SHB ............................................. 47

gain of zero- and first-order modes. 234gain saturation........................... 14, 206Gaussian beam waist....................... 150general MMI ................................... 127

299

geometrySee "dimension"See "structure"

group delay difference .................... 235group velocity ................................... 61

HHakki-Paoli method .................... 82, 85harmonic mode-locking .................... 11Hermite-Gauss mode .............. 179, 186hi-fi music........................................... 8high-definition TV images.................. 8

Iindex of refraction

second order................................. 54InGaAsP material parameters ... 81, 276input image

antisymmetric ............................ 129symmetric .................................. 129

integratedfiltering mechanism ................... 249signal separation ........................ 225

intensityin terms of electric fields ........... 281in terms of photon density ......... 281

interband-relaxation times ................ 57interferometric config. ...... 15, 196, 225interleaved MZI-configuration ....... 235internal losses.................................... 88internal quantum efficiency ...... 64, 119intraband-relaxation time.......... 58, 270

KKane-dipole matrix element ..... 55, 268Kerr effect ................................... 14, 18Kramers-Krönig relation........... 39, 100k-selection rule................................ 268

Llaser diode

structure ....................................... 83

lattice match condition.................... 276law of addition of recipr. lifetimes.... 57lifetime

differential carrier ................ 57, 118effective carrier............................ 57law of reciprocal addition ............ 57See also relaxation time

linear electrooptic effect ................... 50lineshape function ........................... 270linewidth-broadening

function ................................ 42, 270of modes ...................................... 63

linewidth-enhancement factor... 62, 103losses

internal ......................................... 88parametrization ..................... 112

mirror ..................................... 85, 88total .............................................. 88

MMach-Zehnder interferometer... 15, 225

asymmetric................................. 196dual-order mode......................... 230interleaved ........................... 16, 235

materialKerr .............................................. 13parameters of InGaAsP........ 81, 276second-order nonlinear ................ 13SOA ............................................. 13

material dispersion.......................... 141material gain ................................... 266

calculated ..................................... 42comparis. exp theory.................... 91measured ...................................... 90of InGaAsP .................................. 82parametrization .......................... 108

material parameterInGaAsP....................... 81, 276, 280

matrix elementdipole ......................................... 266Kane..................................... 55, 268

300

millimeter-wave systems .................... 8mirror losses ............................... 85, 88MMI................................................ 125

butterfly ..................................... 160converter-combiner.... 148, 177, 230converter-combiner generalized 167coupling length .................. 127, 171design guidelines ....................... 132design tolerances ....................... 177filter ........................... 156, 170, 230length ......................................... 171

expression..................... 127, 139figure of merit....................... 139

mode decompositions ................ 132non-uniform power-split. ratio .. 158phase relations ........................... 129structure ............................. 133, 162theory......................................... 126trap waveguide........................... 135types

general MMI......................... 127overlap MMI......................... 129restricted interference MMI.. 130symmetric interference MMI 130

WDM......................................... 136WDM add-drop device .............. 145width

consideration ........................ 132equivalent ............................. 127geometric .............................. 127

modal gain ........................................ 84mode

first-order ................... 150, 177, 230zero-order .................. 148, 177, 230

mode conversion..................... 147, 150generalized................................. 167

mode decoupler............................... 156mode filter, spatial .......................... 170mode separation.............................. 156mode spacing

longitudinal................................ 102

mode splitterTE/TM ....................................... 147

mode-analysis calculation 132, 142, 180mode-locked laser diode ................... 11monolithically integr. signal separ.. 225multi-channel operation .................... 17multimode-interference................... 125

See also "MMI"multiplexer ................................ 12, 228MZI-SOA

asymmetric configurations ........ 205dual-order mode......................... 230generalized all-optical switch .... 199interleaved ......................... 235, 251transfer function........................... 71

Nnet-modal gain .................................. 85noise

figure............................................ 10from polarization fluctuations ... 194from power fluctuations............. 194reduction ...................................... 16signal-shot.................................. 194signal-spontaneous-beat............. 194spontaneous-emission shot ........ 194spontaneous-spontaneous beat... 194

nondegenerate fourwave mixing....... 52nonlinear

effects........................................... 40gain compression ......................... 47gain-suppression parameter ....... 274optics...................................... 13, 36second-order refr. index............... 14wave

equation .................................. 36solution ................................... 37

Nonlinear Optical Loop Mirror ........ 18nonlinearity ....................................... 14

intraband ...................................... 16second order................................. 50

301

third-order.................................... 52nonparabolicity of bands................. 268

Ooccupation probabilities in bands ... 269on-off ratio ...................................... 237

definition.................................... 219opening window........ 12, 217, 241, 242optical

bandwidthfibre .......................................... 8MMI...................................... 177SOA ........................................ 16

Kerr effect.................................... 52logic ............................................. 12space-division multiplexing........... 9susceptibility................................ 37time-division multiplexing ............ 9

optical phase conjugation ............... 249output power versus current.... 116, 119output spectrum ........................ 84, 252overlap

of zero- and first-order modes ... 233overlap MMI................................... 129

Pparametrization ............................... 108

alpha factor ................................ 114internal-loss ............................... 112material-gain.............................. 108refractive-index change ............. 113

penalty-free ....................................... 16performance of all-optical devices.. 240phase change equation ...................... 66phase conjugation ..................... 16, 249phase offset ..................................... 200phased array .............................. 10, 136phase-shift

induced in SOA ......................... 201photon transition energy ................. 266plasma effect ..................................... 46

Pockels effect .................................... 50polarization

independent all-optical ........ 17, 244independent FWM ..................... 249insensitive .................................... 16

polarization of a material .................. 37population-inversion factor....... 70, 269power

control signal 214, 243, 245, 246, 247input signal ........................ 216, 246switching.................................... 235

pump power ...................................... 17

Qquadratic-electrooptic effect ............. 52quantum efficiency

external ........................................ 64internal ................................. 64, 118

quasi-Fermi levels..................... 70, 269

Rrate equation

carrier........................................... 68for local

carriers .................................. 273temperature ........................... 273

photon density.............................. 69real-time media ................................... 8recombination term of

amplified signals.......................... 69ASE.............................................. 69Auger ........................................... 69Schockley-Read-Hall................... 68spontaneous band-to-band ........... 68

refractive indexcomplex ....................................... 38effective ....................................... 61effective group..................... 61, 102real ............................................... 38

refractive index of InGaAsP ........... 278

302

refractive-index changea.c. quadratic Stark effect ............ 54all effects ..................................... 59bandgap shrinkage ....................... 45carrier-related ........................ 14, 97gain compression ......................... 49linear electrooptic ........................ 50parametrization .......................... 113Plasma effect ............................... 47temperature-related.................... 101

refractive-index change expressiondue to control signal .................... 72due to the current bias.................. 72intensity dependent ...................... 54second-order .......................... 50, 54

refractive-index change in InGaAsP. 95relaxation time

interband ...................................... 57intraband .............................. 58, 270See also "lifetime"temperature ................................ 273

required transmission capacity ........... 8reshaping........................................... 17restricted interference MMI............ 130

SSagnac interferometer....................... 18satellites .............................................. 8saturation

absorption .................................... 41gain .............................................. 41

scatteringBrillouin................................. 13, 40carrier-carrier ............................... 47carrier-phonon ............................. 48Raman.................................... 13, 40Rayleigh....................................... 40

Schockley-Read-Hall recombination 68second-harmonic generation (SHG) . 51self images ...................................... 126self-phase modulation................. 40, 52

semiconductor optical amplifierequations...................................... 61structure ....................................... 83

separationefficiency ................................... 237

simulation ....................................... 108MMI........................................... 157

single-pass gain................. 67, 200, 281spectral hole burning........... 14, 47, 273spectrum

of SOA......................................... 84split-off energy of InGaAsP............ 277spontaneous band-to-band recomb. .. 68spontaneous-emission term............... 69state-of-the-art all-optical devices .... 13storage function ................................ 12structure

all-optical devices ...................... 237MMI........................... 137, 162, 178MMI-filter.................................. 173MZI-SOA switch ....................... 214zinc-blende................................. 276

sub-picosecond range...................... 226sum-frequency generation .......... 40, 51suppression of higher-order mode .. 170susceptibility

effective ....................................... 53optical .................................... 37, 39second-order .............................. 276

switch .............................................. 228switching........................................... 12

power ......................................... 235window

See "opening window"symmetric interference MMI.......... 130symmetry

centre ......................................... 276modes................................. 147, 230

TTalbot effect.................................... 126

303

technological landmarks ................... 13temperature

carrier......................................... 273cavity ......................................... 115correction for device.................. 115effect on bandgap ........................ 45function for local ....................... 273lattice ......................................... 273refr.-index change...................... 101sensor ......................................... 115

terabit-per-second transmission .......... 8theory on

all-optical devices ........................ 35MMI........................................... 126

lengths .................................. 126MMI-converter-combiner..... 148MMI-WDMs ........................ 137optical bandwidth ................. 178phases ................................... 127

MZI extinction ratios ................. 203MZI-SOA transfer function ....... 199nonlinearities ............................... 36semiconductor optical amplifier .. 61

thermal resistance of device............ 115third-harmonic generation .......... 40, 52time constants of

nonlinear effect ............................ 56total losses......................................... 88transceiver....................................... 136transfer function

MZI-SOA ............................ 71, 200transfer matrix........................... 72, 199transmission

losses in MMI-WDMs............... 144probabilities in switches ............ 200probabilities of splitters ............. 201

transparent networks......................... 12trap waveguide................................ 135TV images........................................... 8two-photon absorption ...................... 48

two-section SOA............................. 235

UUrbach tail....................................... 270

VVarshni equation ............................. 277versatile all-optical devices............. 228video camera images....................... 164video images ....................................... 8virtual states ...................................... 40voice signal ......................................... 8voltage-versus-current curve........... 116

Wwave equation ............................. 36, 65wavelength conversion ....... 12, 16, 228wavelength of

control signal ............................. 247input signal ................................ 245

wavelength-division multiplexing ...... 9width

equivalent MMI ......................... 127geometric MMI.......................... 127

Wireless video-broadcast .................... 8

XXGM ............................................... 225

See also cross-gain modulation . 225XPM................................................ 225

See also cross-phase modulation 225

YY-junction, asymmetrical........ 136, 147

Zzinc-blende structure....................... 276