Birefringent Thin Films BffiLPolarizing

Elements

Ian J Hodgkinson Qi hong Wu Department of Physics University of Otago New Zealand

Vfe World Scientific * f l l Sinqapore» New Jersey *L Singapore • New Jersey • London • Hong Kong

Contents

List of Tables xv

List of Figures xvii

Glossary xxiii

1 Introduction 1

1.1 Structural Classification of Crystals 2 1.2 Optical Classification of Crystals 2 1.3 Structure of Birefringent Films 4 1.4 Optical Classification of Birefringent Films 5 1.5 Layout of the Book 7

1 Propagation in Biaxial Media 9

2 Propagation Equations 11 2.1 Maxwell's Equations 12 2.2 Propagation in Free Space. Mathematical Methods 12

2.2.1 SI units 16 2.3 Propagation in Isotropic Media 17 2.4 Propagation in Anisotropic Media 18 2.5 Energy Flow 21 2.6 Notation for Biaxial Media 22

2.6.1 Material Axes 22 2.6.2 Propagation Axes 24 2.6.3 Rotations 24 2.6.4 Computations 26

2.7 Propagation in a Common Direction in a Biaxial Medium . . . . 26 2.7.1 Maxwell's Equations 26 2.7.2 Eresnel's Equation 27

vii

viii CONTENTS

2.7.3 Eigenequations for Normalized Fields 28

3 Basis Vectors 31 3.1 Partially Coherent States 32

3.1.1 Coherence 32 3.1.2 Stokes Parameters 33 3.1.3 Stokes Vectors 34 3.1.4 Degree of Polarization 35 3.1.5 Unpolarized Light 35 3.1.6 Partially Polarized Light 35 3.1.7 Polarized Light 36 3.1.8 Basis Vectors 36

3.2 Coherent States 38 3.2.1 Jones Vectors 38 3.2.2 Elliptical Polarization 38 3.2.3 Circular Polarization 39 3.2.4 Linear Polarization 40 3.2.5 Basis Vectors 40 3.2.6 Photons 41 3.2.7 Ellipsometric Parameters 41

3.3 Propagation in Layered Biaxial Media 43 3.3.1 Fresnel's Quartic Equation 43 3.3.2 Propagation in the Deposition Plane 44 3.3.3 Uniaxial Media 45 3.3.4 Isotropic Media 45 3.3.5 Basis Travelling Wave Fields 46 3.3.6 Power 48

4 Transfer Matrices 49 4.1 Mueller Calculus 50

4.1.1 Rotated Elements 50 4.1.2 Elements in Series 51 4.1.3 Mueller Calculus Computations 52

4.2 Jones Calculus 52 4.2.1 Linear Polarizer 54 4.2.2 Retardation Plate 54 4.2.3 Quarter-Wave Plate 54 4.2.4 Rotated Elements 55 4.2.5 Elements in Series 55 4.2.6 Periodic Arrangements 56

CONTENTS ix

4.2.7 Jones Calculus Computations 56 4.3 Relationship of Mueller and Jones Calculus 56 4.4 Berreman Calculus 57

4.4.1 Field Matrix F 57 4.4.2 Field Coefficients a 58 4.4.3 Total Field m 59 4.4.4 Phase Matrix Ad 59 4.4.5 Characteristic Matrix M 60 4.4.6 System Matrix A 62 4.4.7 Properties of M 62 4.4.8 Computation of Film Parameters from M 63

4.5 Abeles and Heavens Calculus 65 4.5.1 Isotropic Layer 65 4.5.2 Deposition Plane 67 4.5.3 Berreman Calculus Computations 69

4.6 Relationship of Jones and Berreman Calculus 74 4.6.1 Jones Matrix with Interference 74 4.6.2 Jones Matrix with Reflections but without Interference . 75

5 Reflection and Transmission 77 5.1 General Case - All Media Biaxial 78

5.1.1 Crystal-Crystal Interface 81 5.2 Sorting Columns of F 81 5.3 Isotropic Cover and Substrate 85

5.3.1 Amplitude Reflection and Transmission Coefficients . . . 87 5.3.2 Irradiance Reflectance Coefficients 88

5.4 All Media Isotropic 89 5.4.1 Phase Changes on Reflection and Transmission 90

5.5 Computations Using the BTF Toolbox 90 5.5.1 General Birefringent Coating 90 5.5.2 PS Coatings 91

6 Guided Waves 93 6.1 Modal Condition 94

6.1.1 General Case 94 6.1.2 Isotropic Cover and Substrate 95 6.1.3 Uncoupled Modes 97 6.1.4 Poles of R 98 6.1.5 Examples 98

6.2 Modal Cutoffs 100

x CONTENTS

6.3 Modal Contours 100 6.4 Modal Field Structure 104 6.5 Modal Polarization 106 6.6 Modal Overlap 108 6.7 Modal Order I l l 6.8 Power Flow I l l 6.9 Prism Couplers 112

II Characterization of Anisotropic Films 115

7 Deposition of Microstructures 117 7.1 Vacuum Deposition 118

7.1.1 Apparatus 118 7.1.2 Deposition Parameters 119

7.2 Columnar Structures and Effective Media 119 7.2.1 Uniaxial Media 119 7.2.2 Biaxial Media 119 7.2.3 Effective Anisotropic Media 123 7.2.4 Zig-Zag and Wavy Anisotropic Media 123 7.2.5 Helical Microstructures 125

7.3 Computer Modelling of Deposition 127 7.3.1 Serial Deposition of Hard Spheres 127 7.3.2 Visual Analysis of Simulations 127 7.3.3 Radial Distribution Function 128 7.3.4 Two-Dimensional Angular Distribution 131 7.3.5 Column Angle 131 7.3.6 Birefringence 131 7.3.7 Conclusions from Simulations of Deposition 131

8 Form Birefringence 135 8.1 Perpendicular Incidence Ellipsometry 136

8.1.1 Computation of Ellipsometric Parameters 136 8.1.2 Characteristic Ellipsometric Curves 138 8.1.3 Experimental Values 144

8.2 Measurement of Principal Refractive Indices 144 8.2.1 In Situ Measurements 145 8.2.2 Use of Narrowband Filters 146 8.2.3 Photometric Method 147 8.2.4 Waveguide Method 147

CONTENTS xi

8.3 Modelling Form Birefringence 149 8.3.1 Bragg-Pippard Equations 149 8.3.2 Inversion of the Bragg-Pippard Equations 151

9 Effective Media 153 9.1 Herpin Indices for Isotropic Layers 154 9.2 Biaxial Layers with a Common Deposition Plane 155

9.2.1 A and В Normal Columnar 157 9.2.2 A and В Parallel, Tilted Columnar 159 9.2.3 A and В Coplanar, Tilted Columnar with фА = —фв • • 162

9.3 Biaxial Layers Deposited in Different Planes 164

10 Anisotropic Scatter 167 10.1 Scatter into the Air 168 10.2 Scatter From Stress-Related Cracks 169 10.3 Scatter Patterns Formed on the Film 174 10.4 Scatter into the Substrate 175 10.5 In Situ Measurement of Scatter 175

10.5.1 Dependence of Haze on Д 177 10.5.2 Haze from Herring-Bone Stacks 177

10.6 Simple Theory of Scatter 180

11 Fluid Transport 183 11.1 Fluid Patches 184

11.1.1 Recording Fluid Patches 185 11.1.2 MDM Narrowband Filters 185

11.2 Scatter from Fluid Patches 188 11.2.1 Scatter Anisotropy 188 11.2.2 Theory of Scatter 191 11.2.3 General AR Coating 194 11.2.4 High Reflectance Coating 194 11.2.5 Narrowband Interference Filter 194

11.3 Influence on Birefringence 197 11.3.1 Change of Birefringence in Fluid Patches 197 11.3.2 Principal Refractive Indices 201 11.3.3 Cooling and Venting 202

12 Metal Films 203 12.1 Growth and Post-Deposition Sputter Etching 204 12.2 Direct Recording of Optical Anisotropies 206

12.2.1 Silver and Gold 209

xii CONTENTS

12.2.2 Aluminium 209 12.2.3 Aging 216 12.2.4 Argon Ion Sputter Etching 217

12.3 Computer Modelling of Anisotropy in Metals 217 12.3.1 Bulk Metals 221 12.3.2 Depolarization Factors 222 12.3.3 Isotropic Resonance 223 12.3.4 Anisotropic Resonance 225

12.4 Modelling Deposition and Etching 227 12.4.1 Simulated Deposition of Gold 228 12.4.2 Simulated Deposition of Silver 231 12.4.3 Simulated Deposition of Aluminium 231 12.4.4 Simulated Deposition / Etch Paths 231

12.5 Summary 236

III Applications of Birefringent Media 237

13 Linear polarizers 239 13.1 Real Polarizers 240 13.2 Dichroic Polarizers 241 13.3 Tilted Plate and Thin Film Polarizers 243

13.3.1 Plate Polarizers 243 13.3.2 Coated-Plate Polarizers 244 13.3.3 Embedded Thin Film Polarizers 246 13.3.4 Birefringent Fabry-Perot Polarizing Filter 247

13.4 Crystalline Prism Polarizers 251 13.4.1 Glan-Foucault Prism 252 13.4.2 Wollaston Prism 253 13.4.3 Rochon Prism 253

14 Phase Retarders 255 14.1 Crystalline Wave Plates 256

14.1.1 Quartz and Magnesium Fluoride 256 14.1.2 Multiple-Order Wave Plates 257 14.1.3 Zero-Order Wave Plate 259 14.1.4 Achromatic Wave Plates 259 14.1.5 Wide-Field Elements 264 14.1.6 Variable Phase Compensators 265

14.2 Birefringent Thin Film Analogues 267

xiii

14.2.1 Thin Film Wave Plates 267 14.2.2 Thin Film Babinet Compensator 270 14.2.3 Thin Film Soleil-Babinet Compensator 270 14.2.4 Thin Film Berek Compensator 271

15 Birefringent Filters 273 15.1 Polarization State Filters 274

15.1.1 Linear Polarizer 274 15.1.2 Circular Polarizer 274 15.1.3 Rotator 274 15.1.4 Depolarizer 275

15.2 Wavelength Filters 276 15.2.1 Lyot-Ohman Filter 276 15.2.2 Sole Filters 279 15.2.3 Filters for Tuning Dye Lasers 282

16 Birefringent Coatings 289 16.1 Isotropic Coatings 290 16.2 General Birefringent Coating 290 16.3 PS Coatings 294 16.4 Design Considerations for PS Coatings 294

16.4.1 Making an Anisotropic Version 296 16.4.2 Replacing an Intermediate Index 299 16.4.3 Identical Response Profiles Separated in Wavelength . . 299 16.4.4 Spoiling the s-response 299

16.5 Normal and Hybrid Monitoring 300 16.6 PS Sampler 302

16.6.1 Anisotropic Antireflection Coating 302 16.6.2 Anisotropic Reflector 304 16.6.3 Anisotropic-Phase Reflector 306 16.6.4 Achromatic Antireflection Coating 308 16.6.5 Achromatic Fifty Percent Reflector 310 16.6.6 Single-Cavity Narrowband Filter 312 16.6.7 Multi-Cavity Narrowband Filter 314 16.6.8 Edge Filter 316 16.6.9 Common-Index Thin Film Polarizer 318 16.6.10 Multi-Cavity Linear Polarizer 320

xiv CONTENTS

A Birefringent Thin Films Toolbox 323 A.l Quick Reference 324 A.2 Commands and Functions 327

Notes and References 365

Index 371