Table of Contents

Series Editor's Foreword xiii

Preface xv

Acknowledgments xix

List of Abbreviations xxi

About the Companion Website xxiii

1 Polarization of Monochromatic Waves. Background of the Jones Matrix Methods. The Jones Calculus 1

1.1 Homogeneous Waves in Isotropic Media 1

1.1.1 Plane Waves 1

1.1.2 Polarization. Jones Vectors 3

1.1.3 Coordinate Transformation Rules for Jones Vectors. Orthogonal Polarizations. Decomposition of a Wave into Two Orthogonally Polarized Waves 9

1.2 Interface Optics for Isotropic Media 14

1.2.1 Fresnel's Formulas. Snell's Law 14

1.2.2 Reflection and Transmission Jones Matrices for a Plane Interface between Isotropic Media 20

1.3 Wave Propagation in Anisotropic Media 23

1.3.1 Wave Equations 23

1.3.2 Waves in a Uniaxial Layer 25

1.3.3 A Simple Birefringent Layer and Its Principal Axes 30

1.3.4 Transmission Jones Matrices of a Simple Birefringent Layer at Normal Incidence 32

1.3.5 Linear Retarders 36

1.3.6 Jones Matrices of Absorptive Polarizers. Ideal Polarizer 38

1.4 Jones Calculus 41

1.4.1 Basic Principles of the Jones Calculus 42

1.4.2 Three Useful Theorems for Transmissive Systems 46

1.4.3 Reciprocity Relations. Jones's Reversibility Theorem 50

1.4.4 Theorem of Polarization Reversibility for Systems Without Diattenuation 53

1.4.5 Particular Variants of Application of the Jones Calculus. Cartesian Jones Vectors for Wave Fields in Anisotropic Media 55

References 57

2 The Jones Calculus: Solutions for Ideal Twisted Structures and Their Applications in LCD Optics 59

2.1 Jones Matrix and Eigenmodes of a Liquid Crystal Layer with an Ideal Twisted Structure 59

2.2 LCD Optics and the Gooch–Tarry Formulas 64

2.3 Interactive Simulation 67

2.4 Parameter Space 69

References 73

3 Optical Equivalence Theorem 75

3.1 General Optical Equivalence Theorem 75

3.2 Optical Equivalence for the Twisted Nematic Liquid Crystal Cell 77

3.3 Polarization Conserving Modes 77

3.3.1 LP1 Modes 78

3.3.2 LP2 Modes 79

3.3.3 LP3 Modes 80

3.3.4 CP Modes 81

3.4 Application to Nematic Bistable LCDs 82

3.4.1 2pi Bistable TN Displays 82

3.4.2 Pi Bistable TN Displays 83

3.5 Application to Reflective Displays 84

3.6 Measurement of Characteristic Parameters of an LC Cell 86

3.6.1 Characteristic Angle Omega 86

3.6.2 Characteristic Phase Gamma 87

References 87

4 Electro-optical Modes: Practical Examples of LCD Modeling and Optimization 91

4.1 Optimization of LCD Performance in Various Electro-optical Modes 91

4.1.1 Electrically Controlled Birefringence 91

4.1.2 Twist Effect 101

4.1.3 Supertwist Effect 109

4.1.4 Optimization of Optical Performance of Reflective LCDs 116

4.2 Transflective LCDs 119

4.2.1 Dual-Mode Single-Cell-Gap Approach 119

4.2.2 Single-Mode Single-Cell-Gap Approach 122

4.3 Total Internal Reflection Mode 124

4.4 Ferroelectric LCDs 131

4.4.1 Basic Physical Properties 131

4.4.2 Electro-optical Effects in FLC Cells 135

4.5 Birefringent Color Generation in Dichromatic Reflective FLCDs 145

References 149

5 Necessary Mathematics. Radiometric Terms. Conventions. Various Stokes and Jones Vectors 153

5.1 Some Definitions and Relations from Matrix Algebra 153

5.1.1 General Definitions 153

5.1.2 Some Important Properties of Matrix Products 160

5.1.3 Unitary Matrices. Unimodular Unitary 2 x 2 Matrices. STU Matrices 160

5.1.4 Norms of Vectors and Matrices 163

5.1.5 Kronecker Product of Matrices 166

5.1.6 Approximations 167

5.2 Some Radiometric Quantities. Conventions 167

5.3 Stokes Vectors of Plane Waves and Collimated Beams Propagating in Isotropic Nonabsorbing Media 169

5.4 Jones Vectors 171

5.4.1 Fitted-to-Electric-Field Jones Vectors and Fitted-to-Transverse-Component-of-Electric-Field Jones Vectors 171

5.4.2 Fitted-to-Irradiance Jones Vectors 172

5.4.3 Conventional Jones Vectors 175

References 176

6 Simple Models and Representations for Solving Optimization and Inverse Optical Problems. Real Optics of LC Cells and Useful Approximations 177

6.1 Polarization Transfer Factor of an Optical System 178

6.2 Optics of LC Cells in Terms of Polarization Transport Coefficients 182

6.2.1 Polarization-Dependent Losses and Depolarization. Unpolarized Transmittance 185

6.2.2 Rotations 187

6.2.3 Symmetry of the Sample 190

6.3 Retroreflection Geometry 192

6.4 Applications of Polarization Transport Coefficients in Optimization of LC Devices 195

6.5 Evaluation of Ultimate Characteristics of an LCD that can be Attained by Fitting the Compensation System. Modulation Efficiency of LC Layers 207

References 216

7 Some Physical Models and Mathematical Algorithms Used in Modeling the Optical Performance of LCDs 217

7.1 Physical Models of the Light–Layered System Interaction Used in Modeling the Optical Behavior of LC Devices. Plane-Wave Approximations. Transfer Channel Approach 217

7.2 Transfer Matrix Technique and Adding Technique 237

7.2.1 Transfer Matrix Technique 238

7.2.2 Adding Technique 242

7.3 Optical Models of Some Elements of LCDs 246

References 248

8 Modeling Methods Based on the Rigorous Theory of the Interaction of a Plane Monochromatic Wave with an Ideal Stratified Medium. Eigenwave (EW) Methods. EW Jones Matrix Method 251

8.1 General Properties of the Electromagnetic Field Induced by a Plane Monochromatic Wave in a Linear Stratified Medium 252

8.1.1 Maxwell's Equations and Constitutive Relations 252

8.1.2 Plane Waves 256

8.1.3 Field Geometry 259

8.2 Transmission and Reflection Operators of Fragments (TR Units) of a Stratified Medium and Their Calculation 275

8.2.1 EW Jones Vector. EW Jones Matrices. Transmission and Reflection Operators 275

8.2.2 Calculation of Overall Transmission and Overall Reflection Operators for Layered Systems by Using Transfer Matrices 281

8.3 Berreman’s Method 283

8.3.1 Transfer Matrices 283

8.3.2 Transfer Matrix of a Homogeneous Layer 285

8.3.3 Transfer Matrix of a Smoothly Inhomogeneous Layer. Staircase Approximation 287

8.3.4 Coordinate Systems 289

8.4 Simplifications, Useful Relations, and Advanced Techniques 291

8.4.1 Orthogonality Relations and Other Useful Relations for Eigenwave Bases 291

8.4.2 Simple General Formulas for Transmission Operators of Interfaces 297

8.4.3 Calculation of Transmission and Reflection Operators of Layered Systems by Using the Adding Technique 303

8.5 Transmissivities and Reflectivities 304

8.6 Mathematical Properties of Transfer Matrices and Transmission and Reflection EW Jones Matrices of Lossless Media and Reciprocal Media 311

8.6.1 Properties of Matrix Operators for Nonabsorbing Regions 311

8.6.2 Properties of Matrix Operators for Reciprocal Regions 313

8.7 Calculation of EW 4 x 4 Transfer Matrices for LC Layers 319

8.8 Transformation of the Elements of EW Jones Vectors and EW Jones Matrices Under Changes of Eigenwave Bases 322

8.8.1 Coordinates of the EW Jones Vector of a Wave Field in Different Eigenwave Bases 322

8.8.2 EW Jones Operators in Different Eigenwave Bases 326

References 328

9 Choice of Eigenwave Bases for Isotropic, Uniaxial, and Biaxial Media 331

9.1 General Aspects of EWB Specification. EWB-generating routines 331

9.2 Isotropic Media 338

9.3 Uniaxial Media 342

9.4 Biaxial Media 352

References 365

10 Efficient Methods for Calculating Optical Characteristics of Layered Systems for Quasimonochromatic Incident Light. Main Routines of LMOPTICS Library 367

10.1 EW Stokes Vectors and EW Mueller Matrices 368

10.2 Calculation of the EW Mueller Matrices of the Overall Transmission and Reflection of a System Consisting of "Thin" and "Thick" Layers 375

10.3 Main Routines of LMOPTICS 384

10.3.1 Routines for Computing 4 x 4 Transfer Matrices and EW Jones Matrices 384

10.3.2 Routines for Computing EW Mueller Matrices 388

10.3.3 Other Useful Routines 391

References 392

11 Calculation of Transmission Characteristics of Inhomogeneous Liquid Crystal Layers with Negligible Bulk Reflection 393

11.1 Application of Jones Matrix Methods to Inhomogeneous LC Layers 394

11.1.1 Calculation of Transmission Jones Matrices of LC Layers Using the Classical Jones Calculus 394

11.1.2 Extended Jones Matrix Methods 404

11.2 NBRA. Basic Differential Equations 409

11.3 NBRA. Numerical Methods 420

11.3.1 Approximating Multilayer Method 421

11.3.2 Discretization Method 427

11.3.3 Power Series Method 428

11.4 NBRA. Analytical Solutions 430

11.4.1 Twisted Structures 430

11.4.2 Nontwisted Structures 432

11.4.3 NBRA and GOA. Adiabatic and Quasiadiabatic Approximations 434

11.5 Effect of Errors in Values of the Transmission Matrix of the LC Layer on the Accuracy of Modeling the Transmittance of the LCD Panel 437

References 438

12 Some Approximate Representations in EWJones Matrix Method and Their Application in Solving Optimization and Inverse Problems for LCDs 441

12.1 Theory of STUM Approximation 442

12.2 Exact and Approximate Expressions for Transmission Operators of Interfaces at Normal Incidence 447

12.3 Polarization Jones Matrix of an Inhomogeneous Nonabsorbing Anisotropic Layer with Negligible Bulk Reflection at Normal Incidence. Simple Representations of Polarization Matrices of LC Layers at Normal Incidence 463

12.4 Immersion Model of the Polarization-Converting System of an LCD 466

12.5 Determining Configurational and Optical Parameters of LC Layers With a Twisted Structure: Spectral Fitting Method 474

12.5.1 How to Bring Together the Experiment and Unitary Approximation 476

12.5.2 Parameterization and Solving the Inverse Problem 480

12.5.3 Appendix to Section 12.5 489

12.6 Optimization of Compensation Systems for Enhancement of Viewing Angle Performance of LCDs 490

References 504

13 A FewWords About Modeling of Fine-Structure LCDs and the Direct Ray Approximation 507

13.1 Virtual Microscope 508

13.2 Directional Illumination and Diffuse Illumination 513

References 516

A LCD Modeling Software MOUSE-LCD Used for the HKUST Students Final Year Projects (FYP) from 2003 to 2011 517

A.1 Introductory Remarks 517

A.2 Fast LCD 517

A.2.1 TN Cell 517

A.2.2 Effect of d/p Ratio 519

A.2.3 Effect of K22/K11 520

A.2.4 Effect of K33/K11 520

A.2.5 Effect of delta 521

A.2.6 Effect of gamma 521

A.2.7 Effect of Anchoring Strength W 523

A.2.8 Optimized TN Cell With Fast Response Time 523

A.2.9 Other LC Modes 524

A.3 Color LCD 524

A.3.1 The Super-Twisted Nematic Cell 524

A.3.2 STN Birefringent Colors in Transmissive and Reflective Modes 525

A.4 Transflective LCD 525

A.4.1 Vertical Aligned Nematic Cell 525

A.5 Switchable Viewing Angle LCD 535

A.6 Optimal e-paper Configurations 535

A.7 Color Filter Optimization 536

References 536

B Some Derivations and Examples 537

B.1 Conservation Law for Energy Flux 537

B.2 Lorentz’s Lemma 538

B.3 Nonexponential Waves 538

B.4 To the Power Series Method (Section 11.3.3) 540

B.5 One of the Ways to Obtain the Explicit Expressions for Transmission Jones Matrices of an Ideal Twisted LC Layer 541

Reference 543

Index 545

About the Author

Dmitry A. Yakovlev, Saratov State University, Russia
Dr Yakovlev is a senior researcher in the Department of Physics at Saratov State University, Russia. He is the head developer of commercial software MOUSE-LCD (MOdeling Universal System of Electrooptics of LCDs), developed in cooperation with HKUST, and the author of a number of efficient methods for computer modeling and optimization of LCDs used within many research projects performed in cooperation with Center Display Research of Hong Kong University of Science and Technology, ROLIC Research Ltd (Switzerland), TechnoDisplay AS (Norway. He has authored 30 refereed journal papers.

Vladimir G. Chigrinov, Hong Kong University of Science and Technology, Hong Kong
Professor Chigrinov is a member of the department of electrical and electronic engineering at Hong Kong University of Science and Technology. He is the author of 3 books, including Photoalignment of Liquid Crystalline Materials (with Professor Kwok), published by Wiley (2008). He has authored more than 150 refereed journal papers and holds 56 patents in the field of liquid crystals. He is a member of the editorial board of Liquid Crystal Today and Associate Editor of the Journal of SID. Prof. Chigrinov is Vice-President of the Russian SID chapter and a SID Fellow.

Hoi Sing Kwok, Hong Kong University of Science and Technology, Hong Kong
Professor Kwok is a member of the department of electrical and electronic engineering at Hong Kong University of Science and Technology. He is a fellow of the IEEE, Optical Society of America and the Hong Kong Institution of Engineers. Prof. Kwok is the co-author of Photoalignment of Crystalline Materials (Wiley, 2008) with Prof. Chigrinov and Vladimir M. Kozenkov, and has authored over 300 refereed journal papers.