advanced understanding of the optical …...advanced understanding of the optical properties in...

ADVANCED UNDERSTANDING OF THE OPTICAL PROPERTIES IN

PHASE COMPENSATED LIQUID CRYSTAL DEVICES

A dissertation submitted to Kent State University in partial

fulfillment of the requirements for the degree of Doctor of Philosophy

by

Yong-Kyu Jang

August 2007

ii

Approved by

Chair, Doctoral Dissertation Committee

Philip J. Bos , Professor, Chemical Physics Interdisciplinary Program

Members, Doctoral Dissertation Committee

Deng-Ke Yang , Professor, Chemical Physics Interdisciplinary Program

Liang-Chy Chien , Professor, Chemical Physics Interdisciplinary Program

John L. West , Professor, Chemistry Department

Robert Twieg , Professor, Chemistry Department

Accepted by

Oleg D. Lavrentovich , Director, Chemical Physics Interdisciplinary Program

Jerry Feezel , Dean, College of Arts and Sciences

Dissertation written by Yong-Kyu Jang

B.A., Chungnam National University, 1991 M.S., Korea University, 1993

Ph.D., Kent State University, 2007

iii

TABLE OF CONTENTS

LIST OF FIGURES……………………………………………………………………....xi

LIST OF TABLES……………………………………………………………………..xxiv

ACKNOWLEDGEMENTS………………………………………………………….....xxv

CHAPTER 1. INTRODUNTION

1-1. Motivations and objectives…………………………………………………...1

1-2. Overview of the dissertation………………………………………………….3

CHAPTER 2. ANALYSIS OF THE MULTI-REFLECTION EFFECTS IN

COMPENSATED LIQUID CRYSTAL DEVICES

2-1. Introduction…………………………………………………………………..7

2-2. Analytical calculations of multi-reflection effects in LCDs………………….9

2-3. Multi-reflection in an isotropic layer………………………………………..16

2-4. Multi-reflection in a liquid crystal device with no residual retardation in the

dark state (ECB-type LCD)………………………………………………....17

2-4-1. Phase analysis of the e-ray and o-ray………………………………...20

2-4-2. Analysis of the angle of light polarization…………………………...22

2-4-3. Analysis of transmittance without considering the dispersion properties

of refractive indices in ECB type LCDs……………………………..26

2-4-4. Analysis of transmittance with considering the dispersion properties of

refractive indices in ECB type LCDs………………………………...28

iv

2-5. Multi-reflection in a liquid crystal device with residual retardation in the dark

state (Pi-cell type LCD)……………………………………………………..31

2-5-1. Analysis of transmittance without considering the dispersion properties

of refractive indices in Pi-cell type LCDs……………………………35


refractive indices in Pi-cell type LCDs………………………………37

2-5-3. Analysis of high and low frequency interference modes and their

effects on the optical properties of LCDs……………………………41

2-5-4. Refractive index mismatching effects in Pi-cell type LCDs…………47

2-6. Experimental and numerical calculation results of a Pi-cell………………..50

2-6-1. Compensating a Pi-cell by using a uniaxial crystal compensator……50

2-6-2. Compensating a Pi-cell by using hybrid-aligned negative C-plates…54

2-6-2-1. Multi-reflection effects in a real Pi-cell…………………….56

2-6-2-2. Multi-reflection effects in a real Pi-cell with additional crystal

polarizers……………………………………………………60

2-6-2-3. Refractive index mismatching effects in a Pi-cell………….63

2-6-3. Numerical calculations of the multi-reflection effects in a Pi-cell…...66

2-7. Summary…………………………………………………………………….70

CHAPTER 3. UNIVERSAL OFF-AXIS LIGHT TRANSMISSION PROPERTIES

OF THE BRIGHT STATE IN COMPENSATED LIQUID CRYSTAL

DEVICES

3-1. Introduction…………………………………………………………………71

v

3-2. Optical properties of general liquid crystal devices………………………...72

3-2-1. Calculation methods and compensations of the dark states of the LCDs

………………………………………………………………………..75

3-2-2. Optical parameters of the LCDs……………………………………...76

3-2-3. Calculation results of the off-axis light transmittance in dark states...79

3-2-4. Calculation results of the off-axis light transmittance in bright states.83

3-3. Universal simple models……………………………………………………87

3-3-1. Simple dark and bright state modeling……………………………….87

3-3-2. “3-layer” modeling…………………………………………………...90

3-4. Calculations of the optical properties of the simple models………………...92

3-4-1. Basic calculations…………………………………………………….92

3-4-2. Analyses of the effective phase retardation of the simple models…...95

3-4-2-1. Effective phase retardation in the director plane…………...96

3-4-2-2. Effective phase retardation out of the director plane……...101

3-4-3. Analyses of the off-axis light transmittance of the simple models…105

3-4-3-1. Off-axis light transmittance in a dark state………………..105

3-4-3-2. Off-axis light transmittance in a bright state………………105

3-5. Comparison of the viewing angle properties in all directions……………..109

3-6. Summary…………………………………………………………………...116

CHAPTER 4. OPTIMIZATION OF THE BRIGHT STATE DIRECTOR

CONFIGURATION IN COMPENSATED PI-CELL DEVICES

4-1. Introduction………………………………………………………………..118

vi

4-2. Optical Properties of Different Bright States in Pi-cells…………………..120

4-2-1. Optical compensations of the dark states of the Pi-cells……………120

4-2-2. Optical parameters of the Pi-cells…………………………………..120

4-2-3. Numerical calculations of the off-axis light transmittance in dark states

………………………………………………………………………125

4-2-4. Numerical calculations of the off-axis light transmittance in bright

states………………………………………………………………...130

4-3. Universal Bright State Model of Pi-cells…………………………………..138

4-3-1. Dark and bright state modeling……………………………………..138

4-3-2. Calculations…………………………………………………………141

4-3-3. Analyses of the effective birefringence in the director plane……….143

4-3-4. Analyses of the transmittance………………………………………149

4-3-5. Detailed analyses of the transmittance out of the director plane…...154

4-3-5-1. Contribution from the positive A-plates…………………..154

4-3-5-2. Contribution from the negative C-plates…………………..156

4-4. Summary…………………………………………………………………...163

CHAPTER 5. LUMINANCE AND COLOR PROPERTIES OF THE

COMPENSATED LIQUID CRYSTAL DEVICES IN THEIR

BRIGHT STATES

5-1. Introduction………………………………………………………………..165

5-2. Color Calculations…………………………………………………………166

5-2-1. Tristimulus values and color matching functions…………………..166

vii

5-2-2. Chromaticity diagrams……………………………………………...169

5-2-3. Color difference……………………………………………………..172

5-2-4. Illuminants…………………………………………………………..173

5-3. Analyses of the luminous transmittance (Y )……………………………...177

5-3-1. Thickness effects on luminous transmittance and cell parameters…177

5-3-2. Off-axis luminous transmittance……………………………………183

5-3-2-1. Luminous transmittance in the bright state model………...183

5-3-2-2. Luminous transmittance in real LCDs…………………….184

5-3-2-3. Viewing angle properties of luminous transmittances in all

directions…………………………………………………..185

5-4. Analyses of the color properties…………………………………………...197

5-4-1. Thickness effects on color properties……………………………….197

5-4-2. Chromaticity coordinates of the bright state model in the off-axis

viewing angles……………………………………………………...204

5-4-3. Chromaticity coordinates of the common bright state LCDs in the

off-axis viewing angles……………………………………………..213

5-4-4. Color difference of the bright state model in the off-axis viewing

angles……………………………………………………………….227

5-4-5. Color difference of the common bright state LCDs in the off-axis

viewing angles……………………………………………………...229

5-4-5-1. Detailed analyses of the color difference of the Pi-cell…...229

5-4-5-2. Detailed analyses of the color difference of the TN mode..230

viii

5-5. Summary…………………………………………………………………...241

5-5-1. Luminous transmittance of the bright state LCDs………………….241

5-5-2. Color properties of the bright state LCDs…………………………..242

CHAPTER 6. THE CONDITIONS AND THE LIMITATIONS OF THE PERFECT

PHASE COMPENSATION IN LIQUID CRYSTAL DISPLAYS

6-1. Introduction………………………………………………………………..244

6-2. Calculations of the phase difference……………………………………….244

6-2-1. “Complete” method…………………………………………………245

6-2-2. Approximate method………………………………………………..248

6-2-3. Comparison of both methods for calculating the phase difference…250

6-3. Conditions and limitations for the perfect phase compensation in the simple

director configurations……………………………………………………..253

6-3-1. Relationship of the parameters……………………………………...253

6-3-2. Thickness ratio for the perfect compensation………………………260

6-3-3. Thickness ratio for the different director configurations…………...261

6-3-4. Transmittances in the compensation system………………………..267

6-4. Applications………………………………………………………………..280

6-4-1. Uniform director configurations…………………………………….280

6-4-2. Non-uniform director configurations……………………………….300

6-4-2-1. A Pi-cell that has different bright state voltages

(thickness effects)……………………………………...…300

6-4-2-2. A Pi-cell that has different pretilt angles

ix

(thickness and director tilt angle effects)…………………301

6-5. Summary…………………………………………………………………...305

CHAPTER 7. CONCLUSIONS

7-1. Analysis of the multi-reflection effects in compensated liquid crystal devices

……………………………………………………………………………..308

7-2. Universal off-axis light transmission properties of the bright state in perfectly

compensated liquid crystal devices………………………………………...309

7-3. Optimization of the bright state director configuration for perfectly

compensated pi-cell devices……………………………………………….310

7-4. Luminance and color properties of the compensated liquid crystal devices in

their bright states…………………………………………………………...312

7-4-1. Luminous transmittance of the bright state LCDs………………….312

7-4-2. Color properties of the bright state LCDs ………………………….313

7-5. The conditions and the limitations of the perfect phase compensation in

liquid crystal displays……………………………………………………...315

APPENDIX A. CALCULATION OF THE 1-DIMENSIONAL LIQUID CRYSTAL

DIRECTOR CONFIGURATION BY THE VECTOR METHOD BASED

ON THE RELAXATION TECHNIQUE

A-1. Calculations……………………………………………………………….318

A-2. Program source codes (Matlab)…………………………………………...324

APPENDIX B. CALCULATION OF THE OPTICS IN LIQUID CRYSTAL

DEVICES BY EXTENDED 2×2 JONES MATRIX

x

B-1. Calculations………………………………………………………………..337

B-2. Program source codes (Matlab)…………………………………………...345

APPENDIX C. PROGRAM OVERVIEW (“LC Optics”)

C-1. “LC Optics” main screen………………………………………………….350

C-2. Popup menu of each layer…………………………………………………351

C-3. Layer properties…………………………………………………………...352

C-4. Variable rearrangement and plotting……………………………………...353

C-5. Plotting options……………………………………………………………354

C-6. Example of the 2-D Plotting………………………………………………355

C-7. Example of the 3-D Plotting………………………………………………356

C-8. Example of the Conoscopy Plotting………………………………………357

C-9. Example of the LC director configuration………………………………...358

REFERENCES………………………………………………………………………...359

xi

LIST OF FIGURES

Fig. 2-1. Simple layout of a compensated liquid crystal device and the definition of each

axis…….................................................................................................................15

Fig. 2-2. Director configurations of the liquid crystal in an ECB cell…………………...19

Fig. 2-3. Calculation result of the phases of 'eEr

and 'oEr

in ECB type LCDs……………21

Fig. 2-4. Effects on the angle [ )(λθ ] of (a) the index mismatching ( airo nn − ) and

(b) the relative difference of refractive indices ( oe nn − )……………………...24

Fig. 2-5. Effects on the transmittance (light leakage) of (a) the index mismatching

( airo nn − ) and (b) the relative difference of refractive indices ( oe nn − )…….25

Fig. 2-6. Transmittances of each mode and the total transmittance with the angle

variation in the dark state of an ECB type LCD without considering the dispersion

of refractive indices………………………………………………………………27

Fig. 2-7. Light wavelength dispersion of refractive indices ( on , en ) of a commercialized

liquid crystal (LC53)……………………………………………………………..29

Fig. 2-8. Transmittances of each mode and the total transmittance with the angle

variation in the dark state of an ECB type LCD with considering the dispersion of

refractive indices…………………………………………………………………30

Fig. 2-9. Director configurations of the liquid crystal in a Pi-cell……………………….33

Fig. 2-10. Calculation results of the effective birefringence at the normal direction

xii

in a Pi-cell………………………………………………………………………..34

Fig. 2-11. Transmittances of each mode and the total transmittance in the dark state

of a Pi-cell-type LCD without considering dispersion of refractive indices…….36

Fig. 2-12. Effective refractive indices of the liquid crystal (LC53) for the thickness

of 5.53 µm and effective residual birefringence of 100 nm……………………...39

Fig. 2-13. Transmittances of each mode and the total transmittance in the dark state

of a Pi-cell-type LCD with considering dispersion of refractive indices………...40

Fig. 2-14. Thickness effect on the total transmittance of a Pi-cell-type LCD in the dark

state………………………………………………………………………………43

Fig. 2-15. Phase retardation effect on the total transmittance of a Pi-cell-type LCD in the

dark state…………………………………………………………………………44

Fig. 2-16. Effective refractive indices ( effen ) of the liquid crystal (LC53) with different

thicknesses and a constant phase retardation ( effndΔ =100 nm)…………………45

Fig. 2-17. Effective refractive indices ( effen ) of the liquid crystal (LC53) with a constant

thickness ( d =5.53 μm) and different phase retardations………………………..46

Fig. 2-18. The effects on the dark state transmittance due to the refractive index

mismatching ( airo nn − )…………………………………………………………49

Fig. 2-19. Measurement setup of a Pi-cell compensation system and the cell structure

of the Pi-cell……………………………………………………………………...52

Fig. 2-20. Measured transmittance and the extracted minimum transmittance of the dark

state Pi-cell compensated with the compensator at each wavelength separately...53

xiii

Fig. 2-21. Basic structure of the compensation scheme of a Pi-cell by using WV-Film...55

Fig. 2-22. Optical stack configuration of a compensated real Pi-cell and the measurement

setup. All layers are combined as a single unit without any air gaps……………58

Fig. 2-23. Measurement results of the transmittance of a compensated real Pi-cell…….59

Fig. 2-24. Transmittances of the crossed sheet polarizers and the parallel sheet polarizers.

………………………………………………………………………………...61

Fig. 2-25. Measurement results of the transmittance of a compensated Pi-cell

with additional crystal polarizers and the experimental setup…………………...62

Fig. 2-26. Measurement results of the transmittances of a compensated Pi-cell

with additional crystal polarizers and air gaps, and the experimental setup……..65

Fig. 2-27. Measurement data of the real and imaginary parts of refractive indices……..68

Fig. 2-28. Measurement results and the numerical calculation results of the transmittance

of the Pi-cell compensated with the hybrid aligned negative-discotic films

for different applied voltages…………………………………………………….69

Fig. 3-1. Director configurations of the most common liquid crystal display modes…...74

Fig. 3-2. Stack configuration of the common liquid crystal devices…………………….78

Fig. 3-3. Numerical calculation results of the off-axis light transmission properties

of the common liquid crystal devices in their dark states………………………..80

Fig. 3-4. Director tilt angles of the dark state-liquid crystal layers……………………...81

Fig. 3-5. Effective phase retardation of the dark state-liquid crystal layers in the director

plane……………………………………………………………………………...82

Fig. 3-6. Numerical calculation results of the off-axis light transmission properties

xiv

of the common liquid crystal devices in their bright state……………………….84

Fig. 3-7. Director tilt angles of the bright state-liquid crystal layers…………………….85

Fig. 3-8. Effective phase retardation of the bright state-liquid crystal layers in the director

plane……………………………………………………………………………...86

Fig. 3-9. (a) Simplification of the liquid crystal layers of the various devices, (b) optical

compensation of the simplified liquid crystal layer and (c) simple dark and bright

state models………………………………………………………………………89

Fig. 3-10. (a) “3-layer” modeling of the most common LCD modes and (b) simple bright

state model……………………………………………………………………….91

Fig. 3-11. Projections of the optic axes of A and C-plates onto the plane perpendicular to

the light propagation vector ( Kr

) that lies (a) in the director plane (x-z plane) and

(b) out of the director plane (y-z plane)………………………………………….98

Fig. 3-12. Birefringence analyses of the bright state model of mode 1 in the director plane.

………………………………………………………………………………...99

Fig. 3-13. Birefringence analyses of the bright state model of mode 2 in the director

plane…………………………………………………………………………….100

Fig. 3-14. Birefringence analyses of the bright state model of mode 1 out of the director

plane…………………………………………………………………………….103


plane…………………………………………………………………………….104

Fig. 3-16. Numerical calculation results of the off-axis light transmission properties of the

simple models in their dark states………………………………………………107

xv

Fig. 3-17. Off-axis light transmission properties of the bright state models…………...108

Fig.3-18. Viewing angle properties of the common liquid crystal devices in their dark

states…………………………………………………………………………….110

Fig. 3-19. Viewing angle properties of the dark state models………………………….112

Fig. 3-20. Viewing angle properties of the common liquid crystal devices in their bright

states…………………………………………………………………………….113

Fig. 3-21. Viewing angle properties of the bright state models………………………...115

Fig. 4-1. Stack configuration of a Pi-cell……………………………………………….124


dark state Pi-cells that have different bright state voltages (1.0-3.0V)…………126

Fig. 4-3. Director tilt angles of the dark state Pi-cells that have different bright state

voltages (1.0-3.0V)……………………………………………………………..127


dark state Pi-cells that have different pretilt angles (2.0-30.0º)………………...128

Fig. 4-5. Director tilt angles of the dark state Pi-cells that have different pretilt angles

(2.0-30.0º)………………………………………………………………………129


bright state Pi-cells that have different bright state voltages (1.0-3.0V)……….132

Fig. 4-7. Director tilt angles of the bright state Pi-cells that have different bright state

voltages (1.0-3.0V)……………………………………………………………..133

Fig. 4-8. Conoscopic properties of the bright state transmittances with different bright

state voltages of (a) 1.0V, (b) 1.3V, (c) 2.0V, (d) 2.5V and (e) 3.0V…………..134

xvi


bright state Pi-cells that have different pretilt angles (2.0-30.0º)………………135

Fig. 4-10. Director tilt angles of the bright state Pi-cells that have different pretilt angles

(2.0-30.0º)………………………………………………………………………136

Fig. 4-11. Conoscopic properties of the bright state transmittances with different pretilt

angles of (a) 2.0º, (b) 5.5º, (c) 10.0º, (d) 20.0º and (e) 30.0º…………………...137

Fig. 4-12. (a) A Pi-cell whose dark state is perfectly compensated using hybrid-negative

C-plates and (b) simple dark and bright state models of the Pi-cell……………140

Fig. 4-13. Effective birefringence of each layer of the bright state model in the director

plane with a tilt angle (θ) of 50º………………………………………………...146

Fig. 4-14. Effective birefringence of the positive A-plates (PA1+PA2) of the bright state

model in the director plane……………………………………………………..147

Fig. 4-15. Effective birefringence of the total layers of the bright state model in the

director plane…………………………………………………………………...148

Fig. 4-16. Off-axis light transmission properties of the bright state model with different

tilt angles………………………………………………………………………..151

Fig. 4-17. Conoscopic properties of the transmittance of the bright state model with tilt

angle (a) 10º, (b) 30º, (c) 50º, (d) 65º, and (e) 70º……………………………...152

Fig. 4-18. Off-axis light transmission properties of the dark state model……………...153

Fig. 4-19. Off-axis light transmittances contributed separately from each layer of the

bright state model (tilt angle of 70º) out of the director plane………………….157

Fig. 4-20. Apparent azimuth angle (γ ) of the optic axis of each layer in the bright state

xvii

model with tilt angle 70º out of the director plane……………………………...158

Fig. 4-21. Director angle definition in (a) lab coordinate system (x’, y’, z’) and

(b) incident light frame ( KSPrrr

,, )………………………………………………159

Fig. 4-22. Difference of the apparent azimuth angle ( γΔ ) between the PA1 and

the PA2 in bright state model with various tilt angles (10º-70º)……………….160

Fig. 4-23. Effective birefringence of a positive A-plate (PA1 or PA2) out of the director

plane in the bright state model………………………………………………….161

Fig. 4-24. Subtotal effective birefringence of the negative C-plates (NC1+NC2) out of the

director plane in the bright state model…………………………………………162

Fig. 5-1. CIE 1931 2º color matching functions………………………………………..168

Fig. 5-2. CIE 1931 2º chromaticity diagram (2º viewing angle)……………………….170

Fig. 5-3. CIE 1976 ''vu chromaticity diagram (2º viewing angle)……………………..171

Fig. 5-4. Spectral power distributions of CIE illuminants……………………………...175

Fig. 5-5. Simple dark and bright state models (mode 1)………………………………..179

Fig. 5-6. Thickness effects on the luminous transmittance of the bright state model at the

normal direction………………………………………………………………...180

Fig. 5-7. Stack configuration of the common liquid crystal devices…………………...182

Fig. 5-8. Off-axis luminous transmittances (Y ) of the bright state model with different

Y values at the normal direction………………………………………………..187

Fig. 5-9. Off-axis luminous transmittances (Y ) of the bright state ECB mode with

different Y values at the normal direction………………………………………188

Fig. 5-10. Off-axis luminous transmittances (Y ) of the bright state VA mode with

xviii


Fig. 5-11. Off-axis luminous transmittances (Y ) of the bright state Pi-cell mode with


Fig. 5-12. Off-axis luminous transmittances (Y ) of the bright state TN mode with


Fig. 5-13. Viewing angle properties of the luminous transmittances (Y ) in the bright state

model with different Y values at the normal direction…………………………192


ECB mode with different Y values at the normal direction…………………….193


VA mode with different Y values at the normal direction……………………...194


Pi-cell mode with different Y values at the normal direction…………………..195


TN mode with different Y values at the normal direction……………………...196

Fig. 5-18. ( 'u , 'v ) chromaticity coordinates of the bright state model in terms of

thicknesses at the normal direction……………………………………………..199

Fig. 5-19. Transmittances of the bright state model at the three major colors

[blue (λ =450 nm), green (λ =550 nm), and red (λ =650 nm)] at the normal

direction……………………………………………………………………...…200

Fig. 5-20. Phase retardation of the bright state model at the three major colors

[blue (λ =450 nm), green (λ =550 nm), and red (λ =650 nm)] at the normal

xix

direction………………………………………………………………………...201

Fig. 5-21. Lightness ( *L ) of the bright state model in function of thickness at the normal

direction………………………………………………………………………...202

Fig. 5-22. Color difference ( uvE *Δ ) of the bright state model in function of thickness at

the normal direction…………………………………………………………….203

Fig. 5-23. ( 'u , 'v ) chromaticity coordinates of the bright state model in functions of

viewing angles………………………………………………………………….206

Fig. 5-24. ( *u , *v ) chromaticity coordinates of the bright state model in functions of

viewing angles………………………………………………………………….207

Fig. 5-25. Transmittances of the bright state model with Y (%)=100 in the three major

colors (blue: 450 nm, green: 550 nm, red: 650 nm)…………………………….208







Fig. 5-29. Off-axis phase retardation of the bright state model with Y (%)=80 in the three

major wavelengths (blue: 450 nm, green: 550 nm, red: 650 nm)………………212

Fig. 5-30. ( 'u , 'v ) chromaticity coordinates of the ECB in functions of viewing angles.

……………………………………………………………………………….215

Fig. 5-31. ( 'u , 'v ) chromaticity coordinates of the VA in functions of viewing angles.

xx

……………………………………………………………………………….216

Fig. 5-32. ( 'u , 'v ) chromaticity coordinates of the Pi-cell in functions of viewing angles.

……………………………………………………………………………….217

Fig. 5-33. ( 'u , 'v ) chromaticity coordinates of the TN in functions of viewing angles..218

Fig. 5-34. ( *u , *v ) chromaticity coordinates of the ECB in functions of viewing angles.

……………………………………………………………………………….219

Fig. 5-35. ( *u , *v ) chromaticity coordinates of the VA in functions of viewing angles.

……………………………………………………………………………….220

Fig. 5-36. ( *u , *v ) chromaticity coordinates of the Pi-cell in functions of viewing

angles…………………………………………………………………………...221

Fig. 5-37. ( *u , *v ) chromaticity coordinates of the TN in functions of viewing angles.

……………………………………………………………………………….222

Fig. 5-38. Transmittances of the TN mode with Y (%)=100 in the three major colors...223

Fig. 5-39. Transmittances of the TN mode with Y (%)=90 in the three major colors….224



Fig. 5-42. Off-axis color difference ( uvE *Δ ) of the bright state model………………..228

Fig. 5-43. Off-axis color difference ( uvE *Δ ) of the ECB mode……………………….233

Fig. 5-44. Off-axis color difference ( uvE *Δ ) of the VA mode………………………...234

Fig. 5-45. Off-axis color difference ( uvE *Δ ) of the Pi-cell mode……………………...235

Fig. 5-46. Off-axis color difference ( uvE *Δ ) of the TN mode…………………………236

xxi

Fig. 5-47. Transmittances of the Pi-cell with Y (%)=100 in the three major colors……237

Fig. 5-48. Transmittances of the Pi-cell with Y (%)=90 in the three major colors……..238



Fig. 6-1. Angle definitions with a uniaxial slab………………………………………...247

Fig. 6-2. Angle definitions with a uniaxial slab in the approximation method………...249

Fig. 6-3. Effective phase retardations of the nematic slabs (θ ,φ ) calculated by the

approximation method (“Simple”) and complete method

(“Recursive” and “2×2 matrix”)………………………………………………..251

Fig. 6-4. Angle definitions of a simple compensation system………………………….256

Fig. 6-5. Total effective phase retardation in functions of 'en and 'd in the compensation

system…………………………………………………………………………..257

Fig. 6-6. A pair of parameters ( 'en , 'd ) for the perfect compensation ( 0=Δ Totaleffnd ) of

the uniaxial slab………………………………………………………………...259

Fig. 6-7. Thickness ratio ( dd /' ) for the perfect compensation of the uniaxial slab

(θ =0, 30, 60, and 90º; d =1.0 μm) with the light wavelength of 550 nm……...263

Fig. 6-8. Three-dimensional thickness ratio ( dd /' ) for the perfect compensation of the

uniaxial slab (θ =0, 30, 60, and 90º; d =1.0 μm) with the light wavelength

of 550 nm……………………………………………………………………….265

Fig. 6-9. Off-axis light transmission properties of the compensation system with the

azimuth angle (φ ) of 0º (director plane)………………………………………..268

xxii


azimuth angle (φ ) of 90º (out of the director plane)…………………………...270


azimuth angle (φ ) of 45º……………………………………………………….272

Fig. 6-12. Three-dimensional off-axis light transmission properties of the compensation

system with the azimuth angle (φ ) of 0º (director plane)……………………...274


system with the azimuth angle (φ ) of 90º (out of the director plane)………….276


system with the azimuth angle (φ ) of 45º……………………………………...278

Fig. 6-15. Total phase retardation ( TotaleffndΔ ) of the compensation systems (θ =0º) that are

perfectly compensated at the normal direction for the light wavelength of 550 nm.

……………………………………………………………………………….286

Fig. 6-16. Total phase retardation ( TotaleffndΔ ) of the compensation systems (θ =30º) that

are perfectly compensated at the normal direction for the light

wavelength of 550 nm…………………………………………………………..288


are perfectly compensated at the normal direction for the light

wavelength of 550 nm…………………………………………………………..290

Fig. 6-18. Off-axis light transmittances of the compensation systems (θ =0º) that are


xxiii

……………………………………………………………………………….292



……………………………………………………………………………….294



……………………………………………………………………………….296

Fig. 6-21. Off-axis light transmittances in functions of the thickness and the director tilt

angle of the liquid crystal layer in the compensation systems that are perfectly

compensated at the normal direction for the light wavelength of 550 nm……...298


dark state Pi-cells that have different pretilt angles (2.0-50.0º)………………...303


(2.0-50.0º)………………………………………………………………………304

xxiv

LIST OF TABLES

Table 3-1. Cell parameters of the common liquid crystal (LC) devices we used………..77

Table 4-1. Cell parameters of the Pi-cells with different bright state voltages…………122

Table 4-2. Cell parameters of the Pi-cells with different pretilt angles………………...123

Table 4-3. Thickness of each layer of the bright state model with different tilt angles (θ).

………………………………………………………………………………145

Table 5-1. Tristimulus values and chromaticity coordinates of illuminants in

fields of view 2º………………………………………………………………...176

Table 5-2. Cell parameters of the common liquid crystal (LC) devices we used………181

Table 5-3. Thicknesses of the bright state model and the common LCDs……………..181

Table 6-1. Thicknesses of the uniformly aligned perfect compensation systems………285

Table 6-2. Cell parameters of the Pi-cells with different pretilt angles………………...302

xxv

ACKNOWLEDGEMENTS

First of all, I am sincerely thankful to my adviser, Dr. Philip J. Bos, for his great

dedication to my dissertation. Without his help, I would not been able to finish my

research at LCI, Kent State University. He always gave me the right directions to go

when I was stuck in something and encouraged me in difficult situations. He has not only

helped in the research, but has also taken care of my first American life and made me

very comfortable in a new culture. I really appreciate his efforts and considerations for

my family and me.

I would like to thank LCI faculty and staff members, Dr. Oleg Lavrentovich, Dr.

Philip Bos, Dr. Peter Palffy-Muhoray, Dr. Deng-Ke Yang, Dr. L. –C. Chien, Dr. Tony

Jakli, Dr. John West, Dr. Jack Kelly, Dr. Sergij Shiyanovskii, Dr. David Allender, Dr.

Satyendra Kumar, and Dr. Samuel Sprunt for their enthusiastic and very helpful lectures

about liquid crystals and various kind assistance. The knowledge that I have learned from

their classes will be the reference and the lighthouse in my future works.

I also want to thank to all my colleagues including my classmates at LCI,

especially Bohdan Senyuk, John Harden, Christopher Bailey, Clinton Braganza, Xiaoli

Zhou, Jeremy Neal, and Shouping Tang for their kind help and valuable discussions. I

also really thank to all BOSLAB members, particularly Dr. Bin Wang, Dr. Cheng Chen,

and Dr. Yanli Zhang for their kindness and diverse help.

xxvi

I could study here by virtue of Samsung Electronics Co., Ltd., especially S. M. D.

Hyung-Guel Kim, because they gave me this fabulous opportunity to study more in USA,

which was, actually, my dream come true. I know it was a hard decision for them to make,

but they did it for me. I am very grateful for their decision and efforts. Additionally, there

are lots of colleagues at my company supporting this opportunity, and I really appreciate

their efforts.

This dissertation is dedicated to my honorable parents, Chan-gak Jang and Bok-

gum Lee, my lovely wife, Sook-Ja Jang, my cute kids, Hayeri Jang and Byeol-Ha (Daisy)

Jang, and my brother, sisters and other relatives. Without their persistent patience,

support and encouragement, this dissertation would not have been possible. I deeply

appreciate their endeavors and much assistance.

1

CHAPTER 1

INTRODUCTION

1-1. Motivations and objectives

The viewing angle dependence of light transmittance in liquid crystal displays

(LCDs) is a well-known feature and one of the biggest problems. Especially, the dark

(black) state is very important because it is critical to the contrast ratio of the device or

the visibility of the images which are displayed on a screen. On this account, a great

amount of research has been done to reduce the dark state transmittance and its variation

at all viewing angles.

To minimize the dark state light leakage, the effective phase retardation at all

viewing angles should be minimized. There are several ways for the compensation of the

phase retardation in a dark state, such as the use of negative O-plates, positive O-plates,

and biaxial films. Although the methods are different, their final goals are the same, i.e.

making the total effective phase retardation of the dark state zero in all viewing directions.

Among the methods, using the negative O-plates, or recently it is called polymerized

discotic material (PDM) or hybrid aligned negative C-plates, is the most popular one

because of its effectiveness. By using this method, we can almost perfectly compensate

the phase retardation of the dark state in a LCD at the normal direction. Therefore, we

could obtain the dark transmittance that is limited only by the polarizer and the analyzer

used in conjunction with the liquid crystal device if there is no other effects.

2

However, in our calculations and experiments, the dark transmittance is much

larger than that of the crossed polarizers and changes the color of the dark state, even

though we optically compensate the dark state perfectly at the normal direction. We

analyzed this situation and found that it is related to the multi-reflected light from internal

interfaces within the liquid crystal device, and the multi-reflection strongly depends on

the residual birefringence of the dark state regardless of the phase compensation.

On the other hand, the bright (white) state determines the luminance and the

spectral variations of the transmittance as a function of the angle of the incident light.

This bright state is particularly important in the application of large size displays such as

LCD monitors, TVs, and signs. Therefore, this should be another vital factor when

evaluating the optical properties of the LCDs. However, we have not seen those studies

dealing with the off-axis light transmission properties (viewing angle properties) of the

bright state of general liquid crystal devices such as ECB, VA, TN, and Pi-cell.

Additionally, the viewing angle dependences of the luminance and the color properties

are essential in the optical properties of the devices. We have investigated those optical

properties of the bright states of the common LCDs, whose dark states are optically

optimized to give the minimum transmittance at the normal direction by using the hybrid

aligned negative C compensators, and found that there is universality in the optical

properties of the bright states no matter what liquid crystal modes are used.

Another important topic we want to consider is the phase compensation of the

dark states in LCDs. Recently, most of the commercialized liquid crystal displays use

phase compensators to improve the contrast ratio, viewing angle, and color properties of

3

the devices. However, it has not been reported what the exact conditions and limitations

are for the phase compensation of a liquid crystal layer. Also, the parameter relationship

affecting the limitations is obviously significant when we optically design a LCD.

As mentioned above, the hybrid aligned negative C-plate is suitable for the optical

compensation of a liquid crystal layer with the same optic axes. Seemingly, we could

perfectly compensate a uniformly aligned positive uniaxial slab by using a tilted negative

C-plate with adjusting the optic axis, refractive indices, and thickness. However,

according to our investigation, there is a limitation for the compensation of a uniaxial

layer even in the most simple director configuration. Additionally, we found very

important parameters that strongly affect the limitation of the phase compensation.

1-2. Overview of the dissertation

In Chapter 2, we consider perfectly compensated liquid crystal devices at the

normal direction and show the analytical and numerical calculation results and

experimental data showing the effects of the multi-reflections on the optical properties of

dark state LCDs. Specifically, we describe the contribution of the residual birefringence

and index mismatching to the extinction ratio and its wavelength dependence. According

to the analyses, there are two types of the interference. The first type of interference,

which has higher frequency, could affect the dark level almost equivalently for all visible

wavelengths, so it does not affect the color of the dark state. On the other hand, the

second type, which is closely related to the residual birefringence of the dark state, could

4

cause not only increasing the dark level but also a color shift because of the lower

frequency pattern of the interference in the wavelength space.

In Chapter 3, we investigate the off-axis light transmission characteristics of the

bright state of common liquid crystal device modes. The dark state of these device modes

is optically compensated to have minimum light transmittance at the normal direction.

Our research shows there is an unexpected universal shape of the off-axis light

transmission value in its bright state, regardless of what liquid crystal mode is used. To

understand this surprising fact, we build simple dark and bright state models that can be

applied to general liquid crystal devices and analyze them in terms of the effective

retardation and transmittance.

In Chapter 4, we study the off-axis light transmission properties of the bright state

in Pi-cell devices as a function of the bright state director configuration that is determined

by the applied bright state voltage and the pretilt angle of the device. We find that below

certain values of the voltage or pretilt angle, the off-axis light transmission properties are

insensitive to these parameters and can be described by a previously considered simple

model. However, above a critical pretilt angle or bright state voltage, the light

transmittance is a much stronger function of the incident angle of light. To understand the

facts, we develop a new model that explains this result and provides a description of the

basic issues affecting the optics of these types of devices.

In Chapter 5, we investigate the luminance and color properties of the bright state

simple model and the common LCDs such as ECB, VA, Pi-cell, and TN modes. We use

the same bright state simple model that we built in Chapter 3. The dark states of the

5

model and the common LCDs are optically compensated by using hybrid aligned-passive

type negative C-plates, as the same way that we used in Chapter 3. We analyze the

luminous transmittance of our bright state model in functions of the cell thickness and the

viewing angle, and after that we compare those results with that of real common LCDs.

These results will confirm the universality of the bright state viewing angle properties

and give an important conclusion: single domain LCDs can not escape from the

anisotropic shape of the bright state viewing angle properties. We also calculate and

analyze the off-axis color properties including color difference of the bright state model

and the common LCDs. From these results, we know that there are common features of

the color properties in the bright state of common LCDs.

In Chapter 6, we will study the conditions and the limitations of the phase

compensation in dark state LCDs. As an example compensator, we will use hybrid

aligned negative C-plates. We discuss the difference between an approximate method and

the “complete” method for the calculation of the phase retardation in a uniaxial material.

We analyze the relationship among the parameters ( ',',' dnn oe ) of a compensator for the

perfect compensation of a uniaxial slab. With this parameter relationship, we calculate

the thickness ratio of the compensator to uniaxial slab for the perfect compensation. We

also calculate and analyze the total effective phase retardations and the off-axis light

transmittances in functions of the director tilt angle, effective phase retardation, and the

thickness in the uniformly and non-uniformly aligned liquid crystal layers.

All calculations in this dissertation are fulfilled by our own GUI program, “LC

Optics”, which was made for the purpose of calculating the director configuration of a

6

liquid crystal layer and the optical properties of general LCDs such as transmittance,

reflectance, luminance, lightness, and color analyses. In Appendices A and B, we give the

basic calculations and program source codes related to the director calculation and the

optics calculation. Due to the huge size of the program, it is hard to attach the whole

source codes on this dissertation. Finally in Appendix C, we introduce the overview of

our program, “LC Optics”.

7

CHAPTER 2

ANALYSIS OF THE MULTI-REFLECTION EFFECTS IN COMPENSATED

LIQUID CRYSTAL DEVICES

2-1. Introduction

Many modes used in liquid crystal devices particularly those with an untwisted

structure1 have residual birefringence in their dark state due to the anchoring energy at

the alignment layers. For these devices, to obtain a true dark state, a passive phase

compensator is needed. Ideally, we can compensate the residual birefringence so that the

total effective birefringence will be zero at all wavelengths, and we can achieve an

extinction ratio that is limited by the polarizer and analyzer used in conjunction with the

liquid crystal device.

However, in our calculations and experiments, the dark transmittance is much

larger than that of the crossed polarizers and depends on the wavelength. We analyzed

this situation and found that it is related to multi-reflected light from internal interfaces

within the liquid crystal device. These reflections result in two types of interference. The

first is the interference of the extraordinary ray (e-ray) and ordinary ray (o-ray) by

themselves and the second one is the interference between e-ray and o-ray. The first type

has higher frequency in the wavelength space and is related to the optical path lengths of

the e-ray and o-ray independently. The second type has a lower frequency and depends

on the residual birefringence of the dark state. So, as the residual birefringence increases,

8

there is a greater contribution from the second type of interference. Considering the

optical properties of the liquid crystal display, the first type interference could affect the

dark level almost equivalently for all visible wavelengths, so it does not affect the color

of the dark state. On the other hand, the second type could cause a color shift because of

the lower frequency pattern of the interference in the wavelength space.

The numerical calculation of the light intensity including multi-reflection effects

in a reflection-mode image transducer utilizing a nematic liquid crystal with a 45° twist

has been reported2. That work covers the liquid crystal thickness dependency of the

output intensity with and without isotropic layers for a single wavelength. However, we

were not able to find reports concerning the interference effects related to the residual

birefringence and their contribution to the optical properties including the wavelength

dependence of the extinction ratio.

In this chapter, we will consider perfectly compensated liquid crystal devices at

the normal direction and show the analytical and numerical calculation results and

experimental data showing the effects of the multi-reflections on the optical properties.

Specifically, we will describe the contribution of the residual birefringence to the

extinction ratio and its wavelength dependence.

In Sec. 2-2, we will describe the effect of the multi-reflection analytically in a

dark state of a simple uniaxial type liquid crystal device that has one pair of interfaces,

but this could be expanded to a more complicated configuration easily using the same

concept. In Secs. 2-3~5 we will show the calculation results of light transmission when

multi-reflections are considered and analyze the effects. In Sec. 2-6, we will give the

9

experimental data with the numerical calculation including all necessary layers in an

example device.

2-2. Analytical calculations of multi-reflection effects in LCDs

In this section, we will consider a simple-uniaxial-type liquid crystal device with

its dark state compensated perfectly at the normal direction by using passive optical

retarders. Light is assumed to be incident at the normal direction and is reflected only at

the one pair of surfaces; on either side of the liquid crystal layer in air. [In actual liquid

crystal devices, there are no air layers inside the devices. However, in an actual LCD

there can be reflections from the multiple interfaces. Those reflections can provide a

similar (but perhaps smaller) effect as will be shown in Sec. 2-6] Figure 2-1 shows this

configuration with two crossed polarizers, and in this figure, δ and Γ are the phase

changes of the light at the liquid crystal and compensator, respectively. The electric field

after the polarizer is expressed as

)](exp[2

1ˆ wtkziExE in −=r

, (2-1)

where inE is the amplitude of electric field of incident light, and x is the unit vector

along the x-axis. In this equation, we assumed the polarizer and analyzer are ideal, i.e. the

incident light that is polarized along the transmittance axis of the polarizer can transmit it

perfectly, but the perpendicular component of incident light is blocked by polarizer. This

10

electric field ( )Er

that has wavelength λ in free space can be split into two eigenmodes,

extraordinary mode (e-mode) and ordinary mode (o-mode), in the liquid crystal layer, and

they will propagate with different phases. Each wave is also multi-reflected at the liquid

crystal–air interface and has the following form after the liquid crystal layer (time related

terms have been dropped here for simplicity):

epaeeeee

eapine tiriritEeE ...])5exp()3exp()[exp(

21ˆ 42 +++= δδδ

r (2-2)

...])4exp()2exp(1)[exp(21ˆ 42 +++= eeeee

epa

eapin iririttEe δδδ (2-3)

)2exp(1

1)exp(21ˆ

2ee

eepa

eapin ir

ittEeδ

δ−

= (2-4)

)2exp(1)exp()1(ˆ

2

2

ee

eee ir

irEe

δδ

−−

≡ . (2-5)

opaooooo

oapino tiriritEoE ...])5exp()3exp()[exp(

21ˆ 42 +++−= δδδ

r (2-6)

)2exp(1)exp()1(ˆ

2

2

oo

ooo ir

irEo

δδ

−−

−≡ , (2-7)

where e and o are the unit vectors along the electric field directions of the extraordinary

ray (e-ray) and ordinary ray (o-ray) [Fig. 2-1 (b)], respectively. The eapt and e

pat are the

transmission coefficients of the e-mode at the interface, air-liquid crystal, and er and

or are the reflection coefficients of the e-mode and o-mode, respectively at the same

11

interface. The eδ and oδ are the spatial phase changes of the e-mode and o-mode at the

liquid crystal layer. Those parameters are expressed using the refractive indices of e-ray

and o-ray ( oe nn , ) and the thickness ( d ) of the liquid crystal layer when light is incident

at a normal direction3,

22

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

=aire

airee nn

nnr , (2-8)

22

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

=airo

airoo nn

nnr , (2-9)

λπ

δdne

e2

= , (2-10)

λπ

δdno

o2

= . (2-11)

During the calculation, we used the fact that,

21 eepa

eap rtt −= , (2-12)

21 oopa

oap rtt −= , (2-13)

and we set,

inoe EEE21

≡= . (2-14)

12

After the liquid crystal layer, the e-ray and o-ray enter the compensator (refractive

indices '' , oe nn , thickness 'd ) and will have the phase change eΓ and oΓ , respectively. The

electric fields after the compensator are,

)](exp[)2exp(1

)1(ˆ2

2'

oeee

eee i

irr

EeE Γ+−

−= δ

δ

r, (2-15)

)](exp[)2exp(1

)1(ˆ2

2'

eooo

ooo i

irr

EoE Γ+−

−−= δ

δ

r, (2-16)

where

λπ ''2 dne

e =Γ , (2-17)

λπ ''2 dno

o =Γ . (2-18)

After the analyzer, the total electric field and transmittance can be calculated as follows:

)(2

1ˆ ''oeout EEyE +=

r, (2-19)

2

2

in

out

E

ET r

r

= (2-20)

eooe TTT ++≡ , (2-21)

13

where

)2cos(21)1(

21

24

222

eee

eee rr

rET

δ−+−

= , (2-22)

)2cos(21)1(

21

24

222

ooo

ooo rr

rET

δ−+−

= , (2-23)

)1)(1(21 22

oeoeeo rrEET −−−=

⎭⎬⎫

⎩⎨⎧

−−−+

−−−×

)]2exp(1)][2exp(1[1

)]2exp(1)][2exp(1[1

2222ooeeooee iriririr δδδδ

, (2-24)

where we used the fact that the difference of the phase between e-mode and o-mode at

the liquid crystal layer ( oe δδ − ) is the same with that of the compensator layer ( oe Γ−Γ )

because we assume the dark state of the liquid crystal device (LCD) is compensated

perfectly. We also set 1=inE for simplicity.

Equations (2-20~24) show that there are three contributions to the total

transmittance in a LCD compensation system; eT is the transmittance related to the

interference of the pure e-mode by itself due to the multi-reflection in a LCD, and oT is

the transmittance that came from the interference of the pure o-mode with the same

reason. The last term eoT is a coupled term between the e-mode and the o-mode. So, eT

and oT depend on the absolute light path lengths (λdne ,

λdno ) of the e-mode and the o-

14

mode, respectively, but the eoT is related to the relative difference of the light path length

(λλdndn oe − ) between the e-mode and the o-mode.

15

(a) Simple layout of a compensated liquid crystal device

(b) Definition of each axis

Fig. 2-1. Simple layout of a compensated liquid crystal device and the definition of each

axis. The compensator is designed to compensate the phase of the dark state perfectly in

the visible wavelength range.

45º

x

eo

y

16

2-3. Multi-reflection in an isotropic layer

In the simplest case, let us consider the liquid crystal layer is isotropic. Then, e-

ray and o-ray are not distinguishable. Therefore, we have these relationships among the

optical parameters,

δδδ ≡= oe , 0≡Γ=Γ oe , rrr oe ≡= ,

and can write the electric fields at the position between the liquid crystal and the analyzer

( 'eEr

, 'oEr

) as follows from Eqs. 2-15, 16:

)exp()2exp(1

)1(ˆ2

2' δ

δi

irrEeE ee −

−=

r, (2-25)

)exp()2exp(1

)1(ˆ2

2' δ

δi

irrEoE oo −

−−=

r. (2-26)

These two fields have the exactly same magnitude and phase. Consequently, in the

optical viewpoint, their vector summation is linearly polarized, and its direction is always

parallel to the direction of the absorption axis of the analyzer [ x -axis in Fig. 2-1(b)]

regardless of light wavelengths. Mathematically, from Eqs. 2-14, 19, the total electric

field after the analyzer is

0=outEr

, (2-27)

17

and the total transmittance is

0=T , (2-28)

for all light wavelengths.

2-4. Multi-reflection in a liquid crystal device with no residual retardation in the

dark state (ECB-type LCD)

Electrically controlled birefringence (ECB) is one of the well-known liquid

crystal display mode because of its simple structure. Figure 2-2 shows the illustration of

the director configurations of an ECB cell. The dark and bright states are exchangeable

according to the relative angles between the directions of liquid crystal molecules and

polarizers. In this illustration, we set the high voltage state as a dark state because this is a

usual way in real devices. In an ideal ECB cell, the residual birefringence at the high

voltage (dark state) is very low because most of the liquid crystal molecules align along

the external electric field if applied voltage is high enough. However, the molecules near

the surfaces are not completely aligned due to the anchoring energy at the interface

between the liquid crystal and the alignment layer. We assume here that there is no

residual birefringence at the high voltage for simplicity, but still the device has different

reflection coefficients ( oe rr , ) of the e-mode and o-mode at the interfaces,

18

δδδ ≡= oe , 0=Γ=Γ oe , oe rr ≠ .

Then, the electric fields (e-ray and o-ray) just before the analyzer, Eqs. 2-15 and 2-18, are

simplified to be

)exp()2exp(1

)1(ˆ2

2' δ

δi

irr

EeEe

eee −

−=

r, (2-29)

)exp()2exp(1

)1(ˆ2

2' δ

δi

irr

EoEo

ooo −

−−=

r. (2-30)

19

Fig. 2-2. Director configurations of the liquid crystal in an ECB cell.

Dark State Bright State

Voltage

20

2-4-1. Phase analysis of the e-ray and o-ray

We calculated the phases of the electric fields, 'eEr

and 'oEr

in function of light

wavelength (λ ), and the Fig. 2-3 shows the result. During the calculation, we set en and

on of liquid crystal at the air-liquid crystal interfaces are 1.656 and 1.5, respectively in all

wavelengths, and the thickness of the liquid crystal layer is 5.53 μm. As in Fig. 2-3, 'eEr

and 'oEr

are almost in the same phase, but their magnitudes depend on light wavelength

(λ ) from Eqs. 2-10, 11.

21

380 480 580 680 780-300

-200

-100

0

100

200

300 Ee' E

o'

Wavelength (nm)

Angl

e (d

eg.)

Fig. 2-3. Calculation result of the phases of 'eEr

and 'oEr

in ECB type LCDs.

22

2-4-2. Analysis of the angle of light polarization

The phases between e-ray and o-ray after the compensator are almost in phase in

ECB-type LCDs as in the result of the previous section. Therefore, the total electric field

( ''oe EErr

+ ) in front of the analyzer is almost linearly polarized at all wavelengths, but the

direction of the total field varies from the absorption axis of the analyzer periodically

with the same phase of 'eEr

and 'oEr

in wavelength space, and the angle variation [ )(λθ ]

between them is calculated from the formula 2-29 and 2-30,

)2exp(1)2exp(1

11

tan4

)( 2

2

2

21

δδπλθ

irir

rr

e

o

o

e

−−

×−−

−= − . (2-31)

When the direction of the total field is in the absorption axis of the analyzer ( 0=θ ), it

gives zero transmittance, and as the angle (θ ) increases, light leakage becomes bigger.

The amplitude of this angle variation is not only proportional to the relative

difference of the reflection coefficients ( oe rr − ), but also depends on the absolute value

of each of them. Therefore, from Eqs. 2-8, 9, and 2-31, the light leakage of a dark state

LCD becomes larger as the refractive index mismatching or the relative difference of the

refractive indices ( oe nn − ) of the e-mode and o-mode increases at the interfaces. Figure

2-4 is the calculation results of the angle [ )(λθ ] in order to see the effect of refractive

indices on the light leakage, and Fig. 2-5 is the corresponding calculation of

transmittance. In Figs. 2-4(a) and 2-5(a), we used the refractive index of air ( airn ) as a

23

variable ( airn =1.0~1.5, on =1.5, en =1.656, d =5.53 µm, and ndΔ =0), and in Fig. 2-4(b)

and 2-5(b), refractive index of extraordinary ray ( en ) was the variable ( airn =1.0, on =1.5,

en =1.5~2.0, d =5.53 µm, and ndΔ =0) while the other parameters are fixed values. From

Fig. 2-4(a), we can see that as the refractive index of air ( airn ) is close to the refractive

index of an ordinary ray ( on ) or the refractive index of an extraordinary ray ( en ), the

amplitude of the angle [ )(λθ ] decreases because the magnitude of index mismatching

( airo nn − ) is going down. This variation of the angle leads to the variation of the light

leakage as in Fig. 2-5(a). Figure 2-4(b) shows that as the relative difference of the

refractive indices ( oe nn − ) rises, the amplitude of the angle [ )(λθ ] increases because

the relative difference of the reflection coefficients ( oe rr − ) increases, and the light

leakage also has the same trend as in Fig. 2-5(b).

On the other hand, the amplitude of the )(λθ is independent of light wavelength

if there is no dispersion of the refractive indices, so the amplitude of the light leakage of a

dark state is the same in whole wavelength region, and it does not affect the dark color.

24

380 480 580 680 7800.0

0.5

1.0

1.5

2.0 n

air=1.0 n

air=1.1 n

air=1.2

nair

=1.3 nair

=1.4 nair

=1.5

Wavelength (nm)

θ (d

eg.)

(a) airn =1.0~1.5, on =1.5, en =1.656, d =5.53 µm, and ndΔ =0

380 480 580 680 780

0

1

2

3

4

5 n

e=1.5 n

e=1.6 n

e=1.7

ne=1.8 n

e=1.9 n

e=2.0

Wavelength (nm)

θ (d

eg.)

(b) airn =1.0, on =1.5, en =1.5~2.0, d =5.53 µm, and ndΔ =0

Fig. 2-4. Effects on the angle [ )(λθ ] of (a) the index mismatching ( airo nn − ) and (b) the

relative difference of refractive indices ( oe nn − ).

25

380 480 580 680 780

0.0000

0.0001

0.0002

0.0003 n

air=1.0 n

air=1.1 n

air=1.2

nair

=1.3 nair

=1.4 nair

=1.5

Wavelength (nm)

Tran

smitt

ance

(a) airn =1.0~1.5, on =1.5, en =1.656, d =5.53 µm, and ndΔ =0

380 480 580 680 780

0.000

0.001

0.002

0.003 n

e=1.5 n

e=1.6 n

e=1.7

ne=1.8 n

e=1.9 n

e=2.0

Wavelength (nm)

Tran

smitt

ance

(b) airn =1.0, on =1.5, en =1.5~2.0, d =5.53 µm, and ndΔ =0

Fig. 2-5. Effects on the transmittance (light leakage) of (a) the index mismatching

( airo nn − ) and (b) the relative difference of refractive indices ( oe nn − ).

26

2-4-3. Analysis of transmittance without considering the dispersion properties of

refractive indices in ECB type LCDs

Figure 2-6 shows the calculation results of the separate transmittance ( eT , oT , eoT )

and the total transmittance T with the angle variation (θ ) in a parameter condition:

airn =1.0, on =1.5, en =1.656 at the interface, d =5.53 µm, and ndΔ =0. As we mentioned

above, the e-mode and o-mode are almost in phase, and the valley and ridge points of the

total transmittance (T ) are constant as the wavelength changes because we assumed that

refractive indices are constant at all wavelengths. Also, the angle variation (θ ) is in the

same phase with that of the total transmittance exactly. Therefore, the minimum

transmittance at the dark state is limited by the amplitude of the interference pattern (T ),

and the dark color is hardly affected in this type of interference.

27

380 480 580 680 7800.00

0.04

0.08

0.12

0.16 Te To Teo

Wavelength (nm)

Te, T

o

-0.28

-0.24

-0.20

-0.16

-0.12

Teo

(a) Transmittances of each mode

380 480 580 680 7800.0000

0.0002

0.0004

0.0006

0.0008

0.0010

T θ

Wavelength (nm)

Tran

smitt

ance

0

1

2

3

4

Angl

e (d

eg.)

(b) Total transmittance and the angle variation

Fig. 2-6. Transmittances of each mode and the total transmittance with the angle variation

in the dark state of an ECB type LCD without considering the dispersion of refractive

indices. (Δnd = 0 nm, d=5.53 μm, ne=1.656, and no=1.5 at the surface).

28


refractive indices in ECB type LCDs

In real situations, most materials, including liquid crystals, have wavelength

dispersion properties in their refractive indices. Figure 2-7 shows the dispersion property

of refractive indices of the commercialized liquid crystal material (LC53, Chisso Co.),

and we use this material in this dissertation. With considering the dispersion property of

this liquid crystal, we calculated the separate transmittance ( eT , oT , eoT ) and the total

transmittance T with the angle variation (θ ) as we did in Fig. 2-6, and the Fig. 2-8 is the

result. During the calculation, we used the same parameters as in Fig. 2-6 except the

refractive indices of liquid crystal. Comparing the results with Fig. 2-6, the amplitude of

the total transmittance depends on light wavelength, but the valley points of the

transmittance are still zero as the same as in Fig. 2-6. Therefore, if we consider the

dispersion property of refractive indices, the minimum transmittance of dark state could

be different for each light wavelength, and it could lead to the color shift of a dark state

even though the magnitude of the color shift is very small as we may neglect.

29

380 480 580 680 7801.45

1.50

1.55

1.60

1.65

1.70

1.75 n

e n

o

Wavelength (nm)

Ref

ract

ive

Indi

ces

Fig. 2-7. Light wavelength dispersion of refractive indices ( on , en ) of a commercialized

liquid crystal (LC53).

30

380 480 580 680 7800.00

0.04

0.08

0.12

0.16 Te To Teo

Wavelength (nm)

Te, T

o

-0.28

-0.24

-0.20

-0.16

-0.12

Teo


380 480 580 680 7800.0000

0.0002

0.0004

0.0006

0.0008

0.0010

T θ

Wavelength (nm)

Tran

smitt

ance

0

1

2

3

4

Angl

e (d

eg.)

(b) Total transmittance with the angle variation

Fig. 2-8. Transmittances of each mode and the total transmittance with the angle variation

in the dark state of an ECB type LCD with considering the dispersion of refractive

indices. (Δnd = 0 nm, d=5.53 μm, LC: LC53).

31

2-5. Multi-reflection in a liquid crystal device with residual retardation in the dark

state (Pi-cell type LCD)

Most of the liquid crystal device modes have residual birefringence in their dark

states even though we apply very high voltage because of the surface anchoring energy at

the liquid-alignment layer interface. As an example mode, we will consider a Pi-cell4, 5

mode LCD. Figure 2-9 shows the illustration of the director configuration of liquid

crystal in a Pi-cell. The director configuration of the Pi-cell has bend structure in both

dark and bright states as in Fig. 2-9. Usually, other display modes such as an ECB, and a

twisted nematic (TN) have splay structure in their dark states. Commonly, the elastic

energy of a bend structure is higher than that of a splay structure because it has a bigger

elastic constant. For example, the elastic constants of the liquid crystal (LC53) that we

are using are as follows:

Splay ( 11K ): 10.5 pN ,

Twist ( 22K ): 7.2 pN ,

Bend ( 33K ): 15.7 pN .

Therefore, we have to apply a much bigger voltage in Pi-cells to achieve the same level

of residual birefringence of the other modes. That means the dark state of Pi-cells has

bigger residual birefringence than the other modes in a given similar condition, and that

fact requires the use of a passive type optical retardation film to compensate it. Figure 2-

10 shows the effective birefringence at normal direction with the incident light

32

wavelengths, 450, 550, and 650nm, in a Pi-cell [liquid crystal: LC53; dielectric constant

( εΔ of LC) of 9.4; cell thickness of 5.53 μm]. As in this figure, the Pi-cell still has ~100

nm retardation at around the dark voltage (5~6 V). Therefore, we have these relationships

in a Pi-cell,

oe δδ ≠ , oe Γ≠Γ , oe rr ≠ .

So, from Eqs. (2-15) and (2-26), we know that the e-mode and o-mode components of the

electric fields ( 'eEr

, 'oEr

) after the compensator are out of phase in time and wavelength

space. Therefore, the light after compensator is elliptically polarized, and the ellipticity

and the angle of the major axis are functions of the light wavelength. This causes the total

transmittance to have a beat frequency behavior after the analyzer.

33

Fig. 2-9. Director configurations of the liquid crystal in a Pi-cell.

Dark State

Voltage

Bright State

34

2 4 6 8 10

0

100

200

300

400 450nm 550nm 650nm

LC Voltage (V)

Δnd

(nm

)

Fig. 2-10. Calculation results of the effective birefringence at the normal direction in a Pi-

cell.

35

2-5-1. Analysis of transmittance without considering the dispersion properties of

refractive indices in Pi-cell type LCDs

Figure 2-11 shows the calculation results of the transmittance ( eT , oT , eoT )

contributed from each mode and the total transmittance T at the dark state in a Pi-cell,

respectively. The magnitude of the residual birefringence we used here is 100 nm at all

light wavelengths, and the compensator compensates the value exactly at all wavelengths.

The cell thickness in this calculation is 5.53 μm, and the refractive indices, on and en of

the liquid crystal at the reflection interfaces are 1.5 and 1.656 with no wavelength

dispersion, respectively. As seen in Fig. 2-11(a), eT and oT are not in phase and have

different amplitudes, which depend on the magnitude of reflection coefficients ( oe rr , ). In

Fig. 2-11(b), the minimum values (valley points) of the transmittance curve are not zero,

and their values vary as the wavelength changes. These features are unlike that of the

ideal ECB-type LCDs. These phenomena make the dark level of LCDs higher, and not

only limit the contrast ratio (luminance of bright state / luminance of dark state) that the

displays can reach (here 500-1000:1) but also lead to the color shift of the dark state of

the LCDs.

36

380 480 580 680 7800.00

0.04

0.08

0.12

0.16 Te To Teo

Wavelength (nm)

Te, T

o

-0.28

-0.24

-0.20

-0.16

-0.12

Teo


380 480 580 680 7800.000

0.001

0.002

0.003

Wavelength (nm)

Tran

smitt

ance

(b) Total transmittance

Fig. 2-11. Transmittances of each mode and the total transmittance in the dark state of a

Pi-cell-type LCD without considering dispersion of refractive indices. ( en =1.656 and

on =1.5 at the reflection interfaces).

37


refractive indices in Pi-cell type LCDs

For understanding the effects of wavelength dispersion properties of refractive

indices, we calculated transmittances in the similar condition as the previous section. The

magnitude of effective residual birefringence that we used here is 100 nm at a wavelength

550 nm of incident light, and the value is compensated by a compensator precisely at all

wavelengths. The cell thickness ( d ) in this calculation is 5.53 μm, and the liquid crystal

is LC53 that has refractive indices, )(λen and )(λon of extraordinary and ordinary rays,

respectively. We fully considered the dispersion of the refractive indices of the liquid

crystal with adjusting the effective value of the refractive index of extraordinary ray

( effen ) to meet the effective residual birefringence ( effndΔ ),

dnnnd oeffeeff )]()([)( λλλ −=Δ , (2-32)

at a wavelength 550 nm. During this adjusting process, we keep the ratio of

)( oe nnn −Δ to )( oeffeeff nnn −Δ constant at all wavelengths. However, when we calculate

the reflectance coefficients, we used real refractive indices ( en , on ) because the liquid

crystal molecules near the surfaces hardly change their orientations due to anchoring

energy. Figure 2-12 shows the effective refractive indices of the liquid crystal (LC53)

with the condition that has a thickness of 5.53 µm and effective residual birefringence

( effndΔ ), 100 nm at an incident light wavelength of 550 nm. By using these parameters,

38

we calculated the transmittances as in Fig. 2-13. As the same as in Fig. 2-11, eT and oT

are out of phase and have different amplitudes which depend on the magnitude of

reflection coefficients ( oe rr , ). In Fig. 2-13(b), the minimum transmittances at each

wavelength also are not zero, and their values vary as the wavelength changes. These

features are the same with in Fig. 2-11. However, additionally if we consider the

properties of wavelength dispersion of refractive indices, we can notice that the

amplitude of the transmittance depends on the light wavelength as it is in ECB-type

LCDs in the previous section, i.e. as the wavelength decreases, the amplitude increases.

This can be understandable because as the wavelength decreases, the refractive indices

increase as in Fig. 2-12, and the magnitude of refractive index mismatching at the liquid

crystal-air interfaces increases, and this causes the amplitude of the transmittance

increases as similar as in Fig. 2-5. From comparing these results with the result in Fig. 2-

11, we see that the effect of wavelength dispersion is relatively not big as much as it can

govern the color of the dark states of the LCDs, although it could aggravate the color

shift that caused by other reasons.

39

380 480 580 680 780

1.50

1.55

1.60

1.65

1.70

1.75 n

e n

o n

eeff

Wavelength (nm)

Ref

ract

ive

Indi

ces

Fig. 2-12. Effective refractive indices of the liquid crystal (LC53) for the thickness of

5.53 µm and effective residual birefringence of 100 nm.

40

380 480 580 680 7800.00

0.04

0.08

0.12

0.16 Te To Teo

Wavelength (nm)

Te, T

o

-0.28

-0.24

-0.20

-0.16

-0.12

Teo


380 480 580 680 7800.000

0.001

0.002

0.003

Wavelength (nm)

Tran

smitt

ance

(b) Total transmittance

Fig. 2-13. Transmittances of each mode and the total transmittance in the dark state of a

Pi-cell-type LCD with considering dispersion of refractive indices.

41

2-5-3. Analysis of high and low frequency interference modes and their effects on

the optical properties of LCDs

The total transmittance [Fig. 2-13(b)] of a “Pi-cell type LCD” has two frequency

modes, roughly, high frequency and low frequency. In order to analyze the source of the

modes, we calculated the total transmittance at the two conditions. Firstly, the same phase

difference ( effndΔ = 100 nm) but different cell thicknesses ( d = 5.53, 10 μm), and the

result is shown in Fig. 2-14. Secondly, the constant cell thickness ( d = 5.53 μm) with

several different phase retardations ( effndΔ = 10, 50, 100, 300 nm), and Fig. 2-15 shows

the results. During the calculations, we considered fully the dispersion properties of the

refractive indices of liquid crystal (LC53) as the same as we did in Fig. 2-13, and the

corresponding effective refractive indices are in Figs. 2-16 and 2-17. From these results,

we know that the higher frequency mode is coming from the interference of the e-mode

and o-mode independently in a LCD and depends on the absolute light path lengths of

each mode. The lower frequency mode is caused by the interference between e-mode and

o-mode and affected by the relative phase difference of them. Considering the optical

properties, the lower frequency mode is much more critical, so we need to reduce the

residual birefringence to improve the dark quality of a LCD irrespective of the

compensation films. Another important thing is that the lower frequency mode is

governed by the optically anisotropic layers, so at which interface the multi-reflection

takes place is not so important in a real LCD because other layers except the liquid

crystal are mostly isotropic material. Therefore, the spacing between the interfaces

42

causing the reflections affects only on the higher frequency mode, as in Fig. 2-14 and

does not effect the color shift.

43

380 480 580 680 7800.000

0.001

0.002

0.003

d=5.53 μm d=10.0 μm

Wavelength (nm)

Tran

smitt

ance

Fig. 2-14. Thickness effect on the total transmittance of a Pi-cell-type LCD in the dark

state. Light path lengths of the e-mode and o-mode affect the high frequency mode.

( effndΔ =100 nm fixed).

44

380 480 580 680 7800.000

0.001

0.002

0.003

Δnd=0 nm Δnd=50 nm Δnd=100 nm Δnd=300 nm

Wavelength (nm)

Tran

smitt

ance

Fig. 2-15. Phase retardation effect on the total transmittance of a Pi-cell-type LCD in the

dark state. The low frequency mode is related to the phase difference between the e-mode

and o-mode. ( d =5.53 μm fixed).

45

380 480 580 680 780

1.50

1.55

1.60

1.65

1.70

1.75 n

e n

o

neeff (d=5.53μm)

neeff (d=10.0μm)

Wavelength (nm)

Ref

ract

ive

Indi

ces

Fig. 2-16. Effective refractive indices ( effen ) of the liquid crystal (LC53) with different

thicknesses and a constant phase retardation ( effndΔ =100 nm).

46

380 480 580 680 780

1.50

1.55

1.60

1.65

1.70

1.75 ne

no

neeff (Δnd=50 nm)

neeff (Δnd=100 nm)

neeff (Δnd=300 nm)

Wavelength (nm)

Ref

ract

ive

Indi

ces

Fig. 2-17. Effective refractive indices ( effen ) of the liquid crystal (LC53) with a constant

thickness ( d =5.53 μm) and different phase retardations.

47

2-5-4. Refractive index mismatching effects in Pi-cell type LCDs

There are about 15 to 20 layers in a real active matrix liquid crystal display

(AMLCD). Each of the interfaces between them can cause multi-reflection. Refractive

index mismatching at those interfaces are the source of multi-reflection in LCDs. We

calculated the effects of index mismatching on the light transmittance in dark states of

LCDs. The cell thickness ( d ) is 5.53 µm, and the effective residual retardation of the cell

( effndΔ ) is 100 nm at 550 nm of incident light wavelength. The liquid crystal is LC53,

and we fully considered the dispersion of the refractive indices of the liquid crystal as we

did in Sec. 2-5-2, and the refractive indices are in Fig. 2-12. However, when we calculate

the reflectance coefficients, we used real refractive indices ( en , on ) as the same reason.

Figure 2-18 shows the calculated results of the dark state transmittances in terms

of the magnitudes of index mismatching ( airo nn −)(λ ). As we can see in this figure, the

index mismatching strongly affects not only on the amplitude of interference pattern but

also on the absolute value of light leakage in dark states at all wavelengths. As expected,

the light leakage and the amplitude of interference due to multi-reflections sharply

decrease as the magnitude of index mismatching ( airo nn − ) falls.

As a rough calculation, let us say that we want to make a Pi-cell type LCD that

has at least a contrast ratio of 1000 (luminance of bright state / luminance of dark state).

We assume that polarizers are ideal, and there are no other sources that could give light

leakage in the dark state of the LCD. We also assume that the LCD is optically designed

to have transmittance of 0.35 at a normal direction. That means we have to achieve less

48

than 0.00035 of dark state transmittance. The refractive index of ordinary ray ( on ) of the

liquid crystal (LC 53) is about 1.5 at a wavelength of 550 nm. Therefore, from Fig. 2-18

we can say that we should control the magnitude of index mismatching ( airo nn − ) under

0.3. If the Pi-cell is contacting air interfaces directly, the magnitude of index mismatching

( airo nn − ) is about 0.5. Therefore, the maximum contrast ratio will be about 350 from

Fig. 2-18.

49

380 480 580 680 780

0.000

0.001

0.002

0.003 n

air=1.0

nair

=1.1 n

air=1.2

nair

=1.3 n

air=1.4

nair

=1.5

Wavelength (nm)

Tran

smitt

ance

Fig. 2-18. The effects on the dark state transmittance due to the refractive index

mismatching ( airo nn − ). ( d =5.53 μm, effndΔ =100 nm)

50

2-6. Experimental and numerical calculation results of a Pi-cell

2-6-1. Compensating a Pi-cell by using a uniaxial crystal compensator

To confirm the calculation results, we did experiments using a Pi-cell (liquid

crystal: LC53; cell thickness of 5.53 μm, pretilt angle of 5.5º on both sides). We used a

commercialized uniaxial compensator to compensate the phase retardation of the dark

state Pi-cell (applied voltage of 5.15 V) at the normal direction. The compensated and

measured wavelength range was 400-700 nm, and the step was 10 nm. However, the

phase difference of the LCD is hard to compensate perfectly over all of the wavelength

range simultaneously. Therefore, for each wavelength, we changed the retardation values

of the compensator to have minimum transmittance where the phase difference of the Pi-

cell is compensated exactly. Figure 2-19 (a) is the measurement layout, and Fig. 2-19 (b)

shows the structure of the Pi-cell. We used crystal polarizers to achieve high extinction

ratio in this experiment (Melles Griot, model 03PTO003/A, extinction ratio < 1/100 000).

The spectrophotometer (Model: USB2000 Ocean Optics) was used as a light detector.

Each optical component in the setup is separated physically, and the surfaces of the

crystal polarizers and the compensator are treated for anti-reflection in visible light

wavelength range. Therefore, the main sources of the multiple reflections are the either

sides of the Pi-cell and the layers inside the Pi-cell such as substrates, electrodes,

alignment layers, and liquid crystal.

Figure 2-20 (a) shows the measurement result of the dark state transmittance of

the Pi-cell whose phase retardations are compensated separately for each wavelength, and

Fig. 2-20 (b) is the minimum transmittance we take from the experimental raw data [Fig.

51

2-20 (a)]. As we can see in Fig. 2-20, there is a big light leakage in all of the wavelengths,

and especially in the blue region. This means the maximum contrast ratio of the Pi-cell is

limited by the multi-reflection effect, and also that effect could cause the blue shift in the

color of the dark state. In case of this experiment, the calculation results of the change of

the color coordinates [1931 CIE (International Commission on Illumination)] for the

standard D65 light source are

x = 0.3127, y = 3291 → x = 0.2722, y = 0.2802.

52

(a) Measurement setup of a Pi-cell compensation system

(b) Cell structure of the Pi-cell

Fig. 2-19. Measurement setup of a Pi-cell compensation system and the cell structure of

the Pi-cell. [Applied voltage of 5.15 V (dark state)].

Indium Tin Oxide (ITO) Polyimide

Liquid Crystal

Polyimide Indium Tin Oxide (ITO)

Glass

Glass

53

400 450 500 550 600 650 7000

1

2

3

4

5

Wavelength (nm)

Tran

smitt

ance

(%)

(a) Measured transmittance of the dark state Pi-cell

400 450 500 550 600 650 7000.00

0.01

0.02

Wavelength (nm)

Tran

smitt

ance

(b) Extracted minimum transmittance of the dark state Pi-cell

Fig. 2-20. Measured transmittance and the extracted minimum transmittance of the dark

state Pi-cell compensated with the compensator at each wavelength separately. Reference

(T=1): parallel crystal polarizers; LC volt: 5.15 V (dark state).

54

2-6-2. Compensating a Pi-cell by using hybrid-aligned negative C-plates

Hybrid-aligned negative C-plates or recently it is called polymerized discotic

material (PDM) was introduced by Mori et al.6, 7, 8 in 1997 to enlarge the viewing angle

and to reduce the driving voltage of a Pi-cell. The basic his idea is that the effect of each

layer of a positive birefringence material in a liquid crystal device can be optically

compensated by a layer of negative birefringence materials with the same optic axis

orientation. The PDM9 layer combined with a biaxial film [they are called wide view

(WV)-film or Fuji-film) is widely used now in many liquid crystal display modes

including twisted nematic (TN) and Pi-cells, and Fig. 2-21 shows the basic structure

when it is applied to a Pi-cell. As the figure, the PDM layers have negative birefringence

[ 0)( <−Δ oe nnn ] for optical compensation of the liquid crystal layer which has positive

birefringence [ 0)( >−Δ oe nnn ]. The optic axes of the PDM layer have similar angle

distribution compared with that of liquid crystal directors.

55

Fig. 2-21. Basic structure of the compensation scheme of a Pi-cell by using WV-Film.

The PDM layers have negative birefringence for optical compensation of the liquid

crystal that has positive birefringence. The optic axes of the PDM layer have similar

angle distribution compared with that of liquid crystal directors.

Pi-Cell Δn > 0

Biaxial Film

Polarizer

Polarizer

Biaxial Film

PDM Δn < 0

PDM Δn < 0

WV-Film

WV-Film

56

2-6-2-1. Multi-reflection effects in a real Pi-cell

We used the WV-film that is designed for a Pi-cell to compensate the phase

retardation of the dark state of a Pi-cell. The optical stack configuration of the

compensated real Pi-cell and the measurement setup are shown in Fig. 2-22. In this

experiment, we used sheet polarizers in place of the crystal polarizers with the same

angles. The reference of transmittance that gives 100% of transmittance was the

transmittance of the parallel sheet polarizers. All films and the Pi-cell are combined as a

single unit so that we can achieve minimum index mismatching at the interfaces unlike

the previous experiment. We applied voltage ranging from 4.9 to 5.7 V to liquid crystal

layer to get the minimum transmittance at each wavelength where the phase retardation

of the liquid crystal is compensated by the compensation films (PDM layers) perfectly

because the Pi-cell and the films are acting like the crossed uniaxial layers.

Figure 2-23 shows the measurement results of transmittances. From this

experiment, we see that there is a big multi-reflection effect in the dark state of the real

Pi-cell, and its effect is much more severe in the blue region. In this figure, we also see

that there is light leakage above the light wavelength of 700 nm. This is coming from the

light leakage of the crossed sheet polarizers as we can see in Fig. 2-24(a). An important

fact to note is that the minimum transmittances we achieved at each wavelength are

bigger than the transmittance of the crossed polarizers, and that difference increases in

the blue region even though we compensated the phase of the liquid crystal separately in

each wavelength. This experimental fact agrees well with our calculation results in Sec.

57

2-5. Therefore, this situation should lead to the blue shift of the dark color of the Pi-cell,

and the maximum contrast ratio we can get is around 1000 in this condition.

58

(a) Optical stack configuration of a compensated real Pi-cell

(b) Measurement setup

Fig. 2-22. Optical stack configuration of a compensated real Pi-cell and the measurement

setup. All layers are combined as a single unit without any air gaps.

Halogen Lamp

Pi-cell combined with all films

Spectro-photometer

Indium Tin Oxide (ITO) Polyimide

Liquid Crystal Polyimide

Indium Tin Oxide (ITO) Glass

Compensator (PDM) Biaxial film Polarizer

Triacetyl Cellulose (TAC)

Triacetyl Cellulose (TAC) Polarizer

Biaxial film Compensator (PDM)

Glass

Light Source

59

400 450 500 550 600 650 700 7500.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0 4.9 V 5.0 V 5.1 V 5.2 V 5.3 V 5.4 V 5.5 V 5.6 V 5.7 V

Wavelength (nm)

Tran

smitt

ance

(%)

Fig. 2-23. Measurement results of the transmittance of a compensated real Pi-cell. All

films including polarizers and compensators are attached to the Pi-cell as a single unit.

Reference transmittance (T=100%): transmittance of parallel sheet polarizers.

60

2-6-2-2. Multi-reflection effects in a real Pi-cell with additional crystal polarizers

To remove the effects of sheet polarizers that have relatively large transmittance

in their crossed state, we did another experiment. We used the same setup and Pi-cell but

inserted additional crystal polarizers (with the same angles of sheet polarizers, Melles

Griot, model 03PTO003/A, extinction ratio < 1/100 000 at visible wavelengths) outside

the sheet polarizers to get rid of the light leakage that takes place in crossed sheet

polarizers. In this experiment, the reference of transmittance that gives 100% of

transmittance was the parallel crystal polarizers. It means the measurement results

include the absorption effect of sheet polarizers. Figure 2-25 (a), (b) show the

measurement result and experimental setup, respectively. In this result, there in no big

light leakage above the light wavelength of 700 nm unlikely in Fig. 2-23, and this is

because we used almost ideal additional polarizers. Another different point from Fig. 2-

23 is that the light leakage in blue wavelengths (about 400-480 nm) decreases as the

wavelength decreases, and this is opposite direction compared with the result in Fig. 2-23.

This result is understandable if we see the transmittance curve of parallel sheet polarizers

[Fig. 2-24(b)]. As we mentioned above, the transmittance in Fig. 2-25 includes the

absorption effects of sheet polarizers, and they have large absorption in blue region

regardless of the polarization of incident light as we can see in Fig. 2-24(b). From this

experiment, we can say that there is light leakage in a Pi-cell even though we compensate

the phase retardation of the dark state perfectly and use ideal polarizers. This light

leakage is coming from the multiple reflection effect in a LCD, and it depends on the

light wavelength. Therefore, it could affect the color of the dark state of a LCD.

61

400 450 500 550 600 650 700 7500.00

0.05

0.10

0.15

0.20

0.25

0.30

Wavelength (nm)

Tran

smitt

ance

(%)

(a) Crossed sheet polarizers

400 450 500 550 600 650 700 7500

20

40

60

80

100

Wavelength (nm)

Tran

smitt

ance

(%)

(b) Parallel sheet polarizers

Fig. 2-24. Transmittances of the crossed sheet polarizers and the parallel sheet polarizers.

Reference transmittance (T=100%): transmittance in air.

62

400 450 500 550 600 650 700 7500.0

0.2

0.4

0.6

0.8

1.0 4.9 V 5.0 V 5.1 V 5.2 V 5.3 V 5.4 V 5.5 V 5.6 V 5.7 V

Wavelength (nm)

Tran

smitt

ance

(%)

(a) Measurement results of the transmittances

(b) Experimental setup

Fig. 2-25. Measurement results of the transmittances of a compensated Pi-cell with

additional crystal polarizers and the experimental setup. Reference transmittance

(T=100%): transmittance of parallel crystal polarizers.

Halogen Lamp

Crystal Polarizer

Pi-cell combined with

all films

Spectro-photometer

Crystal Polarizer

63

2-6-2-3. Refractive index mismatching effects in a Pi-cell

As we described in Sec. 2-5-4, refractive index mismatching is a very important

factor in multi-reflections. In order to see the effects experimentally, we did another

similar experiment. All methods and samples including a Pi-cell and compensation films

are the same things that we used in the previous section. The difference is that we

separated the compensation films from the Pi-cell to make big index mismatching

between the Pi-cell and compensation films by inserting an air gap. The measurement

results and the experimental setup are shown in Fig. 2-26 (a), (b), respectively.

Comparing with Fig. 2-25, overall interference patterns are the same, but the magnitude

of light leakage increases more than three times. This trend agrees with our analytical

calculation results (Fig. 2-18). Therefore, we can say that the refractive index

mismatching should be one of the important factors when we design a LCD; otherwise, it

could not only increase the light leakage but also make color shift (possible blue shift) of

a dark state.

In Fig. 2-25, the maximum contrast ratio that we can achieve is about 100/0.1 =

1000 as a rough calculation at an incident light wavelength of 550nm. The refractive

index of the ordinary ray ( on ) of the liquid crystal (LC 53) is about 1.5 at a wavelength

of 550 nm. The layer that has the maximum refractive index in LCD layers is Indium Tin

Oxide (ITO) that has about 1.8 of refractive index at a wavelength of 550nm. It means the

magnitude of refractive index mismatching ( ITOo nn − ) is about 0.3. This result of a

rough calculation based on the experimental data agrees well with that of the analytical

calculation in Sec. 2-5-4.

64

If we do similar calculation in Fig. 2-26 that is measured with bigger refractive

index mismatching, the maximum contrast ratio is about 100/0.3≈330 at a wavelength of

550nm. In this case, the magnitude of index mismatching ( airo nn − ) is about 0.5. This

experimental result is also very consistent with that of the analytical calculation in Sec. 2-

5-4.

65

400 450 500 550 600 650 700 7500.0

0.2

0.4

0.6

0.8

1.0 4.9 V 5.0 V 5.1 V 5.2 V 5.3 V 5.4 V 5.5 V 5.6 V 5.7 V

Wavelength (nm)

Tran

smitt

ance

(%)

(a) Measurement results of the transmittances

(b) Experimental setup

Fig. 2-26. Measurement results of the transmittances of a compensated Pi-cell with

additional crystal polarizers and air gaps, and the experimental setup. Reference

transmittance (T=100%): transmittance of parallel crystal polarizers.

Halogen Lamp

Crystal Polarizer

Pi-cell

Spectro-photometer

Crystal Polarizer

Sheet polarizers and compensators (WV-films)

66

2-6-3. Numerical calculations of the multi-reflection effects in a Pi-cell

We simulated these experiments by using a numerical relaxation technique to

calculate the director field in the Pi-cell and by using the Berreman 4×4 method10, 11, 12,

13 to calculate the optical properties. One of the problems when we use the Berreman 4x4

method is that there is a Fabry-Perot effect in the calculation results due to the multiple

reflections in a cell. This effect is coming from the assumption that the incident light has

an infinite coherence length14. However, most of light sources have a finite coherence

length in real situations. To remove the Fabry-Perot effect, there are several methods

such as spectrum averaging of transmission light, apodization method15, 16, restricting the

number of multi-reflections 17, and changing the refractive index of outside media 18.

During our optical calculations, we used the spectrum averaging method with 1.0 nm of

averaging bandwidth and 0.01 nm of calculation interval. We made this calculation

program by ourselves.

In these calculations, we used exactly the same optical stack as in Fig. 2-22 and

considered the dispersion of the refractive indices of all layers. The optical parameters of

the layers that we did not have from the manufacturer such as the glass, ITO (Indium Tin

Oxide), and polyimide, were measured by ourselves at a resolution of 1 nm by using a

spectroscopic ellipsometry (model: WVASE 32, J.A. Woollam. Co., Inc.), and the results

are in Fig. 2-27.

Figure 2-28 (a), (b) shows the measurement and numerical calculation results,

respectively. The only measured parameter that was adjusted to acquire Fig. 2-28 (b) was

the thickness of the liquid crystal layer. The thickness of the liquid crystal layer was

67

measured before filling the liquid crystal as 5.53 µm, but we adjusted the value to 5.18

µm during the calculation to achieve the best fitting to the experimental data. We think

this is acceptable because the thickness of the liquid crystal layer could be changed

during the cell-making process.

68

380 480 580 680 7801.0

1.5

2.0

2.5 Glass ITO Polyimide

Wavelength (nm)

Ref

ract

ive

Inde

x, n

(a) Real parts of the refractive indices

380 480 580 680 780

0.000

0.004

0.008

0.012

0.016

0.020 Glass ITO Polyimide

Wavelength (nm)

Ref

ract

ive

Inde

x, k

(b) Imaginary parts of the refractive indices

Fig. 2-27. Measurement data of the real and imaginary parts of refractive indices.

Instrument: WVASE 32 spectroscopic ellipsometry (J.A. Woollam. Co., Inc.)

69

400 450 500 550 600 650 7000.000

0.005

0.010

0.015

0.020 5.0V 5.2V 5.4V 5.6V

Wavelength (nm)

Tran

smitt

ance

(a) Measurement results

400 450 500 550 600 650 7000.000

0.005

0.010

0.015

0.020 5.0V 5.2V 5.4V 5.6V

Wavelength (nm)

Tran

smitt

ance

(b) Numerical calculation results

Fig. 2-28. Measurement results and the numerical calculation results of the transmittance

of the Pi-cell compensated with the hybrid aligned negative-discotic films for different

applied voltages. Reference (T=1): parallel sheet polarizers.

70

2-7. Summary

We calculated the multi-reflection effects analytically and numerically in the dark

state of a compensated liquid crystal device and compared the results with the measured

transmittance of an example device.

According to our analysis, there are two types of interference in devices with

significant residual retardation in the dark state that is compensated by a passive optical

retarder. The first one is due to the pure e-ray and pure o-ray by themselves, and the

second one is coming from the coupling between the e-mode and o-mode. The first type

has higher frequency in the wavelength space and is related to the optical path length of

the e-ray and o-ray and is independent of their difference. Most of the modes used in

liquid crystal devices have this type of interference. The second type of interference has

lower frequency than that of the first one and depends on the residual birefringence of

dark state. So, as the residual birefringence increases, the second type of interference

becomes more significant. In the viewpoint of the optical properties of a liquid crystal

device, the first type of interference could affect the dark level and extinction ratio almost

equivalently for visible wavelength region. On the other hand, the second type could

cause a wavelength dependence of the extinction ratio, or a color shift of the dark state

because of the lower frequency pattern of the interference in the wavelength space.

71

CHAPTER 3

UNIVERSAL OFF-AXIS LIGHT TRANSMISSION PROPERTIES OF THE

BRIGHT STATE IN COMPENSATED LIQUID CRYSTAL

DEVICES

3-1. Introduction

The viewing angle dependence of the light transmittance in liquid crystal devices

is a well-known feature and one of the biggest problems. The dark (black) state is

especially important because it is critical to the contrast ratio of the device. On this

account, a great deal of research has been done to reduce the dark state transmittance and

its variation at all viewing angles.

On the other hand, the bright (white) state determines the luminance and the

spectral variations of the transmittance as a function of the angle of the incident light.

This bright state is especially important in the applications of large size displays such as

LCD monitors, TVs, and signs. Therefore, this should be another vital factor when

deciding the optical properties of the devices. However, we have not seen those studies

dealing with the off-axis light transmission properties (viewing angle properties) of the

bright state of general liquid crystal devices.

In this work, we investigate the off-axis light transmission properties (viewing

angle properties) of the bright state in the most common liquid crystal devices whose

dark states are optically compensated to have minimum transmittance at the normal

72

direction. According to our results, there is an interesting universality in the off-axis light

transmission properties of the bright state of liquid crystal devices, which is independent

of the display modes. In order to explain these interesting facts, we make simple dark and

bright state models that can be applied to general liquid crystal devices and analyze them

in terms of the effective retardation and transmittance.

In Sec. 3-2, we will show the off-axis light transmission properties of the bright

state in the most commonly used liquid crystal devices. In Sec. 3-3, we will introduce

simple dark and bright state models, and in Sec. 3-4, we will calculate the angular

dependencies of the effective retardation and transmittance of the models and then

compare the results with those of Sec. 3-2. Finally, we will compare the viewing angle

properties of our models with those of the real liquid crystal devices in all viewing

directions in Sec. 3-5.

3-2. Optical properties of general liquid crystal devices

Figure 3-1 shows the cartoons of director configurations of the most common

liquid crystal display (LCD) modes, such as the electrically controlled birefringence

(ECB), vertical alignment (VA), Pi-cell19, 20 (or OCB a, 21, 22), and symmetric splay-cell.

With the twisted nematic (TN), these modes cover most of the popular concepts of LCDs

that use nematic liquid crystal materials. The bright and dark states of each mode can be

a Pi-cell was invented in 1983 by Bos. Subsequently, Uchida used it with a single biaxial film as a

compensator with normally black mode and called it OCB in 1993. Their structures of the liquid crystal and

polarizers are same.

73

exchangeable by changing the optical design-parameters such as polarizer angles and

compensation films. In this dissertation, we set the bright and dark states as in Fig. 3-1

because those methods are widely used now to improve the dark state qualities.

74

Fig. 3-1. Director configurations of the most common liquid crystal display modes. The

dark and bright states are exchangeable by optical design.

Dark Bright Dark Bright

ECB VA

Pi-cell Symmetric splay

Dark Bright Bright Dark

75

3-2-1. Calculation methods and compensations of the dark states of the LCDs

We calculated the off-axis light transmission (or viewing angle) properties of

these LCD modes in their dark and bright states with an incident light wavelength of 550

nm. During the calculation, we used the numerical relaxation technique to get the director

field of the liquid crystal layer. We also used the 2×2-matrix method23, 24, 25 to calculate

the optical properties of the devices. We made these calculation programs by ourselves.

The dark state of each mode is optically compensated via two (above and below

the liquid crystal layer) compensators with the hybrid-aligned negative C plates as

discussed in Sec. 2-6-2. They have exactly the same angular distribution as the directors

of the liquid crystal layer of the dark state, and their extraordinary and ordinary refractive

indices are the same as the ordinary and extraordinary indices of the liquid crystal,

respectively. This is in accordance with Mori’s argument26, 27 that the effect of each layer

of a positive birefringence material in a liquid crystal device can be optically

compensated by a layer of negative birefringence materials with the same optic axis

orientation. Other methods, for example, using positive O-plates28, 29 or biaxial material30

can be used for optical compensation of the dark state. However, their final destinations

are the same, i.e. we want to make the total effective retardation of liquid crystal and

compensator layers zero in all viewing directions and wavelengths if we use ideal

polarizers. Therefore, the more we perfectly compensate the dark state, the more the final

compensator effects are the same, no matter what compensation schemes are used.

76

3-2-2. Optical parameters of the LCDs

The thicknesses of the liquid crystal layer and the compensators of each display

mode are determined by meeting two conditions. Firstly, it gives the minimum

transmittance at all viewing angles in the dark state. Secondly, the transmittance of the

bright state at the normal direction has a specific value ( oT ). This value can be chosen

from zero to maximum transmittance that we can achieve under the crossed polarizers,

but a bit smaller than the maximum value is usually used to achieve high transmittance

and to escape gray scale inversions and color shift in the off axis viewing directions. In

this dissertation, we set the value to ¾ of the maximum transmittance because this is a

similar condition in real liquid crystal devices, and this value is not a critical factor in

determining the physical concept of devices.

The detailed cell specifications and the stack configuration are in Table 3-1 and

Fig. 3-2, respectively. Where, the liquid crystal we used is LC53, and we assumed the

polarizers are ideal, i.e. the light that polarized along the absorption axis of a polarizer is

absorbed completely, but the light that is polarized along the transmission axis of a

polarizer is transmitted perfectly except surface reflections. The refractive indices of the

glass, indium tin oxide (ITO), and polyimide are in Fig. 2-27, and their thicknesses are

0.7 mm, 400 Å, and 500 Å. We want to make clear that the specific numbers are only for

the purposes of producing a graph that demonstrates the features of the general concepts

considered here.

77

Table 3-1. Cell parameters of the common liquid crystal (LC) devices we used. These

specific numbers are only for the purposes of producing a graph that demonstrates the

features. (LC: en =1.6644 and on =1.5070 at λ=550 nm; Δε=-9.4 for VA, +9.4 for other

devices).

Thickness (µm)

Easy axis (°)

Voltage (V)

Pretilt angle(°) Device

modes LC Compensator Top Bottom Bright Dark LC Pi-cell 5.037 2.255 90 90 1.3 5.0 5.5 ECB 1.423 0.637 90 -90 0.0 5.0 5.5 VA 1.455 0.541 90 -90 5.0 0.0 84.5 TN 2.038 0.832 45 -45 0.0 5.0 5.5

Splay-cell 1.87 0.934 90 90 5.0 0.0 5.5

78

Fig. 3-2. Stack configuration of the common liquid crystal devices.

ITO Polyimide

Liquid Crystal

Polyimide ITO

Glass

Compensator (PDM)

Polarizer (+45º)

Polarizer (-45º)

Compensator (PDM)

Glass

Light Source

79

3-2-3. Calculation results of the off-axis light transmittance in dark states

Figure 3-3 depicts the numerical calculation results of the off-axis light

transmission properties of the common liquid crystal devices in their dark states, whose

phase retardations are compensated as described in Sec. 3-2-1. We added the off-axis

light transmittance curves of the just crossed polarizers for the comparison. We also

calculated the director tilt angles and effective phase retardation of the dark state-liquid

crystal layer of each device, as in Fig. 3-4 and Fig. 3-5, respectively. These figures show

that the dark state transmittances of the common LCDs have the similar level of the

crossed polarizers in main viewing directions. Therefore, we can say that the effective

phase retardation of each dark state-liquid crystal layer of the common LCDs is optically

compensated well.

Another thing we want to note here is that there is relatively big light leakage at

the off-axis viewing angles, as in Fig. 3-3 (a) and (b), although we used ideal crossed

polarizers. This light leakage is caused by the fact that the absorption planes of the

crossed polarizers are not perpendicular with each other at off-axis viewing angles except

the planes that include the absorption or transmission axis of the one of the polarizers.

This fact was firstly pointed out by Chen et al.31 in 1998, and the problem can be curable

by using an A-plate and a C-plate or single biaxial film. To improve the dispersion

properties of light wavelength, using two-biaxial films methods were introduced32, 33, 34,

and, recently, wide viewing-angle polarizers having relaxed manufacturing tolerances

were presented35.

80

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03 Pi-cell ECB VA TN Splay-cell Crossed polarizers

Viewing angle (deg.)

Tran

smitt

ance

(a) Out of the director plane (Φ=0º)

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02



Tran

smitt

ance

(b) Director plane (Φ=90º)

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02



Tran

smitt

ance

(c) Φ=135º (Most of the curves are overlapped.)


common liquid crystal devices in their dark states.

81

0.0 0.2 0.4 0.6 0.8 1.0

0

50

100

150

200 Pi-cell ECB VA Splay-cell

Normalized cell thickness

Tilt

angl

e (d

eg.)

Fig. 3-4. Director tilt angles of the dark state-liquid crystal layers.

82

-80 -60 -40 -20 0 20 40 60 80-50

0

50

100

150

200

250

300

350

400

450



Effe

ctiv

e Δn

d (n

m)

Fig. 3-5. Effective phase retardation of the dark state-liquid crystal layers in the director

plane. (λ=550 nm).

83

3-2-4. Calculation results of the off-axis light transmittance in bright states

We numerically calculated the off-axis light transmission properties of the

common liquid crystal devices in their bright states, and the Fig. 3-6 shows the results.

From these transmittance figures, we notice that even though the director configurations

and effective birefringence of the liquid crystal layers are completely different from each

other, as in Fig. 3-7 and Fig. 3-8, the off-axis light transmission properties of the

compensated liquid crystal devices, amazingly, have unified shapes. (We cannot define

the birefringence out of the director plane because the optic axes of the directors are

apparently twisted, so we calculated it only in the director plane. For the same reason,

only the off-axis light transmission properties are calculated in TN mode.) In the viewing

angle of the director plane [Fig. 3-6 (a)], all the transmittance curves have similar “bell”

shapes, i.e. the transmittance constantly decreases as the viewing angle increases from the

normal direction no matter what liquid crystal modes are used. On the other hand, the

transmittance out of the director plane [Fig. 3-6 (b)] rises as the viewing angle increases

from the normal direction and then falls after passing the specific angles (about ± 50° in

these calculations), regardless of the director configurations of the liquid crystal in their

bright states. These results show that the off-axis light transmission properties of the

bright state in single domain liquid crystal devices have a common shape, independent of

their modes, as long as their dark states are optically compensated to give the lowest

transmittance for all viewing angles. This surprising fact motivated us to investigate the

reason for this in the next section.

84

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5 Pi-cell ECB VA TN Splay-cell


Tran

smitt

ance

(a) Director plane

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Pi-cell ECB VA TN Splay-cell


Tran

smitt

ance

(b) Out of the director plane


common liquid crystal devices in their bright state.

85

0.0 0.2 0.4 0.6 0.8 1.0-100

-50

0

50

100

150



Tilt

angl

e (d

eg.)

Fig. 3-7. Director tilt angles of the bright state-liquid crystal layers.

86

-80 -60 -40 -20 0 20 40 60 800

100

200

300

400



Effe

ctiv

e Δn

d (n

m)

Fig. 3-8. Effective phase retardation of the bright state-liquid crystal layers in the director

plane. (λ=550 nm).

87

3-3. Universal simple models

Because we have seen universality in the bright state of many display modes that

have an almost perfectly compensated dark state, we are looking for a very basic

explanation of the bright state viewing angle properties that does not include highly

detailed considerations. In this section, we will go to two different ways, but finally, we

will achieve the same simple bright state model.

3-3-1. Simple dark and bright state modeling

Let us consider the liquid crystal director configurations of several non-twisted

common liquid crystal devices (LCDs) such as ECB, VA, Pi-cell, and symmetric splay-

cell. Figure 3-9 (a) shows cartoons of the liquid crystal director configurations of each

device. Each of the devices has two states, state 1 and state 2, depending on applied

voltage, and either of them is bright state and the other is dark state. In these two states,

we can notice that there are simply two parts in a liquid crystal layer: the static part

(empty director shape in the figure) and dynamic part (filled director shape). In the static

part, most of the liquid crystal directors only slightly change their orientation between the

two states. On the contrary, the directors in the dynamic part are very sensitive to the

applied voltage, so the orientation of the directors is completely different between the

state 1 and state 2 of each device as in Fig. 3-4 and Fig. 3-7. Considering the director-tilt

angles, both states have vertical and horizontal components, but state 1 has a larger

vertical component than state 2, and state 2 has a larger horizontal component than state 1.

We can conceptually think of each liquid crystal device as being composed of two types

88

of layers, static and dynamic, as shown in Fig. 3-9 (a). The director configurations of the

static layers are the same in both state 1 and 2. Therefore, the only difference between the

two states is in the dynamic layers.

We consider compensating one of the two states perfectly to make it the dark state

by using passive, negative type optical compensators (negative A and C-plates). In order

to do that, we set the direction of the optic axes of the negative plates to be exactly the

same as those of the positive plates of the dark state. There are two ways: mode 1 and

mode 2. In mode 1, we use the state 1 as the dark state and the state 2 as the bright state,

and they are reversed in mode 2, as shown in Fig. 3-9 (b). In this figure, we can see that

the static layers of liquid crystal in mode 1 and mode 2 are always optically completely

compensated in both dark and bright states, simultaneously by their corresponding

components (empty shapes in the figure) of the compensators. Therefore, the contribution

of the static layers to the optical transmittance is canceled by the compensators, and

finally we achieve the simple dark and bright state models of mode 1 and 2 as in Fig. 3-9

(c). In these simple models, the dark states are optically neutralized in both modes, but

the bright states have a net birefringence that is a function of the viewing direction.

89

Fig. 3-9. (a) Simplification of the liquid crystal layers of the various devices, (b) optical

compensation of the simplified liquid crystal layer and (c) simple dark and bright state

models. The long axis of the ellipses indicates the optic axis orientation of the liquid

crystal with a positive birefringence. The symmetry axis of the disks indicates the optic

axis of the compensator with a negative birefringence. The shaded ellipses represent

directors that change, going from state 1 to state 2, and the shaded disks represent the

component of the compensator that compensates for them in one of the states.

90

3-3-2. “3-layer” modeling

As the second way, we will consider a simple “3-layer” model. This type of

model was used in early calculation of LCDs before detailed calculations were easy to

perform. The “3-layer” model of the common display modes is shown in Fig. 3-10 (a). In

each case, the display mode is shown along with a compensator that is assumed to

compensate the dark state of the device. The director distribution of the compensator is

shown as outlined directors to indicate that the material has a negative birefringence.

From this figure, we can see that the net optical effect of the compensator and the

bright state of the liquid crystal layer is only seen in the mid-layer of each device.

Furthermore, the difference between the compensator and the bright state of the liquid

crystal is the same in all display modes. As a result, we get the simple bright state model

composed of a positive A-plate and a negative C-plate as in Fig. 3-10 (b). This simple

model is exactly the same as the mode 1 in Fig. 3-9 (c).

From this bright state model, we can see that simple interaction between a

uniform negative C-plate and a positive A-plate might be expected to describe the

viewing angle properties of the bright state of most compensated liquid crystal devices.

This implies that the physics of the device is very simple and might be able to be

explained by a very simple and intuitive optical model.

91

Fig. 3-10. (a) “3-layer” modeling of the most common LCD modes and (b) simple bright

state model.

(b)

Bright Dark

Bright

ECB

Dark Bright Dark

Pi-cell

VA TN(a)

92

3-4. Calculations of the optical properties of the simple models

3-4-1. Basic calculations

The light transmittance (T ) of the simple model [Fig. 3-9 (c)] under ideal crossed

polarizers can be written as follows:

2sin

2sin

21 2

max2 Γ

≡Γ

= TTTT paap , (3-1)

where apT and paT are the transmittances with considering the surface reflections at the

interfaces between air and polarizers in the incident and exit media, respectively. They

can be calculated through the Fresnel equations36,

2

coscos

ijii

jjij t

nn

Tθθ

= , (3-2)

where in and jn are the refractive indices of the incident and refracted media, iθ and jθ

are the incident and refraction angles, respectively, and ijt is the transmission coefficient.

The p -polarization and s -polarization components of ijt are expressed in Eqs. (3-3) and

(3-4), respectively,

93

jiij

iipij nn

nt

θθθcoscos

cos2+

= , (3-3)

jjii

iisij nn

nt

θθθcoscos

cos2+

= . (3-4)

The Γ in Eq. (3-1) is the phase difference between e-ray and o-ray in the anisotropic

layers and written in terms of incident light wavelength (λ ) and effective retardation

( effndΔ ) of media when we assume the e-ray and o-ray have the same path in the media,

effndΔ=Γλπ2 . (3-5)

[The exact expression of the phase difference (Γ ) is dKK ozez )( −=Γ , where ezK and

ozK are z components of the wave vectors of the e-ray and o-ray, respectively. For

simplicity, we assume the e-ray and o-ray have the same path in this dissertation, which

is reasonable in a real situation.] In the normal direction, let us say the transmittance of

the bright state is oT , and the corresponding phase difference is oΓ . Then, as indicated by

Eq. (3-1),

21

max

1sin2 ⎟⎟⎠

⎞⎜⎜⎝

⎛=Γ −

TTo

o . (3-6)

94

This phase difference is purely caused by the positive A-plate (bottom layer) of the bright

state model in mode 1 of Fig. 3-9 (c) because the optical axis of the negative C-plate (top

layer) is in the normal direction. With similar reasoning, the phase difference is

exclusively related to the negative A-plate (top layer) of the bright state model in mode 2.

From these facts and Eqs. (3-5) and (3-6), we can determine the thicknesses of the

positive A(C)-plate (bottom layer) in mode 1 and the negative A-plate (top layer) in mode

2 that gives the bright state transmittance oT for a given material in the normal direction.

In mode 1, the incident light-angle-dependence of the effective retardation

( effndΔ ) of the positive A(C)-plate (bottom layer) and negative C-plate (top layer) are

calculated in Eqs. (3-7) and (3-8), respectively,

απλ

ψψ cos2sincos 2222 nn

nn

nnnd o

o

oe

oepositiveeff Δ

Γ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

+=Δ , (3-7)

'cos''

'sin''cos'

''2222 αψψ

dnnn

nnnd o

oe

oenegativeeff ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛−

+=Δ , (3-8)

where )'( ee nn and )'( oo nn are the refractive indices of the e-ray and o-ray, )'(ψψ is the

angle between the optic axis of the director and light propagation vector, )'(αα is the

polar angle of incident light in the medium, and 'd is the thickness of the negative C-

plate. In a given assumption that the transmittance of the dark state should be minimized,

95

the total retardation of the dark state model ( darkΔ ) should be zero at a given light

incident angle,

0=Δ+Δ=Δdark

negativeeffdark

positiveeffdark ndnd . (3-9)

From Eqs. (3-7), (3-8) and (3-9), the thickness of the negative C-plate ( 'd ) of mode 1

can be decided. In the same way, we can also determine the thickness of the positive

A(C)-plate (bottom layer) in mode 2.

3-4-2. Analyses of the effective phase retardation of the simple models

In this section, we are going to calculate the total effective retardation of the

bright state by way of the formula in the previous section in the two viewing planes:

director plane [x-z plane in Fig. 3-9 (c)] and out of the director plane [y-z plane in Fig. 3-

9 (c)]. The extraordinary and ordinary refractive indices of the positive (negative) plates

are 1.6 (1.5) and 1.5 (1.6) at the incident light wavelength of 550 nm, respectively. The

thicknesses of the positive (bottom) and negative (top) plates in mode 1 are 1.852 and

1.736 µm and are 1.852 µm in both plates of mode 2. These values were chosen to meet

the ideal dark and bright state conditions in Sec. 3-2-2 under crossed polarizers. We use

these parameters to calculate the phase retardation and transmittance of our simple

models in this chapter. All calculations are performed at the incident light wavelength of

550 nm.

96

3-4-2-1. Effective phase retardation in the director plane

With the viewing direction in the director plane of our bright state model in mode

1 [Fig. 3-9 (c)], the projection of the optic axes of the positive A-plate and negative C-

plate onto the plane perpendicular to the light propagation vector ( Kr

) are parallel to each

other, as in Fig. 3-11 (a). Consequently, the total effective phase retardation ( ||whiteΔ ) of

the bright state is the summation of both values,

negativeeff

positiveeffwhite ndnd Δ+Δ=Δ|| . (3-10)

Figure 3-12 illustrates how the effective retardation changes, as calculated from Eqs. (3-

7), (3-8) and (3-10), in the director plane of our bright state model in mode 1. During the

calculations, we considered the refraction of the incident light at the air interface. In the

normal direction, the birefringence of the negative C-plate is zero as expected, so it does

not contribute to the total retardation. As the viewing angle increases from the normal

direction, the effective retardation of the positive A-plate and the negative C-plate always

decreases. The total effective retardation of both plates then decreases steeply as the

viewing angle increases in the director plane.

Figure 3-13 is the corresponding figure for mode 2. It is almost the mirror image

of Fig. 3-12. Intuitively, this feature of mirror symmetry is expected from the director

structures in mode 1 and 2, i.e. positive A-plate in mode 1 to negative A-plate in mode 2

and negative C-plate in mode 1 to positive C-plate in mode 2. In this mode, unlike mode

1, the effective retardation of each layer (positive C-plate and negative A-plate) and their

97

total increase as the viewing angles increase from the normal direction, but the sign of the

total effective birefringence is negative. Consequently, the absolute value of the total

effective retardation decreases as the viewing angle increases, and this is the same trend

as in mode 1.

98

Fig. 3-11. Projections of the optic axes of A and C-plates onto the plane perpendicular to

the light propagation vector ( Kr

) that lies (a) in the director plane (x-z plane) and (b) out

of the director plane (y-z plane).

99

-80 -60 -40 -20 0 20 40 60 80-200

-100

0

100

200

300

400

PA NC PA & NC


Effe

ctiv

e Δn

d (n

m)

Fig. 3-12. Birefringence analyses of the bright state model of mode 1 in the director plane.

(PA: Positive A-plate; NC: Negative C-plate).

100

-80 -60 -40 -20 0 20 40 60 80-400

-300

-200

-100

0

100

200 PC NA PC & NA


Effe

ctiv

e Δn

d (n

m)

Fig. 3-13. Birefringence analyses of the bright state model of mode 2 in the director

plane. (PC: Positive C-plate; NA: Negative A-plate).

101

3-4-2-2. Effective phase retardation out of the director plane

Unlike the director plane, for viewing angles out of the director plane of our

bright state model in mode 1, the projection of the optic axes of the positive A-plate and

negative C-plate onto the plane perpendicular to the light propagation vector ( Kr

) are

perpendicular to each other, as in Fig. 3-11 (b). Therefore, the total effective retardation

( ⊥Δwhite ) of the bright state in this direction should be the difference between them,

negativeeff

positiveeffwhite ndnd Δ−Δ=Δ⊥ . (3-11)

The retardation variations of the bright state model in mode 1 [Fig. 3-9 (c)] out of the

director plane are calculated from Eqs. (3-7), (3-8) and (3-11) and shown in Fig. 3-14.

The contribution of the negative C-plate is the same as when in the director plane.

However, in this direction, the retardation of the positive A-plate rises continuously as

the viewing angle increases due to the increase in effective thickness. The total effective

retardation of the bright state is the difference between the retardation values of the

positive A-plate and negative C-plate, as mentioned above. Consequently, the value

increases as the viewing angle increases from the normal direction.

Figure 3-15 shows the calculation results for mode 2 by using the same method as

mode 1. This one is also a near-mirror image of the figure in mode 1 [Fig. 3-14]. The

total effective retardation decreases as the viewing angle increases, and it has a negative

sign. Therefore, the absolute value of the total effective retardation always increases as

the viewing angle increases out of the director plane. We saw this same result in mode 1,

102

so we can expect that the optical properties of the bright states are similar between mode

1 and mode 2, even though their director structures are different.

103

-80 -60 -40 -20 0 20 40 60 80-200

-100

0

100

200

300

400 PA NC PA & NC


Effe

ctiv

e Δn

d (n

m)


plane. (PA: Positive A-plate; NC: Negative C-plate).

104

-80 -60 -40 -20 0 20 40 60 80-400

-300

-200

-100

0

100

200 PC NA PC & NA


Effe

ctiv

e Δn

d (n

m)


plane. (PC: Positive C-plate; NA: Negative A-plate).

105

3-4-3. Analyses of the off-axis light transmittance of the simple models

We numerically calculated the off-axis light transmittance of our simple model in

dark and bright states. In these calculations, we inserted the dark and bright state models

[Fig. 3-9 (c)] between two crossed ideal polarizers. The optic axes of the positive A-plate

in mode 1 and negative A-plate in mode 2 made 45° relative to the one of the polarizer

axes. We used the incident light wavelength of 550 nm. All other parameters are the same

as we used when we calculated the effective phase retardation of our models, as in Sec. 3-

4-2.

3-4-3-1. Off-axis light transmittance in a dark state

Figure 3-16 shows the numerical calculation results of the off-axis light

transmission properties of our simple models in their dark states, whose phase

retardations are compensated as described in Sec. 3-3. We added the off-axis light

transmittance of the crossed polarizers for the comparison. These figures show that the

dark state transmittances of the simple models have almost the same level of the crossed

polarizers in main viewing directions. Therefore, we can say that the effective phase

retardations of the dark state-simple models are optically compensated well.

3-4-3-2. Off-axis light transmittance in a bright state

From the analyses of effective phase retardation in Sec. 3-4-2, we know that

although we use an ideal compensator for perfect optical compensation of the dark state,

it adversely changes the retardation of the bright state for all viewing angles. Figure 3-17

106

shows the numerical calculation results of the off-axis light transmission properties of the

bright state model in mode 1 and 2 in the director and out of the director planes. As

expected from the retardation analyses and Eq. (3-1), the transmittance curves of mode 1

and mode 2 are almost the same in the director plane and out of the director plane, even

though their detailed structures are different. Another thing to be noted is that the

transmittance in the director plane decreases as the viewing angle increases from the

normal direction, and it is consistent with the retardation analyses. On the other hand, the

transmittance out of the director plane rises until the total effective retardation reaches the

half wavelength of the incident light and then falls because it has passed the maximum

transmittance point (± 50° in this calculation, and it depends on oT ). The properties of

fallen transmittance in the large viewing angles in both viewing planes are also partially

coming from surface reflection effects.

When we compare the results of our bright state model to the optical properties of

the real common liquid crystal devices in Sec. 3-2, Fig. 3-6, they have very similar curve

shapes for both main viewing-angle-directions. In other words, the transmittance of the

bright state falls as the viewing angle increases in the director plane; while, out of the

director plane, the transmittance rises first and then falls after it reaches the peak point if

the liquid crystal layer is optically designed so that the transmittance of the normal

direction is lower than the maximum value.

107

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03 Mode 1 Mode 2 Crossed polarizers


Tran

smitt

ance

(a) Out of the director plane (Φ=0º)

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02



Tran

smitt

ance

(b) Director plane (Φ=90º)

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02



Tran

smitt

ance

(c) Φ=135º


simple models in their dark states. (Most of the curves are overlapped.)

108

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5 Mode 1 Mode 2


Tran

smitt

ance

(a) Director plane

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Mode 1 Mode 2


Tran

smitt

ance

(b) Out of the director plane (mode 1 and mode 2 are overlapped)

Fig. 3-17. Off-axis light transmission properties of the bright state models.

109

3-5. Comparison of the viewing angle properties in all directions

Until now, we described the optical properties only in the two main viewing

planes: in the director and out of the director planes, but the similarities of the optical

properties between our simple model and the real liquid crystal devices are maintained

over all viewing angles. We numerically calculated the transmittances in all viewing

directions at an incident light wavelength of 550 nm. The calculation methods and optical

parameters are the same that we used in previous sections.

Figures 3-18 and 3-19 show the conoscopic figures of the dark state

transmittances of the real liquid crystal devices and our simple models, respectively.

They have almost the same figure shapes, and this is manifest because we compensated

the phase retardation of their dark states almost perfectly in all viewing angles.

Figures 3-20 and 3-21 illustrate the conoscopic figures of the bright state

transmittances of the real, common liquid crystal devices and our simple models,

respectively. Though the symmetries are a little bit different depending on the detailed

director angle distribution of each display mode, the overall figure shapes are the same

and have the common features, as explained above.

110

(a) Pi-cell (b) ECB

(c) VA (d) TN

(e) Symmetric splay-cell: next page

Fig.3-18. Viewing angle properties of the common liquid crystal devices in their dark

states. (Azimuth angles: 0 ~ 360º; Polar angle: 0 ~ 80º).

111

(e) Symmetry splay cell

112

(a) Mode 1

(b) Mode 2

Fig. 3-19. Viewing angle properties of the dark state models.

(Azimuth angles: 0 ~ 360º; Polar angle: 0 ~ 80º).

113

(a) Pi-cell (b) ECB

(c) VA (d) TN

(e) Symmetric splay-cell: next page

Fig. 3-20. Viewing angle properties of the common liquid crystal devices in their bright

states. (Azimuth angles: 0 ~ 360º; Polar angle: 0 ~ 80º).

114

(e) Symmetry splay cell

115

(a) Mode 1

(b) Mode 2

Fig. 3-21. Viewing angle properties of the bright state models.

(Azimuth angles: 0 ~ 360º; Polar angle: 0 ~ 80º).

116

3-6. Summary

We calculated the off-axis light transmission properties of the bright state of most

common liquid crystal devices, such as ECB, VA, TN, Pi-cell, and symmetric splay-cell,

whose dark states were optically compensated to have minimum transmittances for all

viewing angles. From the results of these calculations, we found that their bright states

have a universal viewing angle shape in spite of completely different director structures

in their liquid crystal layers.

In order to understand this strange phenomenon, we made simple dark and bright

state models describing general liquid crystal devices and analyzed them in terms of

effective retardation and transmittance. In accordance with these analyses, the total

effective retardation in the director plane constantly falls as the viewing angle increases

(“bell shape”). On the contrary, the total effective retardation out of the director plane

consistently rises in the same situation (“reversed bell shape”). These retardation

changes cause the transmittance changes. In the director plane, the transmittance

decreases as the viewing angle becomes larger because the birefringence decreases in that

direction. On the other hand, the transmittance out of the director plane increases first and

then falls after the specific viewing angle if the liquid crystal layer is optically designed

so that the transmittance of the normal direction is lower than the maximum value.

These viewing angle features of our bright state models agree well with the

properties of most common liquid crystal devices, not only in the two main viewing

planes, but also for all viewing directions. Therefore, we can say that our simple model

can reasonably describe the optical properties of the real liquid crystal devices considered

117

here. Accordingly, our simple model can be used to analytically understand and predict

the optical properties, such as transmittance, luminance distribution and color analyses of

current LCDs or possible candidates of new display modes because usually analytical

methods for optical calculations are almost impossible in real devices.

Based on these results, we can say that the single domain LCD modes, considered

here, whose dark states are optically compensated to give minimum transmittance,

inevitably have asymmetric shapes of the off-axis light transmission properties between

the director and out of the director planes in their bright states. Therefore, in order to

achieve isotropic shapes of the bright state viewing angle properties, multi-domain liquid

crystal modes are necessary.

118

CHAPTER 4

OPTIMIZATION OF THE BRIGHT STATE DIRECTOR CONFIGURATION

IN COMPENSATED PI-CELL DEVICES

4-1. Introduction

One of the popular modes in liquid crystal displays (LCDs) is the twisted nematic

mode (TN), and it has been used in many applications because of its simple and low cost

manufacturing process. In spite of those merits, it is hard to use the mode in large size

monitor or TV applications due to its slow response time and limited viewing angle

properties. To overcome these problems, various other LCD modes based on multi-

domain technology such as patterned vertical alignment (PVA), multi-domain vertical

alignment (MVA), in-plane switching (IPS), and advanced super view (ASV) have been

developed, commercialized and shown outstanding improvement recently. However,

their transmittances are sacrificed to improve viewing angle properties, and their response

times are not satisfactory to drive fast moving pictures.

To achieve the desired characteristics of wide viewing angle and fast response,

the optimization of the Pi-cell 37 , 38 mode is currently of high interest. (Pi-cell was

invented in 1983 by Bos. Subsequently, Uchida used it with a single biaxial film as a

compensator with normally black mode and called it OCB39, 40 in 1993. Their structures

of the liquid crystal and polarizers are the same.). The dark state of Pi-cell is especially

very important to improve visibility of the display, so a great amount of work has been

119

done to achieve high contrast ratio without color variation at all viewing angles41, 42, 43, 44,

45, 46. On the other hand, the bright state determines the luminance of the display at all

angles of the incident light. Therefore, it must be another crucial factor when evaluating

the optical properties of a device. However, we have not seen those studies dealing with

the off-axis light transmission properties of the bright states of Pi-cells.

In this chapter, we investigate the off-axis light transmission properties of the

differently director configured bright states in Pi-cells whose dark states are optically

compensated almost perfectly. To make various director configurations of the bright state,

we applied different bright state voltages and used diverse pretilt angles. During the

calculations, we adjusted the thicknesses of the liquid crystal layer and compensators to

achieve the same transmittance of the bright state at normal direction. According to our

results, their bright states have a curious universality in the off-axis light transmission

properties and are insensitive to the parameters if the bright state voltage or pretilt angle

is not so big, and these features agree well with our fist simple bright state model47.

However, in bright state with high tilt angle of director configuration, the off-axis light

transmittance fluctuates as the incident light angle varies, and it makes the viewing angle

properties of a Pi-cell worse. In order to explain these interesting facts, we make a new

dark and bright state model and analyze it in terms of effective birefringence,

transmittance and angular distribution of directors.

In Sec. 4-2, we will show the off-axis light transmission properties of the bright

states in Pi-cells that have various director configurations. In Sec. 4-3, we will build a

dark and bright state model, calculate the angular dependencies of the effective

120

birefringence and transmittance of the model and then analyze them after comparing the

results with those of the Sec. 4-2.

4-2. Optical Properties of Different Bright States in Pi-cells

We numerically calculated the off-axis light transmission properties of Pi-cells at

an incident light wavelength of 550 nm. Their bright states are implemented using

different applied voltages ranging from 1 to 3 V and pretilt angles of 2-30º with all the

same dark voltage, 5 V. During the calculations, we used numerical relaxation techniques

to get the director field in the liquid crystal layer and the 2×2-matrix method 48 to

calculate the optical properties of the devices.

4-2-1. Optical compensations of the dark states of the Pi-cells

The dark state of each mode is optically compensated via two (top and bottom of

the liquid crystal layer) compensators with the hybrid-negative C structure. The hybrid-

negative C structure has exactly the same angular distribution as the directors of liquid

crystal layer of dark state, and their extraordinary, ordinary refractive indices are the

same as the ordinary, extraordinary indices of the liquid crystal, respectively. We used

this same method in chapter 3 for the compensation of the dark state in a LCD.

4-2-2. Optical parameters of the Pi-cells

The thicknesses of the liquid crystal layer and the compensator of each display

mode are determined to meet two conditions. Firstly, it gives the minimum transmittance

121

at all viewing angles in the dark state. Secondly, the transmittance of the bright state at

normal direction has a specific value ( oT ). This value can be chosen from zero to

maximum transmittance that we can achieve under the crossed polarizers, but a bit

smaller than the maximum value is usually used to achieve high transmittance and to

escape color shift at the off axis viewing directions. In this chapter, we set the value to ¾

of the maximum transmittance because this is a similar condition in real liquid crystal

devices and is not a critical factor in determining physical concept of the devices. The

detailed specifications are in Table 4-1 and Table 4-2 for the different bright state

voltages and pretilt angles, respectively. The liquid crystal (LC) we used here is LC53

( en =1.6644, on =1.5070 at λ=550 nm; Δε=9.4), and the cell layout is shown in Fig. 4-1.

Other optical parameters are the same as we used in Sec. 3-2.

122

Table 4-1. Cell parameters of the Pi-cells with different bright state voltages.

(Pretilt angle of 5.5°, dark state voltage of 5.0 V).

Thickness (µm) Bright state voltage (V) LC Compensator 1.0 4.598 2.06 1.3 5.037 2.255 2.0 6.875 3.078 2.5 9.273 4.151 3.0 13.25 5.932

123

Table 4-2. Cell parameters of the Pi-cells with different pretilt angles.

(Bright state voltage of 1.3 V, dark state voltage of 5.0 V).

Thickness (µm) Pretilt angle (°) LC Compensator 2.0 4.745 2.131 5.5 5.037 2.255 10.0 5.474 2.438 20.0 6.769 2.963 30.0 8.785 3.752

124

Fig. 4-1. Stack configuration of a Pi-cell.

ITO Polyimide

Liquid Crystal

Polyimide ITO

Glass

Compensator (PDM)

Polarizer (-45º)

Polarizer (+45º)

Compensator (PDM)

Glass

Light Source

125

4-2-3. Numerical calculations of the off-axis light transmittance in dark states

Figure 4-2 shows the numerical calculation results of the off-axis light

transmission properties of the dark state Pi-cells, whose bright states are optimized to

have several different voltages, but have the same dark state voltage of 5.0 V and pretilt

angle of 5.5º. The phase retardations of their dark states are compensated separately, as

described in Sec. 4-2-1. We added the off-axis light transmittance curves of the just

crossed polarizers for the comparison. We also calculated the director tilt angles of the

dark state-liquid crystal layer, as in Fig. 4-3.

We did similar calculations for the Pi-cells that have different pretilt angles, but

they have the same dark and bright states voltages of 5.0 V and 1.3 V, respectively.

Figures 4-4 and 4-5 illustrate the results of off-axis light transmission properties and the

director configurations of liquid crystal layers in their dark states.

These figures show that the dark state transmittances of the Pi-cells are not far

from the level of the crossed polarizers in main viewing directions, although there is a

relatively large difference at the viewing angle of about ±70º when the bright state

voltages or the pretilt angles are big enough. Actually, these results hint that there might

be a limitation in the phase compensation of a dark state by using the layers of hybrid-

negative C structure. We will deal with this topic in chapter 6.

From these numerical calculation results, we can say that the effective phase retardation

of each dark state-liquid crystal layer of the Pi-cells is optically compensated reasonably.

126

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1.0 V 1.3 V 2.0 V 2.5 V 3.0 V Crossed polarizers


Tran

smitt

ance

(a) Director plane

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1.0 V 1.3 V 2.0 V 2.5 V 3.0 V Crossed polarizers


Tran

smitt

ance



dark state Pi-cells that have different bright state voltages (1.0-3.0V).

(Dark state voltage: 5.0V; pretilt angle: 5.5º).

127

0.0 0.2 0.4 0.6 0.8 1.0

0

50

100

150

200


Tilt

angl

e (d

eg.)

Fig. 4-3. Director tilt angles of the dark state Pi-cells that have different bright state

voltages (1.0-3.0V). (Dark state voltage: 5.0V; pretilt angle: 5.5º).

128

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

2.0o 5.5o

10.0o 20.0o

30.0o Crossed polarizers


Tran

smitt

ance

(a) Director plane

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

2.0o 5.5o

10.0o 20.0o

30.0o Crossed polarizers


Tran

smitt

ance



dark state Pi-cells that have different pretilt angles (2.0-30.0º).

(Bright state voltage: 1.3 V; dark state voltage: 5.0 V).

129

0.0 0.2 0.4 0.6 0.8 1.0

0

50

100

150

200 2.0o 5.5o

10.0o 20.0o

30.0o


Tilt

angl

e (d

eg.)


(2.0-30.0º). (Bright state voltage: 1.3 V; dark state voltage: 5.0V).

130

4-2-4. Numerical calculations of the off-axis light transmittance in bright states

We numerically calculated the off-axis light transmission properties of the bright

state Pi-cells that have several different bright state voltages (1.0-3.0 V), but they have

the same dark state voltage of 5.0 V and pretilt angle of 5.5º. Figures 4-6 and 4-7 show

the results of the transmittances and the director tilt angles of liquid crystal layer

calculated in the same conditions, and Fig. 4-8 shows the conoscopic figures of the bright

state transmittances of the Pi-cells.

We also did similar calculations for the Pi-cells that have different pretilt angles

(2-30º), but the same dark and bright states voltages of 5.0 V and 1.3 V, respectively.

Figures 4-9 and 4-10 depict the results of off-axis light transmission properties in two

main directions and the director configurations of the liquid crystal layers in their bright

states, respectively. Figure 4-11 shows the conoscopic figures of the bright state

transmittances of the Pi-cells, which are calculated in the same conditions.

From these transmittance figures, we can notice that the off-axis light

transmission properties of the bright state in a compensated Pi-cell, amazingly, have

unified shape and are relatively less sensitive to the parameters if the bright state voltage

or pretilt angle is below certain value (about 2.0 V and 20º in these figures). In the

viewing angle of the director plane [Fig. 4-6 (a) and Fig. 4-9 (a)], all the transmittance

curves have similar “bell” shapes i.e. the transmittance constantly decreases as the

viewing angle increases from the normal direction no matter what director configurations

(bright state voltage or pretilt angle) of liquid crystal are used as a bright state. The

transmittance out of the director plane [Fig. 4-6 (b) and Fig. 4-9 (b)] rises as the viewing

131

angle increases from the normal direction and then falls monotonically after passing the

specific angles (about ± 50° in these calculations) if the Pi-cell has the director

configuration of low tilt angle (low bright state voltage or low pretilt angle). These

features of universality and insensitiveness of transmittance to the director configuration,

exactly, comply with our first simple bright state model, as described in chapter 3 and

give room to the optical design of a Pi-cell.

However, in the high tilted director configuration of the bright state out of the

director plane, the transmittance is a much strong function of incident light angle, and it

leads to narrow viewing angle properties. Therefore, it does not follow our first simple

bright state model anymore. Additionally, these results mean that the more the director

configuration of the bright state becomes similar to that of dark state, the worse the off-

axis light transmission property of the bright state when its dark state is optically

optimized. We analyzed these interesting facts by building another more general bright

state model that can not only include our first simple model but also stand for all director

configurations of bright states in Pi-cells, and the results are in next section.

132

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5 1.0 V 1.3 V 2.0 V 2.5 V 3.0 V


Tran

smitt

ance

(a) Director plane

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

1.0 V 1.3 V 2.0 V 2.5 V 3.0 V


Tran

smitt

ance



bright state Pi-cells that have different bright state voltages (1.0-3.0V).

(Dark state voltage: 5.0V; pretilt angle: 5.5º).

133

0.0 0.2 0.4 0.6 0.8 1.00

50

100

150

200 1.0 V 1.3 V 2.0 V 2.5 V 3.0 V


Tilt

angl

e (d

eg.)

Fig. 4-7. Director tilt angles of the bright state Pi-cells that have different bright state

voltages (1.0-3.0V). (Dark state voltage: 5.0V; pretilt angle: 5.5º).

134

Fig. 4-8. Conoscopic properties of the bright state transmittances with different bright

state voltages of (a) 1.0V, (b) 1.3V, (c) 2.0V, (d) 2.5V and (e) 3.0V. (Dark state voltage:

5.0V; pretilt angle: 5.5º).

135

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5 2.0o 5.5o 10.0o

20.0o 30.0o


Tran

smitt

ance

(a) Director plane

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

2.0o 5.5o

10.0o 20.0o

30.0o


Tran

smitt

ance

(b) Out of director plane


bright state Pi-cells that have different pretilt angles (2.0-30.0º).

(Bright state voltage: 1.3 V; dark state voltage: 5.0V).

136

0.0 0.2 0.4 0.6 0.8 1.0

0

50

100

150

200

2.0o 5.5o

10.0o 20.0o

30.0o


Tilt

angl

e (d

eg.)

Fig. 4-10. Director tilt angles of the bright state Pi-cells that have different pretilt angles

(2.0-30.0º). (Bright state voltage: 1.3 V; dark state voltage: 5.0V).

137

Fig. 4-11. Conoscopic properties of the bright state transmittances with different pretilt

angles of (a) 2.0º, (b) 5.5º, (c) 10.0º, (d) 20.0º and (e) 30.0º. (Bright state voltage: 1.3 V;

dark state voltage: 5.0V).

138

4-3. Universal Bright State Model of Pi-cells

4-3-1. Dark and bright state modeling

Let us consider a Pi-cell whose dark state is optically optimized to give the

minimum transmittance at the normal direction by means of passive type phase

compensators, hybrid-negative C-plates. The angle distribution of the optic axes of the

negative C plates should be the same as that of the liquid crystal directors for the best

compensation of the phase difference between ordinary ray (o-ray) and extraordinary ray

(e-ray) in the liquid crystal layer. Figure 4-12 (a) shows the cartoon of this director

configuration of the device. In this figure, we can notice that there are simply two parts in

a liquid crystal layer: static part (empty director shape in the figure) and dynamic part

(filled director shape). In the static part, most of the liquid crystal directors hardly change

their orientations, or the changes are very small if they do between the dark and bright

state because of the anchoring energy at the liquid crystal–alignment layer interfaces. On

the contrary, the directors in the dynamic part are relatively sensitive to the applied

voltage, so the orientations of the directors are completely different between the dark and

bright state. In the aspect of the total system composed of liquid crystal and passive type

compensators, the static part of the liquid crystal layer is optically neutralized by the

corresponding part of the compensators (empty director shape in the figure) in both dark

and bright state, simultaneously, and it does not contribute to the optical properties of the

device. Therefore, we can simplify the compensated device, conceptually, and get a

simple dark and bright state model, which made up of two symmetrically tilted positive

139

A-plates (PA1, 2. In this chapter, an O-plate is referred to as a tilted A-plate, as is

common usage) and two same normal negative C-plates (NC1, 2), as Fig. 4-12 (b). In this

simple model, the dark state is still optically neutralized, but the bright state has net

birefringence in any viewing directions. Intuitively, we can also see that the tilt angle (θ )

of a positive A-plate in the bright state model is related to the director configuration of

the bright state in real liquid crystal device, e.g. as the voltage or pretilt angle of bright

state in Pi-cell increases, the tilt angle (θ ) of the A-plate also increases.

140

Fig. 4-12. (a) A Pi-cell whose dark state is perfectly compensated using hybrid-negative

C-plates and (b) simple dark and bright state models of the Pi-cell.

141

4-3-2. Calculations

The light transmittance (T ) of the simple model [Fig. 4-12 (b)] under the ideal

crossed polarizers can be written as follows:

2sin

2sin

21 2

max2 Γ

≡Γ

= TTTT paap , (4-1)

where apT and paT are the transmittances with considering the surface reflections at the

interfaces between air and polarizers in the incident and exit media, respectively. The

Γ in Eq. (4-1) is the phase difference between e-ray and o-ray in the anisotropic layers

and written in terms of the incident light wavelength (λ ) and effective birefringence

( effndΔ ) of the media when we assume the e-ray and o-ray have the same path in the

media,

effndΔ=Γλπ2 . (4-2)

[The exact expression of the phase difference (Γ ) is dKK ozez )( −=Γ , where ezK and

ozK are the z components of the wave vectors of the e-ray and o-ray, respectively. For

simplicity in this chapter, we assume the e-ray and o-ray have the same path, and this is

very reasonable in a real situation. We will compare both methods in Chap. 6.] The

142

incident light angle dependence of the effective birefringence ( effndΔ ) in each tilted

positive A-plate and negative C-plate is calculated in Eqs. (4-3) and (4-4), respectively,

αψψ cossincos 2222

dnnn

nnnd o

oe

oepositiveeff ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛−

+=Δ , (4-3)

'cos''

'sin''cos'

''2222 αψψ

dnnn

nnnd o

oe

oenegativeeff ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛−

+=Δ , (4-4)

where )'( ee nn and )'( oo nn are the refractive indices of e-ray and o-ray, )'(ψψ is the

angle between the optic axis of the director and light propagation vector, )'(αα is the

polar angle of incident light in the medium, and )'(dd is the thickness of a tilted positive

A-plate (negative C-plate). Where, d is determined so that oTT = in normal direction. In

a given assumption that the transmittance of the dark state should be minimized, the total

birefringence of the dark state model ( darkΔ ) should be zero at a given light incident

angle,

022 =Δ×+Δ×=Δdark

negativeeffdark

positiveeffdark ndnd . (4-5)

From Eqs. (4-3), (4-4) and (4-5), the thickness of a negative C-plate ( 'd ) can be decided.

143

4-3-3. Analyses of the effective birefringence in the director plane

We cannot define the birefringence out of the director plane [x-z plane in Fig. 4-

12 (b)] in the bright state model or in a real Pi-cell, because the projection of the directors

onto a plane perpendicular to the light propagation vector ( Kr

) are not co-linear.

Therefore, in this section, we are going to analyze the total effective birefringence of the

bright state model only in the director plane [y-z plane in Fig. 4-12 (b)]. All calculations

are performed at an incident light wavelength of 550 nm. During the calculations, we

considered the refraction of the incident light at the air interface.

The extraordinary and ordinary refractive indices of the positive (negative) plates

are 1.6 (1.5) and 1.5 (1.6), respectively. The thickness of each positive (PA1, 2) and

negative (NC1, 2) plate for the various tilt angles of the positive A-plate are on Table 4-3.

These values were taken to meet the ideal dark and the bright state conditions under

crossed polarizers, as described in Sec. 4-2-2. We use these parameters in all calculations

of our model in this chapter.

In the viewing direction on the director plane of our bright state model, the

projections of the optic axes of the tilted positive A-plates (PA1, 2) and negative C-plates

(NC1, 2) onto the plane perpendicular to the light propagation vector ( Kr

) are parallel to

each other [Fig. 3-11 (a)]. Therefore, the total effective birefringence of the bright state

model in the director plane ( ||brightΔ ) is the summation of each value,

2121|| NCeff

NCeff

PAeff

PAeffbright ndndndnd Δ+Δ+Δ+Δ=Δ . (4-6)

144

Figure 4-13 shows the effective birefringence of each layer calculated from Eqs. (4-3),

(4-4) and (4-6), in the director plane of our bright state model with a tilt angle (θ) of 50º.

In the normal direction, the birefringence of the negative C-plates (NC1, 2) is zero as

expected, so it does not contribute to the total birefringence. As the viewing angle

increases from the normal direction, the effective birefringence of the negative C-plates

always decreases because of its molecular structure. Meanwhile, the effective

birefringence of the tilted positive A-plates (PA1, 2) depends on the tilt angles (θ) and

symmetrically varies each other.

Figure 4-14 illustrates the summation of effective birefringence contributed from

the positive A-plates (PA1+PA2), which have tilt angles of 10-70º. The curve shape of

this summation depends on the tilt angle (θ) of the positive A-plate as the figure. As we

can see in this figure, if the tilt angle is small, the subtotal effective birefringence

(PA1+PA2) decreases as the viewing angle increases. On the contrary, if the tilt angle is

relatively large, the subtotal value increases as the viewing angle increases from the

normal direction.

Figure 4-15 is the total effective birefringence of the bright state model, which is

composed of the tilted positive A-plates (PA1, 2) and negative C-plates (NC1, 2) with

various tilt angles ranging from 10º to 70º. Amazingly, the curves are all “bell” shapes

and their detailed values are very similar to each other no matter what tilt angles are used

in the bright state model. This universality of the birefringence in the director plane

agrees well with our first simple bright state model.

145

Table 4-3. Thickness of each layer of the bright state model with different tilt angles (θ).

Thickness (µm) Tilt angle (°) PA1,2 NC1,2

10.0 0.958 0.898 30.0 1.267 1.188 50.0 2.376 2.228 65.0 5.618 5.267 70.0 8.627 8.088

146

-80 -60 -40 -20 0 20 40 60 80-300

-200

-100

0

100

200

300

400

[PA1+PA2]PA2

Total

PA1

NC1,2[NC1+NC2]


Effe

ctiv

e Δn

d (n

m)

Fig. 4-13. Effective birefringence of each layer of the bright state model in the director

plane with a tilt angle (θ) of 50º.

147

-80 -60 -40 -20 0 20 40 60 800

200

400

600

800

1000

1200

10o 30o 50o

65o 70o


Effe

ctiv

e Δn

d (n

m)

Fig. 4-14. Effective birefringence of the positive A-plates (PA1+PA2) of the bright state

model in the director plane. [tilt angle (θ): 10, 30, 50, 65, and 70º]

148

-80 -60 -40 -20 0 20 40 60 800

50

100

150

200

250 10o 30o 50o

65o 70o


Effe

ctiv

e Δn

d (n

m)

Fig. 4-15. Effective birefringence of the total layers of the bright state model in the

director plane. [tilt angle (θ): 10, 30, 50, 65, and 70º]

149

4-3-4. Analyses of the transmittance

Figure 4-16 is the numerical calculation results showing the off-axis light

transmission properties of the bright state model with various tilt angles (10-70º) in the

director and out of the director planes, and Fig. 4-17 shows their conoscopic properties

under the same situation. We also calculated the off-axis light transmission properties of

the dark state model to make clear that we compensated the phase retardation of the dark

state completely, and the results are in Fig. 4-18. In these calculations, we inserted the

bright state models [Fig. 4-12 (b)] between two crossed polarizers, and the azimuth angle

of an optic axis of a tilted positive A-plate made 45° angle relative to the one of the

polarizer axes, as usual liquid crystal devices. We used a light wavelength of 550 nm

during the calculations.

In the director plane [Fig. 4-16 (a)], the light transmittance curves are almost the

same regardless of the tilt angles of positive A-plate in the bright state model, and their

values decrease as the viewing angle increases from the normal direction. These results

are consistent with the birefringence analyses and are the same shapes as the

transmittance curves of real Pi-cells [Fig. 4-6 (a) and Fig. 4-9 (a)]. This is the universal

viewing angle properties of liquid crystal devices whose dark states are optically

optimized.

On the other hand, the light transmittance out of the director plane [Fig. 4-16 (b)]

looks somewhat complicated, but their shapes are very similar to those of the real Pi-cells

[Fig. 4-6 (b) and Fig. 4-9 (b)]. Therefore, we can say our new bright state model can

describe the basic properties of bright states of real Pi-cells well. With a low tilt angle of

150

the positive A-plate in the bright state model, the transmittance rises until the total

effective retardation reaches a half-wavelength of the incident light and then falls because

it has passed the maximum transmittance point (± 50° in this calculation and it depends

on oT ). These features with a low tilt angle (low bright state voltage or low pretilt angle)

are the same as the properties of our first simple bright state model, as in Chap. 3, and

can be understandable because, in this case, the projections of the directors onto a plane

perpendicular to the light propagation vector ( Kr

) are roughly co-linear. In a different

way, the transmittance of the bright state with a high tilt angle fluctuates as the viewing

angle varies, and it makes viewing angle properties worse. In the hope of understanding

the detailed reasons for the fluctuation, we analyze the transmittance out of the director

plane in next section.

151

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

10o 30o 50o

65o 70o


Tran

smitt

ance

(a) Director plane

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

10o 30o

50o 65o

70o


Tran

smitt

ance


Fig. 4-16. Off-axis light transmission properties of the bright state model with different

tilt angles.

152

Fig. 4-17. Conoscopic properties of the transmittance of the bright state model with tilt

angle (a) 10º, (b) 30º, (c) 50º, (d) 65º, and (e) 70º.

153

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

10o 30o

50o 65o

70o Crossed polarizers


Tran

smitt

ance

(a) Director plane

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

10o 30o

50o 65o

70o Crossed polarizers


Tran

smitt

ance

(b) Out of director plane

Fig. 4-18. Off-axis light transmission properties of the dark state model. All curves are

overlapped each other, and their values are the same level of the crossed polarizers.

154

4-3-5. Detailed analyses of the transmittance out of the director plane

We calculated two more transmittance curves contributed separately from each

layer (PA: positive A-plates only and NC: negative C-plates only) of the bright state

model (tilt angle of 70º) out of the director plane, as in Fig. 4-19. Roughly, we can say

from this figure that the fluctuation of the total transmittance is mainly coming from the

effects of each separate layer. Therefore, we are going to analyze the fluctuations of each

separate transmittance to understand the total transmittance.

4-3-5-1. Contribution from the positive A-plates

Firstly, we want to know what causes the fluctuation of the transmittance

contributed from positive A-plates only. Figure 4-20 shows the apparent azimuth angle

(γ ) of the optic axis of each layer with respect to the direction of P-polarization ( Pr

) of

the incident light. In this calculation, the bright state model has a tilt angle of 70º, and the

light incident direction is out of the director plane. The detailed angle definitions are in

Fig. 4-21, and the angles, γ and ψ are calculated from Eqs.49 (4-7) and (4-8),

θαφθαφθγ

sinsincoscoscossincostan−

= , (4-7)

θαφθαψ sincoscoscossincos += , (4-8)

where θ and φ are the tilt and azimuth angles of a director, α is the light incident angle

in a medium, γ is the angle between Pr

and er , and ψ is the angle from the light

155

propagation vector ( Kr

) to the director ( nr ). From Fig. 4-20, we can see that as the

viewing angle increases from the normal direction, the azimuth angle difference ( γΔ )

between PA1 and PA2 in incident light frame ( KSPrrr

,, ) varies from zero to bigger than

90º. The maximum angle difference it can reach depends on the magnitude of the tilt

angle (θ ) of the positive A-plates, as in Fig. 4-22. If the angle difference ( γΔ ) is 90º, the

total effective birefringence of the positive A-plates is canceled regardless of their

magnitudes of birefringence, so they do not contribute to the transmittance. This is the

reason why the transmittance contributed from the A-plates is almost zero at the viewing

angle around 33º (1st minimum point of PA in Fig. 4-19). After this first minimum point,

the transmittance increases and then falls again. This feature can be understood from Fig.

4-23 that shows the effective birefringence of a positive A-plate (PA1 or PA2) out of the

director plane in the bright state model, which is calculated from Eqs. (4-3) and (4-8).

The magnitudes of the birefringence of the PA1 and PA2 are the same at all viewing

angles out of the director plane because of their symmetry. However, their main

directions are different, so they can not be added. As we can see in this figure, the

birefringence of the A-plate (PA1 or PA2) whose tilt angle (θ) is 70º goes to one

wavelength of incident light ( λ =550 nm) as the viewing angle increases to 80º.

Therefore, the transmittance contributed from the A-plates (PA1, PA2) goes to zero due

to Eqs. (4-1) and (4-2), and it makes second minimum point at the viewing angle around

80º, as in Fig. 4-19.

156

4-3-5-2. Contribution from the negative C-plates

Secondly, we are going to analyze the fluctuation of the transmittance coming

from the negative C-plates (NC in Fig. 4-19) of the bright state model [Fig. 4-12 (b)] with

70º tilt angle of positive A-plates. In contrast to the positive A-plates, the optic axes of

the negative C-plates are in the same direction, so the total effective birefringence of

them is summation of each magnitude. Additionally, the angles between their optic axes

and transmission axes of the polarizers remain almost ±45º at all off-axes out of the

director plane, as in Fig. 4-20. It means only the effective birefringence of the negative C-

plates affects the transmittance of the bright state model. Figure 4-24 shows the subtotal

effective birefringence of the negative C-plates (NC1 and NC2) out of the director plane

with light wavelength (λ ) of 550 nm. In this figure, with tilt angle 70º of A-plates, the

absolute value of effective birefringence increases as viewing angle increases, and it

passes through the first maximum point ( 2λ=Δ effnd around 35º of viewing angle), the

first minimum point ( λ=Δ effnd around 55º) and the second maximum point

( 23λ=Δ effnd around 70º) of transmittance, which are calculated from Eqs. (4-1) and (4-

2). This variation of the birefringence causes the fluctuation of the transmittance

contributed from the negative C-plates (NC in Fig. 4-19). In the viewing angles out of the

director plane, we can say conclusively that these features of each separate transmittance

make the total transmittance fluctuate in large tilt angles of positive A-plates in our bright

state model (or director configuration of bright state with high voltage or high pretilt

angle in a real Pi-cell).

157

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5 PA NC Total


Tran

smitt

ance

Fig. 4-19. Off-axis light transmittances contributed separately from each layer of the

bright state model (tilt angle of 70º) out of the director plane.

(PA: positive A-plates only and NC: negative C-plates only).

158

-80 -60 -40 -20 0 20 40 60 80-50

0

50

100

150

200

PA1

Polarizer

Analyzer

PA2

NC1,2


γ (d

eg.)

Fig. 4-20. Apparent azimuth angle (γ ) of the optic axis of each layer in the bright state

model with tilt angle 70º out of the director plane.

159

Fig. 4-21. Director angle definition in (a) lab coordinate system ( ',',' zyx ) and (b)

incident light frame ( KSPrrr

,, ).

160

-80 -60 -40 -20 0 20 40 60 800

50

100

150

70o

65o

50o

30o

10o


Δγ (d

eg.)

Fig. 4-22. Difference of the apparent azimuth angle ( γΔ ) between the PA1 and the PA2

in bright state model with various tilt angles (10º-70º).

161

-80 -60 -40 -20 0 20 40 60 800

100

200

300

400

500

600

10o 30o 50o

65o 70o


Effe

ctiv

e Δn

d (n

m)

Fig. 4-23. Effective birefringence of a positive A-plate (PA1 or PA2) out of the director

plane in the bright state model. [Tilt angle of A-plates (θ): 10º~70º; Light wavelength (λ):

550 nm].

162

-80 -60 -40 -20 0 20 40 60 80-1200

-1000

-800

-600

-400

-200

0

200

10o 30o

50o 65o

70o


Effe

ctiv

e Δn

d (n

m)

Fig. 4-24. Subtotal effective birefringence of the negative C-plates (NC1+NC2) out of the

director plane in the bright state model. [Tilt angle of A-plates (θ): 10º~70º; Light

wavelength (λ): 550 nm].

163

4-4. Summary

We calculated the off-axis light transmission properties of the differently director

configured bright states in Pi-cells whose dark states were optically compensated to have

minimum transmittances at the normal direction. From the results of these calculations,

we found that the off-axis light transmission properties of the bright states surprisingly

had unified shapes and were relatively insensitive to the variation of the parameters

considered when the voltage of the bright state or the pretilt angle were below some

particular value. On the other hand, the transmittance in highly tilted director

configuration of bright state out of the director plane varies more significantly and makes

the viewing angle properties of a Pi-cell worse.

In order to understand these curious phenomena, we made a new dark and bright

state model describing Pi-cells and analyzed it in terms of effective birefringence,

transmittance and angular distribution of directors. According to the analyses, the total

effective birefringence in the director plane constantly falls as the viewing angle

increases (“bell shape”) regardless of the director configuration, and this result is

consistent with transmittance analyses and describes the optical properties of real Pi-cells

well.

Out of the director plane, we cannot define the total birefringence because the

projection of the directors on the plane perpendicular to the light propagation vector ( Kr

)

are not co-linear, so we analyzed transmittance in this viewing plane. In the director

configuration of low tilt angle in bright state model, the transmittance rises first and then

falls after the specific viewing angle if the liquid crystal layer is optically designed so that

164

the transmittance of the normal direction is lower than the maximum value. On the other

hand, the transmittance with a high tilt director configuration fluctuates as the viewing

angle varies as in real Pi-cells. To understand the causes of the fluctuation of the total

transmittance, we calculated the transmittances, which are contributed separately from

positive A-plates and negative C-plates of the bright state model, and analyzed them

using the apparent angle distribution of directors in incident light coordinate system and

the birefringence of each layer. In accordance with the analyses, both the apparent angle

distribution of directors and the birefringence of each layer play a key role in the

fluctuation of the total transmittance.

165

CHAPTER 5

LUMINANCE AND COLOR PROPERTIES OF THE COMPENSATED LIQUID

CRYSTAL DEVICES IN THEIR BRIGHT STATES

5-1. Introduction

In this chapter, we investigate the luminance and color properties of the bright

state simple model and the common LCDs such as ECB, VA, Pi-cell, and TN modes. We

use the same bright state simple model that we built in Chapter 3. The dark states of the

model and the common LCDs are optically compensated almost perfectly by using hybrid

aligned-passive type negative C-plates, as the same way that we used in Chapter 3.

During the calculation, we use the numerical relaxation technique to get the director field

of the liquid crystal layer and the 2×2-matrix method to calculate the light transmittances

for each wavelength, ranging from 380 to 780 nm. After calculating the transmittance at

each wavelength, we calculate and analyze the luminous transmittances and the color

properties of the devices.

In Sec. 5-2, we give the basic calculations for the luminance and color analyses of

the LCDs. Based on these calculations, we analyze the luminous transmittance of our

bright state model in functions of the cell thickness and the viewing angle, and after that

we compare those results with that of real common LCDs in Sec. 5-3. In Sec. 5-4, we

calculate and analyze the off-axis color properties including color difference of the bright

state model and the common LCDs.

166

5-2. Color Calculations

In this dissertation, we will use the CIE (International Commission on

Illumination or Commission International de l’Éclairage) standard colorimetric system to

describe the color properties of LCDs because it is commonly used in industrial

companies and many other quantitative applications.

5-2-1. Tristimulus values and color matching functions

The tristimulus values ( X , Y , Z ) are kinds of roots for color calculations.

According to the CIE, the values are defined as follows50:

∫= λλλλ dxTSkX )()()( , (5-1)

∫= λλλλ dyTSkY )()()( , (5-2)

∫= λλλλ dzTSkZ )()()( , (5-3)

where λ is the light wavelength, )(λT is the transmission spectrum of a LCD, )(λS is

the light source spectrum, and k is the proportionality associated with choice of units.

For example, if we use 683 lumens per watt, known as the maximum luminous

efficiency 51 , as a k value, then Y is a luminance having a unit, 2/ mcd . Another

conventional way is using k as a normalization constant defined as

167

∫=

λλλ dySk

)()(100 . (5-4)

This definition of k means the Y tristimulus value is 100 for a perfect transmission

system in all light wavelengths, i.e. 1)( =λT or for a light source. In this case, the

tristimulus value Y is called the luminance factor or the luminous transmittance for a

transmission light and has no unit. We will use this latter definition in this dissertation. In

Eqs. (5-1), (5-2), and (5-3), the integration is taken in the visible wavelength region

(about 380-780 nm), and the x , y , and z are the color matching functions. There are

two types of the color matching functions depending on fields of view: 2º and 10º. The

former was introduced in 1931 and recommended by the CIE when the viewing angle is

1-4º, and the latter was introduced in 1964 and recommended for a viewing angle

exceeding 4º. In this dissertation, we will use 2º fields of view, and the Fig. 5-1 shows the

corresponding color matching functions.

168

380 480 580 680 7800.0

0.5

1.0

1.5

2.0

x y z

Wavelength (nm)

Tris

timul

us v

alue

s

Fig. 5-1. CIE 1931 2º color matching functions

169

5-2-2. Chromaticity diagrams

There are several chromaticity diagrams to indicate a color on a two-dimensional

plane, such as xy , uv , and ''vu diagrams. The xy chromaticity diagram (Fig. 5-2) was

firstly introduced in 1931 by the CIE and has been used until now, but the uniformity of

the diagram is not good. In order to improve the uniformity, the uv diagram was

proposed by MacAdam and recommended by the CIE in 1960. In 1976, the CIE modified

the uv chromaticity diagram according to Eastwood’s report, and recommended a new

''vu chromaticity diagram 52 (Fig. 5-3). In this dissertation, we use ''vu chromaticity

diagram because it has the best uniformity so far and widely used. These all chromaticity

diagrams are expressed in terms of the tristimulus values ( X , Y , Z ) as follows:

)/( ZYXXx ++= , (5-5)

)/( ZYXYy ++= , (5-6)

)315/(4 ZYXXu ++= , (5-7)

)315/(6 ZYXYv ++= , (5-8)

)315/(4' ZYXXu ++= , (5-9)

)315/(9' ZYXYv ++= . (5-10)

170

Fig. 5-2. CIE 1931 2º chromaticity diagram (2º viewing angle).

171

Fig. 5-3. CIE 1976 ''vu chromaticity diagram (2º viewing angle).

172

5-2-3. Color difference

In 1976, the CIE recommended CIE 1976 *** vuL Color Space (abbreviated

CIELUV) for applications such as TVs. That is expressed by the following three-

dimensional orthogonal coordinates, *L (lightness), *u (redness-greenness), and *v

(yellowness-blueness) as follows:

16)/(116* −= nYYfL , (5-11)

)''(*13* nuuLu −= , (5-12)

)''(*13* nvvLv −= , (5-13)

where

3/1)/()/( nn YYYYf = if 3)116/24(/ >nYY , (5-14)

116/16)/)(108/841()/( += nn YYYYf if 3)116/24(/ ≤nYY , (5-15)

where Y , 'u , and 'v represent the tristimulus value Y of the object under examination

and the chromaticity coordinates obtained according to Eqs. (5-9) and (5-10). nY , 'nu ,

and 'nv represent the tristimulus value Y and the chromaticity coordinates 'u and 'v of a

suitable reference under the same illuminant. The values are normalized so that 100=nY .

For investigating the angular dependence of the color in LCDs, 'u , 'v , Y are the ( 'u , 'v )

coordinates and luminous transmittance of the pixel under examination, and 'nu , 'nv , nY

173

are the ( 'u , 'v ) coordinates and luminous transmittance of the backlight at normal

direction53. In this dissertation, we use the backlight power distribution as a reference

when we calculated the 'nu , 'nv , and nY .

In order to quantitatively compare the color difference between a pair of samples or two

interesting points, ( *1L , *1u , *1v ), ( *2L , *2u , *2v ), the color difference ( uvE *Δ ),

defined in the CIE 1976 *** vuL Color Space, is usually used, and the expression is

212

212

212 *)*(*)*(*)*(* vvuuLLE uv −+−+−=Δ . (5-16)

For analyzing the angular dependence of the color in LCDs, *2u , *2v , 2Y are the ( *u ,

*v ) chromaticity coordinates and luminous transmittance of the pixel under examination

and *1u , *1v , 1Y are the ( *u , *v ) chromaticity coordinates and luminous transmittance

of the same pixel at the normal direction.

5-2-4. Illuminants

A number of spectral power distributions have been defined by the CIE for use in

describing color, such as illuminant A , equivalent to a blackbody radiator with a color

temperature of 2856 K , illuminant C , and 65D . Natural daylight is defined by the D

illuminants. In this dissertation, we use 65D , CIE daylight with a correlated color

temperature of 6500 K . Figure 5-4 shows the spectral power distribution of those CIE

174

illuminants, and Table 5-1 shows the tristimulus values and chromaticity coordinates of

the illuminants.

175

380 480 580 680 7800.0

0.5

1.0

1.5

2.0

A C D65

Wavelength (nm)

Rel

ativ

e po

wer

dis

tribu

tion

Fig. 5-4. Spectral power distributions of CIE illuminants.

176

Table 5-1. Tristimulus values and chromaticity coordinates of illuminants in fields of

view 2º.

Tristimulus values Chromaticity coordinates Illuminants X Y Z x y 'u 'v

A 109.850 100.0 35.585 0.4476 0.4074 0.2560 0.5243 C 98.07 100.0 118.23 0.3101 0.3163 0.2009 0.4610 65D 95.047 100.0 108.883 0.3127 0.3291 0.1978 0.4684

177

5-3. Analyses of the luminous transmittance (Y )

Figure 5-5 shows the simple dark and bright state models (mode 1) that we built

in Chapter 3. In this chapter, we use the same model for analysis of the luminous

transmittance (Y ) and color properties of LCDs. The dark states of the model and LCDs

were compensated almost perfectly by using passive type negative C-plates, as the same

way that we used in Chapter 3. During the calculation, we used the numerical relaxation

technique to get the director field of the liquid crystal layer and the 2×2-matrix method to

calculate the light transmittances for each wavelength, ranging from 380 to 780 nm. After

calculation of the transmittance at each wavelength, we calculated the luminous

transmittance (Y ) of the devices according to the equations in Sec. 5-2.

5-3-1. Thickness effects on luminous transmittance and cell parameters

We calculated the luminous transmittance (Y ) of the bright state model at the

normal direction in function of the thickness of the positive A-plate, and the result is in

Fig. 5-6. In these calculations, the bright state model is located between ideal crossed

polarizers. Figure 5-6 (b) shows the normalized luminous transmittance ( Y %) with

respect to the maximum value of Y . We also did the same calculations for the common

real liquid crystal devices such as ECB, VA, Pi-cell, and TN modes. During the

calculations, the dark states of each different thickness are compensated separately for

each thickness. The detailed cell specifications, thickness information and the stack

configuration are in Table 5-2, Table 5-3, and Fig. 5-7, respectively. Where, the liquid

crystal we used is LC53 whose refractive indices and elastic constants are shown in Fig.

178

2-7 and Sec. 2-5, respectively. We assumed the polarizers are ideal, i.e. the light that

polarized along the absorption axis of a polarizer is absorbed completely, but the light

that is polarized along the transmission axis of a polarizer is transmitted perfectly except

surface reflections. The refractive indices of the glass, indium tin oxide (ITO), and

polyimide are shown in Fig. 2-27, and their thicknesses are 0.7 mm, 400 Å, and 500 Å.

We want to make clear that the specific numbers are only for the purposes of producing a

graph that demonstrates the features of the general concepts considered here.

179

Fig. 5-5. Simple dark and bright state models (mode 1)

Bright Dark

Positive A-plate

Negative C-plate

180

0.0 0.5 1.0 1.5 2.00

10

20

30

40

50

Thickness (μm)

Y

(a) Raw data

0.0 0.5 1.0 1.5 2.00

20

40

60

80

100

120

Thickness (μm)

Y (%

) of M

ax. Y

(b) Normalized data

Fig. 5-6. Thickness effects on the luminous transmittance of the bright state model at the

normal direction.

181

Table 5-2. Cell parameters of the common liquid crystal (LC) devices we used.

(LC: LC53; Δε=-9.4 for VA, +9.4 for other devices).

Easy axis (°)

Voltage (V)

Pretilt angle (°) Device

modes Top Bottom Bright Dark LC Pi-cell 90 90 1.3 5.0 5.5 ECB 90 -90 0.0 5.0 5.5 VA 90 -90 5.0 0.0 84.5 TN 45 -45 0.0 5.0 5.5

Table 5-3. Thicknesses of the bright state model and the common LCDs.

[PA: positive A-plate; NC: negative C-plates (compensator); LC: liquid crystal]

Model ECB VA Pi-cell TN Y (%) PA NC LC NC LC NC LC NC LC NC 100 1.742 0.780 2.055 0.919 2.1 0.780 7.274 3.255 3.187 1.28190 1.381 0.618 1.628 0.728 1.665 0.619 5.766 2.581 2.393 0.96280 1.222 0.547 1.442 0.645 1.474 0.548 5.106 2.285 2.089 0.83970 1.093 0.489 1.29 0.577 1.318 0.490 4.567 2.044 1.851 0.744

182

Fig. 5-7. Stack configuration of the common liquid crystal devices.

ITO Polyimide

Liquid Crystal

Polyimide ITO

Glass

Compensator (PDM)

Polarizer (+45º)

Polarizer (-45º)

Compensator (PDM)

Glass

Illuminant ( 65D )

183

5-3-2. Off-axis luminous transmittance

We numerically calculated the off-axis luminous transmittance (Y ) of the bright

state model and the common LCDs (ECB, VA, Pi-cell, and TN) for different cell

thicknesses, which have four different normalized Y (%) values at the normal direction:

100, 90, 80, and 70 %. The thicknesses of the liquid crystal and compensator layers,

which give the almost perfect compensation in their dark states, are tabulated in Table 5-

3.

5-3-2-1. Luminous transmittance in the bright state model

Figure 5-8 shows the results of the bright state model in two main directions: the

director and the out of the director planes. These results are consistent with that of

Chapter 3, where we calculated and analyzed the effective phase retardations and the off-

axis light transmittances at a single wavelength of 550 nm and at a single thickness of a

liquid crystal layer corresponding about 75% of Y (%) in this chapter.

In the director plane, the luminous transmittances (Y ) always decrease as the

viewing angle increases. This is because, as the viewing angle increases, the total

effective retardation at each wavelength decreases monotonically (“bell shape”), as we

analyzed in Chapter 3.

On the other hand out of the director plane, the luminous transmittances (Y )

increase first and then fall after specific viewing angles when the cell is optically

designed so that Y (%) value is less than 100% at the normal direction. If the cell is

designed as Y (%) has 100% at the normal direction, the luminous transmittance

184

decreases constantly as the viewing angle increases, as in Fig. 5-8 (b). These results also

agree well with the analyses in Chapter 3, i.e. as the viewing angle increases, the total

effective retardation out of the director plane always increases in all wavelengths

(“reversed bell shape”).

5-3-2-2. Luminous transmittance in real LCDs

We did the same calculations for the real LCDs (ECB, VA, Pi-cell, and TN) and

the results are in Fig. 5-9~12. All figures are, amazingly, very similar to that of the bright

state model except Y 100% of the Pi-cell out of the director plane, as in Fig. 5-11(b). In

this case, the luminous transmittance (Y ) fluctuates as the viewing angle increases from

the normal direction. We remember these kinds of the fluctuations in Chapter 4.

According to the results, in the highly tilted director configuration of a bright state, the

off-axis light transmittance varies significantly as the viewing angle increases out of the

director plane, and it makes the viewing angle properties of a Pi-cell worse. In

accordance with the analyses in Chapter 4, both the apparent angle distribution of

directors and the effective phase retardation of each layer play a key role in the

fluctuation of the total transmittance. We think the fluctuation of the luminous

transmittance in Fig. 5-11 (b) is also coming from the similar reasons. This is reasonable

because, as Y (%) value increases at the normal direction, the magnitudes of the effective

retardations of both liquid crystal and compensator layers increase at off-axis viewing

angles, and this is the similar situation in highly tilted director configurations. If we use

the highly tilted director configuration in bright state, the cell thickness of the device

185

should be increased to maintain the magnitude of effective retardation at the normal

direction, when we use the same liquid crystal material. It means the magnitudes of

effective retardations of both liquid crystal and compensator layers increase at off-axis

viewing angles.

5-3-2-3. Viewing angle properties of luminous transmittances in all directions

We also numerically calculated the luminous transmittances of the bright state

model and the real LCDs in all viewing directions, and the Fig. 5-13~17 show the

conoscopic figures. As we can notice in these results, the figure shapes are the same as

that of in Chapter 3 if the Y (%) value is less than 100%. That means, although we

calculated the off-axis light transmittance at a single wavelength of 550 nm in Chapter 3,

it is reasonable and can describe the luminance properties of LCDs well. Another thing

we earned in this section is that the analyses of the phase retardation in Chapter 3 can

predict the luminance properties of LCDs, excellently, even though the cell thicknesses

are changed.

All the conoscopic figures look like similar as we saw in two main directions in

previous section. It confirms again that if the dark states of LCDs are optically optimized

to have minimum transmittance, there is universality in the optical properties of the bright

state regardless of the display modes.

Another important thing is that all figures are anisotropic shapes, although the

figure of Y 100% of each display mode is similar to isotropic shape. Usually, we

optically design a LCD to have a little lower luminance than its maximum value at the

186

normal direction on the purposing of escaping gray scale inversions and to minimize

color shift at off-axis viewing angles. With this in mind, we can say that the single

domain LCD modes, considered here, whose dark states are optically compensated to

give the minimum luminous transmittance, inevitably have asymmetric shapes of the off-

axis luminous transmittances between the director and out of the director planes in their

bright states. Conclusively, we can say that in order to achieve isotropic shapes of the

bright state viewing angle properties, multi-domain methods or asymmetric distribution

of backlight intensity is mandatory.

187

-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50 Y100% Y90% Y80% Y70%


Y

(a) Director plane

-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50

Y100% Y90% Y80% Y70%


Y


Fig. 5-8. Off-axis luminous transmittances (Y ) of the bright state model with different

Y values at the normal direction.

188

-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50 Y100% Y90% Y80% Y70%


Y

(a) Director plane

-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50

Y100% Y90% Y80% Y70%


Y


Fig. 5-9. Off-axis luminous transmittances ( Y ) of the bright state ECB mode with

different Y values at the normal direction.

189

-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50 Y100% Y90% Y80% Y70%


Y

(a) Director plane

-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50

Y100% Y90% Y80% Y70%


Y


Fig. 5-10. Off-axis luminous transmittances ( Y ) of the bright state VA mode with


190

-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50 Y100% Y90% Y80% Y70%


Y

(a) Director plane

-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50

Y100% Y90% Y80% Y70%


Y


Fig. 5-11. Off-axis luminous transmittances (Y ) of the bright state Pi-cell mode with


191

-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50 Y100% Y90% Y80% Y70%


Y

(a) Director plane

-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50

Y100% Y90% Y80% Y70%


Y


Fig. 5-12. Off-axis luminous transmittances ( Y ) of the bright state TN mode with


192

(a) Y =100% (b) Y =90%

(c) Y =80% (d) Y =70%


model with different Y values at the normal direction.

193

(a) Y =100% (b) Y =90%

(c) Y =80% (d) Y =70%


ECB mode with different Y values at the normal direction.

194

(a) Y =100% (b) Y =90%

(c) Y =80% (d) Y =70%


VA mode with different Y values at the normal direction.

195

(a) Y =100% (b) Y =90%

(c) Y =80% (d) Y =70%


Pi-cell mode with different Y values at the normal direction.

196

(a) Y =100% (b) Y =90%

(c) Y =80% (d) Y =70%


TN mode with different Y values at the normal direction.

197

5-4. Analyses of the color properties

In this section, we calculate and analyze the color properties of the bright state

model (Fig. 5-5) and the real LCD modes (ECB, VA, PI-cell, and TN). The dark states of

these devices are almost perfectly compensated by using passive type negative C-plates,

as the same way that we used in Chapter 3. During the calculation, we used the numerical

relaxation technique to get the director field of the liquid crystal layer and the 2×2-matrix

method to calculate the light transmittances for each wavelength, ranging from 380 to

780 nm. After calculation of the transmittance at each wavelength, we calculated the

color properties of the devices according to the equations in Sec. 5-2 with the illuminant

(or light source), 65D whose specifications are in Fig. 5-4 and Table 5-1.

5-4-1. Thickness effects on color properties

In order to investigate the thickness (or phase retardation) effects on the color

properties of LCDs, we calculated the chromaticity coordinates ( 'u , 'v ) of the bright state

model in function of the thickness of the positive A-plate at the normal direction, and the

result is in Fig. 5-18. This figure shows that as the thickness (or phase retardation)

increases from zero, the chromaticity coordinates ( 'u , 'v ) goes from blue color to yellow

color first and then revolve around the point of the light source color ( 'u =0.1978,

'v =0.4684).

To understand this color variation, we calculated the transmittances and phase

retardations (Γ ) of the bright state model at the three major colors [blue (λ =450 nm),

green (λ =550 nm), and red (λ =650 nm)] at the normal direction, and the Figs. 5-19 and

198

5-20 show the results, respectively. As expected from Eqs. (3-1), (3-5), and Fig. 5-20, the

transmittances are sinusoidal functions in thickness space, and the pitch of blue color is

smaller than that of red color, and the pitch of green color is larger than that of blue color.

These different pitches make different colors for different thicknesses (or phase

retardations).

Comparing Figs. 5-18 and 5-19 shows that at a thickness about 2.6 μm, the blue

transmittance is zero, and at the same thickness, the chromaticity diagram gives the

maximum yellow color. At a thickness about 3.6 μm, the green transmittance goes to

zero and it lead to the maximum blue color in the chromaticity diagram, as in Fig. 5-18.

This color variation with the change of lightness ( *L ), as in Fig. 5-21, cause the

color difference ( uvE *Δ ) fluctuate as in Fig. 5-22.

199

Fig. 5-18. ( 'u , 'v ) chromaticity coordinates of the bright state model in terms of

thicknesses at the normal direction. (Thickness of the A-plate: 0.2-5.0 μm, 0.2 μm step)

d=0.2 μm

d=5.0 μm

d=4.0 μm

d=3.0 μm d=2.0 μm

200

0 1 2 3 4 50.0

0.1

0.2

0.3

0.4

0.5 Blue Green Red

Thickness (μm)

Tran

smitt

ance

Fig. 5-19. Transmittances of the bright state model at the three major colors [blue

(λ =450 nm), green (λ =550 nm), and red (λ =650 nm)] at the normal direction.

201

0 1 2 3 4 50.0

0.5

1.0

1.5

2.0 Blue Green Red

Thickness (μm)

Phas

e re

tard

atio

n (λ

)

Fig. 5-20. Phase retardation of the bright state model at the three major colors [blue

(λ =450 nm), green (λ =550 nm), and red (λ =650 nm)] at the normal direction. (Liquid

crystal: LC53)

202

0 1 2 3 4 50

20

40

60

80

Thickness (μm)

L*

Fig. 5-21. Lightness ( *L ) of the bright state model in function of thickness at the normal

direction. (Reference: illuminant 65D )

203

0 1 2 3 4 50

20

40

60

80

100

120

Thickness (μm)

ΔE* uv

Fig. 5-22. Color difference ( uvE *Δ ) of the bright state model in function of thickness at

the normal direction. (Reference point: thickness of 0.2 μm)

204

5-4-2. Chromaticity coordinates of the bright state model in the off-axis viewing

angles

We calculated the off-axis chromaticity coordinates, ( 'u , 'v ) and ( *u , *v ), of the

bright state model that has different Y (%) values (100, 90, 80, and 70 %) at the normal

direction, and the results are in Figs. 5-23 and 5-24, respectively. The illuminant (or light

source) we used is 65D , and the Eqs. 5-9, 5-10, 5-12, and 5-13 are used for the

calculations. The detailed specifications are in Tables 5-1, 5-2, and 5-3.

Firstly, these figures show that the absolute chromaticity coordinates at the

normal direction (red dots in the figures) move from the blue to yellow colors as the

Y (%) value (or cell thickness) increases, and this result is consistent with the thickness

effects of the previous section.

Secondly, as the Y (%) value increases, the total variation of the chromaticity

coordinates (from the normal direction to the viewing polar angle of 80º in these figures)

widens in both director (Φ=90, 270º) and out of the director (Φ=0, 180º) planes. If we

compare the magnitudes of the variations between the director plane and the out of the

director plane, the variation out of the director plane is bigger. This implies that when we

optically design a LCD to have higher transmittance at the normal direction, the display

could have bigger color variation, especially out of the director plane. The reason of this

fact can be understandable from the off-axis light transmittances of the red, green, and

blue colors. Figures 5-25, 5-26, 5-27, and 5-28 show the calculation results of the off-axis

light transmittances of the bright state model, which has Y (%) value of 100, 90, 80, and

70% at the normal direction. In these transmittance curves, we see that as the Y (%) value

205

increases, the transmittances of the blue, green, and red colors more significantly change

at off-axis viewing angles compared with normal direction, and also this phenomenon is

more severe in the viewing angles out of the director plane than in the director plane.

This transmittance variation associated with phase retardation causes the color change at

the off-axis viewing angles.

Finally, we can see that as the viewing angle increases, the color coordinates of

the bright state model, initially, move to yellow direction out of the director plane. On the

other hand, the color goes to the blue direction first in the director plane. These facts are

always true regardless of Y (%) value at the normal direction and related to the variation

of the phase retardation at off-axis directions. Figure 5-29 is the calculated result of the

off-axis phase retardation of the bright state model with Y (%)=80. This effective phase

retardation is calculated from Eqs. 3-7, 3-8, 3-10, and 3-11. As we can see in this figure,

the effective phase retardations increase as the viewing angle increases out of the

direction plane for all light wavelengths. On the contrary, the effective phase retardations

fall as the angle of incident light increases for all light colors. This universality is

consistent with the result of the Chapter 3. With these facts and the analysis of the

thickness effects on the color properties in Sec. 5-4-1, we can understand the reasons. As

the viewing angle increases out of the director plane, the effective phase retardation of

the bright state model increases, and this leads to color shift to yellow initially. In

contrast in the director plane, the effective phase retardation falls as the viewing angle

increases, and this initially causes a color shift to blue.

206

(a) Y (%)=100 (b) Y (%)=90

(c) Y (%)=80 (d) Y (%)=70

Fig. 5-23. ( 'u , 'v ) chromaticity coordinates of the bright state model in functions of

viewing angles. [0, 180: out of the director plane (overlapped each other); 90, 270:

director plane (overlapped each other); red dot: normal direction; viewing polar angles

are spanned from 0 to 80º with 5º step]

207

(a) Y (%)=100 (b) Y (%)=90

(c) Y (%)=80 (d) Y (%)=70

Fig. 5-24. ( *u , *v ) chromaticity coordinates of the bright state model in functions of

viewing angles. [0, 180: out of the director plane (overlapped each other); 90, 270:

director plane (overlapped each other); red dot: normal direction; viewing polar angles

are spanned from 0 to 80º with 5º step]

208

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance

(a) Out of the director plane

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance

(b) Director plane


colors (blue: 450 nm, green: 550 nm, red: 650 nm).

209

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance


-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance

(b) Director plane



210

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance


-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance

(b) Director plane



211

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance


-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance

(b) Director plane



212

-80 -60 -40 -20 0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0 Blue Green Red


Phas

e re

tard

atio

n (λ

)


-80 -60 -40 -20 0 20 40 60 800.0

0.2

0.4

0.6

0.8

1.0 Blue Green Red


Phas

e re

tard

atio

n (λ

)

(b) Director plane

Fig. 5-29. Off-axis phase retardation of the bright state model with Y (%)=80 in the three

major wavelengths (blue: 450 nm, green: 550 nm, red: 650 nm).

213

5-4-3. Chromaticity coordinates of the common bright state LCDs in the off-axis

viewing angles

We did the similar color calculations for the real LCDs (ECB, VA, Pi-cell, and

TN modes), which are optically designed to have several different Y (%) values (100, 90,

80, and 70 %) at the normal direction. We used 65D illuminant again, and the detailed

cell specifications are in Tables 5-1, 5-2, and 5-3. Figures 5-30~33 and 5-34~37 show the

calculation results of the off-axis chromaticity coordinates, ( 'u , 'v ) and ( *u , *v ),

respectively. Overall, the figure shapes are in good agreement with that of our simple

bright state model except the TN mode.

The absolute chromaticity coordinates at the normal direction move from blue to

yellow as Y (%) value increases in all display modes.

As Y (%) value increases, the total variation of the chromaticity coordinates

increases in both main viewing planes, and the magnitudes of the variations out of the

director plane are bigger than that of the director plane.

As the viewing angle increases, the color coordinates of the bright state move to

the yellow direction first in the viewing angles out of the director plane, but the color

shifts to the blue direction first in the director plane.

In the TN mode, the magnitude of the color variation out of the director plane

(refer to the mid-layer of directors) is the similar level with that of other display modes.

However, in the director plane, the color variation is much bigger than that of other

display modes, and the variation becomes large as Y (%) value decreases. This is the

opposite direction compared with our bright state model and other display modes

214

considered here. To understand these interesting facts, we calculated the off-axis

transmittances of TN mode for three light wavelengths: blue (450 nm), green (550 nm),

and red (650 nm), and the results are in Figs. 5-38~41. From these figures, we can notice

that as the Y (%) value decreases, the off-axis transmittances, especially red and green,

sharply go to zero at large viewing angles compared with that of the bright state model

(Figs. 5-25~28). This makes the bright state of a TN mode have a strong bluish color at

large viewing angle in the director plane (refer to mid-layer). We think this kind of

behavior of TN modes is related to the twist effect of liquid crystal directors. From these

results, we can say that the conventional “3-layer” model for the TN mode, which is

considered in Chapter 3, is partially incorrect when we calculate the optical properties of

the device.

215

(a) Y (%)=100 (b) Y (%)=90

(c) Y (%)=80 (d) Y (%)=70

Fig. 5-30. ( 'u , 'v ) chromaticity coordinates of the ECB in functions of viewing angles.

[0, 180: out of the director plane (almost overlapped each other); 90, 270: director plane

(almost overlapped each other); red dot: normal direction; viewing polar angles are

spanned from 0 to 80º with 5º step]

216

(a) Y (%)=100 (b) Y (%)=90

(c) Y (%)=80 (d) Y (%)=70

Fig. 5-31. ( 'u , 'v ) chromaticity coordinates of the VA in functions of viewing angles. [0,

180: out of the director plane (almost overlapped each other); 90, 270: director plane



217

(a) Y (%)=100 (b) Y (%)=90

(c) Y (%)=80 (d) Y (%)=70

Fig. 5-32. ( 'u , 'v ) chromaticity coordinates of the Pi-cell in functions of viewing angles.

[0, 180: out of the director plane; 90, 270: director plane (almost overlapped each other);

red dot: normal direction; viewing polar angles are spanned from 0 to 80º with 5º step]

218

(a) Y (%)=100 (b) Y (%)=90

(c) Y (%)=80 (d) Y (%)=70

Fig. 5-33. ( 'u , 'v ) chromaticity coordinates of the TN in functions of viewing angles. [0,

180: out of the director plane (almost overlapped each other); 90, 270: director plane



219

(a) Y (%)=100 (b) Y (%)=90

(c) Y (%)=80 (d) Y (%)=70

Fig. 5-34. ( *u , *v ) chromaticity coordinates of the ECB in functions of viewing angles.




220

(a) Y (%)=100 (b) Y (%)=90

(c) Y (%)=80 (d) Y (%)=70

Fig. 5-35. ( *u , *v ) chromaticity coordinates of the VA in functions of viewing angles.




221

(a) Y (%)=100 (b) Y (%)=90

(c) Y (%)=80 (d) Y (%)=70

Fig. 5-36. ( *u , *v ) chromaticity coordinates of the Pi-cell in functions of viewing

angles. [0, 180: out of the director plane; 90, 270: director plane (almost overlapped each

other); red dot: normal direction; viewing polar angles are spanned from 0 to 80º with 5º

step]

222

(a) Y (%)=100 (b) Y (%)=90

(c) Y (%)=80 (d) Y (%)=70

Fig. 5-37. ( *u , *v ) chromaticity coordinates of the TN in functions of viewing angles.

[0, 180: out of the director plane (almost overlapped each other); 90, 270: director plane;

red dot: normal direction; viewing polar angles are spanned from 0 to 80º with 5º step]

223

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance


-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance

(b) Director plane

Fig. 5-38. Transmittances of the TN mode with Y (%)=100 in the three major colors

(blue: 450 nm, green: 550 nm, red: 650 nm).

224

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0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance


-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance

(b) Director plane

Fig. 5-39. Transmittances of the TN mode with Y (%)=90 in the three major colors (blue:

450 nm, green: 550 nm, red: 650 nm).

225

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0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance


-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance

(b) Director plane


450 nm, green: 550 nm, red: 650 nm).

226

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0.1

0.2

0.3

0.4

0.5

Blue Green Red


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ance


-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance

(b) Director plane


450 nm, green: 550 nm, red: 650 nm).

227

5-4-4. Color difference of the bright state model in the off-axis viewing angles

In this section, we investigate the viewing angle dependence of the color

difference ( uvE *Δ ) in the bright state model. We chose the color coordinates of the

normal direction as the reference point ( *1L , *1u , *1v ) for the calculations. Figure 5-42

shows the results calculated from Eq. 5-16.

In the viewing directions out of the director plane, as the Y (%) value increases,

the off-axis color difference ( uvE *Δ ) increases. This result is consistent with the color

variation as a function of Y (%) value in Sec. 5-4-2, and coming from the facts that the

off-axis transmittances of the red, green, and blue are different more and more as the

Y (%) value increases.

In the viewing angles of the director plane, the off-axis color difference ( uvE *Δ )

increases as the Y (%) value increases, but the variation is very small except the

condition,Y (%)=100. This fact can be understandable from the transmittance curves of

red, green, and blue in Figs. 5-25~28. As we see from these figures, the transmittances of

red, green, and blue smoothly change as the Y (%) value increases, compared with that of

out of the director plane. In the cell condition, Y (%)=100, the color difference sharply

increases at the off-axis angles, and this is coming from the sort of the fluctuation of the

blue transmittance, as we can see in Fig. 5-25 (b). These features of transmittances are

correlated to the effective phase retardation at the off-axis angles (Fig. 5-29).

228

-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50

60

70 Y100% Y90% Y80% Y70%


ΔE* uv


-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50

60

70 Y100% Y90% Y80% Y70%


ΔE* uv

(b) Director plane

Fig. 5-42. Off-axis color difference ( uvE *Δ ) of the bright state model.

(Reference point: normal direction).

229

5-4-5. Color difference of the common bright state LCDs in the off-axis viewing

angles

We did the same calculations for the common LCDs (ECB, VA, Pi-cell, and TN

modes), and the Figs. 5-43~46 show the results. Comparing the results with that of the

bright state model in previous section, the figure shapes are very similar each other but

the TN mode.

In the directions out of the director plane, as the Y (%) value increases, the off-

axis color difference ( uvE *Δ ) increases, as the same way of the bright state model. In the

director plane, the off-axis color difference ( uvE *Δ ) very slowly increases as the Y (%)

value increases if the LCD is designed to have less than 100% of Y (%) at the normal

direction. In the condition that the Y (%) value approaches 100, the color difference

sharply increases at the off-axis viewing angles, with the same reason described in

previous section.

5-4-5-1. Detailed analyses of the color difference of the Pi-cell

The color difference ( uvE *Δ ) of the Pi-cell out of the director plane fluctuates at

large viewing angles when the cell is designed to have high transmittance at the normal

direction. We think this is one of the strong points of the Pi-cell because we can suppress

the color shift of the device better than other LCD modes at large viewing angles. This

sort of the fluctuation of the color difference can be imagined from the analyses of the

chromaticity coordinates of the Pi-cell in Sec. 5-4-3. If we see the Figs. 5-30~37, the

chromaticity coordinates of the Pi-cell turn back to their start position as the viewing

230

angle increases out of the director plane. This feature of the Pi-cell could provide the

optical design of a LCD with room to achieve high transmittance at the normal direction

without big color shift at large viewing angles.

The reasons for this phenomenon in the Pi-cell can be understood from the

separated transmittances of the red, green, and blue colors. Figures 5-47~50 show the off-

axis transmittance curves calculated under the same conditions. In the director plane, the

transmittances of the red, green, and blue colors vary smoothly as the Y (%) value

increases, and this gives the similar color differences for diverse Y (%) values, as in Fig.

5-45 (b). On the contrary, out of the director plane, the transmittances of the red, green,

and blue colors more significantly change as the Y (%) value increases, and these results

in various curve shapes of the color difference, as in Fig. 5-45 (a). Another important

thing, in the viewing direction out of the director plane, is that the transmittances,

especially the blue color, fluctuate as the Y (%) value increases at large viewing angles.

We think this fluctuation leads to the fluctuation of the color difference, and it suppresses

the color shift at large viewing angles.

5-4-5-2. Detailed analyses of the color difference of the TN mode

Another interesting fact we can notice from the calculations of the color

differences in the bright state LCDs is in the TN mode (Fig. 5-46). As we can see from

this figure, the color difference ( uvE *Δ ) of the TN mode is relatively insensitive to the

Y (%) value in both main viewing planes. Especially, the color difference in the condition,

Y (%)=100, is excellent and even better than the others. This surprising feature in TN

231

mode is very noticeable because, in our bright state model and in other display modes

considered here, the color differences ( uvE *Δ ) are the worst in the condition that has

100% of Y (%) value.

This feature of the TN mode can be understood from the transmission properties

of the major wavelengths in Fig. 5-38. As we see, the transmittance curves have the

different shapes, especially for the blue wavelength, compared with that of the bright

state model (Fig. 5-25). All three major colors (blue: 450 nm, green: 550 nm, and red:

650 nm) have very similar transmittances not only in the normal direction, but also in the

off-axis angles out of the director plane. This phenomenon becomes much stronger as the

Y (%) value increases.

These features of the TN mode remind us the Mauguin condition54, 55,

ndΔ<<Φλπ2 , (5-17)

where Φ is the total twist angle, d is the thickness of the liquid crystal layer, nΔ is the

birefringence of the liquid crystal material, and λ is the light wavelength. The Mauguin

condition states that if the director configuration of the liquid crystal satisfies the

condition, the liquid crystal acts like a wave-guide for incident linear polarized light,

whose direction is along the easy axis of the liquid crystal. This means the light

transmittance is insensitive to the light wavelengths.

232

With this knowledge in mind, now we can understand the surprising facts of the

TN modes. As the Y (%) value increases, and as the viewing angle increases out of the

director plane, and as the light wavelength decreases, the effective retardation becomes

larger, and there are much more chances to meet the Mauguin condition. Therefore, the

light transmittance becomes insensitive to the incident light wavelengths, and this reduces

the color difference ( uvE *Δ ) of the device. One of the possible problems, which might

happen, is the gray scale inversions in TN mode when we design the device to have the

maximum luminance at the normal direction.

233

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10

20

30

40

50

60

70 Y100% Y90% Y80% Y70%


ΔE* uv


-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50

60

70 Y100% Y90% Y80% Y70%


ΔE* uv

(b) Director plane

Fig. 5-43. Off-axis color difference ( uvE *Δ ) of the ECB mode. (Reference point: normal

direction).

234

-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50

60

70 Y100% Y90% Y80% Y70%


ΔE* uv


-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50

60

70 Y100% Y90% Y80% Y70%


ΔE* uv

(b) Director plane

Fig. 5-44. Off-axis color difference ( uvE *Δ ) of the VA mode. (Reference point: normal

direction).

235

-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50

60

70 Y100% Y90% Y80% Y70%


ΔE* uv


-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50

60

70 Y100% Y90% Y80% Y70%


ΔE* uv

(b) Director plane

Fig. 5-45. Off-axis color difference ( uvE *Δ ) of the Pi-cell mode. (Reference point:

normal direction).

236

-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50

60

70 Y100% Y90% Y80% Y70%


ΔE* uv


-80 -60 -40 -20 0 20 40 60 800

10

20

30

40

50

60

70 Y100% Y90% Y80% Y70%


ΔE* uv

(b) Director plane

Fig. 5-46. Off-axis color difference ( uvE *Δ ) of the TN mode. (Reference point: normal

direction).

237

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance


-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance

(b) Director plane

Fig. 5-47. Transmittances of the Pi-cell with Y (%)=100 in the three major colors (blue:

450 nm, green: 550 nm, red: 650 nm).

238

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance


-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance

(b) Director plane


450 nm, green: 550 nm, red: 650 nm).

239

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

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ance


-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance

(b) Director plane


450 nm, green: 550 nm, red: 650 nm).

240

-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance


-80 -60 -40 -20 0 20 40 60 800.0

0.1

0.2

0.3

0.4

0.5

Blue Green Red


Tran

smitt

ance

(b) Director plane


450 nm, green: 550 nm, red: 650 nm).

241

5-5. Summary

We numerically calculated and analyzed the off-axis luminous transmittance (Y )

and color properties of the bright state model with the common LCDs (ECB, VA, Pi-cell,

and TN). Their cell thicknesses are adjusted to have four different Y (%) values at the

normal direction: 100, 90, 80, and 70 %. Their dark states are almost perfectly

compensated by using the hybrid aligned negative C-plates.

5-5-1. Luminous transmittance of the bright state LCDs

According to the results, in both the bright state simple model and the real LCDs,

the luminous transmittances (Y ) always decrease as the viewing angle increases in the

director plane. On the other hand out of the director plane, the luminous transmittances

(Y ) increase first and then fall after specific viewing angles when the cell is optically

designed so that Y (%) value is less than 100% at the normal direction. These results are

in good agreement with the birefringence analyses.

We also calculated the luminous transmittances (Y ) in all viewing directions, and

compared them by using conoscopic figures. All the figures look similar. It confirms

again that if the dark states of LCDs are optically optimized, there is universality in the

optical properties of the bright state regardless of the display modes.

Another important thing is that all conoscopic figures are anisotropic shapes. This

means that the single domain LCD modes, considered here, inevitably have asymmetric

shapes of the luminance between the director and out of the director planes in their bright

states. Therefore, we can say that in order to achieve isotropic shapes of the bright state

242

viewing angle properties, multi-domain methods or asymmetric distribution of backlight

intensity is mandatory.

5-5-2. Color properties of the bright state LCDs

Firstly, we investigated the thickness (or phase retardation) effects on the color

properties of LCDs. These calculations show that as the thickness (or phase retardation)

increases from zero, the chromaticity coordinates ( 'u , 'v ) initially goes from blue to

yellow colors and then revolve around the point of the light source color. We analyzed

this fact by using the transmittances and the phase retardations of the three major colors.

Secondly, we calculated the off-axis chromaticity coordinates of the bright state

model and real LCDs such as ECB, VA, PI-cell, and TN modes. Overall figure shapes of

the real LCDs are in good agreement with that of our simple bright state model except the

TN mode. The absolute chromaticity coordinates at the normal direction move from blue

to yellow as Y (%) value increases in all display modes. As Y (%) value increases, the

total variation of the chromaticity coordinates increases in both main viewing planes, and

the magnitudes of the variations out of the director plane are bigger than that of the

director plane. As the viewing angle increases, the color coordinates of the bright state

move to the yellow direction first in the viewing angles out of the director plane, but the

color shifts to the blue direction first in the director plane.




243



considered here.

Finally, we investigate the viewing angle dependence of the color difference

( uvE *Δ ) in the bright state model and the real LCDs. When we compare the results, the

color differences of the real LCDs have the same curve shapes as that of the bright state

model except the TN mode. In the viewing directions out of the director plane, as the

Y (%) value increases, the off-axis color difference ( uvE *Δ ) increases, and this is coming

from the facts that the off-axis transmittances of the red, green, and blue are different

more and more as the Y (%) value increases. In the viewing angles of the director plane,

the off-axis color difference ( uvE *Δ ) increases as the Y (%) value increases, but the

variation is very small. This result is coming from the properties of the blue, green, and

red transmittances.

In the Pi-cell, out of the director plane, the color difference ( uvE *Δ ) fluctuates at



the color shift of the device better than other LCD modes at large viewing angles.

In the TN mode, the color difference ( uvE *Δ ) is relatively insensitive to the

Y (%) value. Especially, the color difference in the condition, Y (%)=100, is excellent

and even better than the others. This surprising feature in TN mode is related to the

Mauguin condition.

244

CHAPTER 6

THE CONDITIONS AND THE LIMITATIONS OF THE PERFECT PHASE

COMPENSATION IN LIQUID CRYSTAL DISPLAYS

6-1. Introduction

In this chapter, we will study the conditions and the limitations of the phase

compensation for the dark state of LCDs. As an example compensator, we will use hybrid

aligned negative C-plates.

In Sec. 6-2, we discuss the difference between an approximate method and a

“complete” method for the calculation of the phase retardation in a uniaxial material. In

Sec. 6-3, we analyze the relationship among the parameters ( ',',' dnn oe ) of a

compensator for the perfect compensation of a uniaxial slab. With this parameter

relationship, we discuss the thickness ratio of the compensator to uniaxial slab for the

perfect compensation. In Sec. 6-4, we calculate and analyze the total effective phase

retardation and the off-axis light transmittances as a function of the director tilt angle and

the thickness in the uniformly and non-uniformly aligned liquid crystal layers.

6-2. Calculations of the phase difference

In this section, we investigate the methods for calculating the phase difference of

the extraordinary and ordinary rays in a uniaxial medium.

245

6-2-1. “Complete” method

Let us consider a uniaxial slab (ordinary and extraordinary refractive indices of

on and en , respectively) whose optic axis or director ( nr ) is aligned to have the tilt angle

of θ and the azimuth angle of φ in the laboratory coordinate system ( zyx ,, ). Incident

light with the wavelength of λ in free space propagates in the zx − plane. Figure 6-1

shows the definition of these quantities. The phase difference ( Γ ) between the

extraordinary ray ( e -ray) and the ordinary ray ( o -ray) after propagating the uniaxial slab

is expressed as follows56, 57:

dKK ozez )( −=Γ , (6-1)

where ezK and ozK are the z components of the light propagation vectors of the e -ray

and the o -ray, respectively, and they are written as,

'cos2e

effeez nK θ

λπ

= , (6-2)

'cos2oooz nK θ

λπ

= , (6-3)

where effen is the effective refractive index of the e -ray and expressed as58,

246

ψψ 2222 sincos oe

oeeffe

nn

nnn

+= , (6-4)

where ψ is the angle between the propagation vector of the e -ray and the director ( nr )

of the uniaxial slab, and it can be calculated from59,

θθφθθψ sin'coscoscos'sincos ee += , (6-5)

where 'eθ is the angle of the light propagation vector from z -axis in the medium.

247

(a)

(b)

Fig. 6-1. Angle definitions with a uniaxial slab.

θ

'eθ

eKr

ψ

z

),( φθnr

x

φ y

d'oθ

'eθ

e -ray

oα

o -ray

oKr

Uniaxial slab

( oe nn , ) x

z

248

6-2-2. Approximate method

Let us say that the refractive indices of the e -ray and the o -ray in the uniaxial

slab are similar, i.e. oe nn ≅ . Then, we can assume that the optical paths of the e -ray and

the o -ray are the same in the medium, as in Fig. 6-2. Therefore, we have this relationship,

αθθ ≡= '' oe , (6-6)

where α is the angle of the light propagation vector ( e -ray and the o -ray) from the z -

axis. With this assumption, the phase difference (Γ ) between the e -ray and the o -ray

after propagating the uniaxial slab is written as follows60:

effndΔ=Γλπ2

αλ

πcos

)(2 dnn oeffe −= , (6-7)

where effen is the effective refractive index of the e -ray as expressed in Eq. (6-4), and the

ψ is rewritten as,

θαφθαψ sincoscoscossincos += . (6-8)

249

(a)

(b)

Fig. 6-2. Angle definitions with a uniaxial slab in the approximation method.

dα

e -ray

oα

o -ray

oKr

Uniaxial slab

( oe nn , ) x

z

θ

α

Kr

ψ

z

),( φθnr

x

φ y

250

6-2-3. Comparison of both methods for calculating the phase difference

In order to see the difference between two methods, described in previous

sections, we calculate the effective phase retardation ( effndΔ ) of a nematic slab by using

both methods. The ordinary ( on ) and extraordinary ( en ) refractive indices of the slab are

1.5 and 1.6, respectively, and the thickness is 2.75 μm. The light is incident in zx −

plane with the wavelength of 550 nm, and the incident angle ( oα ) in air is range from -

80º to +80º. In both methods, the refraction is considered at the air interface. In the

simple approximation method, the average refractive index is used to calculate the

refraction angle. In the “complete” method, we have to know the effen and 'eθ ,

simultaneously. To find both values, we tried two ways. First, we used the recursive

program. Second, we used the eigenvalues of the 2×2 matrix method.

Figure 6-3 shows the results for the several different director configurations

(θ ,φ ). As we can see, they are very similar, and the two ways used in the “complete”

method are the same. At the normal direction, the approximation method and the

“complete” method are the same as expected. As the incident light angle ( oα ) increases,

the difference between two methods increases, but the magnitudes of the differences are

extremely small. Therefore, in this chapter, without hurting the generality, we will use the

simple approximation method to calculate and analyze the effective phase retardations of

the uniaxial type anisotropic materials.

251

-80 -60 -40 -20 0 20 40 60 800

50

100

150

200

250

300

350

400 Simple Recursive 2x2 matrix

Light incident angle, αo (deg.)

Effe

ctiv

e Δn

d (n

m)

(a) θ =0º, φ =0º (director plane)

-80 -60 -40 -20 0 20 40 60 800

50

100

150

200

250

300

350

400

Simple Recursive 2x2 matrix


Effe

ctiv

e Δn

d (n

m)

(b) θ =0º, φ =90º (out of the director plane)

(c) and (d): next page

Fig. 6-3. Effective phase retardations of the nematic slabs (θ ,φ ) calculated by the

approximation method (“Simple”) and complete method (“Recursive” and “2×2 matrix”).

“Recursive” and “2×2 matrix” are overlapped each other in all the figures.

252

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50

100

150

200

250

300

350

400



Effe

ctiv

e Δn

d (n

m)

(c) θ =90º, φ =0º

-80 -60 -40 -20 0 20 40 60 800

50

100

150

200

250

300

350

400



Effe

ctiv

e Δn

d (n

m)

(d) θ =45º, φ =45º

253

6-3. Conditions and limitations for the perfect phase compensation in the simple

director configurations

6-3-1. Relationship of the parameters

Let us consider a simple compensation system, as in Fig. 6-4. We have a uniaxial

slab that has the thickness of d , the refractive indices of ( en , on ) and the director angles

of (θ , φ ). We are going to compensate the uniaxial slab with a layer of negative type

compensator, which has the thickness of 'd , the refractive indices of ( 'en , 'on ) and the

same director angles of the uniaxial slab. Using the same director angles is intuitively

reasonable.

The effective phase retardations of the uniaxial slab ( LCeffndΔ ) and the

compensator ( NCeffndΔ ) are expressed, respectively as follows:

αψψ cossincos 2222

dnnn

nnnd o

oe

oeLCeff ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛−

+=Δ , (6-9)

'cos''

'sin''cos'

''2222 αψψ

dnnn

nnnd o

oe

oeNCeff ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛−

+=Δ . (6-10)

The total effective phase retardation ( TotaleffndΔ ) of the uniaxial slab and the compensator

is the summation of both values because their optic axes are parallel with each other,

254

NCeff

LCeff

Totaleff ndndnd Δ+Δ=Δ . (6-11)

For the perfect compensation of the uniaxial slab, the total effective phase retardation

( TotaleffndΔ ) should be zero,

0=Δ+Δ=Δ NCeff

LCeff

Totaleff ndndnd . (6-12)

If we think about the parameters that we need for the perfect compensation of a uniaxial

slab, we have to decide the values, 'en , 'on , and 'd of the compensator. To understand

the relationship between the parameters, we calculate the total effective retardation of a

compensation system. The uniaxial slab has the ordinary and extraordinary refractive

indices of 1.5 and 1.6, respectively, and its thickness is 1.0 μm. The director tilt angle (θ )

of the slab split to 0, 30, and 60º, but their azimuth angles (φ ) are the same, 0º. We fixed

the ordinary refractive index ( 'on ) of the compensator as 1.6. The extraordinary index

( 'en ) and the thickness ( 'd ) of the compensator are spanned as 1.4~1.54 and 0.4~1.6 μm,

respectively. The light is incident with the wavelength of 550 nm, and the incident angle

( oα ) is zero.

Figure 6-5 shows the calculated total effective retardation ( TotaleffndΔ ) in functions of 'en

and 'd of the compensation system. From this figure, we know that for a given value of

255

'en , we can always choose the corresponding 'd that gives zero total effective

retardation. Figure 6-6 shows a pair of parameters ( 'en , 'd ) for the perfect compensation

( 0=Δ Totaleffnd ) of the uniaxial slab. These results show that only two variables among 'en ,

'on , and 'd are independent for the perfect compensation of a given uniaxial slab. For

example, if 'en , 'on are given, we can perfectly compensate a uniaxial slab in a given

viewing angle by adjusting only the thickness ( 'd ) of the negative type compensator.

This means that we do not have to think about the detailed values of 'en and 'on .

256

(a)

(b)

Fig. 6-4. Angle definitions of a simple compensation system.

Compensator ( ',' oe nn )

d

'd

θ

),( φθnr

Uniaxial slab ( oe nn , )

θ

α

Kr

ψ

z

),( φθnr

x

φ y

257

(a) θ =0º

(b) and (c): next page

Fig. 6-5. Total effective phase retardation in functions of 'en and 'd in the compensation

system. (φ =0º, oα =0º, en =1.6, on =1.5, d =1.0 μm, 'on =1.6)

258

(b) θ =30º

(c) θ =60º

259

1.40 1.42 1.44 1.46 1.48 1.50 1.52 1.54

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

θ=0o

θ=30o

θ=60o

ne'

d' (μ

m)

Fig. 6-6. A pair of parameters ( 'en , 'd ) for the perfect compensation ( 0=Δ Totaleffnd ) of

the uniaxial slab. (φ =0º, oα =0º, en =1.6, on =1.5, d =1.0 μm, 'on =1.6)

260

6-3-2. Thickness ratio for the perfect compensation

With these facts in mind, let us set the ordinary and extraordinary refractive

indices of the negative type compensator the same as the extraordinary and ordinary

refractive indices of the uniaxial slab, respectively, i.e. oe nn =' , eo nn =' . Then, the Eq.

(6-10) is written as,

αψψ cos'

sincos 2222

dnnn

nnnd e

eo

oeNCeff ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛−

+=Δ . (6-13)

From the fact that the total effective phase retardation ( TotaleffndΔ ) should be zero for the

perfect compensation of a uniaxial slab [Eq. (6-12)], we have this thickness relationship

that satisfies the condition for the perfect phase compensation of a uniaxial slab,

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

+−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−

+=

e

eo

oe

o

oe

oe

nnn

nn

nnn

nn

dd

ψψ

ψψ

2222

2222

sincos

sincos' (6-14)

effNC

effLC

nnΔ−Δ

≡ .

This equation means that in a given director configuration of a uniaxial slab (θ , φ ), we

can always compensate it completely in an incident light condition (α , λ ). However, the

261

magnitude of dd /' varies in functions of θ , φ , and α . Therefore, it implies that it is

almost impossible to compensate an anisotropic system exactly for all light wavelengths

and viewing angles, simultaneously. If the anisotropic system is composed of the layers

that have different director configurations, the perfect compensating system becomes

much more difficult. This means we need to adjust the thickness of each sub-layer of the

compensator, separately, with considering the director configuration of each sub-layer of

the uniaxial slab for better phase compensation.

6-3-3. Thickness ratio for the different director configurations

In this section, we calculate the thickness ratio ( dd /' ) for the perfect

compensation of the uniaxial slab (Fig. 6-4), which has several different director

configurations. We use the refractive indices of the commercialized liquid crystal, LC53

(Fig. 2-7), as the refractive indices of the uniaxial slab. We fixed the thickness of the

uniaxial slab to be 1.0 μm for the convenience. We followed the description of the

previous sections for the refractive indices and the director angle of the compensator. All

angle definitions are complied with that in Fig. 6-4. Figure 6-7 shows the calculated

thickness ratios ( dd /' ) for the perfect compensation of the uniaxial slab (θ =0, 30, 60,

and 90º) with the light wavelength of 550 nm, and Fig. 6-8 shows the three-dimensional

figure for the same light wavelength. These figures show that as stated in the previous

section, the thickness ratio ( dd /' ) for the perfect compensation of the uniaxial slab is the

functions of θ , φ , and α for a given material. The light wavelength (λ ) dependence on

the thickness ratio is relatively small if the wavelength dispersion of the uniaxial slab is

262

not so big. However, the thickness ratio is very sensitive to the director configuration (θ ,

φ ) and the light incident angle ( oα ).

263

-80 -60 -40 -20 0 20 40 60 80

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2 θ=0o

θ=30o

θ=60o

θ=90o


d'/d

(a) φ =0º (director plane), λ =550 nm


Fig. 6-7. Thickness ratio ( dd /' ) for the perfect compensation of the uniaxial slab (θ =0,

30, 60, and 90º; d =1.0 μm) with the light wavelength of 550 nm.

264

-80 -60 -40 -20 0 20 40 60 80

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2 θ=0o

θ=30o

θ=60o

θ=90o


d'/d

(b) φ =90º (out of the director plane), λ =550 nm

-80 -60 -40 -20 0 20 40 60 80

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2 θ=0o

θ=30o

θ=60o

θ=90o


d'/d

(c) φ =45º, λ =550 nm

265

(a) φ =0º (director plane), λ =550 nm


Fig. 6-8. Three-dimensional thickness ratio ( dd /' ) for the perfect compensation of the

uniaxial slab (θ =0~90º; d =1.0 μm) with the light wavelength of 550 nm.

266

(b) φ =90º (out of the director plane), λ =550 nm

(c) φ =45º, λ =550 nm

267

6-3-4. Transmittances in the compensation system

In this section, we numerically calculate the off-axis light transmittance of the

compensation system that we used in the previous section (Sec. 6-3-3). We insert the

system between the ideal crossed polarizers, where the transmission axis of the bottom

polarizer makes -45º angles relative to the easy axis of the uniaxial slab. We calculate the

off-axis light transmittance for three different thickness ratios ( dd /' ), 0.8, 0.9, and 1.0

with the light wavelength of 550 nm. In this calculation, we also set the thickness of the

uniaxial slab as 1.0 μm for all different tilt angles (θ ). Figures 6-9, 6-10, and 6-11 show

the results for the different azimuth angles of the uniaxial slab (φ ), 0, 90, and 45º,

respectively, and Figs. 6-12~14 are the corresponding three-dimensional figures. In these

figures, the transmittance is 1 if the transmitted light has the same intensity as that of

incident light. In the viewing directions of the director plane (φ =0º) and the out of the

director plane (φ =90º), the light transmittances include the light leakage that comes from

the geometrical reasons of the crossed polarizers, as described in Sec. 3-2-3. However,

the transmittance in the direction, φ =45º, does not include the geometrical polarizer

effects. With this knowledge and the results in previous section in mind, we can see that

as the thickness ratio ( dd /' ) varies, the points of the perfect compensation (or the

minimum light transmittance) move toward the points that satisfy the condition, dd /' . In

the meanwhile, in the points that does not satisfy the condition, dd /' , the light

transmittance (or leakage) could increase depending on the director configuration and the

light incident angle. These transmittance results are in good agreement with the results in

Sec. 6-3-3.

268

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05 θ=0o

θ=30o

θ=60o

θ=90o


Tran

smitt

ance

(a) dd /' =0.8



azimuth angle (φ ) of 0º (director plane).

(θ =0, 30, 60, and 90º; d =1.0 μm, λ =550 nm).

269

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05 θ=0o

θ=30o

θ=60o

θ=90o


Tran

smitt

ance

(b) dd /' =0.9

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05 θ=0o

θ=30o

θ=60o

θ=90o


Tran

smitt

ance

(c) dd /' =1.0

270

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05 θ=0o

θ=30o

θ=60o

θ=90o


Tran

smitt

ance

(a) dd /' =0.8



azimuth angle (φ ) of 90º (out of the director plane).

(θ =0, 30, 60, and 90º; d =1.0 μm, λ =550 nm).

271

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05 θ=0o

θ=30o

θ=60o

θ=90o


Tran

smitt

ance

(b) dd /' =0.9

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05 θ=0o

θ=30o

θ=60o

θ=90o


Tran

smitt

ance

(c) dd /' =1.0

272

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05

θ=0o

θ=30o

θ=60o

θ=90o


Tran

smitt

ance

(a) dd /' =0.8



azimuth angle (φ ) of 45º.

(θ =0, 30, 60, and 90º; d =1.0 μm, λ =550 nm).

273

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05

θ=0o

θ=30o

θ=60o

θ=90o


Tran

smitt

ance

(b) dd /' =0.9

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05

θ=0o

θ=30o

θ=60o

θ=90o


Tran

smitt

ance

(c) dd /' =1.0

274

(a) dd /' =0.8



system with the azimuth angle (φ ) of 0º (director plane).

(θ =0~90º; d =1.0 μm, λ =550 nm).

275

(b) dd /' =0.9

(c) dd /' =1.0

276

(a) dd /' =0.8



system with the azimuth angle (φ ) of 90º (out of the director plane).

(θ =0~90º; d =1.0 μm, λ =550 nm).

277

(b) dd /' =0.9

(c) dd /' =1.0

278

(a) dd /' =0.8



system with the azimuth angle (φ ) of 45º.

(θ =0~90º; d =1.0 μm, λ =550 nm).

279

(b) dd /' =0.9

(c) dd /' =1.0

280

6-4. Applications

6-4-1. Uniform director configurations

As an application, let us consider a uniformly aligned liquid crystal layer. The tilt angle

(θ ) of the liquid crystal director is split into 3 types, 0, 30, and 60º. The thickness of each

uniformly tilted liquid crystal layer is determined so that each liquid crystal layer has four

different effective phase retardations ( LCeffndΔ : 50, 100, 200, and 300 nm at a light

wavelength of 550 nm) at the normal direction. Now, we want to compensate the liquid

crystal layers perfectly at the normal direction with the light wavelength of 550 nm. We

use a negative type compensator for the phase compensation of the liquid crystal layer

with the same optic axis as that of liquid crystal. The thickness of the compensator is

determined to meet the thickness ratio ( dd /' ) for the perfect phase compensation at the

normal direction. The thickness ratios, here we use, are the same as the values of the

normal direction in Fig. 6-7. The detailed thicknesses are calculated as in Table 6-1.

Where, we used the optical parameters of the commercialized liquid crystal, LC53 (Fig.

2-7), for the liquid crystal layer of the compensation system. The ordinary and

extraordinary refractive indices of the compensator are the extraordinary and ordinary

refractive indices of the liquid crystal, respectively, as we did in Sec. 6-3.

Firstly, we calculate and analyze the total effective phase retardation ( TotaleffndΔ ) of

the compensation systems. Figures 6-15, 6-16, and 6-17 show the calculation results for

the different tilt angles (θ ) of 0, 30, and 60º, respectively. Each figure contains the total

281

effective phase retardation in the viewing directions (φ ) of 0º (director plane), 90º (out of

the director plane), and 45º.

As we can see in these figures, the total effective phase retardation at the normal

direction is zero at all figures because we compensated perfectly at the normal direction

for each director configuration. All figure shapes are consistent with the analyses of the

thickness ratio ( dd /' ) in Fig. 6-7. If we compare between the director plane (φ = 0º) and

the out of the director plane (φ = 90º), the variation of the total effective retardation in the

director plane is bigger than that of the out of the director plane. This is because the

variation of the thickness ratio ( dd /' ) in the director plane [Fig. 6-7 (a)] is larger than

that of the out of the director plane [Fig. 6-7 (b)]. With the similar reason, all curve

shapes of the total effective retardations out of the director plane are symmetric.

In a given tilt angle (θ ) of the liquid crystal director, as the effective phase

retardation of the liquid crystal layer ( LCeffndΔ ) increases, the magnitude of the off-axis

total effective retardation ( TotaleffndΔ ) also increases. On the other hand, with the constant

effective phase retardation of the liquid crystal layer, the magnitude of the off-axis total

effective retardation increases as the tilt angle of the liquid crystal director increases.

These important facts cause the light leakage at the off-axis angles.

Secondly, we calculate the off-axis light transmittances of the compensation

systems with the ideal crossed polarizers. The transmission axis of the bottom polarizer

makes -45º angles with the easy axis of the liquid crystal layer. Figures 6-18, 6-19, and 6-

282

20 are the numerical calculation results of the transmittances for the light wavelength of

550 nm. We used 2×2 matrix method for the calculations.

In these results, the effects of the light leakage coming from the geometrical

reason of the crossed polarizers are added to the off-axis transmittances in the director

(φ =0º) and out of the director (φ =90º) planes, as described in Sec. 3-2-3. With the

similar geometrical reason, the apparent relative angle between the transmission axis of

the polarizer and the director of the liquid crystal layer varies as the viewing polar angle

( oα ) increases in the viewing azimuth angles (φ ) of 45º and 90º (out of the director

plane). These additional effects make the analyses of the off-axis transmittance, based on

the birefringence analyses, complicated. As an example, if we compare Fig. 6-16(b) with

Fig. 6-19(b) and Fig. 6-17(b) with Fig. 6-20(b), which have the same set of the effective

retardation of the liquid crystal layer ( LCeffndΔ ), the transmittance out of the director plane

is not the same order as that of the total effective phase retardation ( TotaleffndΔ ) at the off-

axis angles, i.e. the transmittance out of the director plane does not exactly follow the

total effective phase retardation. We think this phenomenon in the viewing direction out

of the director plane is coming from the geometrical reasons.

In other viewing directions, as expected, the off-axis light transmittances agree

well with the analyses of the total effective phase retardation ( TotaleffndΔ ). In a given

director tilt angle ( θ ), the off-axis light transmittance increases as the effective

retardation of the liquid crystal layer ( LCeffndΔ ) at the normal direction increases. These

results mean that as the residual birefringence increases with the constant director

283

configuration, the dark state light leakage of a LCD increases. In a similar way, as the tilt

angle of the director increases with the constant phase retardation of the liquid crystal

layer, the off-axis light leakage rises.

These facts contradict each other. According to the results, if we want to have low

light leakage in a dark state, the effective phase retardation of the liquid crystal layer at

the normal direction should be reduced regardless of the director tilt angle. In addition,

the director tilt angle of the liquid crystal layer should be decreased for better phase

compensation. However, if the director tilt angle decreases, it makes the liquid crystal

layer have higher phase retardation. Therefore, the effective phase retardation of the

liquid crystal layer competes with the director tilt angle for the best performance of the

phase compensation of the liquid crystal layer.

In order to make this issue clear, we calculate the off-axis light transmittances as a

function of the thickness and the director tilt angle of the liquid crystal layer with the

light wavelength of 550 nm. In this calculation, we compensate the liquid crystal layer

perfectly at the normal direction at each thickness and tilt angle, as the previous

calculation. We set the light incident angle ( oα ) as -70º because this direction is the most

sensitive to the transmittance, as in Figs.6-18~20. Figure 6-21 shows the results. In this

figure also, the geometrical polarizer effects are in the director (φ =0º) and out of the

director (φ =90º) planes, and the geometrical director angle effects are out of the director

plane ( φ =90º) and the diagonal plane ( φ =45º). The fluctuation of the off-axis

transmittance out of the director plane in Fig. 6-21 (b) is coming from the same effects.

284

This calculation result confirms that as the cell thickness with a given liquid

crystal material (or effective phase retardation) decreases, the light leakage of a dark state

LCD decreases independent of the director configuration. In a given cell thickness, the

light leakage is the function of the director tilt angle of the liquid crystal layer. In this

calculation, as the tilt angle approaches both sides, θ =0º and θ =90º, the light leakage

falls in all conditions of the cell thickness. This is the result of the competition between

the effective phase retardation and the director tilt angle for the best phase compensation

of the liquid crystal layer, as described above.

285

Table 6-1. Thicknesses of the uniformly aligned perfect compensation systems (at the

normal direction). (unit: μm; LC: liquid crystal; NC: negative C-type compensator;

λ =550 nm).

θ =0º ( dd /' =1.0) θ =30º ( dd /' =0.9281) θ =60º ( dd /' =0.7998)LCeffndΔ

(nm) LC NC LC NC LC NC 50 0.3175 0.3175 0.4405 0.4088 1.4225 1.1377 100 0.6351 0.6351 0.8809 0.8176 2.8451 2.2754 200 1.2701 1.2701 1.7618 1.6351 5.6901 4.5509 300 1.9052 1.9052 2.6428 2.4527 8.5352 6.8263

286

-80 -60 -40 -20 0 20 40 60 80

-30

-20

-10

0

10

20

30

40 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tota

l Effe

ctiv

e Δn

d (n

m)

(a) φ =0º (director plane)


Fig. 6-15. Total phase retardation ( TotaleffndΔ ) of the compensation systems (θ =0º) that are


287

-80 -60 -40 -20 0 20 40 60 80

-30

-20

-10

0

10

20

30

40 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tota

l Effe

ctiv

e Δn

d (n

m)

(b) φ =90º (out of the director plane)

-80 -60 -40 -20 0 20 40 60 80

-30

-20

-10

0

10

20

30

40 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tota

l Effe

ctiv

e Δn

d (n

m)

(c) φ =45º

288

-80 -60 -40 -20 0 20 40 60 80

-30

-20

-10

0

10

20

30

40 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tota

l Effe

ctiv

e Δn

d (n

m)




are perfectly compensated at the normal direction for the light wavelength of 550 nm.

289

-80 -60 -40 -20 0 20 40 60 80

-30

-20

-10

0

10

20

30

40 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tota

l Effe

ctiv

e Δn

d (n

m)


-80 -60 -40 -20 0 20 40 60 80

-30

-20

-10

0

10

20

30

40 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tota

l Effe

ctiv

e Δn

d (n

m)

(c) φ =45º

290

-80 -60 -40 -20 0 20 40 60 80

-50

0

50

100

150

200

250 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tota

l Effe

ctiv

e Δn

d (n

m)




are perfectly compensated at the normal direction for the light wavelength of 550 nm.

291

-80 -60 -40 -20 0 20 40 60 80

-50

0

50

100

150

200

250 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tota

l Effe

ctiv

e Δn

d (n

m)


-80 -60 -40 -20 0 20 40 60 80

-50

0

50

100

150

200

250 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tota

l Effe

ctiv

e Δn

d (n

m)

(c) φ =45º

292

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tran

smitt

ance





293

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tran

smitt

ance


-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tran

smitt

ance

(c) φ =45º

294

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tran

smitt

ance





295

-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tran

smitt

ance


-80 -60 -40 -20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tran

smitt

ance

(c) φ =45º

296

-80 -60 -40 -20 0 20 40 60 80

0.00

0.05

0.10

0.15

0.20

0.25

0.30 Δnd

effLC=50 nm

Δndeff

LC=100 nm Δnd

effLC=200 nm

Δndeff

LC=300 nm


Tran

smitt

ance





297

-80 -60 -40 -20 0 20 40 60 80

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Δndeff

LC=50 nm Δnd

effLC=100 nm

Δndeff

LC=200 nm Δnd

effLC=300 nm


Tran

smitt

ance


-80 -60 -40 -20 0 20 40 60 80

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Δndeff

LC=50 nm Δnd

effLC=100 nm

Δndeff

LC=200 nm Δnd

effLC=300 nm


Tran

smitt

ance

(c) φ =45º

298

(a) Viewing direction: φ =0º (director plane), oα =-70º


Fig. 6-21. Off-axis light transmittances in functions of the thickness and the director tilt

angle of the liquid crystal layer in the compensation systems that are perfectly

compensated at the normal direction for the light wavelength of 550 nm.

299

(b) Viewing direction: φ =90º (out of the director plane), oα =-70º

(c) Viewing direction: φ =45º, oα =-70º

300

6-4-2. Non-uniform director configurations

In Chapter 4, Sec. 4-2, we numerically calculated the off-axis light transmission

properties of Pi-cells, whose dark states are optically optimized at the normal direction by

using hybrid aligned negative C-plates. Their bright states are implemented by using

different applied voltages and different pretilt angles with all the same dark voltage. In

this section, we will investigate their dark state transmittances in the same situations,

based on our knowledge in this chapter. The detailed information related to the phase

compensation, the cell parameters, and the structure is in Sec. 4-2.

6-4-2-1. A Pi-cell that has different bright state voltages (thickness effects)

Figure 4-2 is the numerical calculation results of the off-axis light transmission

properties of the dark state Pi-cells, whose bright states are optimized to have several

different voltages, but they have the same dark state voltage of 5.0 V and pretilt angle of

5.5º. Therefore, their dark states have the same director configuration, as in Fig. 4-3, but

their cell thicknesses (or effective phase retardation at the normal direction) are different,

as in Table 4-1. Figure 4-2 shows that as the bright state voltage (or effective phase

retardation at the normal direction) increases, the off-axis light transmittance of the dark

state increases. This result is in good agreement with the results in Sec. 6-4-1 where we

considered uniformly aligned director configurations. Consequently, we can say that as

the effective phase retardation of the liquid crystal layer increases, the perfect phase

compensation becomes harder, and the dark state light leakage increases at the off-axis

301

viewing directions. This is true in both uniformly and non-uniformly aligned liquid

crystal displays.

6-4-2-2. A Pi-cell that has different pretilt angles (thickness and director tilt angle

effects)

We did similar calculations for the Pi-cells that have different pretilt angles, but

they have the same dark and bright states voltages of 5.0 V and 1.3 V, respectively. In

addition to the results in Sec. 4-2, we calculated two more pretilt angles, 40.0º and 50.0º

for more investigation. The detailed cell parameters are in Table 6-2. Figures 6-22 and 6-

23 show the results of the off-axis light transmission properties with the light wavelength

of 550 nm and the director configurations of the liquid crystal layers in their dark states,

respectively. In this situation, as the pretilt angle increases, the director tilt-angle as well

as the cell thickness increases, as in Table 6-2 and Fig. 6-23. Figure 6-22 shows that as

the pretilt angle (or cell thickness) increases, the off-axis light transmittance of the dark

state increases very much. In other words, as the pretilt angle increases in a Pi-cell, the

average tilt angle and the cell thickness increase to meet the same transmittance of the

bright state at the normal direction, and this makes the phase compensation of the dark

state of the liquid crystal layer difficult at the off-axis viewing angles. This result also

agrees well with the results in Sec. 6-4-1.

302

Table 6-2. Cell parameters of the Pi-cells with different pretilt angles.

(Bright state voltage of 1.3 V, dark state voltage of 5.0 V).

Thickness (µm) Pretilt angle (°) LC Compensator 2.0 4.745 2.131 10.0 5.474 2.438 20.0 6.769 2.963 30.0 8.785 3.752 40.0 12.150 5.036 50.0 18.366 7.371

303

-80 -60 -40 -20 0 20 40 60 80

0.00

0.05

0.10

0.15

0.20

0.25

Pretilt 2o

Pretilt 10o

Pretilt 20o

Pretilt 30o

Pretilt 40o

Pretilt 50o

Viewing angle, αo (deg.)

Tran

smitt

ance

(a) Director plane

-80 -60 -40 -20 0 20 40 60 80

0.00

0.05

0.10

0.15

0.20

0.25

Pretilt 2o

Pretilt 10o

Pretilt 20o

Pretilt 30o

Pretilt 40o

Pretilt 50o

Viewing angle, αo (deg.)

Tran

smitt

ance



dark state Pi-cells that have different pretilt angles (2.0-50.0º).

(Bright state voltage: 1.3 V; dark state voltage: 5.0 V; liquid crystal: LC53).

304

0.0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

120

140

160

180 Pretilt 2o Pretilt 30o

Pretilt 10o Pretilt 40o

Pretilt 20o Pretilt 50o


Tilt

angl

e (d

eg.)


(2.0-50.0º). (Bright state voltage: 1.3 V; dark state voltage: 5.0V; liquid crystal: LC53).

305

6-5. Summary

We compared the calculation methods of the phase difference. At the normal

direction, the approximate method and the “complete” method are the same as expected.

As the light incident angle increases, the difference between them increases, but the

magnitudes of the differences are very small.

We analyzed the parameters for the perfect phase compensation of a uniaxial slab.

To understand the relationship between the parameters, we calculated the total effective

phase retardation of a compensation system. From this, we knew that only two variables

among 'en , 'on , and 'd are independent for the perfect compensation of a given uniaxial

slab.

We found that there is a constant thickness ratio ( dd /' ) for the perfect phase

compensation of a uniaxial slab. In a given director configuration of a uniaxial slab (θ ,

φ ), we can always compensate it completely in an incident light condition (α , λ ).

However, the magnitude of dd /' varies in functions of θ , φ , and α . Therefore, it

implies that it is almost impossible to compensate an anisotropic system exactly for all

light wavelengths and viewing angles, simultaneously. If the anisotropic system is

composed of the layers that have different director configurations, the perfect

compensating the system becomes much more difficult. This means we need to adjust the

thickness of each sub-layer of the compensator, separately, with considering the director

configuration of each sub-layer of the uniaxial slab for better phase compensation.

The light wavelength dependence on the thickness ratio is relatively small if the

wavelength dispersion of the uniaxial slab is not so big. However, the thickness ratio is

306

very sensitive to the director configuration and the light incident angle. We confirmed

these facts by calculating the off-axis transmittance of the compensation systems.

As the first application, we considered a uniformly aligned liquid crystal layer

that had different tilt angles and effective phase retardations. We calculated the off-axis

phase retardation of the compensation system, which was perfectly compensated at the

normal direction. According to the results, in a given tilt angle of the liquid crystal

director, as the effective phase retardation of the liquid crystal layer increases, the

magnitude of the off-axis total effective retardation also increases. On the other hand,

with the constant effective phase retardation of the liquid crystal layer, the magnitude of

the total effective retardation increases as the tilt angle of the liquid crystal director

increases. These results mean that as the residual birefringence increases with the

constant director configuration, the dark state light leakage of a LCD increases. In a

similar way, as the tilt angle of the director increases with the constant phase retardation

of the liquid crystal layer, the off-axis light leakage rises.

Another important fact we found is that the effective phase retardation of the

liquid crystal layer competes with the director tilt angle for the best compensation of the

liquid crystal layer. To make this fact clear, we calculated the off-axis light

transmittances in functions of the thickness and the director tilt angle of the liquid crystal

layer. These calculations confirm that as the cell thickness decreases with a given liquid

crystal material, the light leakage of a dark state LCD decreases independent of the

director configuration. In a given cell thickness, the light leakage is the function of the

307

director tilt angle of the liquid crystal layer. As the tilt angle approaches both sides, θ =0º

and θ =90º, the light leakage falls in all conditions of the cell thickness.

As the second application, we considered a non-uniformly aligned Pi-cell that has

several different bright state voltages and pretilt angles. According to the results, as the

cell thickness (or effective phase retardation at the normal direction) or the director tilt

angle increase, the phase compensation of the dark state becomes harder, and this makes

the off-axis light leakage increases in the dark state. These results are consistent with the

analyses of the uniformly aligned compensation system.

308

CHAPTER 7

CONCLUSIONS

7-1. Analysis of the multi-reflection effects in compensated liquid crystal devices

We calculated the multi-reflection effects analytically and numerically in the dark

state of a compensated liquid crystal device and compared the results with the measured

transmittance of an example device. According to our analysis, there are two types of

interference in devices with significant residual retardation in the dark state that is

compensated by a passive optical retarder. The first one is due to the pure e-ray and pure

o-ray by themselves, and the second one is coming from the coupling between the e-

mode and o-mode. The first type has higher frequency in the wavelength space and is

related to the optical path length of the e-ray and o-ray and is independent of their

difference. Most of the modes used in liquid crystal devices have this type of interference.

The second type of interference has lower frequency than that of the first one and

depends on the residual birefringence of dark state. So, as the residual birefringence

increases, the second type of interference becomes more significant. In the viewpoint of

the optical properties of a liquid crystal device, the first type of interference could affect

the dark level and extinction ratio almost equivalently for visible wavelength region. On

the other hand, the second type could cause a wavelength dependence of the extinction

ratio, or a color shift of the dark state because of the lower frequency pattern of the

interference in the wavelength space.

309

7-2. Universal off-axis light transmission properties of the bright state in

compensated liquid crystal devices

We calculated the off-axis light transmission properties of the bright state of most

common liquid crystal devices, such as ECB, VA, TN, Pi-cell, and symmetric splay-cell,

whose dark states were optically compensated to have minimum transmittances for all

viewing angles. From the results of these calculations, we found that their bright states

have a universal viewing angle shape in spite of completely different director structures

in their liquid crystal layers.

In order to understand this strange phenomenon, we made simple dark and bright

state models describing general liquid crystal devices and analyzed them in terms of

effective retardation and transmittance. In accordance with these analyses, the total

effective retardation in the director plane constantly falls as the viewing angle increases

(“bell shape”). On the contrary, the total effective retardation out of the director plane

consistently rises in the same situation (“reversed bell shape”). These retardation

changes cause the transmittance changes. In the director plane, the transmittance

decreases as the viewing angle becomes larger because the birefringence decreases in that

direction. On the other hand, the transmittance out of the director plane increases first and

then falls after the specific viewing angle if the liquid crystal layer is optically designed

so that the transmittance of the normal direction is lower than the maximum value.

These viewing angle features of our bright state models agree well with the

properties of most common liquid crystal devices, not only in the two main viewing

planes, but also for all viewing directions. Therefore, we can say that our simple model

310

can reasonably describe the optical properties of the real liquid crystal devices considered

here. Accordingly, our simple model can be used to analytically understand and predict

the optical properties, such as transmittance, luminance distribution and color analyses of

current LCDs or possible candidates of new display modes because usually analytical

methods for optical calculations are almost impossible in real devices.

Based on these results, we can say that the single domain LCD modes, considered

here, whose dark states are optically compensated to give minimum transmittance,

inevitably have asymmetric shapes of the off-axis light transmission properties between

the director and out of the director planes in their bright states. Therefore, in order to

achieve isotropic shapes of the bright state viewing angle properties, multi-domain liquid

crystal modes are necessary.

7-3. Optimization of the bright state director configuration in compensated pi-cell

devices

We calculated the off-axis light transmission properties of the differently director

configured bright states in Pi-cells whose dark states were optically compensated

perfectly to have minimum transmittances in all viewing angles. From the results of these

calculations, we found that the off-axis light transmission properties of the bright states

surprisingly had unified shapes and were relatively insensitive to the variation of the

parameters considered when the voltage of the bright state or the pretilt angle were below

some particular value. On the other hand, the transmittance in highly tilted director

311

configuration of bright state out of the director plane varies more significantly and makes

the viewing angle properties of a Pi-cell worse.

In order to understand these curious phenomena, we made a new dark and bright

state model describing Pi-cells and analyzed it in terms of effective birefringence,

transmittance and angular distribution of directors. According to the analyses, the total

effective birefringence in the director plane constantly falls as the viewing angle

increases (“bell shape”) regardless of the director configuration, and this result is

consistent with transmittance analyses and describes the optical properties of real Pi-cells

well.

Out of the director plane, we cannot define the total birefringence because the

projection of the directors on the plane perpendicular to the light propagation vector ( Kr

)

are not co-linear, so we analyzed transmittance in this viewing plane. In the director

configuration of low tilt angle in bright state model, the transmittance rises first and then

falls after the specific viewing angle if the liquid crystal layer is optically designed so that

the transmittance of the normal direction is lower than the maximum value. On the other

hand, the transmittance with a high tilt director configuration fluctuates as the viewing

angle varies as in real Pi-cells. To understand the causes of the fluctuation of the total

transmittance, we calculated the transmittances, which are contributed separately from

positive A-plates and negative C-plates of the bright state model, and analyzed them

using the apparent angle distribution of directors in incident light coordinate system and

the birefringence of each layer. In accordance with the analyses, both the apparent angle

312

distribution of directors and the birefringence of each layer play a key role in the

fluctuation of the total transmittance.

7-4. Luminance and color properties of the compensated liquid crystal devices in

their bright states

We numerically calculated and analyzed the off-axis luminous transmittance (Y )

and color properties of the bright state model with the common LCDs (ECB, VA, Pi-cell,

and TN). Their cell thicknesses are adjusted to have four different Y (%) values at the

normal direction: 100, 90, 80, and 70 %. The dark states of them are almost perfectly

compensated by using the hybrid aligned negative C-plates.

7-4-1. Luminous transmittance of the bright state LCDs

According to the results, in both the bright state simple model and the real LCDs,

the luminous transmittances (Y ) always decrease as the viewing angle increases in the

director plane. On the other hand out of the director plane, the luminous transmittances

(Y ) increase first and then fall after specific viewing angles when the cell is optically

designed so that Y (%) value is less than 100% at the normal direction. These results are

in good agreement with the birefringence analyses.

We also calculated the luminous transmittances (Y ) in all viewing directions, and

compared them by using conoscopic figures. All the figures look like similar. It confirms

again that if the dark states of LCDs are optically compensated perfectly, there is

universality in the optical properties of the bright state regardless of the display modes.

313

Another important thing is that all conoscopic figures are anisotropic shapes. This

means that the single domain LCD modes, considered here, inevitably have asymmetric

shapes of the luminance between the director and out of the director planes in their bright

states. Therefore, we can say that in order to achieve isotropic shapes of the bright state

viewing angle properties, multi-domain methods or asymmetric distribution of backlight

intensity is mandatory.

7-4-2. Color properties of the bright state LCDs

Firstly, we investigated the thickness (or phase retardation) effects on the color

properties of LCDs. These calculations show that as the thickness (or phase retardation)

increases from zero, the chromaticity coordinates ( 'u , 'v ) initially goes from blue to

yellow colors and then revolve around the point of the light source color. We analyzed

this fact by using the transmittances and the phase retardations of the three major colors.

Secondly, we calculated the off-axis chromaticity coordinates of the bright state

model and real LCDs such as ECB, VA, PI-cell, and TN modes. Overall figure shapes of

the real LCDs are in good agreement with that of our simple bright state model except the

TN mode. The absolute chromaticity coordinates at the normal direction move from blue

to yellow as Y (%) value increases in all display modes. As Y (%) value increases, the

total variation of the chromaticity coordinates increases in both main viewing planes, and

the magnitudes of the variations out of the director plane are bigger than that of the

director plane. As the viewing angle increases, the color coordinates of the bright state

314

move to the yellow direction first in the viewing angles out of the director plane, but the

color shifts to the blue direction first in the director plane.






considered here.

Finally, we investigate the viewing angle dependence of the color difference

( uvE *Δ ) in the bright state model and the real LCDs. When we compare the results, the

color differences of the real LCDs have the same curve shapes as that of the bright state

model except the TN mode. In the viewing directions out of the director plane, as the

Y (%) value increases, the off-axis color difference ( uvE *Δ ) increases, and this is coming

from the facts that the off-axis transmittances of the red, green, and blue are different

more and more as the Y (%) value increases. In the viewing angles of the director plane,

the off-axis color difference ( uvE *Δ ) increases as the Y (%) value increases, but the

variation is very small. This result is coming from the properties of the blue, green, and

red transmittances.

In the Pi-cell, out of the director plane, the color difference ( uvE *Δ ) fluctuates at



the color shift of the device better than other LCD modes at large viewing angles.

315

In the TN mode, the color difference ( uvE *Δ ) is relatively insensitive to the

Y (%) value. Especially, the color difference in the condition, Y (%)=100, is excellent

and even better than the others. This surprising feature in TN mode is related to the

Mauguin condition.

7-5. The conditions and the limitations of the perfect phase compensation in liquid

crystal displays

We compared the calculation methods of the phase difference. At the normal

direction, the approximate method and the “complete” method are the same as expected.

As the light incident angle increases, the difference between them increases, but the

magnitudes of the differences are very small.

We analyzed the parameters for the perfect phase compensation of a uniaxial slab.

To understand the relationship between the parameters, we calculated the total effective

phase retardation of a compensation system. From this, we knew that only two variables

among 'en , 'on , and 'd are independent for the perfect compensation of a given uniaxial

slab.

We found that there is a constant thickness ratio ( dd /' ) for the perfect phase

compensation of a uniaxial slab. In a given director configuration of a uniaxial slab (θ ,

φ ), we can always compensate it completely in an incident light condition (α , λ ).

However, the magnitude of dd /' varies in functions of θ , φ , and α . Therefore, it

implies that it is almost impossible to compensate an anisotropic system exactly for all

light wavelengths and viewing angles, simultaneously. If the anisotropic system is

316

composed of the layers that have different director configurations, the perfect

compensating the system becomes much more difficult. This means we need to adjust the

thickness of each sub-layer of the compensator, separately, with considering the director

configuration of each sub-layer of the uniaxial slab for better phase compensation.

The light wavelength dependence on the thickness ratio is relatively small if the

wavelength dispersion of the uniaxial slab is not so big. However, the thickness ratio is

very sensitive to the director configuration and the light incident angle. We confirmed

these facts by calculating the off-axis transmittance of the compensation systems.

As the first application, we considered a uniformly aligned liquid crystal layer

that had different tilt angles and effective phase retardations. We calculated the off-axis

phase retardation of the compensation system, which was perfectly compensated at the

normal direction. According to the results, in a given tilt angle of the liquid crystal

director, as the effective phase retardation of the liquid crystal layer increases, the

magnitude of the off-axis total effective retardation also increases. On the other hand,

with the constant effective phase retardation of the liquid crystal layer, the magnitude of

the total effective retardation increases as the tilt angle of the liquid crystal director

increases. These results mean that as the residual birefringence increases with the

constant director configuration, the dark state light leakage of a LCD increases. In a

similar way, as the tilt angle of the director increases with the constant phase retardation

of the liquid crystal layer, the off-axis light leakage rises.

Another important fact we found is that the effective phase retardation of the

liquid crystal layer competes with the director tilt angle for the best compensation of the

317

liquid crystal layer. To make clear this fact, we calculated the off-axis light

transmittances in functions of the thickness and the director tilt angle of the liquid crystal

layer. These calculations confirm that as the cell thickness decreases with a given liquid

crystal material, the light leakage of a dark state LCD decreases independent of the

director configuration. In a given cell thickness, the light leakage is the function of the

director tilt angle of the liquid crystal layer. As the tilt angle approaches both sides, θ =0º

and θ =90º, the light leakage falls in all conditions of the cell thickness.

As the second application, we considered a non-uniformly aligned Pi-cell that has

several different bright state voltages and pretilt angles. According to the results, as the

cell thickness (or effective phase retardation at the normal direction) or the director tilt

angle increase, the phase compensation of the dark state becomes harder, and this makes

the off-axis light leakage increases in the dark state. These results are consistent with the

analyses of the uniformly aligned compensation system.

318

APPENDIX A

CALCULATION OF THE 1-DIMENSIONAL LIQUID CRYSTAL DIRECTOR

CONFIGURATION BY THE VECTOR METHOD BASED ON THE

RELAXATION TECHNIQUE

A-1. Calculations

Let us consider a uniaxial liquid crystal cell that has a pair of parallel substrates,

up and down of liquid crystal material. We assume the strong anchoring at the liquid

crystal-substrate interfaces, and apply external voltage (V) across the cell (thickness of

d ) along the z -direction in Cartesian coordinates. The liquid crystal director ( nr )

responds to the applied electric field ( Er

), which is uniform in x , y -directions, and

expressed as follows:

zyx nznynxzn ˆˆˆ)( ++=r , (A-1)

Where x , y , and z are the unit vectors of the directions, x , y , and z , and xn , yn , zn

are the x , y , z components of the director ( nr ), respectively. The director ( nr ) is a

function of z , but independent of x , y -directions.

The elastic energy density or it is called the Flank-Osceen elastic energy density

( OFf . ) of this liquid crystal cell is expressed as,

319

( ) ( ) ( )233

222

211. 2

121

21 nnKqnnKnKf oOF

rrrrr×∇×++×∇•+•∇= , (A-2)

where 11K , 22K , and 33K are the elastic constants of the splay, twist, and bend

deformations of the liquid crystal director, respectively, and oq is written as,

pqo

π2= , (A-3)

where p is the natural twisted pitch of the liquid crystal material and has a positive

(negative) sign for the right (left) handed twisted structure.

The electric energy density ( ef ) of this liquid crystal cell is expressed as follows:

EDfe

rr•=

21 , (A-4)

where Dr

is the electric displacement. With in mind that the applied electric field ( Er

) has

the only z -component ( zE ), Eq. (A-4) can be written as,

])([21

2||

2

zo

ze n

Df

⊥⊥ −+=

εεεε, (A-5)

320

where oε is the electric permittivity in free space, and ||ε and ⊥ε are the parallel and

perpendicular components of the dielectric constant of the liquid crystal material,

respectively. zD is the z -component of the electric displacement vector ( Dr

) and is

constant in the liquid crystal medium from the Maxwell equation,

0=•∇ Dr

. (A-6)

Let us say the liquid crystal layer is divided into N sub layers, and the liquid crystal

molecules are aligned uniformly within each layer. Then, zD can be calculated from this

formula,

∑= ⊥⊥ −+

= N

iiz

oz

ndVN

D

12

|| )(1

1

εεε

ε, (A-7)

where izn is the z -component of the liquid crystal director ( nr ) of the i th sub layer.

The total free energy density ( f ) of this liquid crystal cell consists of the elastic

energy ( OFf . ) and the electric energy ( ef ), and can be written from Eqs. (A-1), (A-2),

and (A-5) as follows at a constant charge (Q ) condition:

2

1121

⎟⎠⎞

⎜⎝⎛=

dzdnKf z

321

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+⎟⎠

⎞⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠

⎞⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠

⎞⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛

+2

22

22

22

2

22

21

ox

yo

yxo

xyyx

xy

yx

qdz

dnnq

dzdn

nqdz

dndz

dnnn

dzdn

ndz

dnn

K

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛+⎟⎠

⎞⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠

⎞⎜⎝

⎛+2

22

22

22

233 2

21

dzdn

ndz

dndz

dnnn

dzdn

ndz

dnn

dzdn

nK yy

yxyx

xx

yz

xz

])([21

2||

2

zo

z

nD

⊥⊥ −++

εεεε. (A-8)

The total free energy ( F ) of the liquid crystal cell will be minimized at an

equilibrium state. We can achieve this equilibrium state via the dynamic equations,

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

=∂∂

−zxx

x

nf

dzd

nf

tn

,

γ , (A-9)

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

=∂

∂−

zyy

y

nf

dzd

nf

tn

,

γ , (A-10)

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂

=∂∂

−zzz

z

nf

dzd

nf

tn

,

γ , (A-11)

where γ is the rotational viscosity of the liquid crystal. If we set the time variation small

enough, Eqs. (A-9), (A-10), and (A-11) can be written as follows with Eq. (A-8):

322

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂Δ

−=Δzxx

x nf

dzd

nftn

,γ

( ) ( )⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛+++

⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟

⎠⎞

⎜⎝⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

Δ=

2

2

22332

22

332

222

33

3322

2

3322

2

3322

22

)2(2

dznd

nnKKdz

ndnKnKnK

dzdn

dzdn

nKdz

dndz

dnnK

dzdn

nKKdz

dnnK

dzdn

qK

t

yyx

xzyx

zxz

yxy

yx

xx

yo

γ, (A-12)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂Δ

−=Δzyy

y nf

dzd

nftn

,γ

( ) ( )⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟

⎟⎠

⎞⎜⎜⎝

⎛+++

⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+

⎟⎠⎞

⎜⎝⎛−−⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛−

Δ=

2

2

22332

22

332

332

22

3322

2

3322

2

3322

22

)2(2

dznd

nnKKdz

ndnKnKnK

dzdn

dzdn

nKdz

dndz

dnnK

dzdn

nKKdz

dnnK

dzdn

qK

t

xyx

yzyx

zyz

yxx

xy

yy

xo

γ, (A-13)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

−∂∂Δ

−=Δzzz

z nf

dzd

nftn

,γ

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−+

−+

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛−

Δ=

⊥⊥

⊥

22||

||2

2

2

11

2

33

2

33

])([

)(

zo

zz

zyz

xz

n

nD

dzndK

dzdn

nKdz

dnnK

t

εεεε

εεγ. (A-14)

323

From these equations, we can calculate the new state-director configuration ( Newjn ) after

time interval, tΔ , from the initial director configuration ( Oldjn ) as follows:

jOldj

Newj nnn Δ+= , (A-15)

with the boundary condition of,

( ) ( ) ( )222 Newz

Newy

Newx

NewjNew

j

nnn

nn

++= , (A-16)

where the subscript, j , represents x , y , and z . By repeating this relaxation process, we

can achieve the director configuration of each sub layer at the equilibrium state. When we

do the calculation under the constant voltage condition, we need recalculate the zD at

each relaxation process by using the given applied voltage and the director configuration

at the instant.

324

A-2. Program source codes (Matlab)

function LcAngle = LcConfigCalVectorFcn(DirectorStruct, InitAngle) %LCCONFIGCALVECTORFCN Calculate LC director configuration by using the VECTOR method %LcAngle: Director configuration return values, format: [Thickness, theta_xy, phi], micro, degree units %DirectorStruct: Structure that includes the parameters needed for the director calculation %DirectorStruct.[K11, K22, K33, EpsPar, EpsPer, Pitch, Thickness, TopRubbing, BotRubbing, TopPretilt, BotPretilt, Twist,... % DirectorTypeRadioButton, Volt, PeripheralSumDoverEps, CalculationMethodRadioButton,... % ConstantModeRadioButton, InitialDirectorRadioButton, LayerNum, MaxIterationNum, StoppingCondition] %InitAngle: Initial director angles, format: [Thickness, theta_xy, phi], micro, degree units, Same format with the "LcAngle" % if InitAngle==[]: Use program default values as an Initial angles to calculate director configuration %theta: LC director polar angle from the Z-axis in RADIAN unit (0 deg.: Homeotropic, 90deg.:In plane Alignment) DURING THE CALCULATION %theta_xy_deg : LC director Tilt angle from the X-Y plane in DEGREE unit (0 deg.: In plane, 90deg.:Homeotropic Alignment) AFTER CALCULATION <= RETURN VALUE %phi : LC azimuthal angle from the X-axis in RADIAN unit <= DURING THE CALCULATION %phi_deg : LC azimuthal angle from the X-axis in DEGREE unit <= RETURN VALUE %Units: MKSA %December 22, 2005 %Yong-Kyu Jang (Liquid Crystal Institute, Kent State University) %Common Parameters eps_0 = 8.8542 * 1e-12; %Permittivity in vacuum (MKSA unit) %LC Parameters K11 = DirectorStruct.K11 * 1e-12; %Splay elastic constant(N) K22 = DirectorStruct.K22 * 1e-12; %Twist elastic constant(N) K33 = DirectorStruct.K33 * 1e-12; %Bend elastic constant(N) eps_par = DirectorStruct.EpsPar; %Dielectric constant (Parallel component to director) eps_per = DirectorStruct.EpsPer; %Dielectric constant (Perpendicular component to director) %Cell Parameters d = DirectorStruct.Thickness * 1e-6; %Cell thickness(m), Positive value (>0) if isempty(DirectorStruct.Pitch) %Infinite Pitch (No Chirality) d_over_p = 0; else

325

d_over_p = d/(DirectorStruct.Pitch * 1e-6); %d/p ("+" sign: Right handed twist, "-" sign: Left handed twist) end top_pretilt_angle = DirectorStruct.TopPretilt * pi/180; %LC pretilt angle from the X-Y plane, Positive value (0 deg.: in plane) bot_pretilt_angle = DirectorStruct.BotPretilt * pi/180;%LC pretilt angle from the X-Y plane, Positive value (0 deg.: in plane) top_rub_angle = DirectorStruct.TopRubbing * pi/180; %Top glass(C/F) rubbing angle from X-axis in assembled LCD bot_rub_angle = DirectorStruct.BotRubbing * pi/180; %Bottom glass(TFT) rubbing angle from X-axis in assembled LCD twist_angle = DirectorStruct.Twist * pi/180; %LC total twist angle in MANUAL mode (+:Right handed twist, -: Left handed twist) V = DirectorStruct.Volt; %LC Voltage peripheral_d_eps = DirectorStruct.PeripheralSumDoverEps;%Sum of the Thickness over Dielectric constant(d/eps) of the adjacent layers of LC that cause LC Voltage drop %Simulation Parameters N = DirectorStruct.LayerNum; %Number of the LC sub layers if DirectorStruct.InitialDirectorRadioButton == 1 twist_mode = 0; %0: AUTO mode to find initial twist angle else if DirectorStruct.DirectorTypeRadioButton == 1 twist_mode = 1; %1: SPLAY/TWIST structure MANUAL mode (Use given twist angle) else twist_mode = 2; %2(else): BEND structure MANUAL mode (Use given twist angle) end end constant_mode = DirectorStruct.ConstantModeRadioButton; %1: Constant VOLTAGE (V) mode, 2(else): Constant CHARGE (Q) mode creep_up = 1.02; %1.02 max_ratio = 1.04; %1.02; max_iteration_num = DirectorStruct.MaxIterationNum; %If iteration number (m) is bigger than this value, relaxation process stop avg_delta_f_limit = DirectorStruct.StoppingCondition; %10^-10 ~ 10^-7 %If average-delta-free energy is less than this value, relaxation process stop %LC director-initialization Parameters theta_0 = 5 * pi/180; %Relaxation starting value of the theta(from X-Y plane) in the mid layer(z=d/2) when pretilt angle = 0 deg %Parameter recalculation & initialziation delta_eps = eps_par - eps_per; %Difference of the dielectric constants delta_d = d/N; %Thickness of the sub layers real_volt = V; %Used to optimize delta_n_limit phi_nat = 2*pi*d_over_p; %Natural twist angle due to the chiral dopant

326

top_pretilt_angle = abs(top_pretilt_angle); %LC pretilt angle from the X-Y plane, Positive value (0 deg.: in plane) bot_pretilt_angle = abs(bot_pretilt_angle); %LC pretilt angle from the X-Y plane, Positive value (0 deg.: in plane) top_rub_angle = rem(top_rub_angle, 2*pi); %Rubbing angle rearrangement: -2*pi < top_rub_angle < +2*pi bot_rub_angle = rem(bot_rub_angle, 2*pi); %Rubbing angle rearrangement: -2*pi < bot_rub_angle < +2*pi q = 2 * pi * d_over_p / d; %2¥ð/P <- Natural pitch, "+" sign: Right handed twist, "-" sign: Left handed twist z = linspace(0, d, N); %Position of the cell, <- Array n = 1 : N; %Array of the LC sub layers d_theta_d_z = zeros(1, N); %1'st derivative of theta to z (d¥È/dz) d2_theta_d_z2 = zeros(1, N); %2'nd derivative of theta to z (d2¥È/dz2) d_phi_d_z = zeros(1, N); %1'st derivative of phi to z (d¥Õ/dz) d2_phi_d_z2 = zeros(1, N); %2'nd derivative of phi to z (d2¥Õ/dz2) dnxdz = zeros(1, N); %1'st derivative of nx to z (dnx/dz) dnydz = zeros(1, N); %1'st derivative of ny to z (dny/dz) dnzdz = zeros(1, N); %1'st derivative of nz to z (dnz/dz) d2nxdz2 = zeros(1, N); %2'st derivative of nx to z (d2nx/dz2) d2nydz2 = zeros(1, N); %2'st derivative of ny to z (d2ny/dz2) d2nzdz2 = zeros(1, N); %2'st derivative of nz to z (d2nz/dz2) delta_t_over_gamma = 0.001; %¥Ät/¥ã <- Determine the period of the relaxation Dz_temp = eps_0 * V; %Temporary Dz real_V_temp = d / eps_0 / N; %Temporary real voltage theta_init = zeros(6, N); %LC director's polar angle from the z-axis phi_init = zeros(6, N); %LC director's azimuthal angle from the X-axis F = zeros(1, 6); %Total free energy per unit area in the LCD cell (J) f_old = zeros(1, N); termination_flag = 0; %0: Not ready to stop relaxation, 1: ready to stop relaxation (avg_delta_f < avg_delta_f_limit) avg_delta_nx_old = 1e10; %Used for stability check avg_delta_ny_old = 1e10; %Used for stability check avg_delta_nz_old = 1e10; %Used for stability check max_delta_nx_old = 1e10; %Used for stability check max_delta_ny_old = 1e10; %Used for stability check max_delta_nz_old = 1e10; %Used for stability check edge_limit_m = 0; %Used to check stopping point old_edge_limit_m = 0; %Used to check stopping point edge_limit_num = 0; %Used to check stopping point if isempty(InitAngle) %if InitAngle==[]: Use program default values as an Initial angles to calculate director configuration %*************************************************************************% %Calculate the top rubbing angle in the coordinate system where the top sub layer's director angle is zero.

327

temp_top_rub_angle = rem((top_rub_angle - (bot_rub_angle + phi_nat)), 2*pi); if temp_top_rub_angle < 0 temp_top_rub_angle = 2*pi + temp_top_rub_angle; end %Calculate the initial twist angle, theta and phi, theta_init: LC polar angle from the Z-axis if twist_mode == 0 %AUTO mode to calculate initial twist angle if (0 <= temp_top_rub_angle) & (temp_top_rub_angle < pi) %Right handed twist(¥Õ>=0), Head to Head (top_pretilt_angle < 0) SPLAY twist_angle = phi_nat + temp_top_rub_angle; theta_init(1, n) = pi/2 - (bot_pretilt_angle + (-top_pretilt_angle - bot_pretilt_angle) * (n-1)/(N-1)); %Used to calculate elastic free energy at initial state phi_init(1, n) = bot_rub_angle + twist_angle * (n-1)/(N-1); %Used to calculate free energy and initial director polar angles %Right handed twist(¥Õ>0), Head to Tail (top_pretilt_angle > 0) HOMOGENEOUS twist_angle = phi_nat + temp_top_rub_angle + pi; theta_init(2, n) = pi/2 - (bot_pretilt_angle + (top_pretilt_angle - bot_pretilt_angle) * (n-1)/(N-1)); phi_init(2, n) = bot_rub_angle + twist_angle * (n-1)/(N-1); %Left handed twist(¥Õ<0), Head to Tail (top_pretilt_angle > 0) HOMOGENEOUS twist_angle = phi_nat + temp_top_rub_angle - pi; theta_init(3, n) = pi/2 - (bot_pretilt_angle + (top_pretilt_angle - bot_pretilt_angle) * (n-1)/(N-1)); phi_init(3, n) = bot_rub_angle + twist_angle * (n-1)/(N-1); %Left handed twist(¥Õ<0), Head to Head (top_pretilt_angle < 0) SPLAY twist_angle = phi_nat + temp_top_rub_angle - 2*pi; theta_init(4, n) = pi/2 - (bot_pretilt_angle + (-top_pretilt_angle - bot_pretilt_angle) * (n-1)/(N-1)); phi_init(4, n) = bot_rub_angle + twist_angle * (n-1)/(N-1); %Right handed twist(¥Õ>=0), Head to Head (top_pretilt_angle > 0) BEND twist_angle = phi_nat + temp_top_rub_angle; theta_init(5, n) = pi/2 - (bot_pretilt_angle + (pi - top_pretilt_angle - bot_pretilt_angle) * (n-1)/(N-1)); phi_init(5, n) = bot_rub_angle + twist_angle * (n-1)/(N-1); %Left handed twist(¥Õ<0), Head to Head (top_pretilt_angle > 0) BEND twist_angle = phi_nat + temp_top_rub_angle - 2*pi; theta_init(6, n) = pi/2 - (bot_pretilt_angle + (pi - top_pretilt_angle - bot_pretilt_angle) * (n-1)/(N-1)); phi_init(6, n) = bot_rub_angle + twist_angle * (n-1)/(N-1);

328

elseif (pi <= temp_top_rub_angle) & (temp_top_rub_angle < 2*pi) %Right handed twist(¥Õ>0), Head to Head (top_pretilt_angle < 0) SPLAY twist_angle = phi_nat + temp_top_rub_angle; theta_init(1, n) = pi/2 - (bot_pretilt_angle + (-top_pretilt_angle - bot_pretilt_angle) * (n-1)/(N-1)); phi_init(1, n) = bot_rub_angle + twist_angle * (n-1)/(N-1); %Right handed twist(¥Õ>=0), Head to Tail (top_pretilt_angle > 0) HOMOGENEOUS twist_angle = phi_nat + temp_top_rub_angle - pi; theta_init(2, n) = pi/2 - (bot_pretilt_angle + (top_pretilt_angle - bot_pretilt_angle) * (n-1)/(N-1)); phi_init(2, n) = bot_rub_angle + twist_angle * (n-1)/(N-1); %Left handed twist(¥Õ<0), Head to Tail (top_pretilt_angle > 0) HOMOGENEOUS twist_angle = phi_nat + temp_top_rub_angle - 3*pi; theta_init(3, n) = pi/2 - (bot_pretilt_angle + (top_pretilt_angle - bot_pretilt_angle) * (n-1)/(N-1)); phi_init(3, n) = bot_rub_angle + twist_angle * (n-1)/(N-1); %Left handed twist(¥Õ<0), Head to Head (top_pretilt_angle < 0) SPLAY twist_angle = phi_nat + temp_top_rub_angle - 2*pi; theta_init(4, n) = pi/2 - (bot_pretilt_angle + (-top_pretilt_angle - bot_pretilt_angle) * (n-1)/(N-1)); phi_init(4, n) = bot_rub_angle + twist_angle * (n-1)/(N-1); %Right handed twist(¥Õ>0), Head to Head (top_pretilt_angle > 0) BEND twist_angle = phi_nat + temp_top_rub_angle; theta_init(5, n) = pi/2 - (bot_pretilt_angle + (pi - top_pretilt_angle - bot_pretilt_angle) * (n-1)/(N-1)); phi_init(5, n) = bot_rub_angle + twist_angle * (n-1)/(N-1); %Left handed twist(¥Õ<0), Head to Head (top_pretilt_angle > 0) BEND twist_angle = phi_nat + temp_top_rub_angle - 2*pi; theta_init(6, n) = pi/2 - (bot_pretilt_angle + (pi - top_pretilt_angle - bot_pretilt_angle) * (n-1)/(N-1)); phi_init(6, n) = bot_rub_angle + twist_angle * (n-1)/(N-1); else beep errordlg('Input rubbing angles ERROR in auto mode', 'Rubbing Angles Error', 'on') LcAngle = []; return end

329

%Calculate elastic free energy to find optimum initial director configuration for p = 1 : 6 d_theta_d_z(2:N-1) = (theta_init(p, 3:N) - theta_init(p, 1:N-2)) / (2 * delta_d); %1'st derivative of the theta d2_theta_d_z2(2:N-1) = (theta_init(p, 3:N) + theta_init(p, 1:N-2) - 2*theta_init(p, 2:N-1)) / (delta_d)^2; %2nd derivative of the theta d_phi_d_z(2:N-1) = (phi_init(p, 3:N) - phi_init(p, 1:N-2)) / (2 * delta_d); %1'st derivative of the phi d2_phi_d_z2(2:N-1) = (phi_init(p, 3:N) + phi_init(p, 1:N-2) - 2*phi_init(p, 2:N-1)) / (delta_d)^2; %2nd derivative of the phi s_t2 = sin(theta_init(p, :)).^2; c_t2 = cos(theta_init(p, :)).^2; %Calculation of the Average free energy density in a layer f = 1/2 * (K11 * s_t2 .* d_theta_d_z .^2 + K22 * (s_t2 .^2 .* d_phi_d_z .^2 - 2 * q * s_t2 .* d_phi_d_z + q^2)... + K33 * (c_t2 .* d_theta_d_z .^2 + c_t2 .* s_t2 .* d_phi_d_z .^2)); %Array of the free energy density in each layers (J/m3) F(p) = sum(f) * delta_d; %Total free energy per unit area in the LCD cell (J/m2) end %Find minimum free energy and initialize the theta & phi by using the minimum free energy condition [min_F, min_index] = min(F); % Find minimum free energy and it's index if (min_index == 1) | (min_index == 4) %Head to Head, SPLAY directer structure theta = pi/2 - (theta_0 * sin(2*pi*(n-1)/(N-1)) + bot_pretilt_angle + (-top_pretilt_angle - bot_pretilt_angle)*(n-1)/(N-1)); %Polar angle of the LC director from the Z-axis elseif (min_index == 2) | (min_index == 3) %Head to Tail, HOMOGENEOUS directer structure theta = pi/2 - (theta_0 * sin(pi*(n-1)/(N-1)) + bot_pretilt_angle + (top_pretilt_angle - bot_pretilt_angle)*(n-1)/(N-1)); %Polar angle of the LC director from the Z-axis else %Head to Head, BEND directer structure theta = theta_init(min_index, :); end phi = phi_init(min_index, :); %Azimuthal angle of the LC director clear theta_init phi_init F %*************************************************************************%

330

elseif twist_mode == 1 %SPLAY or TWIST structure MANUAL mode %Calculate the top rubbing angle in the coordinate system where the bottom rubbing angle is zero. temp_top_rub_angle = rem((top_rub_angle - bot_rub_angle), 2*pi); if temp_top_rub_angle < 0 temp_top_rub_angle = 2*pi + temp_top_rub_angle; end rem_twist_angle = rem(twist_angle, 2*pi); if abs(rem_twist_angle - temp_top_rub_angle)<1e-6 | abs(rem_twist_angle - (temp_top_rub_angle - 2*pi))<1e-6 theta = pi/2 - (theta_0 * sin(2*pi*(n-1)/(N-1)) + bot_pretilt_angle + (-top_pretilt_angle - bot_pretilt_angle)*(n-1)/(N-1)); elseif abs(rem_twist_angle - (temp_top_rub_angle + pi))<1e-6 | abs(rem_twist_angle - (temp_top_rub_angle - pi))<1e-6 | abs(rem_twist_angle - (temp_top_rub_angle - 3*pi))<1e-6 theta = pi/2 - (theta_0 * sin(pi*(n-1)/(N-1)) + bot_pretilt_angle + (top_pretilt_angle - bot_pretilt_angle)*(n-1)/(N-1)); else beep errordlg('Input rubbing angles ERROR in manual mode', 'Rubbing Angles Error', 'on') LcAngle = []; return end phi = bot_rub_angle + twist_angle * (n-1)/(N-1); %*************************************************************************% else %BEND structure MANUAL mode %Calculate the top rubbing angle in the coordinate system where the bottom rubbing angle is zero. temp_top_rub_angle = rem((top_rub_angle - bot_rub_angle), 2*pi); if temp_top_rub_angle < 0 temp_top_rub_angle = 2*pi + temp_top_rub_angle; end rem_twist_angle = rem(twist_angle, 2*pi); if abs(rem_twist_angle - temp_top_rub_angle)<1e-6 | abs(rem_twist_angle - (temp_top_rub_angle - 2*pi))<1e-6 theta = pi/2 - (bot_pretilt_angle + (pi - top_pretilt_angle - bot_pretilt_angle) * (n-1)/(N-1)); else beep errordlg('Input rubbing angles ERROR in manual mode', 'Rubbing Angles Error', 'on') LcAngle = []; return end phi = bot_rub_angle + twist_angle * (n-1)/(N-1); end %End of the initialization

331

%=========================================================================% else %if InitAngle ~= []: Use given values as an Initial angles to calculate director configuration theta = ((90 - InitAngle(:, 2)) * pi/180)'; phi = (InitAngle(:, 3) * pi/180)'; end %Change the initial conditions from theta, phi to nx, ny, nz format nx = sin(theta) .* cos(phi); ny = sin(theta) .* sin(phi); nz = cos(theta); %Used for strong anchoring assumption during the relaxation nx_bot = nx(1); ny_bot = ny(1); nz_bot = nz(1); nx_top = nx(N); ny_top = ny(N); nz_top = nz(N); %=========================================================================% %Start of the relaxation loop m = 1; %Number of the iterations while(1) if m == 1 %Calculation of the Dz eps_n = eps_per + delta_eps .* nz .^2; %z component of the epsilon (epsilon_zz) at each layer, <- Array Dz = Dz_temp / (peripheral_d_eps + d/N*sum(1 ./ eps_n)); %z component of the displacement vector D %Calculation of the nx, ny, nz derivatives dnxdz(2:N-1) = (nx(3:N) - nx(1:N-2)) / (2 * delta_d); %1'st derivative of the nx dnydz(2:N-1) = (ny(3:N) - ny(1:N-2)) / (2 * delta_d); %1'st derivative of the ny dnzdz(2:N-1) = (nz(3:N) - nz(1:N-2)) / (2 * delta_d); %1'st derivative of the nz d2nxdz2(2:N-1) = (nx(3:N) + nx(1:N-2) - 2*nx(2:N-1)) / (delta_d)^2; %2nd derivative of the nx d2nydz2(2:N-1) = (ny(3:N) + ny(1:N-2) - 2*ny(2:N-1)) / (delta_d)^2; %2nd derivative of the ny d2nzdz2(2:N-1) = (nz(3:N) + nz(1:N-2) - 2*nz(2:N-1)) / (delta_d)^2; %2nd derivative of the nz end %Calculation of the delta_nx, delta_ny and delta_nz

332

delta_nx = delta_t_over_gamma.*(2.*K22.*q.*dnydz+K33.*nx.*dnxdz.^2-(2.*K22-K33).*nx.*dnydz.^2+2.*K22.*ny.*dnxdz.*dnydz+2.*K33.*nz.*dnxdz.*dnzdz+(K33.*nx.^2+K22.*ny.^2+K33.*nz.^2).*d2nxdz2+(K33-K22).*nx.*ny.*d2nydz2); delta_ny = delta_t_over_gamma.*(-2.*K22.*q.*dnxdz+K33.*ny.*dnydz.^2-(2.*K22-K33).*ny.*dnxdz.^2+2.*K22.*nx.*dnxdz.*dnydz+2.*K33.*nz.*dnydz.*dnzdz+(K22.*nx.^2+K33.*ny.^2+K33.*nz.^2).*d2nydz2+(K33-K22).*nx.*ny.*d2nxdz2); delta_nz = delta_t_over_gamma.*(-K33.*nz.*dnxdz.^2-K33.*nz.*dnydz.^2+K11.*d2nzdz2+Dz.^2.*delta_eps.*nz./eps_0./(eps_per+delta_eps.*nz.^2).^2); %Calculation of the real LC-Voltage at Constant Charge mode if constant_mode == 2 real_volt = real_V_temp * Dz * sum(1 ./ eps_n); end %Definition of the limit values if real_volt < 1.5 avg_delta_n_limit = 0.001; max_delta_n_limit = 0.002; elseif real_volt >= 1.5 & real_volt < 2 avg_delta_n_limit = 0.002; max_delta_n_limit = 0.003; elseif real_volt >= 2 & real_volt < 3 avg_delta_n_limit = 0.002; max_delta_n_limit = 0.004; else avg_delta_n_limit = 0.005; max_delta_n_limit = 0.01; end %Stability check of the nx, ny, nz %If the maximum delta_nx, delta_ny, delta_nz are larger than the limit, then, reduce the delta_t/¥ã to the 1/2 of the current value. %Max delta_nx and ratio stability check max_delta_nx = max(abs(delta_nx)); max_delta_nx_ratio = max_delta_nx/max_delta_nx_old; while((max_delta_nx > max_delta_n_limit) | (max_delta_nx_ratio > max_ratio)) delta_t_over_gamma = delta_t_over_gamma /2; delta_nx = delta_nx /2; delta_ny = delta_ny /2; delta_nz = delta_nz /2; max_delta_nx = max(abs(delta_nx)); max_delta_nx_ratio = max_delta_nx/max_delta_nx_old; edge_limit_m = m; end max_delta_nx_old = max_delta_nx + (max_delta_nx < eps) * eps; %Max delta_ny and ratio stability check max_delta_ny = max(abs(delta_ny));

333

max_delta_ny_ratio = max_delta_ny/max_delta_ny_old; while((max_delta_ny > max_delta_n_limit) | (max_delta_ny_ratio > max_ratio)) delta_t_over_gamma = delta_t_over_gamma /2; delta_nx = delta_nx /2; delta_ny = delta_ny /2; delta_nz = delta_nz /2; max_delta_ny = max(abs(delta_ny)); max_delta_ny_ratio = max_delta_ny/max_delta_ny_old; edge_limit_m = m; end max_delta_ny_old = max_delta_ny + (max_delta_ny < eps) * eps; %Max delta_nz and ratio stability check max_delta_nz = max(abs(delta_nz)); max_delta_nz_ratio = max_delta_nz/max_delta_nz_old; while((max_delta_nz > max_delta_n_limit) | (max_delta_nz_ratio > max_ratio)) delta_t_over_gamma = delta_t_over_gamma /2; delta_nx = delta_nx /2; delta_ny = delta_ny /2; delta_nz = delta_nz /2; max_delta_nz = max(abs(delta_nz)); max_delta_nz_ratio = max_delta_nz/max_delta_nz_old; edge_limit_m = m; end max_delta_nz_old = max_delta_nz + (max_delta_nz < eps) * eps; %If the average value of the delta_nx, delta_ny, delta_nz are larger than the limit, then, reduce the delta_t/¥ã to the 1/3 of the current value. %Average delta_nx and ratio stability check avg_delta_nx = mean(abs(delta_nx)); %Average value of the delta_nx, sum |new nx - old nx| / N avg_delta_nx_ratio = avg_delta_nx/avg_delta_nx_old; while((avg_delta_nx > avg_delta_n_limit) | (avg_delta_nx_ratio > max_ratio)) delta_t_over_gamma = delta_t_over_gamma /3; delta_nx = delta_nx /3; delta_ny = delta_ny /3; delta_nz = delta_nz /3; avg_delta_nx = mean(abs(delta_nx)); avg_delta_nx_ratio = avg_delta_nx/avg_delta_nx_old; edge_limit_m = m; end avg_delta_nx_old = avg_delta_nx + (avg_delta_nx < eps) * eps; %Average delta_ny and ratio stability check avg_delta_ny = mean(abs(delta_ny)); avg_delta_ny_ratio = avg_delta_ny/avg_delta_ny_old; while((avg_delta_ny > avg_delta_n_limit) | (avg_delta_ny_ratio > max_ratio)) delta_t_over_gamma = delta_t_over_gamma /3;

334

delta_nx = delta_nx /3; delta_ny = delta_ny /3; delta_nz = delta_nz /3; avg_delta_ny = mean(abs(delta_ny)); avg_delta_ny_ratio = avg_delta_ny/avg_delta_ny_old; edge_limit_m = m; end avg_delta_ny_old = avg_delta_ny + (avg_delta_ny < eps) * eps; %Average delta_nz and ratio stability check avg_delta_nz = mean(abs(delta_nz)); avg_delta_nz_ratio = avg_delta_nz/avg_delta_nz_old; while((avg_delta_nz > avg_delta_n_limit) | (avg_delta_nz_ratio > max_ratio)) delta_t_over_gamma = delta_t_over_gamma /3; delta_nx = delta_nx /3; delta_ny = delta_ny /3; delta_nz = delta_nz /3; avg_delta_nz = mean(abs(delta_nz)); avg_delta_nz_ratio = avg_delta_nz/avg_delta_nz_old; edge_limit_m = m; end avg_delta_nz_old = avg_delta_nz + (avg_delta_nz < eps) * eps; %Calculation & update of the new nx, ny, nz values nx = nx + delta_nx; ny = ny + delta_ny; nz = nz + delta_nz; %Assume strong anchoring nx(1) = nx_bot; ny(1) = ny_bot; nz(1) = nz_bot; nx(N) = nx_top; ny(N) = ny_top; nz(N) = nz_top; %Normalization of the LC director sqrt_n = sqrt(nx.^2 + ny.^2 + nz.^2); nx = nx ./ sqrt_n; ny = ny ./ sqrt_n; nz = nz ./ sqrt_n; %Creep up of the delta_t/gamma delta_t_over_gamma = delta_t_over_gamma * creep_up; %1.01 %*********************************************************************% %Calculation of the Average free energy density in a layer %Calculation of the Dz eps_n = eps_per + delta_eps .* nz .^2; %z component of the epsilon (epsilon_zz) at each layer, <- Array

335

if constant_mode == 1 %1: Constant VOLTAGE (V) mode, 2(else): Constant CHARGE (Q) mode Dz = Dz_temp / (peripheral_d_eps + d/N*sum(1 ./ eps_n)); %z component of the displacement vector D end %Calculation of the nx, ny, nz derivatives dnxdz(2:N-1) = (nx(3:N) - nx(1:N-2)) / (2 * delta_d); %1'st derivative of the nx dnydz(2:N-1) = (ny(3:N) - ny(1:N-2)) / (2 * delta_d); %1'st derivative of the ny dnzdz(2:N-1) = (nz(3:N) - nz(1:N-2)) / (2 * delta_d); %1'st derivative of the nz d2nxdz2(2:N-1) = (nx(3:N) + nx(1:N-2) - 2*nx(2:N-1)) / (delta_d)^2; %2nd derivative of the nx d2nydz2(2:N-1) = (ny(3:N) + ny(1:N-2) - 2*ny(2:N-1)) / (delta_d)^2; %2nd derivative of the ny d2nzdz2(2:N-1) = (nz(3:N) + nz(1:N-2) - 2*nz(2:N-1)) / (delta_d)^2; %2nd derivative of the nz %Calculation of the free energy density f = 1./2.*K11.*dnzdz.^2+1./2.*K22.*(nx.^2.*dnydz.^2+ny.^2.*dnxdz.^2-2.*nx.*ny.*dnydz.*dnxdz-2.*q.*nx.*dnydz+2.*q.*ny.*dnxdz+q.^2)+1./2.*K33.*(nz.^2.*dnxdz.^2+nz.^2.*dnydz.^2+nx.^2.*dnxdz.^2+2.*nx.*ny.*dnydz.*dnxdz+ny.^2.*dnydz.^2)+1./2.*Dz.^2./eps_0./(eps_per+delta_eps.*nz.^2); avg_delta_f = mean(abs(f - f_old)); %Average change of the free energy density between the relaxations f_old = f; %Store the current free energy density to the old one %Termination control if avg_delta_f < avg_delta_f_limit %If average-delta-free energy is smaller than avg_delta_f_limit, it's ready to stop termination_flag = 1; end if (termination_flag == 1) & (old_edge_limit_m ~= edge_limit_m) edge_limit_num = edge_limit_num + 1; old_edge_limit_m = edge_limit_m; end if ((termination_flag == 1) & (m == edge_limit_m + 10)) | edge_limit_num >= 10 break end if m >= max_iteration_num %If iteration number (m) is bigger than max_iteration_num, relaxation process stop str = sprintf('Maximum iteration number (%d) OVER!!!', max_iteration_num); h1 = warndlg(str,'Iteration number warning!');

336

%set(h1, 'WindowStyle', 'modal') break end m = m + 1; end %End of the relaxation (while loop) %=========================================================================% %Change the output nx, ny, nz to the theta, phi format theta = acos(nz); phi = acos(nx ./ (sqrt(nx.^2 + ny.^2) + (sqrt(nx.^2 + ny.^2) < eps) * eps)); for p = 1 : N if ny(p) < 0 phi(p) = -phi(p); end end if twist_mode == 2 for p = 1 : N if nx(p) < 0 theta(p) = -theta(p); phi(p) = phi(p) - pi; end end end %Change the definition of the LC theta(theta) theta_xy_deg = (pi/2 - theta) * 180/pi; %Redefine theta again, Angle from X-Y plane in degree unit phi_deg = phi * 180/pi; LcAngle(:, 1) = z * 1e6; LcAngle(:, 2) = theta_xy_deg; LcAngle(:, 3) = phi_deg;

337

APPENDIX B

CALCULATION OF THE OPTICS IN LIQUID CRYSTAL DEVICES BY

EXTENDED 2×2 JONES MATRIX61, 62, 63

B-1. Calculations

Let us consider a uniaxial liquid crystal cell described in Appendix A with a pair

of polarizers; a plane wave is incident on the LCD at an oblique angle of kθ from the

normal direction; and the light propagation vector ( Kr

) lies on the zx − plane. That is,

koko kzkxK θθ cosˆsinˆ +=r

, (B-1)

where λπω /2/ == cko , and λ is the light wavelength in free space. The LCD can be

divided into N layers. The first ( 1=n ) and the last ( Nn = ) layers are the entrance and

exit polarizers, respectively, the second ( 2=n ) and the ( 1−N )th layers are the substrates,

and the liquid crystal layers ( 2~3 −= Nn ) are located between them. Each of these N

layers is characterized by a dielectric constant tensor:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=

zzzyzx

yzyyyx

xzxyxx

εεεεεεεεε

ε , (B-2)

338

where each element of the dielectric constant tensor for the uniaxial material can be

calculated from

jiijij nn)( || ⊥⊥ −+= εεδεε . (B-3)

Let us assume the plane wave in each layer, which is a solution of the Maxwell equation,

as follows:

)](exp[)](exp[),( tzkxkiEtrKiEtrE zxoo ωω −+=−•=rrrrrr

, (B-4)

)](exp[)](exp[),( tzkxkiHtrKiHtrH zxoo ωω −+=−•=rrrrrr

. (B-5)

From Eqs. (B-4), (B-5) and the Maxwell equation, we have the wave equation as below:

0=+⎟⎟⎠

⎞⎜⎜⎝

⎛×× oo

oo

EEkK

kK rr

rr

ε , (B-6)

where oEr

is expressed as,

zoyoxoo EzEyExE ˆˆˆ ++=r

. (B-7)

339

From Eqs. (B-1), (B-6), and (B-7), we obtain these three equations,

022

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛+++⎟

⎟⎠

⎞⎜⎜⎝

⎛+− zoxz

o

zxyoxyxoxx

o

z Ek

kkEE

kk

εεε , (B-8)

02

2

2

2

=+⎟⎟⎠

⎞⎜⎜⎝

⎛+−−+ zoyzyoyy

o

z

o

xxoyx EE

kk

kk

E εεε , (B-9)

02

2

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛+−++⎟

⎟⎠

⎞⎜⎜⎝

⎛+ zozz

o

xyozyxozx

o

zx Ekk

EEk

kkεεε . (B-10)

In these equations, there is a nontrivial solution only when

0

2

2

2

2

2

2

2

22

2

=

+−+

+−−

++−

zzo

xzyzx

o

zx

yzyyo

z

o

xyx

xzo

zxxyxx

o

z

kk

kkk

kk

kk

kkk

kk

εεε

εεε

εεε

. (B-11)

From this equation, we obtain four eigen values for a uniaxial material as follows:

21

2

221

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

o

xo

o

z

kk

nkk

, (B-12)

340

21

2

222

2

222 sincos1

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−−+−=

o

x

e

oezz

zz

eo

o

x

zz

xz

o

z

kk

nnnnn

kk

kk

φθεεε

ε, (B-13)

o

z

o

z

kk

kk 13 −= , (B-14)

o

z

o

z

kk

kk 24 −= , (B-15)

where θ and φ are the director tilt and azimuth angles, and en , on are the refractive

indices of the extraordinary, ordinary rays of the uniaxial material, respectively.

Equations (B-12) and (B-13) are the eigen values corresponding to the transmission eigen

waves, and (B-14) and (B-15) are those of reflection light. In this dissertation, we assume

only the transmitted eigen waves are propagating in the medium.

In Eqs. (B-8), (B-9), and (B-10), only two of them are linearly independent

because there is a constraint of Eq. (B-11). Therefore, we can express two components of

the electric field in terms of the other component. For eigen wave 1, we express 1yE in

functions of 1xE . From Eqs. (B-9) and (B-10), we obtain these equations,

111

111 xy

x

xyy Me

eE

eE == , (B-16)

where

341

zyyzzzo

xyy

o

x

o

zx k

kkk

kk

e εεεε −⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛−+= 2

2

2

2

2

21

1 , (B-17)

⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟

⎟⎠

⎞⎜⎜⎝

⎛−= zx

o

zxyzzz

o

xyxy k

kkkk

e εεεε 21

2

2

1 , (B-18)

)](exp[ 11

10

1

11 tzkxki

eE

eE

M zxx

x

x

xx ω−+== . (B-19)

With the same way, we obtain these equations for the eigen wave 2 from Eqs. (B-8) and

(B-10) as follows:

222

222 yx

y

yxx Me

eE

eE == , (B-20)

where

⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟

⎟⎠

⎞⎜⎜⎝

⎛+−= xz

o

zxzyzz

o

xxyx k

kkkk

e εεεε 22

2

2

2 , (B-21)

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛++⎟

⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛+−= xz

o

zxzx

o

zxzz

o

xxx

o

zy k

kkkkk

kk

kk

e εεεε 22

22

2

2

2

22

2 , (B-22)

)](exp[ 22

20

2

22 tzkxki

eE

eE

M zxy

y

y

yy ω−+== . (B-23)

342

Using equations (B16~23), the E -vector represented by the tangential components of the

total electric field at any point can be expressed in terms of mode vector as follows:

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

2

2

1

1

y

x

y

x

y

x

EE

EE

EE

⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

1

y

x

MM

S , (B-24)

where

⎟⎟⎠

⎞⎜⎜⎝

⎛=

21

21

yy

xx

eeee

S . (B-25)

Using Eqs. (B-19) and (B-23), the mode vector propagates from the bottom of n th layer

to the top of the layer by

0,2

1

,2

1

ny

xn

dnny

x

MM

GMM

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛, (B-26)

where the propagation matrix ( nG ) is expressed as

nz

zn dik

dikG ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

)exp(00)exp(

2

1 . (B-27)

343

From Eqs. (B-24) and (B-26), the E -vector at the top of the n th layer is expressed by the

E -vector at the bottom of the n th layer as,

0,, ny

xn

dnny

x

EE

JEE

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛, (B-28)

where

1−= nnnn SGSJ . (B-29)

With the boundary condition that the tangential components of the electric field are

continuous at each layer interface, we obtain the E -vector relationship between the input

light and the output light of the LCD system composed of N layers as follows:

Iny

x

Outy

x

EE

JEE

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛, (B-30)

where the extended Jones matrix, J is expressed as

121 ...... JJJJJJ nNN −= . (B-31)

344

Until now, we have not considered reflections, especially at surfaces. However, the

reflections at the air interfaces are significant. To take into account the reflections, we

rewrite the extended Jones matrix as follows:

EntnNNExtEntExt JJJJJJJJJJJ 121 ......' −== , (B-32)

where the reflection matrices, ExtJ and EntJ can be calculated from

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+

+=

ppk

k

pkp

p

Ent

n

nJ

θθθ

θθθ

coscoscos2

0

0coscos

cos2

, (B-33)

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

+

+=

kpp

pp

kpp

kp

Ext

nn

nn

J

θθθ

θθθ

coscoscos2

0

0coscos

cos2

, (B-34)

where pn and pθ are the average refractive index and refraction angle in the polarizer (or

outer most layer), respectively.

The total light transmittance (T ) of the LCD system is calculated by

222

222

cos

cosinyk

inx

outyk

outx

EE

EET

θ

θ

+

+= . (B-35)

345

B-2. Program source codes (Matlab)

function T = OpticsCal2by2Fcn(OpticsStruct) %OPTICSCAL2BY2FCN Calculation of the Transmittance by using the EXTENDED JONES MATRIX (2X2 MATRIX) (Liquid Crystals. 1997. Vol. 22. No. 2. 171-175) %T: Transmittance return value %OpticsStruct: Structure that includes the parameters for the Optics calculation %OpticsStruct.[nx, ny, nz, d, Theta, Phi, Psi, Lambda, ni, nt, ThetaV, PhiV, PolarizationAngle, Epi, Esi, Total_N] %X, Y, Z: Lab Coordinate system %x', y', z': Principal Coordinate system %x, y, z: "Incident Light Coordinate system" (Z = z) %Light incidents in x-z plane and %Polar angle: theta_i from the Z(z)-axis (= Viewing Polar Angle if n_i == n_t) %Azimuthal angle: phi_i from the X-axis (=Viewing Azimuthal Angle) %All of the axis of the birefringence material is refered to optic axis (z'-axis) %theta : Tilt angle from the X-Y plane in RADIAN unit (0 deg.: In plane, 90deg.:Homeotropic Alignment) %phi : Azimuthal angle from the X-axis in RADIAN unit %Unit: MKSA %January 02, 2006 %Yong-Kyu Jang (Liquid Crystal Institute, Kent State University) %Simulation parameter lambda = OpticsStruct.Lambda; %Wavelength of the incident light(cm) N = OpticsStruct.Total_N; %The number of the total layers n_i = OpticsStruct.ni; %Refractive index of the incident medium n_t = OpticsStruct.nt; %Refractive index of the transmitted medium theta_t = OpticsStruct.ThetaV; %Transmitted Polar angle of the light from the Z-axis (0 deg.: normal transmittance, radian unit) phi_i = OpticsStruct.PhiV; %Transmitted(=Incident) Azimuthal angle of the light from the X-axis (0 deg.: X-Z plane transmittance, radian unit) sin_theta_i = n_t/n_i * sin(theta_t); %Incident Polar angle of the light from the Z-axis (0 deg.: normal incident, radian unit) cos_theta_i = sqrt(1 - sin_theta_i^2); cos_theta_t = cos(theta_t); if (cos_theta_i <= 0 | cos_theta_t <= 0) beep errordlg('Incident and transmitted light angle should be less than 90 degree', 'Angle definition Error', 'on') T = []; return end

346

if isempty(OpticsStruct.Epi) %Linear polarization Exi = cos(OpticsStruct.PolarizationAngle - phi_i) * cos_theta_i; Eyi = sin(OpticsStruct.PolarizationAngle - phi_i); else Exi = OpticsStruct.Epi * cos_theta_i; %Definition of the polarization of the incident light (X-Y coordinate) Eyi = OpticsStruct.Esi; %Definition of the polarization of the incident light (X-Y coordinate) end %Find the 1'st non-zero thickness layer for p = 1 : N if OpticsStruct.d(p) ~= 0 n_pi = (real(OpticsStruct.nx(p)) + real(OpticsStruct.ny(p)) + real(OpticsStruct.nz(p)))/3; %Average refractive index of the 1'st layer (It is needed to consider the reflection at the interface between air and 1'st layer) break else n_pi = n_t; end end %Find the last non_zero thickness layer for p = N : -1 : 1 if OpticsStruct.d(p) ~= 0 n_pt = (real(OpticsStruct.nx(p)) + real(OpticsStruct.ny(p)) + real(OpticsStruct.nz(p)))/3; %Average refractive index of the last layer (It is needed to consider the reflection at the interface between air and last layer) break else n_pt = n_t; end end cos_theta_pi = sqrt(1 - (n_i/n_pi)^2*sin_theta_i^2); %Polar angle of the light in the 1'st layers (0 deg.: normal incident) cos_theta_pt = sqrt(1 - (n_i/n_pt)^2*sin_theta_i^2); %Polar angle of the light in the last layers (0 deg.: normal incident) K_0 = 2 * pi / lambda; %Magnitude of the wavevector in free space K_x = K_0 * n_i * sin_theta_i; %X component of the wave vector in free space (K_x is conserved in all of the layer) Kx_K0 = n_i * sin_theta_i; Kx_K0_2 = Kx_K0^2; %Calculation of the Entrance & Exit matrix J_ent = zeros(2); %Initialization of the Entrance matrix (Reflection at the entrance interface) J_ext = zeros(2); %Initialization of the Eixt matrix (Reflection at the exit interface)

347

J_ent(1,1) = 2 * n_i * cos_theta_pi / (n_pi * cos_theta_i + n_i * cos_theta_pi); %X-polarization component of the Entrance matrix J_ent(2,2) = 2 * n_i * cos_theta_i / (n_i * cos_theta_i + n_pi * cos_theta_pi); %Y-polarization component of the Entrance matrix J_ext(1,1) = 2 * n_pt * cos_theta_t / (n_t * cos_theta_pt + n_pt * cos_theta_t); %X-polarization component of the Exit matrix J_ext(2,2) = 2 * n_pt * cos_theta_pt / (n_pt * cos_theta_pt + n_t * cos_theta_t); %Y-polarization component of the Exit matrix G_n = zeros(2); %Initialization of the phase matrix %Calculation of the J matrix J = eye(2); %Initialization of the J matrix to the identity matrix for n = 1 : N %Assignment of the optical parameters of each layer nx = OpticsStruct.nx(n); ny = OpticsStruct.ny(n); nz = OpticsStruct.nz(n); h = OpticsStruct.d(n); %cm unit theta = OpticsStruct.Theta(n); %radian unit phi = OpticsStruct.Phi(n); %radian unit psi = OpticsStruct.Psi(n); %radian unit %Trick to escape the uncalculable situation phi_phi_i = phi - phi_i; if abs(sin(phi_phi_i)) < 1e-4 phi_phi_i = phi_phi_i + 1e-4; end if abs(cos(phi_phi_i)) < 1e-6 phi_phi_i = phi_phi_i + 1e-6; end if abs(cos(theta)) < 1e-6 theta = theta + 1e-6; end %Calculation of the dielctric constant tensor s_t = sin(theta); c_t = cos(theta); s_p = sin(phi_phi_i); c_p = cos(phi_phi_i); if isequal(nx, ny) %Isotropic & Uniaxial materials (nx=ny~=nz) nz2_nx2 = nz^2 - nx^2; eps_xx = nx^2 + nz2_nx2 * c_t^2 * c_p^2; eps_xy = nz2_nx2 * c_t^2 * s_p * c_p; eps_xz = nz2_nx2 * s_t * c_t * c_p; eps_yy = nx^2 + nz2_nx2 * c_t^2 * s_p^2; eps_yz = nz2_nx2 * s_t * c_t * s_p; eps_zz = nx^2 + nz2_nx2 * s_t^2;

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%Calculation of the eigenvalues (Kz1_K0, Kz2_K0) Kz1_K0 = sqrt(nx^2 - Kx_K0_2); Kz2_K0 = -eps_xz / eps_zz * Kx_K0 + nx * nz / eps_zz * sqrt(eps_zz - (1 - nz2_nx2 / nz^2 * c_t^2 * s_p^2) * Kx_K0_2); else %Biaxial materials (nx~=ny~=nz) a2 = -sin(psi) * s_t * c_p - cos(psi) * s_p; a3 = c_t * c_p; b2 = -sin(psi) * s_t * s_p + cos(psi) * c_p; b3 = c_t * s_p; c2 = sin(psi) * c_t; c3 = s_t; ny2_nx2 = ny^2 - nx^2; nz2_nx2 = nz^2 - nx^2; eps_xx = nx^2 + ny2_nx2 * a2^2 + nz2_nx2 * a3^2; eps_yy = nx^2 + ny2_nx2 * b2^2 + nz2_nx2 * b3^2; eps_zz = nx^2 + ny2_nx2 * c2^2 + nz2_nx2 * c3^2; eps_xy = ny2_nx2 * a2 * b2 + nz2_nx2 * a3 * b3; eps_xz = ny2_nx2 * a2 * c2 + nz2_nx2 * a3 * c3; eps_yz = ny2_nx2 * b2 * c2 + nz2_nx2 * b3 * c3; %Calculation of the Delta matrix Delta(1,1) = -Kx_K0 * eps_xz/eps_zz; Delta(1,2) = 1 - Kx_K0^2/eps_zz; Delta(1,3) = -Kx_K0 * eps_yz/eps_zz; Delta(2,1) = eps_xx - eps_xz^2/eps_zz; Delta(2,2) = Delta(1,1); Delta(2,3) = eps_xy - eps_xz * eps_yz/eps_zz; Delta(3,4) = 1; Delta(4,1) = Delta(2,3); Delta(4,2) = Delta(1,3); Delta(4,3) = eps_yy - eps_yz^2/eps_zz - Kx_K0^2; %Calculation of the eigenvalues of the Biaxial Film D = eig(Delta); D_temp = real(D); index = find(D_temp > 0); if length(index) == 2 Kz2_K0 = D(index(1)); Kz1_K0 = D(index(2)); else beep errordlg('Can not calculate Eigenvalues of the Biaxial Film', 'Biaxial Film Error', 'on') T = []; return end end %of the Dielectric constant calculation %Calculation of the G_n matrix (Phase matrix) G_n(1,1) = exp(i * K_0 * Kz1_K0 * h); G_n(2,2) = exp(i * K_0 * Kz2_K0 * h);

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%Calculation of the S_n matrix e_x1 = (Kz1_K0^2 + Kx_K0_2 - eps_yy) * (Kx_K0_2 - eps_zz) - eps_yz * eps_yz; e_y1 = eps_xy * (Kx_K0_2 - eps_zz) + eps_yz * (Kx_K0 * Kz1_K0 + eps_xz); e_x2 = eps_xy * (-Kx_K0_2 + eps_zz) - eps_yz * (Kx_K0 * Kz2_K0 + eps_xz); e_y2 = (-Kz2_K0^2 + eps_xx) * (Kx_K0_2 - eps_zz) + (Kx_K0 * Kz2_K0 + eps_xz) * (Kx_K0 * Kz2_K0 + eps_xz); if e_y2 == 0 e_y2 = e_y2 + eps; end if e_x1 == 0 e_x1 = e_x1 + eps; end S_n = [1, e_x2 / e_y2; e_y1 / e_x1, 1]; % while (det(S_n) == 0) % S_n = [S_n(1) + eps, e_x2 / e_y2; e_y1 / e_x1, 1]; % end %Calculation of the J_n matrix J = S_n * G_n / S_n * J; end %End of n loop (Calculation of the J matrix) %Calculation of the J' matrix %J_prime = J; %Neglect reflection at the glass surface J_prime = J_ext * J * J_ent; %Calculation of the Output Electric Field Ext = J_prime(1,1)*Exi + J_prime(1,2)*Eyi; Eyt = J_prime(2,1)*Exi + J_prime(2,2)*Eyi; %Calculation of the Transmittance T = n_t * cos_theta_i * (abs(Ext)^2 + cos_theta_t^2 * abs(Eyt)^2) / (n_i * cos_theta_t * (abs(Exi)^2 + cos_theta_i^2 * abs(Eyi)^2));

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APPENDIX C

PROGRAM OVERVIEW (“LC Optics”)

C-1. “LC Optics” main screen

This program uses the director calculation and the optics calculation explained in

previous two appendices. In this main screen, we build a stack configuration of a LCD by

using “Add”, “Delete”, and “Property” push buttons. This stack configuration including

all information of each layer can be saved in “File” menu as the configuration file (*.cfg).

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C-2. Popup menu of each layer

By the popup menu in each layer of the stack configuration, we can choose a

Polarizer, Glass, ITO, Polyimide, Liquid Crystal, Uniaxial Film, Biaxial Film,

Inhomogeneous Film, Isotropic Layer, Anisotropic Layer, and Metal. These layers cover

most of the possible layers that we can use in LCDs. As similar to the main layers, we

can also choose the light source types such as the standard illuminants A, B, C, D65, E,

and any kind of user light source.

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C-3. Layer properties

The “Property” push button of each layer pops up a window corresponding to the user-

selected layer, which includes all the possible information that we need in calculations.

This information can be saved as a file (*.mat) for the use in other LCDs. As an example,

the bottom figure shows the Liquid Crystal property. We can set the refractive indices by

the three ways and set the material, manufacturing, and simulation parameters. We can

choose the calculation methods such as “Vector”, “Theta-Phi”, “Constant V”, and

“Constant Q”. We can also select “Auto” or “Manual” mode as the initial director

configuration. In “Auto” mode, the program searches the stable director configuration.

Each parameter can be designated as a variable by checking the “Var” check box.

353

C-4. Variable rearrangement and plotting

After building and setting the stack configuration of a LCD, the selected variables show

up in this window automatically. We can set the “Start”, “Step”, “End”, and “Number” of

each variable for the calculation, and each variable can be coupled each other. In this

screen, we can choose the calculation methods, 4×4 or 2 ×2 matrices with the detailed

spectrum averaging methods. We can also calculate the contrast ratio if there is a liquid

crystal layer. After the calculation, we can plot 2-D, 3-D, and conoscopy figures in terms

of the director angle, transmittance, reflectance, luminance, lightness, color coordinates,

and color difference for various color spaces and viewing fields. After calculation, we

can save not only the data but also all the configuration including data also (*.ykj).

354

C-5. Plotting options

If we want to change the default setting of the plotting, we can do that in this

window. We can control the 2-D, 3-D, and conoscopy, separately, and change the

detailed figure shape. We can also do the data interpolation with controlling the method.

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C-6. Example of the 2-D Plotting

356

C-7. Example of the 3-D Plotting

357

C-8. Example of the Conoscopy Plotting

358

C-9. Example of the LC director configuration

359

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