Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology 9781119563778, 9781119563730, 9781119563761, 1119563771

Table of contents :
Cover......Page 1
Title Page......Page 5
Copyright......Page 6
Contents......Page 7
List of Figures......Page 13
List of Tables......Page 35
Preface......Page 37
Acknowledgments......Page 39
Part I Fundamentals......Page 41
Introduction......Page 43
1.1.1 Wave Propagation in Anisotropic Media......Page 45
1.1.3 Index Ellipsoid......Page 46
1.3 Electro‐Optic Effect......Page 48
1.5.1 Index Ellipsoid......Page 51
1.5.1.1 Index Ellipsoid with Applied Electric Field......Page 53
1.5.3 Lithium Niobate......Page 55
1.5.4 KDP‐(KH2PO4)......Page 56
1.6 Concluding Remarks......Page 57
Chapter 2 Photoactive Centers and Photoconductivity......Page 59
2.1 Photoactive Centers: Deep and Shallow Traps......Page 60
2.1.1 Cadmium Telluride......Page 61
2.1.2 Sillenite‐Type Crystals......Page 62
2.1.2.1 Doped Sillenites......Page 65
2.2 Luminescence......Page 68
2.3.1 Localized States: Traps and Recombination Centers......Page 69
2.3.2 Theoretical Models......Page 72
2.3.2.1 One‐Center Model......Page 75
2.3.2.2 Two‐Center/One‐Charge Carrier Model......Page 77
2.4 Photovoltaic Effect......Page 80
2.4.1.2 Some Photovoltaic Nonferroelectric Materials......Page 81
2.4.2 Light Polarization‐Dependent Photovoltaic Effect......Page 83
2.5 Nonlinear Photovoltaic Effect......Page 84
2.5.1 Light‐Induced Absorption and Nonlinear Photovoltaic Effects......Page 86
2.5.2 Deep and Shallow Centers......Page 87
2.6 Light‐Induced Absorption or Photochromic Effect......Page 88
2.7 Dember or Light‐Induced Schottky Effect......Page 91
2.7.1 Dember and Photovoltaic Effects......Page 94
Part II Holographic Recording......Page 95
Introduction......Page 96
Chapter 3 Recording a Space‐Charge Electric Field......Page 97
3.1 Index‐of‐Refraction Modulation......Page 100
3.2 General Formulation......Page 103
3.2.2 Solution for Steady‐State......Page 104
3.3 First Spatial Harmonic Approximation......Page 106
3.3.1 Steady‐State Stationary Process......Page 108
3.3.1.1 Diffraction Efficiency......Page 109
3.3.2 Time‐Evolution Process: Constant Modulation......Page 110
3.4 Steady‐State Nonstationary Process: Running Holograms......Page 112
3.4.1 Running Holograms with Hole‐Electron Competition......Page 116
3.4.1.1 Mathematical Model......Page 118
3.5.1 Uniform Illumination: ∂N/∂x=0......Page 124
3.5.2 Interference Pattern of Light......Page 125
3.5.2.1 Influence of Donor Density......Page 126
4.1 Coupled Wave Theory: Fixed Grating......Page 129
4.1.2 Out of Bragg Condition......Page 131
4.2.1 Combined Phase‐Amplitude Stationary Gratings......Page 132
4.2.1.1 Fundamental Properties......Page 134
4.2.1.2 Irradiance......Page 135
4.2.2.1 Time Evolution......Page 136
4.2.2.2 Stationary Hologram......Page 140
4.2.2.3 Steady‐State Nonstationary Hologram with Wave‐Mixing and Bulk Absorption......Page 146
4.2.2.4 Gain and Stability in Two‐Wave Mixing......Page 150
4.3 Phase Modulation......Page 155
4.3.1.1 Phase Modulation in the Signal Beam......Page 156
4.3.1.2 Output Phase Shift......Page 158
4.4 Four‐Wave Mixing......Page 159
4.5 Conclusions......Page 160
5.1 Coupled‐Wave with Anisotropic Diffraction......Page 161
5.2 Anisotropic Diffraction and Optical Activity......Page 162
5.2.2 Output Polarization Direction......Page 163
6.1 Introduction......Page 165
6.2 Mathematical Formulation......Page 167
6.2.1 Stabilized Stationary Recording......Page 169
6.2.2 Stabilized Recording of Running (Nonstationary) Holograms......Page 170
6.2.2.2 Speed of the Fringe‐Locked Running Hologram......Page 172
6.2.3 Self‐Stabilized Recording with Arbitrarily Selected Phase Shift......Page 173
6.3 Self‐Stabilized Recording in Actual Materials......Page 175
6.3.2 Self‐Stabilized Recording in LiNbO3......Page 176
6.3.2.1 Holographic Recording without Constraints......Page 177
6.3.2.2 Self‐Stabilized Recording......Page 182
Part III Materials Characterization......Page 191
Introduction......Page 192
7.1 Electro‐Optic Coefficient......Page 195
7.2 Light‐Induced Absorption......Page 197
7.3 Dark Conductivity......Page 201
7.4 Photoconductivity......Page 202
7.4.1 Photoconductivity in Bulk Material......Page 203
7.4.2 Alternating Current Technique......Page 204
7.4.3.1 Transverse Configuration......Page 206
7.4.3.2 Longitudinal Configuration......Page 210
7.5.1 Wavelength‐Resolved Photo‐Electric Conversion (WRPC)......Page 213
7.5.1.1 Undoped BTO......Page 214
7.6 Modulated Photoconductivity......Page 215
7.6.1 Quantum Efficiency and Mobility‐Lifetime Product......Page 216
7.7.1 Speckle‐Photo‐Electromotive‐Force (SPEMF) Techniques......Page 218
7.7.1.1 Speckle Pattern onto a Photorefractive Material: Stationary State......Page 219
8.2 Direct Holographic Techniques......Page 229
8.2.1 Energy Coupling......Page 230
8.2.2 Diffraction Efficiency......Page 232
8.2.3 Holographic Sensitivity......Page 233
8.4 Hologram Erasure......Page 235
8.4.1.1 Bulk Absorption......Page 236
8.5.1 Fe‐doped LiNbO3: Hologram Erasure under White Light Illumination......Page 237
8.5.2.1 Undoped BTO under λ=780 nm Illumination......Page 239
8.5.2.2 Bi12TiO20:Pb (BTO:Pb)......Page 240
8.5.2.3 Bi12TiO20:V (BTO:V)......Page 242
8.5.2.4 Holographic Relaxation in the Dark: Dark Conductivity......Page 243
8.6.1 Holographic Sensitivity......Page 245
8.6.2 Holographic Phase‐Shift Measurement......Page 246
8.6.3 Photorefractive Response Time......Page 247
8.6.4 Selective Two‐Wave Mixing......Page 250
8.6.4.1 Amplitude and Phase Effects in GaAs......Page 252
8.6.5 Running Holograms......Page 254
8.7 Holographic Photo‐Electromotive‐Force (HPEMF) Techniques......Page 258
9.1 Holographic Phase Shift......Page 269
9.2.1 Absorbing Materials......Page 272
9.2.2 Characterization of Materials......Page 274
9.2.2.1 Measurements......Page 275
9.2.2.2 Theoretical Fitting......Page 276
9.3 Characterization of LiNbO3:Fe......Page 279
Part IV Applications......Page 283
Introduction......Page 284
10.1 Measurement of Vibration and Deformation......Page 285
10.2 Experimental Setup......Page 286
10.2.2.2 Distribution of Light among Reference and Object Beams......Page 287
10.2.3 Self‐Stabilization Feedback Loop......Page 289
10.2.4 Vibrations......Page 291
10.2.5 Deformation and Tilting......Page 292
10.2.5.1 Applications of PEMF to Mechanical Vibration Measurements......Page 296
11.2 Fixed Holograms in LiNbO3......Page 297
11.2.1.1 Theory......Page 298
11.2.1.2 Experiment: Simultaneous Recording and Compensating......Page 300
12.1 Photoelectric Conversion Efficiency: Dember and Photovoltaic Effects......Page 303
Part V Appendix......Page 305
Introduction......Page 306
A.1.1 Diffraction......Page 307
A.2 Instrumental Detection......Page 308
B.1 Angular Bragg Selectivity......Page 311
B.1.2 Probe Beam......Page 312
B.2 Reversible Holograms......Page 314
B.3 High Index‐of‐Refraction Material......Page 315
Appendix C Effectively Applied Electric Field......Page 319
D.1 Temperature......Page 321
D.1.1 Debye Screening Length......Page 322
D.1.1.1 Debye Length in Photorefractives......Page 323
D.2 Diffusion and Mobility......Page 324
Appendix E Photodiodes......Page 327
E.1 Photovoltaic Regime......Page 328
E.2 Photoconductive Regime......Page 329
E.3 Operational Amplifier......Page 330
Bibliography......Page 331
Index......Page 345
EULA......Page 355

Citation preview

Photorefractive Materials for Dynamic Optical Recording

Photorefractive Materials for Dynamic Optical Recording Fundamentals, Characterization, and Technology

Jaime Frejlich† State University of Campinas Gleb Wataghin Institute of Physics (IFGW) Campinas-SP, Brazil

This edition first published 2020 © 2020 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/ permissions. The right of Jaime Frejlich to be identified as the author of this work has been asserted in accordance with law. Registered Office John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA Editorial Office 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Library of Congress Cataloging-in-Publication Data Names: Frejlich, Jaime, 1946- author. Title: Photorefractive materials for dynamic optical recording : fundamentals, characterization, and technology / Jaime Frejlich State University of Campinas, Gleb Wataghin Institute of Physics (IFGW), Campinas-SP Brazil. Description: First edition. | Hoboken, N.J. : John Wiley & Sons Inc., 2020. | Includes index. Identifiers: LCCN 2019032247 (print) | LCCN 2019032248 (ebook) | ISBN 9781119563778 (hardback) | ISBN 9781119563730 (adobe pdf ) | ISBN 9781119563761 (epub) Subjects: LCSH: Laser recording–Materials. | Photorefractive materials. Classification: LCC TK7882.S3 S67 2020 (print) | LCC TK7882.S3 (ebook) | DDC 621.382/34–dc23 LC record available at https://lccn.loc.gov/2019032247 LC ebook record available at https://lccn.loc.gov/2019032248 Cover Design: Wiley Cover Image: © ArtLight Production/Shutterstock Set in 10/12pt WarnockPro by SPi Global, Chennai, India Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

v

Contents List of Figures xi List of Tables xxxiii Preface xxxv Acknowledgments xxxvii

Fundamentals 1 Introduction 3

Part I

1

Electro-Optic Effect 5

1.1 1.1.1 1.1.2 1.1.3 1.2 1.3 1.4 1.5 1.5.1 1.5.1.1 1.5.2 1.5.3 1.5.4 1.5.5 1.6

Light Propagation in Crystals 5 Wave Propagation in Anisotropic Media 5 General Wave Equation 6 Index Ellipsoid 6 Tensorial Analysis 8 Electro-Optic Effect 8 Perovskite Crystals 11 Sillenite Crystals 11 Index Ellipsoid 11 Index Ellipsoid with Applied Electric Field 13 Other Cubic Noncentrosymmetric Crystals 15 Lithium Niobate 15 KDP-(KH2 PO4 ) 16 Bismuth Tellurium Oxide-Bi2 TeO5 (BTeO) 17 Concluding Remarks 17

2

Photoactive Centers and Photoconductivity

2.1 2.1.1 2.1.2 2.1.2.1 2.1.3 2.1.4 2.2 2.3 2.3.1

19 Photoactive Centers: Deep and Shallow Traps 20 Cadmium Telluride 21 Sillenite-Type Crystals 22 Doped Sillenites 25 Lithium Niobate 28 Bismuth Telluride Oxide: Bi2 TeO5 28 Luminescence 28 Photoconductivity 29 Localized States: Traps and Recombination Centers 29

vi

Contents

2.3.2 2.3.2.1 2.3.2.2 2.3.2.3 2.4 2.4.1 2.4.1.1 2.4.1.2 2.4.2 2.5 2.5.1 2.5.2 2.6 2.6.1 2.7 2.7.1

Theoretical Models 32 One-Center Model 35 Two-Center/One-Charge Carrier Model 37 Dark Conductivity and Dopants 40 Photovoltaic Effect 40 Photovoltaic Crystals 41 Lithium Niobate and Other Ferroelectric Crystals 41 Some Photovoltaic Nonferroelectric Materials 41 Light Polarization-Dependent Photovoltaic Effect 43 Nonlinear Photovoltaic Effect 44 Light-Induced Absorption and Nonlinear Photovoltaic Effects 46 Deep and Shallow Centers 47 Light-Induced Absorption or Photochromic Effect 48 Transmittance with Light-Induced Absorption 51 Dember or Light-Induced Schottky Effect 51 Dember and Photovoltaic Effects 54 Holographic Recording 55 Introduction 56

Part II

3

Recording a Space-Charge Electric Field 57

3.1 3.2 3.2.1 3.2.2 3.3 3.3.1 3.3.1.1 3.3.1.2 3.3.2 3.4 3.4.1 3.4.1.1 3.5 3.5.1 3.5.2 3.5.2.1

Index-of-Refraction Modulation 60 General Formulation 63 Rate Equations 64 Solution for Steady-State 64 First Spatial Harmonic Approximation 66 Steady-State Stationary Process 68 Diffraction Efficiency 69 Hologram Phase Shift 70 Time-Evolution Process: Constant Modulation 70 Steady-State Nonstationary Process: Running Holograms 72 Running Holograms with Hole-Electron Competition 76 Mathematical Model 78 Photovoltaic Materials 84 Uniform Illumination: 𝜕 /𝜕x = 0 84 Interference Pattern of Light 85 Influence of Donor Density 86

4

Volume Hologram with Wave Mixing

4.1 4.1.1 4.1.2 4.2 4.2.1 4.2.1.1 4.2.1.2 4.2.2 4.2.2.1 4.2.2.2 4.2.2.3

89 Coupled Wave Theory: Fixed Grating 89 Diffraction Efficiency 91 Out of Bragg Condition 91 Dynamic Coupled Wave Theory 92 Combined Phase-Amplitude Stationary Gratings 92 Fundamental Properties 94 Irradiance 95 Pure Phase Grating 96 Time Evolution 96 Stationary Hologram 100 Steady-State Nonstationary Hologram with Wave-Mixing and Bulk Absorption 106

Contents

4.2.2.4 4.3 4.3.1 4.3.1.1 4.3.1.2 4.4 4.5

Gain and Stability in Two-Wave Mixing 110 Phase Modulation 115 Phase Modulation in Dynamically Recorded Gratings 116 Phase Modulation in the Signal Beam 116 Output Phase Shift 118 Four-Wave Mixing 119 Conclusions 120

5

Anisotropic Diffraction

5.1 5.2 5.2.1 5.2.2

121 Coupled-Wave with Anisotropic Diffraction 121 Anisotropic Diffraction and Optical Activity 122 Diffraction Efficiency with Optical Activity, 𝜌 123 Output Polarization Direction 123

6

Stabilized Holographic Recording

6.1 6.2 6.2.1 6.2.1.1 6.2.2 6.2.2.1 6.2.2.2 6.2.3 6.3 6.3.1 6.3.2 6.3.2.1 6.3.2.2

125 Introduction 125 Mathematical Formulation 127 Stabilized Stationary Recording 129 Stable Equilibrium Condition 130 Stabilized Recording of Running (Nonstationary) Holograms 130 Stable Equilibrium Condition 132 Speed of the Fringe-Locked Running Hologram 132 Self-Stabilized Recording with Arbitrarily Selected Phase Shift 133 Self-Stabilized Recording in Actual Materials 135 Self-Stabilized Recording in Sillenites 136 Self-Stabilized Recording in LiNbO3 136 Holographic Recording without Constraints 137 Self-Stabilized Recording 142

Materials Characterization 151 Introduction 152

Part III

7

General Electrical and Optical Techniques 155

7.1 7.2 7.3 7.4 7.4.1 7.4.2 7.4.3 7.4.3.1 7.4.3.2 7.5 7.5.1 7.5.1.1 7.6 7.6.1 7.7 7.7.1 7.7.1.1

Electro-Optic Coefficient 155 Light-Induced Absorption 157 Dark Conductivity 161 Photoconductivity 162 Photoconductivity in Bulk Material 163 Alternating Current Technique 164 Wavelength-Resolved Photoconductivity 166 Transverse Configuration 166 Longitudinal Configuration 170 Photo-Electric Conversion 173 Wavelength-Resolved Photo-Electric Conversion (WRPC) 173 Undoped BTO 174 Modulated Photoconductivity 175 Quantum Efficiency and Mobility-Lifetime Product 176 Photo-Electromotive-Force Techniques (PEMF) 178 Speckle-Photo-Electromotive-Force (SPEMF) Techniques 178 Speckle Pattern onto a Photorefractive Material: Stationary State 179

vii

viii

Contents

8

Holographic Techniques 189

8.1 8.2 8.2.1 8.2.2 8.2.2.1 8.2.3 8.2.3.1 8.3 8.4 8.4.1 8.4.1.1 8.4.2 8.4.2.1 8.4.2.2 8.5 8.5.1 8.5.2 8.5.2.1 8.5.2.2 8.5.2.3 8.5.2.4 8.6 8.6.1 8.6.2 8.6.2.1 8.6.3 8.6.4 8.6.4.1 8.6.5 8.7

Holographic Recording and Erasing 189 Direct Holographic Techniques 189 Energy Coupling 190 Diffraction Efficiency 192 Debye Length Dependence on Light Intensity 193 Holographic Sensitivity 193 Computing  195 Hologram Recording 195 Hologram Erasure 195 One Single Photoactive Center Involved 196 Bulk Absorption 196 Two (or More) Photoactive Centers (Localized States) Involved 197 Same Charge Carriers 197 Holes and Electrons on Different Photoactive Centers 197 Materials 197 Fe-doped LiNbO3 : Hologram Erasure under White Light Illumination 197 Bi12 TiO20 (BTO) 199 Undoped BTO under 𝜆 = 780 nm Illumination 199 Bi12 TiO20 :Pb (BTO:Pb) 200 Bi12 TiO20 :V (BTO:V) 202 Holographic Relaxation in the Dark: Dark Conductivity 203 Phase Modulation Techniques 205 Holographic Sensitivity 205 Holographic Phase-Shift Measurement 206 Wave-Mixing Effects 207 Photorefractive Response Time 207 Selective Two-Wave Mixing 210 Amplitude and Phase Effects in GaAs 212 Running Holograms 214 Holographic Photo-Electromotive-Force (HPEMF) Techniques 218

9

Self-Stabilized Holographic Techniques

9.1 9.2 9.2.1 9.2.1.1 9.2.2 9.2.2.1 9.2.2.2 9.3

229 Holographic Phase Shift 229 Fringe-Locked Running Holograms 232 Absorbing Materials 232 Low Absorption Approximation 234 Characterization of Materials 234 Measurements 235 Theoretical Fitting 236 Characterization of LiNbO3 :Fe 239

Applications 243 Introduction 244

Part IV

10

Vibrations and Deformations 245

10.1 10.2

Measurement of Vibration and Deformation 245 Experimental Setup 246

Contents

10.2.1 10.2.2 10.2.2.1 10.2.2.2 10.2.3 10.2.4 10.2.5 10.2.5.1

Reading of Dynamic Holograms 247 Optimization of Illumination 247 Target Illumination 247 Distribution of Light among Reference and Object Beams 247 Self-Stabilization Feedback Loop 249 Vibrations 251 Deformation and Tilting 252 Applications of PEMF to Mechanical Vibration Measurements 256

11

Fixed Holograms 257

11.1 11.2 11.2.1 11.2.1.1 11.2.1.2

Introduction 257 Fixed Holograms in LiNbO3 257 Simultaneous Recording and Compensation 258 Theory 258 Experiment: Simultaneous Recording and Compensating 260

12

Photoelectric Conversion

12.1

263 Photoelectric Conversion Efficiency: Dember and Photovoltaic Effects

Part V Appendix 265 Introduction 266 Reversible Real-Time Holograms 267 Naked-Eye Detection 267 Diffraction 267 Interference 268 Instrumental Detection 268

Appendix A

A.1 A.1.1 A.1.2 A.2

Diffraction Efficiency Measurement 271 Angular Bragg Selectivity 271 In-Bragg Recording Beams 272 Probe Beam 272 Reversible Holograms 274 High Index-of-Refraction Material 275

Appendix B

B.1 B.1.1 B.1.2 B.2 B.3

Appendix C

Effectively Applied Electric Field 279

Physical Meaning of Some Parameters 281 Temperature 281 Debye Screening Length 282 Debye Length in Photorefractives 283 Diffusion and Mobility 284

Appendix D

D.1 D.1.1 D.1.1.1 D.2

Photodiodes 287 Photovoltaic Regime 288 Photoconductive Regime 289 Operational Amplifier 290

Appendix E

E.1 E.2 E.3

Bibliography Index 305

291

263

ix

xi

List of Figures Figure 1

Naturally birefringent uniaxial lithium niobate crystal view under converging white light between crossed polarizers with its c-axis (optical axis) laying perpendicular to the plane (upper) and on the plane (lower). 2

Figure 1.1

Refractive index ellipsoid. 7

Figure 1.2

Refractive indices for a plane wave propagating in an anisotropic medium. 7

Figure 1.3

Crystallographic axes of a sillenite and an applied 3D electric field. 9

Figure 1.4

Structure of an undistorted cubic perovskite structure with general chemical formula ABX3 . The differently shaded spheres represent X atoms (usually oxygens), B atoms (a smaller metal cation, such as Ti4+ ) and A atoms (a larger metal cation, such as Ca2+ ). 10

Figure 1.5

Three-dimensional sillenite structure: darker spheres represent Bi3+ ions and paler gray ones are O2− . Acknowledgments to Prof. Jesiel F. Carvalho, IF/UFG-Goiânia-GO, Brazil. 10

Figure 1.6

Schematic representation of a raw BTO crystal boule with its striations, indicating the way it will be sliced (top left); already sliced crystal with striations not perpendicular to the (011)-face (top right) and ready-to-use crystal with renamed axes (bottom). 11

Figure 1.7

Bi12 TiO20 crystal boule as grown along its [001]-axis. 12

Figure 1.8

Actual undoped sillenite crystals: raw Bi12 TiO20 crystal boule grown along its [001]-axis, showing striations on the lateral surfaces with both opposite (001)-faces cut and polished (left); Bi12 SiO20 crystal showing its (110)-surface cut and polished (center) and Bi12 TiO20 crystal with its larger (110)-face cut and polished with its [001]-axis direction along its longer dimension (right). 12

Figure 1.9

Index-of-refraction of BTO that is formulated by n = 0.00863∕𝜆4 + 0.0199∕𝜆2 + 2.46 [6]. 14

Figure 1.10 Bi12 SiO20 -type cubic crystal and its crystallographic axes X1 , X2 and X3 with an externally electric field E applied along the “x”-direction. 14 Figure 1.11 Principal coordinate axes system 𝜂 − 𝜁 arising by the effect of an electric field E applied along the “x”-axis, as shown in Fig. 1.10. 14 Figure 1.12 Sillenite crystal cut along its principal crystallographic axes, with an electric field along the [001]-axis. 15 Figure 1.13 Lithium niobate crystal with an applied electric field along the photovoltaic c-axis. 16

xii

List of Figures

Figure 1.14 Lithium niobate crystal ellipsoid (black) and its modified (gray) size by the action of an applied field in opposite directions (left and right pictures) along the c-axis. 16 Figure 2.1

Energy diagram for a typical CdTe crystal doped with vanadium, with the Te in the Cd anti-sites at 0.23 eV below the CB and the Cd vacancies 0.4 eV above the VB [19]. 21

Figure 2.2

Dark conductivity measured at various temperatures for a CdTe:V crystal (labeled CdTeBR16B3) produced and measured by Dr. J.C. Launay, ICMCB-Bordeaux, France. From the Arrhenius plot, the energy of the Fermi level EA = 0.83 eV is computed. 22

Figure 2.3

Representation of the sillenite octahedra unit with the lone-electron pair in one corner. 22

Figure 2.4

Octahedra sharing corners. 22

Figure 2.5

Sillenite structure showing (dashed lines) the empty tetrahedra formed by four double-octahedra units. 23

Figure 2.6

Localized states in the Band Gap of nominally undoped Bi12 TiO20 crystal, from [29]. Filled electron-donors are in gray and empty ones in white; the DOS (density of states) is qualitatively represented by the width of the full-line limited levels whereas the dashed-line ones are not. The succession of states close to the VB represents the almost continuous states except the few discrete ones at 2.4 and 2.5 eV. Reproduced from [29]. 26

Figure 2.7

Schematic representation of luminescence effect on a sillenite crystal. 28

Figure 2.8

Photoluminescence in BTO-008. The dashed line is the spectrum of the light of an LED illuminating the BTO crystal sample. The continuous curve is the spectrum of the light measured at the crystal output, very closed to it. A luminescent peak appears at 570 nm (≈ 2.2 eV). 29

Figure 2.9

Intrinsic semiconductor: Fermi level for an intrinsic semiconductor and its “energy vs. occupation-of-states diagram” (right side). 30

Figure 2.10 Doped semiconductor: Fermi level pinned at the position of the dopant in the BG. On the right-hand side is the “energy vs. occupation-of-states” diagram. 30 Figure 2.11 Doped semiconductor: Fermi Ef and quasi-stationary Fermi levels upon illumination. The “energy vs. occupation-of-states” graphics is shown on the right-hand side. 30 Figure 2.12 Recombination centers. 31 Figure 2.13 Traps. 31 Figure 2.14 Schematic representation of a material with one center (one single species) with two valence states (electron donors and electron acceptors) on two correspondingly slightly different localized states in the Band Gap. Electron acceptors are here represented as positively charged so that a nonphotoactive negative ion should be close to it in order to produce electrical neutrality at equilibrium for the as-grown crystal. 32 Figure 2.15 Under the action of light (of adequate wavelength) electrons are excited to the CB, thus increasing the electron density in the CB and therefore increasing the n-type (photo)conductivity. In the CB they diffuse (or are drifted if there is an

List of Figures

Figure 2.16

Figure 2.17

Figure 2.18

Figure 2.19

Figure 2.20 Figure 2.21 Figure 2.22

Figure 2.23

Figure 2.24

externally applied electric field) and are retrapped (on the available acceptors) again and re-excited and so on. 33 In this example, under the action of light, electrons and holes are excited to the CB and VB, respectively, so that the photoconductivity is due to electrons and holes. In this case, electrons do predominate but it could also be the opposite, or even be only holes being excited and the photoconductivity being of the p-type. 34 Under nonuniform light, negative charges (in this case, we assume to be electrons only) accumulate in the darker (less illuminated) regions leaving behind, in the more illuminated regions, opposite (positive here) charges. 34 Photochromic effect and the band-transport model. On the left side, we represent deep photoactive centers (acceptors and donors) and shallower + . In this centers close to the CB, with empty donors (acceptors) only, labeled ND2 figure electron acceptors, both for deep and for shallow centers, are represented as positively charged so that a nonphotoactive negative ion should be close to these charged acceptors to ensure local electric neutrality. On the right side, we see that under the action of light (represented by the arrows), the electrons are excited into the conduction band. Some of the electrons are retrapped to the ND+ + and some others to the ND2 centers. The latter ones, that slowly relax to the + deeper ND centers in the dark, have a higher light absorption coefficient and are therefore responsible for the photochromic darkening effect. 37 Schema for the crystal samples: undoped Bi12 TiO20 (labeled BTO-J40), lead-doped Bi12 TiO20 (labeled BTO-Pb), undoped Bi12 SiO20 (labeled BSO) and photovoltaic iron-doped LiNbO3 (labeled LNb) with the photovoltaic “c” axis parallel to the [110] crystal axis. The light is always incident on the (110) crystal plane. Dimensions for all samples are reported in Fig. 2.20. 42 Crystal samples. 42 Bi2 TeO5 (left) and LiNbO3 :Fe (right) crystal samples showing the [010] and c-axis that are their photovoltaic axes, respectively. 42 Average photovoltaic current density measured along axes [010] and “c”, respectively, on the BTeO and LNbO:Fe crystal samples (depicted on the left side) illuminated with spatially uniform 𝜆 = 532 nm laser light normally incident on their (100) faces, as a function of the intensity I(0) as computed at the input plane inside the material. Reproduced from [12]. Fitting data to Eq. (2.67) with 𝛼 = 5 cm−1 for BTeO [50] and 𝛼 = 7.3 cm−1 for LNbO:Fe [12] it is possible to compute their corresponding 𝜅ph𝑣 , which are reported in Tables 2.1 and 2.2. Reproduced from [12]. 43 Polarization-dependent photovoltaic photocurrent for both BTeO and LNbO:Fe crystal samples, as a function of the polarization direction of the 𝜆 = 532 nm laser light, with the angular position referred to the axes [010] and “c”, respectively, for the incident (onto the (100) crystal faces) intensity (outside the material) I0 = 480 mW/cm2 . Reproduced from [12]. 43 Photocurrent (•) Iph , for undoped Bi12 TiO20 as a function of the angle 𝜃. The photocurrent was measured along the [110]-crystal axis using 𝜆 = 532 nm and incident light intensity I0 = 102 mW/cm2 measured outside the crystal. The initial point, 𝜃 = 0o , corresponds to the polarization parallel to the [110]-axis (see Fig. 2.19). 44

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Figure 2.25 Photovoltaic current versus light intensity I(0) (uniform 𝜆 = 532 nm laser incident on the (110) crystal plane with light polarization direction along [110]) for undoped Bi12 TiO20 (BTO-J40) sample. The ◽ and • represent the photovoltaic current measured along the [001] and [110]-axis, respectively. The continuous line is the best fitting with Eq. (2.78) and the parameters computed from fitting are reported in Table 2.3. 46 Figure 2.26 Photovoltaic current versus light intensity (uniform 𝜆 = 532 nm laser incident on the (110) crystal plane with light polarization direction along [001]) for an undoped Bi12 SiO20 (BSO) sample. The continuous line is the best fitting with Eq. (2.80) and the parameters computed from fitting are reported in Table 2.3. 47 Figure 2.27 Photovoltaic current versus light intensity (uniform 𝜆 = 532 nm laser incident on the (110) crystal plane with light polarization direction along [001]) for a lead-doped Bi12 TiO20 (BTO-Pb) sample. The dashed line is only a guide for the eyes. 47 Figure 2.28 Average photovoltaic current density data, measured along the c-axis, versus light (𝜆 = 532 nm) intensity (light polarization direction along crystal c-axis) for an iron-doped LiNbO3 (LNb) sample show a strict linear behavior with the continuous line being the best fitting with Eq. (2.67). 48 Figure 2.29 Light-induced absorption spots produced in the center of an undoped Bi12 TiO20 crystal by the action of a thin 𝜆 = 532 nm laser line beam; the second spot is due to the beam reflected from the rear crystal face. 50 Figure 2.30 Photochromic relaxation time for Bi12 TiO20 as a function of inverse absolute temperature. Arrhenius data fitting leads to an activation energy of 0.42 ± 0.02 eV. 50 Figure 2.31 Transmitted versus incident power (both measured in the air) for a 8.1 mm thick photorefractive Bi12 TiO20 crystal slab labeled BTO-010 using a 𝜆 = 532 nm Gaussian cross-section intensity laser beam (1.3 mm radius, P = 800 μW corresponding to I ≈ 150 mW/m2 ). Data in the graphics are fitted by a linear equation for the limits Po → 0 (black line) and Po → ∞ (gray line) as shown in the graphics. 51 Figure 2.32 Light-induced Schottky barrier at the illuminated transparent conductive ITO electrode-photorefractive crystal interface. 52 Figure 2.33 Schema of a photorefractive BTO crystal plate between two conductive transparent ITO electrodes including crystal axes and the illuminated front (001) plane. 52 Figure 2.34 Cross-section schema of the ITO-sandwiched BTO plate indicating the photocurrent flow under illumination. 52 Figure 2.35 ITO sandwiched 0.81 mm thick BTO crystal plate with electrodes wired to a lock-in amplifier. 53 Figure 2.36 Measured photocurrent data referred to Fig. 2.35 with •, ◾ and ▾ indicating the front illuminated sample, whereas ⚬, ◽ and ∇ refer to rear plane illumination, with ⚬ and • data refer to the left-side ordinate axis. 53 Figure 2.37 Photovoltaic-based current data (•, ◾ and ▴) computed from curves in Fig. 2.36 are plotted on the left-side ordinate axis, whereas computed Dember-based currents (⚬, ◽ and Δ) are plotted on the right-side ordinate axis. Because of

List of Figures

Figure 3.1

Figure 3.2

Figure 3.3 Figure 3.4 Figure 3.5

Figure 3.6

Figure 3.7 Figure 3.8 Figure 3.9 Figure 3.10 Figure 3.11 Figure 3.12 Figure 3.13 Figure 3.14 Figure 3.15 Figure 3.16 Figure 3.17 Figure 3.18

logarithmic scales, all current are plotted as positive, although Dember and photovoltaic based ones have opposite signs. Data for I0 ≈ 1276 mW/cm2 are represented by ⚬ and • whereas ◾ and ◽ are for I0 ≈ 12.8 mW/cm2 . Data for I0 ≈ 1.02 mW/cm2 are represented by ▴ and Δ. 54 Photoactive centers inside the Band Gap. There are filled traps ND − ND+ (electron-donors), empty traps ND+ (electron-acceptors) and nonphotoactive ions (+) to provide local charge neutrality. 58 Under the action of light the electrons are excited from the traps into the conduction band where they diffuse and are retrapped in the darker regions. A space modulation of electric charge results, with overall positive charge in the illuminated and negative charge in the less illuminated regions. 58 The charge distribution produces a space-charge electric field modulation. 59 The electric field modulation may produce deformations in the crystal lattice. 59 If the photoconductive material is also electro-optic, that is to say it is photorefractive, the space-charge field may produce an index-of-refraction modulation in the crystal volume that is in-phase (or counterphase) with the space-charge field modulation and is π∕2-shifted to the recording pattern of light. 59 Holographic setup: A laser beam is divided by the beamsplitter BS, reflected by mirrors M1 and M2 and interfering with an angle 2𝜃. A sinusoidal pattern of light, as described in the text, is produced in the volume where these two beams interfere. A photorefractive crystal C is placed in the place where this pattern of light is produced. The irradiance of the two interfering beams is measured behind the crystal using photodetectors D1 and D2. Shutters Sh1, Sh2 and Sh3 are used to cut off the main beam and each one of interfering beams, if necessary. 60 Generation of an interference pattern of fringes. 60 Light excitation of electrons to the CB in the crystal. 61 Generation of an electric charge spatial modulation in the material. 61 Generation of a space-charge electric field modulation. 61 The electric field modulation produces a index-of-refraction modulation (volume grating) via electro-optic effect. 61 The recorded grating can be read using one of the recording beams that is transmitted and diffracted. 62 The grating is erased during reading. 62 Until all recording is erased. 62 Space-charge electric field grating being recorded by the 𝜙-shifted sinusoidal pattern of fringes. 63 Space-charge electric field without an externally applied field for a pattern of fringes with modulation m = 0.99 (left), 0.60 (center) and 0.30 (right). 66 Simulated recording (from 0 to 20 au) and erasure (from 20 to 50 au) of a space-charge field with E0 = 0 and 𝜏sc = 10 au. 68 Index-of-refraction modulation arising in the crystal volume. The upper figure shows the pattern of light fringes projected onto the crystal, the middle figure

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Figure 3.19

Figure 3.20

Figure 3.21 Figure 3.22

Figure 3.23

Figure 3.24

Figure 3.25 Figure 3.26 Figure 3.27

Figure 3.28

Figure 4.1 Figure 4.2

shows the resulting charge density and the lower figure shows the spatial-charge field and index-of-refraction modulation in the absence of any externally applied electric field (E0 = 0). All vertical coordinates are in “arbitrary units”. 69 Schematic description of running hologram generation in photorefractives. A moving pattern-of-fringes onto the sample produces a synchronously moving volume hologram that reaches a maximum amplitude at a resonance speed. 72 st 2 Plot of |Esc | ∝ 𝜂 for the assumed parameters: LD = 0.20 μm, lS = 0.02 μm, Φ = 0.5, 𝜔I = 5.1 rad/s, Q ≈ 2 and 𝛼 = 11.5 cm−1 for 𝜆 = 514.5 nm; with the experimental conditions being K = 10 μm−1 , E0 = 106 V/m, and an intensity inside the front crystal plane I(0) = 100 W/m2 . From Eq. (3.21) and K we compute ED = 2.59 × 105 V/m at T = 300 K, from K 2 ls2 and ED in Eq. (3.49) we get Eq = 6.5 × 106 V/m, from E0 and Eq in Eq. (3.50) we compute KlE = 0.15, from Q and 𝜔I in Eq. (3.87) we get 𝜔R = 2.55 rad/s. 74 Plot of 𝜙P from Eq. (3.85) for the same parameters referred to in Fig. 3.20. 75 Plotting of Q as a function of LD (LD-axis) and ls (LS-axis) for E0 = 106 V/m, K = 10 μm−1 , 𝜆 = 514.5 nm with 𝛼 = 11.5 cm−1 , Φ = 0.5 and an intensity inside the front crystal plane I(0) = 100 W/m2 . 76 Plotting of Q as a function of K, from Eq. (3.91), for typical values LD = 0.15 μm, ls = 0.03 μm and different applied electric fields from 5 × 105 , 7 × 105 , 10 × 105 to 15 × 105 V/m, represented by the progressively increasing size of the dashed lines, respectively. 76 st 2 st st | (continuous curve), ℜ{Esc } (long dashing curve) and ℑ{Esc } Plotting of |Esc (short dashing curve) versus K𝑣, for the same parameters referred to in Fig. 3.20. 77 One-species/two-valence/two-charge carrier model contributing to charge transport: one single spatial trap modulation structure is produced. 77 Two-species/two-valence/two-charge carrier model contributing to charge transport: two distinct spatial trap modulation structures are produced. 78 Hole-electron competition on different photoactive centers under the action of low energetic photon recording light: only charge carriers close to the CB (electrons) and to the VB (holes) can be excited, but electrons cannot be excited from the hole-donor level or holes from the electron-donor level, because of energy considerations. In this case, an electron-based hologram is recorded in the level closer to CB, and the same for holes in the level close to VB. However, electrons progressively accumulate in the (deeper) level closer to the VB and holes accumulate in the level close to the CB, where they cannot be re-excited again because the recording light is not energetic enough. The recording is progressively decreasing, because of the decrease in the corresponding charge carriers, until a steady state is achieved because of the exhaustion of any one of the two levels. 78 Short circuit schema using conductive silver glue to electrically connect the opposite faces along the photovoltaic axis c⃗ (left) and open circuit schema, without any electrical connection (right). 85 Reading the recorded hologram with one of the recording beams. 90 Recording a fixed volume index-of-refraction hologram that is phase-shifted by 𝜙 = 𝜙P referred to the recording pattern of fringes with 2𝜃 being the angle inside the material. 90

List of Figures

Figure 4.3 Figure 4.4 Figure 4.5

Figure 4.6

Figure 4.7

Figure 4.8

Figure 4.9

Figure 4.10

Figure 4.11 Figure 4.12 Figure 4.13 Figure 4.14 Figure 4.15 Figure 4.16 Figure 4.17 Figure 4.18

Bragg condition where 𝜌⃗ and 𝛿⃗ are the incident beam and the diffracted beam wavevectors, respectively (or vice versa), and K⃗ is the grating wavevector. 90 Amplitude coupling in two-wave mixing: in this example, the weaker beam receives energy from the stronger, but could also be the other way round. 94 Phase coupling in two-wave mixing: the pattern of fringes and associated grating are progressively shifted by the same amount. The picture shows some degree of amplitude coupling too. 94 Numerical plotting of |S|2 versus the normalized time t∕|𝜏sc |, from Eq. (4.80) for Γd = 1, ℜ{𝜏sc } = 0.4s and ℑ{𝜏sc } = −0.65s with 𝛾z = −0.5, −0.25, 0.25 and 0.5 from the smaller to the larger dashed lines, respectively. 98 Numerical plotting of |S|2 versus the normalized time t∕|𝜏sc |, from Eq. (4.80) for Γd = −1, ℜ{𝜏sc } = 0.4s and ℑ{𝜏sc } = −0.65s with 𝛾z = −0.5, −0.25, 0.25 and 0.5 from the smaller to the larger dashed lines, respectively. 99 Numerical plotting of |S|2 versus the normalized time t∕|𝜏sc |, from Eq. (4.80) for 𝛾d = 1, ℜ{𝜏sc } = 0.4s and ℑ{𝜏sc } = −0.65s with Γz = −0.5, −0.25, 0.25 and 0.5 from the smaller to the larger dashed lines, respectively. 99 Numerical plotting of |S|2 versus the normalized time t∕|𝜏sc |, from Eq. (4.80) for 𝛾d = −1, ℜ{𝜏sc } = 0.4s and ℑ{𝜏sc } = −0.65s with Γz = −0.5, −0.25, 0.25 and 0.5 from the smaller to the larger dashed lines, respectively. 100 Transient effect of a perturbation, in the form of a ramp voltage (thick curve) applied to the PZT-supported mirror in the holographic setup, on the diffraction efficiency (thin curve) of a running hologram recorded in a photorefractive BTO-crystal using the 514.4-nm wavelength. The diffraction efficiency evolution to equilibrium is faster for the negative-gain (lower graphics, with K = 2.55 μm−1 ) than for the positive-gain (upper graphics with K = 4.87 μm−1 ) experiment. In both cases, the applied external field is E0 ≈ 7.5 kV/cm, the total incident irradiance is Io ≈ 22.5 mW/cm2 and the beam ratio is 𝛽 2 ≈ 40. Reproduced from [94]. 100 Computed running hologram 𝜂 as a function of K𝑣 (rad/s) for K = 0.5 μm−1 and different material parameters. 108 Computed running hologram 𝜂 as a function of K𝑣 (rad/s) for K = 2 μm−1 and different material parameters. 108 Computed running hologram 𝜂 as a function of K𝑣 (rad/s) for K = 10 μm−1 and different material parameters. 109 Computed running hologram 𝜂 as a function of K𝑣 (rad/s) for K = 20 μm−1 and different material parameters. 110 Tan 𝜑 versus K𝑣 (rad/s), computed for K = 2 μm−1 and different material parameters, for a typical BTO crystal 2.05 mm thick and 𝛼 = 1165 m−1 . 111 Tan 𝜑 versus K𝑣 (rad/s), computed for K = 11 μm−1 and different material parameters, for a typical BTO crystal 2.05-mm thick and 𝛼 = 1165 m−1 . 112 Tan 𝜑 versus K𝑣 (rad/s), computed for K = 11 μm−1 and different material parameters, for a typical BTO crystal 2.05-mm thick and 𝛼 = 1165 m−1 . 113 Tan 𝜑 versus K𝑣 (rad/s), computed for K = 1 μm−1 and different material parameters, for a typical BTO crystal 2.05 mm thick and a hypothetically low 𝛼 = 1m−1 . 114

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Figure 4.19 Phase modulation setup: BS: beamsplitter, PZT piezoelectric-supported mirror, D: photodetector, LA-Ω and LA-2Ω: lock-in amplifiers tuned to Ω and 2Ω respectively, HV high voltage source for the PZT, OSC oscillator to produce the dithering signal. 115 Figure 4.20 Wave-mixing schema showing the hologram phase shift 𝜙 and the phase shift 𝜑 between the transmitted and diffracted beams at the crystal output. 119 Figure 4.21 Degenerate four-wave mixing showing the signal S and reference R beams interfering to produce a real-time hologram in the nonlinear material (left); then a pump beam P, identical to R but much stronger and counter propagating, is diffracted by the already recorded hologram and the diffracted beam is the conjugate S* of the signal S beam, reflecting back along the same incidence direction. 120 Figure 5.1

Input and output light polarization. 122

Figure 5.2

Input and output polarization referred to actual principal axes coordinates. 122

Figure 5.3

General illustration of the polarization direction of the transmitted and diffracted beams through a crystal with optical activity and anisotropic diffraction. At mid-crystal thickness, the polarization directions of the transmitted and diffracted beams are 10∘ shifted from the [110] and [001] axes, respectively. 124

Figure 5.4

Transmitted and diffracted beams orthogonally polarized at the output through a crystal with optical activity and anisotropic diffraction. Assuming ρd = 20∘ , the incident beam’s polarization direction at the input plane should be −10∘ with reference to the [110]-axis. 124

Figure 5.5

Transmitted and diffracted beams parallel-polarized at the output through a crystal with optical activity and anisotropic diffraction. Assuming 𝜌d = 20∘ , the incident beam’s polarization direction at the input plane should be 35∘ with reference to the [110]-axis. 124

Figure 6.1

Scanning electronic microscopy image of a 1D hollow sleeve structure first recorded on photoresist film, then metallic vacuum deposited and finally washed away from all remaining photoresist to produce hollow metallic structures. Produced and photographed by Lucila Cescato, Laboratório de Óptica, Instituto de Física, Universidade Estadual de Campinas, Brazil. 126

Figure 6.2

Scanning electronic microscopy image of a 2D-array holographically recorded and chemically developed on photoresist film. Produced and photographed by Lucila Cescato, Laboratório e Óptica, Instituto de Física, Universidade Estadual de Campinas, Brazil. 126

Figure 6.3

Scanning electronic microscopy image of a blazed grating made by the holographic recording of the first and the second spatial harmonic components of a sawtooth-shape profile on photoresist film. Produced and photographed at Laboratório de Óptica, Instituto de Física, Universidade Estadual de Campinas, Brazil. Reproduced from [105]. 127

Figure 6.4

Block-diagram of a self-stabilized setup: D photodetector, LA-Ω phase sensitive lock-in amplifiers tuned to Ω, HV voltage source for the phase modulation device PM, OSC oscillator at frequency Ω. The output phase shift, feedback and noise phases are 𝜑, 𝜑f and 𝜑N , respectively. 128

List of Figures

Figure 6.5

Schematic description of the actual self-stabilized holographic recording setup: C photorefractive crystal, D photodetector, LAΩ and LA2Ω phase sensitive lock-in amplifiers tuned to Ω and 2Ω, respectively, HV high voltage source for the piezo-electric supported mirror PZT acting as phase modulator, OSC oscillator at frequency Ω. 128

Figure 6.6

Schematic description of the effect of noise on the two-wave mixing in the holographic setup. 128

Figure 6.7

Block-diagram of fringe-locked running hologram setup: same as for Fig. 6.4 with the addition of an integrator INT at the output of the lock-in amplifier. 130

Figure 6.8

Schematic actual setup for self-stabilized running hologram recording: same as for Fig. 6.4 with the addition of an integrator INT at the output of the lock-in amplifier. 131

Figure 6.9

Fringe-locked running hologram speed: Kv (rad/s) versus feedback amplification 𝜅f (arbitrary units) in a fringe-locked running hologram experiment carried out on an undoped Bi12 TiO20 crystal using the 514.5 nm wavelength, with E0 = 4.7 kV∕cm, IRo = 533 μW/cm2 , ISo = 20 μW/cm2 , Ω∕(2𝜋) = 2.1 kHz, K = 7.55 μm−1 and 𝜓d ≈ 0.011 rad. 133

Figure 6.10 Schema of the self-stabilized setup in Fig. 6.8 modified to operate with arbitrarily selected 𝜑: PM is a generic phase modulation that could also be the ⨂ PZT, BPΩ and BP 2Ω are bandpass filters tuned to Ω and 2Ω, respectively, is a function multiplier, PS is a phase shifter, LA 2Ω is a dual-phase lock-in amplifier tuned to 2Ω with orthogonally shifted outputs X and Y, with all other components as already described in Fig. 6.8. 134 Figure 6.11 Transverse optical configuration for holographic recording on BTO: the incident beams, incidence plane and pattern-of-fringes onto the input crystal face are shown, with the holographic vector K⃗ being perpendicular to the [001]-axis and parallel to the [110]-axis. 136 Figure 6.12 Self-stabilized recording in a Bi12 TiO20 crystal: The upper figure shows the evolution of the VSΩ (thin black line) and the VS2Ω (thick gray line) when the stabilization is off. The lower figure shows the evolution √ of both signals when VSΩ is used as the error signal, in which case VS2Ω ∝ 𝜂. The recording was with 𝜆 = 633 nm with IRo = 0.52 mW/cm2 and ISo = 11 μm/cm2 , interfering with an angle 2𝜃 = 60∘ on a 10-mm-thick crystal with the pattern-of-fringes on the (110) plane and the hologram vector K⃗ perpendicular to the [001]-axis and parallel to [110]. 137 Figure 6.13 Second harmonic evolution during holographic recording in a nominally undoped photorefractive BTO crystal with the self-stabilization off (left) and on (right), for IR0 + IS0 = 12 mmW/cm2 , using the 𝜆 = 514.5 nm laser line and K ≈ 4.5 μm−1 . 137 Figure 6.14 Experimental setup: BS beamsplitter, C: LiNbO3 :Fe crystal, M mirror, PZT pzt-driven mirror, OSC signal generator, HV high voltage source, INT integrator, D1,2 detectors, LA-Ω and LA-2Ω lock-in amplifiers tuned to Ω and 2Ω, respectively. 138 Figure 6.15 Computed 𝜂 as a function of 2𝜅d from Eq. (6.53) for nonstabilized recording in LiNbO3 :Fe with a different degree of oxidation: a reduced sample with

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𝜙 = 𝜋 (top), an oxidized sample with 𝜙 = 2.8 rad (middle) and a still more oxidized sample with 𝜙 = 2.5 rad (bottom). The figures were computed for 𝛽 2 = 1 (thick curve), 2 (thin curve) and 10 (dashed curve). Reproduced from [123]. 140 Figure 6.16 Computed 𝜂 as a function of 2𝜅d and 𝜙, for 𝛽 2 = 1. Reproduced from [123]. 141 Figure 6.17 Computed 𝜂 as a function of 2𝜅d and 𝜙, for 𝛽 2 = 10. The plane 𝜂 = 0.98 superimposed in the lower picture is a guide for the eyes only. Reproduced from [123]. 141 Figure 6.18 Computed IS2Ω (in arbitrary units), with Γ = 0 (that is, 𝜙 = 0,𝜋) as a function of 2𝜅d for 𝛽 2 = 1 (dashed curve), 2 (thin curve) and 10 (thick curve). Reproduced from [123]. 143 Figure 6.19 Computed evolution of 𝜙 (⚬), ISΩ (◽) in arbitrary units and 𝜂 (∇) as functions of 2𝜅d for self-stabilized conditions (IS2Ω = 0) and 𝛽 2 = 1.1. Note that 𝜙 ≈ 𝜋 throughout. Reproduced from [123]. 143 Figure 6.20 Computed evolution of 𝜙 (⚬), ISΩ (◽) in arbitrary units, and 𝜂 (∇) as functions of 2𝜅d for self-stabilized conditions (IS2Ω = 0) and 𝛽 2 = 10. Note that 𝜙 rapidly shifts away from 𝜋 during recording. Reproduced from [123]. 144 Figure 6.21 Self-stabilized recording in the less-oxidized crystal (sample LNB5) with 𝛽 2 ≈ 1 (IR0 = 141.1 W∕m2 and IS0 = 116 W∕m2 ). The evolution of ISΩ during the self-stabilized holographic recording experiment and the error signal I 2Ω are shown both in arbitrary units. At the end of the cycle, 𝜂 = 1 was measured. Reproduced from [123]. 145 Figure 6.22 Self-stabilized recording in an oxidized crystal (sample LNB1) with 𝛽 2 ≈ 1 (IR0 = 113.5 W∕m2 and IS0 = 108.1 W∕m2 ) showing the evolution of the ISΩ (in arbitrary units). The 𝜂 = 1 value by the time ISΩ reached zero was qualitatively verified. Reproduced from [123]. 145 Figure 6.23 Self-stabilized recording in an oxidized crystal (sample LNB1) with 𝛽 2 = 12 (IR0 = 243.2 and IS0 = 20.3 W∕m2 ) showing the evolution of the ISΩ (in arbitrary units). The 𝜂 = 1 value by the time ISΩ reached zero was qualitatively verified. Reproduced from [123]. 145 Figure 6.24 Overall beam IG produced by the interference of the recording beams transmitted and reflected by a thin glassplate G adequately placed close to the photorefractive crystal C being studied. 146 Figure 6.25 Measurement of the running hologram speed for the sample LNB1, 𝛽 2 ≈ 1, IS0 + IR0 ≈ 17 mW∕cm2 and K = 10 per μm. The oscillating shape curve is the interference of the transmitted plus reflected beams in a glassplate fixed close to the sample. Its decreasing amplitude is due to scattering of light in the sample. The filled circles represent the computed pattern-of-fringe speed, corrected from scattering, and the dashed curve is only a guide for the eyes. 146 Figure 6.26 Self-stabilized recording on the same LiNbO3 :Fe sample (LNB3) with ordinarily and extraordinarily polarized 𝜆 = 514.5 nm light simultaneously and 𝛽 2 ≈ 1, all other experimental conditions being similar. Reproduced from [124]. 147 Figure 6.27 Recording setup stabilized on a nearby placed glassplate G, all other elements being the same as described in Fig. 6.14. Reproduced from [123]. 148

List of Figures

Figure 6.28 Glassplate-stabilized experimental data for the recording on an oxidized sample (LNB1) with 𝛽 2 ≈ 1 and ISΩ in arbitrary units. The error signal IGΩ through the glassplate is also shown. At the end of the cycle when ISΩ = 0 it was measured 𝜂 = 0.85. Reproduced from [123]. 148 Figure 6.29 Mathematical simulation of nonself-stabilized recording with 𝛽 2 = 1. The thick curve is 𝜂, the thin curve is ISΩ and the dashed is 𝜑, for tan 𝜙 = 2.8 that seems to qualitatively fit data for LNB1 in Fig. 6.28. Reproduced from [123]. 149 Figure 6.30 Evolution of 𝜂 and scattering PSL during stabilized holographic recording with (figure A) and without (figure B) self-stabilization in LiNbO3 :Fe using 𝜆 = 514.5 nm with IR0 ∕IS0 ≈ 16 and IR0 + IS0 ≈ 4 mW/cm2 . The diffraction efficiency 𝜂 do not consider bulk light absorption and PSL is the scattered light (in %). Reproduced from [115]. 149 Figure 7.1

Schema of the experimental setup for electro-optic coefficient measurement in sillenite crystals as described in [125]: almost monochromatic led (LED), grounded glass plate to improve light uniformity, a lens to collect the light through polarizers and the crystal sample and guide it to the output detector (DET) feeding a lock-in amplifier tuned to the chopper frequency and connected to an oscilloscope for displaying and measurement of the elliptically polarized light at the output. From [125]. 156

Figure 7.2

Evolution of the absorption coefficient in an undoped B12 TiO20 crystal (labeled BTO-010) under uniform illumination of I0 ≈ 2 mW/cm2 at 𝜆 = 532 nm. 159

Figure 7.3

Light-induced absorption: transmitted I t versus incident I0 irradiances measured using an uniform beam of 532 nm wavelength on the same sample BTO-010 as Fig. 7.2. The dashed lines are the best fit at the limit I → 0 (with an angular coefficient of 0.00299) and for saturation with an angular coefficient of 8.78 × 10−4 . Reproduced from [39]. 159

Figure 7.4

Light-induced absorption of undoped Bi12 TiO20 (sample labeled BTO-013) at 𝜆 = 514.5 nm as a function of the incident irradiance measured in the air. The left-hand side graphics is in semi-log scale for detailed view at low irradiances. The continuous curve on the right-hand side graphics is the best fitting to Eq. (2.90) with the following parameters: 𝛼0 = 789 m−1 , a = 1.4 × 10−6 m/(s2 W), b = 4.91 × 10−9 m2 /(W s2 ) and c = 7.48 × 10−9 s−2 . 159

Figure 7.5

Absorption coefficient-thickness 𝛼d measured for three different BTO samples (BTO-8, BTO-Q and BTO-008) as a function of wavelength. BTO-8 and BTO-Q were measured in a standard spectrophotometer whereas BTO-008 was measured with a photodetector placed about 1 cm behind the crystal. 161

Figure 7.6

Arrhenius curve dark conductivity for BTO:V. Data fitting to Eq. (7.11) leads to Ea = 0.89 eV. Reproduced from [30]. 162

Figure 7.7

Frequency-dependence of the absolute value |𝜎dac (f )| in Eq. (7.12) for different temperatures. 163

Figure 7.8

Schematic setup for the electric measurement of photoconductivity. A laser beam is chopped CH at frequency Ω and the beam is filtered and expanded using a spatial filter SF and collimated using a lens L. The chopped expanded and uniform beam shines the sample that produces a photocurrent under the action of a voltage HV. An operational amplifier OA with a feedback resistance R and capacitor C transforms the current into a voltage that is read using a

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Figure 7.9

Figure 7.10

Figure 7.11

Figure 7.12

Figure 7.13 Figure 7.14 Figure 7.15 Figure 7.16

Figure 7.17

Ω-tuned lock-in LA (for the case of photoconductivity) or a simple dc voltmeter (for the case of dark conductivity). Reproduced from [39]. 164 Typical crystal schema, in the so-called “Transverse Configuration”, with the electrodes (H × d area) on the lateral surfaces separated by a distance 𝓁, the thickness (along the light propagation) is d, the height is H and the illuminated surface is H × 𝓁. 164 Photocurrent (in pA) as a function of the incident irradiance on the input plane inside the crystal I(0), measured using a time-modulated uniform 532 nm wavelength laser beam onto crystal sample BTO-010 with 2000 V applied. The corresponding time modulated photocurrent was measured using a lock-in amplifier where (⚬) are data for a sample that has been kept in the dark for a long time and (•) are data for the previously light-saturated crystal. The dashed line for the (•) is the best fitting for the final linear range that gives an angular coefficient of 138 pA.m2 /W. The dashed line for the ( ⚬) in the inset represents the best fitting for the nonexposed sample at the limit I(0) → 0 condition giving an angular coefficient of 69.2 pA.m2 /W. From these data, the values in Table 7.7 were computed for BTO-010. Reproduced from [39]. 166 (Left) Photograph of the wavelength-resolved photoconductivity experimental setup using almost monochromatic LEDs ranging from near infrared to near ultraviolet wavelength and placed on the perimeter of a rotating disc driven by a computer-controlled stepping-motor. The light of the LED is collected by a system of lenses producing a uniform almost monochromatic illumination on the sample placed on a small home-made housing with shielded electrodes connected to an electric voltage source and adequately placed photodetectors to enable the measurement of incident and transmitted intensity on and through the sample. (Right) Schema of the setup with L: lens system, D: photodetectors, BS: beamsplitter, C: crystal sample, V: voltage source and LA: lock-in amplifier. 167 Transverse configuration: coefficient 𝜎 on a logarithmic scale (upper graphics) and on a normal scale (lower graphics) for pre-exposed with h𝜈 = 2.4 eV light (•), normal (◽) and for relaxed (∘) undoped Bi12 TiO20 plotted as a function of h𝜈. Reproduced from [29]. 169 Detailed view of Fig. 7.12 showing a strong increase in 𝜎 for all three curves at about h𝜈 ≈ 2.5 eV. 170 𝜎 (s m/Ω) for thermally relaxed BTO:V (∘) and pre-exposed to h𝜈 = 2.4 eV (▴). Reproduced from [30]. 171 Longitudinal configuration schema showing an externally polarized Bi12 TiO20 crystal plate sandwiched between ITO electrodes. 171 Lateral view of the sandwiched BTO crystal plate showing the light-induced electric potential barriers at both electrodes with a schema of the electric potential distribution at the bottom. 172 Plotting of 𝜂 𝓁 with positive polarization (ranging from 0 to 500 V) both at the front (◽) and at the rear (∘) electrode, as measured on the undoped Bi12 TiO20 crystal plate (labeled BTOJ18L and represented in Fig. (7.15) with d = 0.81 mm and ITO electrodes on the front and rear H𝓁 ≈ 50 mm2 surfaces. The dashed curves are the fitting of both efficiencies near their maximum using a secondorder polynomial. The overall optical absorption coefficient 𝛼 is also shown (▴). Reproduced from [135]. 173

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Figure 7.18 Light-induced photoelectric conversion efficiency 𝜂0 measured (•) on an undoped sandwiched Bi12 TiO20 crystal (labeled BTOJ18L) in the longitudinal configuration together with the light absorption coefficient-thickness 𝛼d (∘). Reproduced from [135]. 174 Figure 7.19 Comparative longitudinal 𝜂0 (without external applied field) (∘) and transverse 𝜎 (•) WRP, respectively, measured on an undoped Bi12 TiO20 crystal. Reproduced from [135]. 175 Figure 7.20 𝜂0 and 𝛼d measured on an ITO-sandwiched BTO with d = 3 mm and d = 0.81 mm under 𝜆 = 532 nm illumination chopped at 200 Hz. 175 Figure 7.21 Modulated photocurrent data of an undoped Bi12 TiO20 crystal, with monochromatic light flux (I∕(h𝜈)) Fdc = 5 × 1014 cm−2 s−1 and Fac = 1013 cm−2 s−1 , where different shades correspond to different temperatures from 130 to 260 K varying in 5 K steps, and different symbols indicating frequencies varying from 12 Hz to 39.9 kHz. After multiple trials, the value Nbe C = 2.5 × 1011 s−1 was chosen, which leads to a rather good superposition of curves for different temperatures and frequencies at the same abscissa indicating a peak at ∣ Ebe − E ∣= 0.29 eV. Reproduced from [29]. 176 Figure 7.22 Plot of the Airy function (left), the equivalent Gaussian function (center) and the superposition of both (right), with x = ∕0 . 179 Figure 7.23 Plotting of Er ∕ED in the xy plane, for d = 0.001 (left) and 0.1 (right). Reproduced from [152]. 181 Figure 7.24 Schematic representation of an ac photocurrent produced by a sinusoidally vibrating (with angular frequency Ω) speckle pattern of light on the surface of a photorefractive crystal with parallel in-plane electrodes (coplanar configuration). Reproduced from [148]. 182 Figure 7.25 Stationary space-charge field arising from a speckle pattern of light vibrating faster than the response time of the space-charge field and slower than the lifetime of the free photoelectrons. 182 Figure 7.26 Plotting of Er ∕ED in the xy plane for a speckle pattern of light vibrating along coordinate x with reduced amplitude 𝛿 = 1 for d = 0.001 (left) and 0.1 (right). Reproduced from [152]. 183 Figure 7.27 Simulation of the first harmonic photocurrent coefficient b1 (in arbitrary units) as a function of 𝛿, for y = 0, ED = 1000 V/m, jD = 1 for d = 0 (•), d = 0.01 (◽) and d = 0.1 (∘). Reproduced from [157]. 185 Figure 7.28 Simulation of the first harmonic photocurrent coefficient b2 as a function of 𝛿, for d = 0, 0.1 and 1. 185 Figure 7.29 Schematic representation of the experimental setup. A laser beam is directed to a vibrating target (commercial loudspeaker with a retroreflecting strip); the backscattered light in the form of an oscillating speckle pattern it is focused onto the photorefractive crystal (BTO with 𝜆 = 532 nm or CdTe with 𝜆 = 1064 nm) fixed on a plate in a metallic housing creating the PEMF effect. The loudspeaker is driven by a function generator FG that also provides the reference signal for the frequency-tuned phase-selective lock-in amplifier LA used for detecting the signal from the photorefractive crystal; the current (iΩ ) from the crystal is converted into a voltage signal by means of a pre-amplifier (an electrometer-class operational amplifier operating in transimpedance mode)

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fixed by the side of the crystal. A home-made laser Doppler vibrometer DV using a 633 nm laser beam is used for independent measurement of the loudspeaker vibration. 186 Figure 7.30 Optical sensor in metallic housing (from Fig. 7.29) showing the separated components, from left to right: adjustable lens, lens adapting ring, main supporting housing with BNC connectors, photorefractive sensor housing. 187 Figure 7.31 Expanded front view of the photorefractive sensor housing (from Fig. 7.30) showing the photorefractive crystal sensor on a fiberglass plate with circuitry. 187 Figure 7.32 First harmonic photocurrent as function of reduced vibration amplitude 𝛿 measured using a BTO crystal under 𝜆 = 532 nm illumination, for frequencies ranging from 50 to 3500 Hz. 187 Figure 7.33 Experimental first harmonic photocurrent I Ω measured on a CdTe:V photorefractive crystal as a function of 𝛿 for Ω = 200 Hz (∘), 400 Hz (◽), 615 Hz (▿), 1300 Hz (•) and 1700 Hz (×), with a I(0) = 3.48 mW/cm2 𝜆 = 1064 nm from a Nd-YAG laser. Reproduced from [157]. 188 Figure 8.1

Holographic setup: a laser beam is divided by the beamsplitter BS, reflected by mirrors M1 and M2 and interfering with an angle 2𝜃. A sinusoidal pattern of light is produced in the volume where these two beams interfere. A photorefractive crystal C is placed in the volume where this pattern of light is produced. The irradiance of the two interfering beams are measured behind the crystal using photodetectors D1 and D2. Shutters Sh1, Sh2 and Sh3 are used to cut off the main and each one of the interfering beams if necessary. 190

Figure 8.2

Energy transfer between interfering 𝜆 = 633 nm beams in the two-wave mixing experiment, represented in Fig. 8.1, on a BTO crystal (2.8 mm thick): The figure shows the overall irradiance at the crystal output with both beams onto the sample (shutters Sh1, Sh2 and Sh3 open) and when one (all open and Sh3 switched off ) and the other (all open and Sh2 switched off ) beam are alternatively switched off. From these data, and knowing the input recording beams irradiance ratio 𝛽 2 = 1.5, it is possible to compute the exponential gain coefficient Γ and also 𝜂. 191

Figure 8.3

Exponential gain coefficient Γ as a function of the external incidence angle 𝜃 measured for a KNSBN:Ti crystal with its optical c-axis parallel to the grating ⃗ Holographic recording is carried out with extraordinarily polarized vector K. (polarization direction along the c-axis) 514.5 nm wavelength laser line. Reproduced from [158]. 192

Figure 8.4

White light hologram erasure in LiNbO3 :Fe: The erasure data (•), measured using one of the 514.5 nm recording beams, adequately fit a single exponential (dashed curve) law as described by Eq. (8.26) with a = 1.06 rad and b = 180 min. 196

Figure 8.5

The graph shows the erasure of holograms in undoped BTO under 10–15 min ≈1 mW/cm2 pre-illumination with light of different wavelengths as indicated in the graph. The recording and erasure were always carried out with 𝜆 = 780 nm. Measurement along the other direction behind the crystal showed similar

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shapes. Erasure curves are artificially shifted in time for better observation. 198 Figure 8.6

Hologram diffraction efficiency (arbitrary units) decay during 𝜆 = 633 nm light erasing of a hologram previously recorded with the same light on a Pb-doped Bi12 TiO20 (BTO:Pb) crystal. Erasure monotonically decreases and adequately fits the double exponential in Eq. (8.27) leading to A1 = 0.37, A2 = 0.28, 𝜏sc1 = 34.0 s, 𝜏sc2 = 5.47 s and background light C = 0.0078. 198

Figure 8.7

Diffraction efficiency (𝜂 in arbitrary units) during erasure of a hologram in a Pb-doped BTO (same sample as in Fig. 8.6) measured along both directions (along the reference beam and along the signal beam) at the crystal output. Both erasure curves (squares and circles) are artificially shifted in time for better observation. The crystal was pre-exposed for a few minutes to a uniform light at 𝜆 = 532 nm. Pre-exposure was switched off immediately before holographic recording started using an He-Ne laser line of 𝜆 = 633 nm. The hologram was erased with one of the in-Bragg recording beams. No external electric field was applied. Experimental data were fitted (continuous curves) with Eq. (8.28) and the resulting parameters reported in Table 8.4. 199

Figure 8.8

Erasure of holograms in Pb-doped BTO (same sample as in Fig. 8.6) recorded over 2 min with a diode laser of 780 nm wavelength, observed along the reference beam direction (left-hand graph) and along the signal beam (right-hand graph) using one of the recording beams. Curves showing a local maximum result from 3 min pre-exposure at 𝜆 = 524 nm (h𝜈 ≈ 1.37 eV) light from a LED and were fitted with Eq. (8.28) leading to a fast grating characteristic f f s time of 𝜏sc ≈ 13 − 16 s and a corresponding value 𝜏sc ≈ 35𝜏sc for the slow grating. The monotonically decreasing curves were not pre-exposed and actually verify a monoexponential law with a 𝜏sc ≈ 100 s. Reproduced from [29]. 200

Figure 8.9

Diffraction efficiency (recorded and measured using 𝜆 = 514.5 nm laser beams [87]) as a function of the applied electric field measured (•) on a V-doped BTO (0.30% V in weight) with hole-electron competition. The continuous curve is the theoretical fit (a single factoring parameter in ordinates was used for data fitting) assuming hole- and electron-charge carriers from different photoactive centers with ls1 = 0.164 μm, 𝜁1 = 0.99, ls2 = 0.163 μm and 𝜁2 = 0.88. The dashed curve is for 𝜁1 = 𝜁2 = 1 (see Eqs. (3.125) and (3.126), which represents holes and electrons at the same position in space, all other parameters unchanged. 202

Figure 8.10 Diffraction efficiency (au) as a function of time (seconds, in logarithmic scale) (•) during erasure with 𝜆 = 514.5 nm light of a hologram recorded on BTO:V using same wavelength coherent laser beams, without externally applied electric field. Curve fitting to Eq. (8.30) leads to: Af = 0.17, As = 0.25, 𝜏f = 0.28 s, 𝜏s = 20 s and background C = 0.011. Reproduced from [30]. 203 Figure 8.11 Hologram relaxation in the dark: exponential time as a function of inverse temperature for hologram relaxation in the dark. The hologram was recorded using 𝜆 = 514.5 nm light onto an undoped BTO sample (BTO-8) approximately 1 mm thick. Diffraction efficiency was measured from time to time using one of the in-Bragg recording beams during a very short time and correcting data for the effect of exposure to light. From the Arrhenius-type curve, an activation energy of 1.04 eV was computed. 204

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Figure 8.12 Photorefractive sensitivity  data (∘) as a function of the external incidence angle 𝜃 for the KNSBN:Ti sample of Table 8.1 in the same optical and recording configuration as in Fig. 8.3. From these data we compute LD = 0.18 μm. 206 Figure 8.13 Second harmonic evolution for KNSBN:Ti for the same sample and experimental conditions as for Fig. 8.12 with 𝜃 = 15o and IS0 + IR0 ≈ 3 mW/cm2 . 206 Figure 8.14 Evolution of the −1∕ tan 𝜙 accounting on self-diffraction effects as described in the text (∘), as a function of the applied field E for a 2 mm thick nominally undoped Bi12 TiO20 sample with K = 7.08 μm−1 , 𝛽 2 = 9 and I0 ≈ 4 mW∕cm2 using the 514.5 nm wavelength laser line. The crystal is in a transverse electro-optical configuration with the (110)-plane perpendicular to the ⃗ Data incidence plane and the [001]-axis perpendicular to the grating vector K. fitting leads to ls = 0.027 μm. In the same figure, (◽), the directly measured tan 𝜑 is plotted. 207 Figure 8.15 Two-wave mixing experiment in a photorefractive GaAs intrinsic crystal with mutually orthogonally polarized diffracted and transmitted beams. The polarization direction is represented by the black arrows: the input and transmitted beam polarization is along the [001]-axis, whereas the diffracted is perpendicular to the [001]-axis. 208 Figure 8.16 Second harmonic response curves for an undoped semi-insulating GaAs crystal illuminated with a 1.06 μm laser wavelength line, with |m| = 1 and an angle 𝛾 = 10∘ between the transmitted beam polarization direction and the grating ⃗ Theoretical fit to data (◽) for K = 3.5 μm−1 and I0 ≈ 118 mW/cm2 lead vector K. to 𝜏sc = 0.22 ms; fit to data (∘) for K = 2.1 μm−1 and I0 ≈ 168 mW/cm2 lead to 𝜏sc = 0.1 ms. 210 Figure 8.17 Two-wave mixing experiment in a photorefractive GaAs intrinsic crystal with incident and transmitted beams polarized along the [001]-axis of the GaAs crystal. The polarization of the diffracted beams (the shorter arrows) at the crystal output depends on the nature of the diffraction grating in the GaAs. A polarizer (P) and two photodetectors with a summation/subtraction device produce the adequate electric signal for TWM processing. 212 Figure 8.18 Two-wave mixing experiment in a photorefractive GaAs intrinsic crystal as for Fig. 8.15, but with a polarizer at the crystal output where its transmitted polarization direction makes an angle 𝛾 with the crystal axis [110]. 213 Figure 8.19 Plot of the first I Ω (Eq. (8.76)) and second I 2Ω (Eq. (8.78)) harmonic terms after fitting the corresponding actual data in GaAs as a function of the polarization angle 𝛾 behind the crystal (see Fig. 8.18) during steady-state multiple nature holograms recorded with 𝜆 = 1064 nm and K = 2.1 μm−1 . 213 Figure 8.20 Experimental setup for the generation and measurement of running holograms. 214 Figure 8.21 Diffraction efficiency (left) and tan 𝜑 (right) as a function of Kv computed with the experimental parameters K = 2.55 μm−1 , 𝛼 = 11.65 cm−1 , 𝜉E0 = 4.55 KV∕cm and I0 = 17.5 mW/cm2 . The material parameters are LD = 0.22 μm, ls = 0.03 μm, 𝛽 2 = 40 and Φ = 0.4 for electrons (continuous curve), whereas for holes they are LDh = 0.16 μm, lsh = 0.15 μm and Φh = 0.004 (dashed curve). The resulting electron-to-hole diffraction efficiency ratio at

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Figure 8.22

Figure 8.23

Figure 8.24

Figure 8.25

Figure 8.26

Figure 8.27 Figure 8.28

Figure 8.29

Figure 8.30

Figure 9.1

Figure 9.2

K𝑣 = 0 is 𝜂e ∕𝜂h ≈ 2.4. The thick continuous curve is the overall result. Reproduced from [191] 216 Diffraction efficiency (left) and tan 𝜑 (right) as a function of Kv computed with K = 11.3 μm−1 . All other experimental and material parameters and the meaning of thick, thin and dashed curves are the same as for Fig. 8.21 with 𝜂e ∕𝜂h ≈ 17 for K𝑣 = 0. Reproduced from [191]. 217 Diffraction efficiency 𝜂 experimental data (spots) as a function of detuning K𝑣 and best theoretical fit (continuous curve) to Eq. (4.145) for 𝜉 = 0.96, K = 2.55 μm−1 , E0 = 7.3 KV/cm, 𝛽 2 = 41.2 and I0 = 22.5 mW/cm2 . The resulting best fitting parameters are LD = 0.14 μm, and Φ = 0.45. Data for K𝑣 < 0 (small spots) were not used for the fit. Reproduced from [191]. 218 Tan 𝜑 experimental data (spots) as a function of K𝑣 for the same conditions as in Fig. 8.23, with data (large spots) fitted to Eq. (4.146) (continuous curve) and the resulting parameter being Φ = 0.41. Data for K𝑣 < 0 (small spots) are also not considered for the fit here. Reproduced from [191]. 218 Holographic photoelectromotive force current setup schema: a laser beam of 514.5 nm wavelength is divided in two, filtered, expanded, collimated and made to interfere over the BTO sample. A piezoelectric-supported mirror PZT in one of the beams is vibrating with angular frequency Ω. A lock-in amplifier measuring current, and schematically represented by the operational amplifier with feedback, is tuned to Ω in order to measure the first harmonic component ⃗ in the sample’s volume. iΩ of the photocurrent along the K-direction Reproduced from [153] 219 |jΩ | (in arbitrary units) as a function of the vibration amplitude KΔ (in radians) for Ω𝜏sc = 1000, 5, 1 and 0.1 rad, from the finest to the coarsest dashed curves, respectively, always without an externally applied field. 224 Computed |jΩ | (in arbitrary units) as a function of Ω𝜏sc in rad for a fixed amplitude KΔ = 1.1 rad. 225 First harmonic component of the holographic current |iΩ | data (spots) as a function of the KΔ for I0 = IRo + ISo = 455 W/m2 . The continuous curves are the best fit to theory, from Ω∕2𝜋 = 980 Hz (thickest continuous) to 3.5 Hz (thinnest dashed). Data for 980, 546 and 349 Hz are omitted because are close to data for 152 Hz. Reproduced from [153]. 225 First harmonic component of the holographic current |jΩ | data (spots) as a function of KΔ for I0 = IRo + ISo = 177 W/m2 . All data fit the same (not shown) curve. Reproduced from [153]. 226 |iΩ | data (spots) plotted as a function of Ω∕2𝜋, for KΔ = 1.1 rad: Ce-doped BTO (thickest curve), for Pb-doped BTO (thinnest curve) and undoped BTO (mid thickness curve). 226 Typical time evolution of the VX and VY signals (dots) at the initial stage of the recording process in Bi12 TiO20 for E = 0 (a) and E = 3.15 kV/cm (b). The ratio between the angular coefficients of the linear fittings (continuous curves) are used to compute 𝜑. The diffraction efficiencies at t = 1.2 s are 𝜂 ≈ 3 × 10−5 (a) and 𝜂 ≈ 5 × 10−5 (b), whereas the minimum detectable signal was estimated to correspond to 𝜂 ≈ 10−7 . Reproduced from [72]. 230 Computed initial tan 𝜑 versus applied electric field data (spots) in Bi12 TiO20 . The best fits to theory are represented by the continuous curves. Curve A represents

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nonstabilized experiments, whereas curves B and C represent stabilized experiments. Experimental parameters and the values for LD and 𝜉 computed from data fitting are reported in Table 9.1. Dashed lines in curve B were plotted for LD = 0.13 μm (upper) and for LD = 0.14 μm (lower) and similarly in curve C for LD = 0.13 μm (upper) and for LD = 0.15 μm (lower), to approximately indicate the precision of the measurement. Reproduced from [72]. 231 Figure 9.3 Output phase-shift 𝜑 versus applied electric field (E0 ) data (circles) for a 2.05 mm thick Bi12 TiO20 crystal and grating-vector K = 5.5 μm−1 for 𝛽 2 = 30, and 532 nm wavelength, with 𝛼 = 8.5 cm−1 . The continuous curve is the best fit to the theoretical equation in Eq. (4.128) that leads to ls = 0.03 with a field factor 𝜉 ≈ 0.74. 232 Figure 9.4 Fringe-locked running hologram speed versus applied electric field for a 1.71 mm thick Bi12 SiO20 crystal with 𝛼 = 3 cm−1 for the 514 nm wavelength with m ≈ 0.3, IS = 12 μW∕cm2 , IR = 440 μW∕cm2 and K = 4.24 μm−1 . Theoretical fit (continuous curve) to experimental data (∘ ) leads to LD = 0.19 μm and 0.46 ≤ Φ ≤ 0.6 ranging from 0.6 to 0.46 with an estimated field factor of 0.87. Reproduced from [205]. 233 Figure 9.5 Fringe-locked running hologram experiment: frequency detuning K𝑣 (measured from the movement of the PZT-supported mirror) versus normalized applied field E0 ∕ED data from a typical fringe-locked running hologram experiment carried out on an undoped Bi12 TiO20 crystal using the 514.5 nm wavelength with K = 7.55 μm−1 , IRo = 21.5 μW/cm2 and ISo = 0.45 μW/cm2 [207]). 235 Figure 9.6 Fringe-locked running hologram experiment on undoped Bi12 TiO20 crystal using the 514.5 nm wavelength with K = 8.5 μm−1 , IRo + ISo = 52 μW∕cm2 and 𝛽 2 = 183: frequency detuning K𝑣 (measured from the interference pattern from an auxiliary glassplate) versus normalized applied field E0 ∕ED data. 236 Figure 9.7 K𝑣 and 𝜂 experimentally measured as function of E0 ∕ED on an undoped Bi12 TiO20 crystal 2.35 mm thick (labeled BTO-013) with IR0 + IS0 = 14 W/m2 , 𝛽 2 ≈ 48, K = 7.55 μm−1 and 𝛼 = 1041 m−1 at 514.5 nm wavelength. 237 Figure 9.8 3D plotting of experimentally measured eta and K𝑣 as function of E0 ∕ED from Fig. 9.07. 237 Figure 9.9 3D surface plotting of 𝜂 and K𝑣 as function of E0 ∕ED from Eq. (9.19) with same experimental data as for Fig. 9.08 showing the best fit theoretical 3D-curve (continuous thick curve) from Fig. 9.08. The resulting best fitting parameters are reported in Table 9.2. 238 Figure 9.10 Characterization of reduced LiNbO3 :Fe (labeled LNB3): self-stabilized holographic recording on a d = 1.39 mm thick crystal (labeled LNB3) using ordinarily and extraordinarily polarized 𝜆 = 514.5 nm light (𝛽 2 ≈ 1 and IR0 +IS0 ≈ 16 mW/cm2 ) with an irradiance absorption 𝛼 = 7.5 cm−1 at this wavelength. The fitting of Eq. (9.25) to experimental I Ω data gives B and 𝜏M as reported in Table 9.3. 240 Figure 9.11 Characterization of reduced LiNbO3 :Fe (labeled LNB5): self-stabilized holographic recording on a d = 0.85 mm thick crystal using extraordinarily polarized 𝜆 = 514.5 nm light with IR0 = 141.1 W/m2 and IS0 = 116 W/m2 . Eq. (9.25) was fitted to data and the resulting parameters reported in Table 9.3. At the end of the cycle when ISΩ = 0, it was measured 𝜂 = 1. From [123] and [124]. 241

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Figure 9.12 Characterization of oxidized LiNbO3 :Fe (labeled LNB1): self-stabilized holographic recording on a d = 1.5 mm thick crystal using extraordinarily polarized 𝜆 = 514.5 nm light (IR0 = 113.5 W/m2 and IS0 = 108.1 W/m2 ) and fitted with Eq. (9.31). The resulting parameters are reported in Table 9.3. Reproduced from [123]. 241 Figure 10.1 Schematic diagram of the experimental holographic setup: PBS: polarizing beamsplitter cube; HWP and QWP: halfwave and quarterwave retardation plates, respectively; M: first surface mirrors; PZT: piezo-electric supported mirror; PLC: path length compensator; EOM: electro-optical modulator; SF: spatial filter; BTO: photorefractive Bi12 TiO20 crystal; D: photodetector; P1 e P2: polarizers; CCD: image detector; LA: lock-in amplifier; INT: integrator; HV: high voltage source for the PZT. 246 Figure 10.2 (a) Lateral view of the holographic setup: CCD camera (1), output polarizer (2), photographic objective lens for imaging the hologram onto the CCD (3), photorefractive crystal in its nylon holder (4), photographic objective lens for imaging the target onto the crystal (5), target painted with retroreflective ink (6) and 633 nm He-Ne laser (7). (b) Detailed view of the photorefractive crystal in its nylon holder, between the two photographic objective lenses and the output polarizer. 248 Figure 10.3 Simplified schema showing the distribution of incident light (I0 ) between reference and object beams: BS, beamsplitter; M mirror; IR1 and IS1 reference and object beams at the BS output; IR0 and IS0 , reference and object beams effectively incident on the crystal. 249 Figure 10.4 Optimization of the target illumination: IRD , diffracted reference beam measured (in arbitrary units) as a function of R = IS1 ∕IR1 (∘), and the best fitting to theory (continuous line). From fitting, we get f ∕𝜁 = 1.15 for our retro-reflective painted loudspeaker membrane. 249 Figure 10.5 Loudspeaker membrane (left) driven at 3.0 kHz and analyzed by the time-average holographic interferometry technique. The brighter areas are those at rest, the first dark fringe indicates a vibration amplitude of 0.12 μm, the second one 0.28 μm, the third one 0.44 μm and so on according to data in the table (right) showing the amplitude d of the vibration associated with the minima (for J0 (x) = 0) and maxima in the pattern of fringes. 251 Figure 10.6 Amplitude of vibration at a point of local maximum in the membrane of a loudspeaker as a function of the applied voltage for a signal of 4.2 kHz. 252 Figure 10.7 Amplitude of vibration at two different points of local maximum in the membrane of a loudspeaker as a function of the applied voltage for a signal of 1.4 kHz. 252 Figure 10.8 Time-average holographic interferometry pattern of a thin phosphorous-bronze metallic plate tightly fixed by its external border to a loudspeaker vibrating at 255 Hz. 253 Figure 10.9 Time-average holographic interferometry pattern of a thin phosphorous-bronze metallic plate tightly fixed by its external border to a loudspeaker vibrating at 600 Hz. 253 Figure 10.10 Time-average holographic interferometry pattern of a thin phosphorous-bronze metallic plate tightly fixed by its external border to a loudspeaker vibrating at 800 Hz. 254

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List of Figures

Figure 10.11 Double exposure holographic interferometry of a tilted rigid plate. The smallest diagonal dimension of the small cells printed in the plate is approximately 4.8 mm. 254 Figure 10.12 Double exposure holographic interferometry of a rigid plate that was less tilted than in Fig. 10.11. The smallest diagonal dimension of the small cells printed in the plate is approximately 4.8 mm. 255 Figure 10.13 Double exposure holographic interferometry of a rigid plate that was more tilted than in Fig. 10.11. The smallest diagonal dimension of the small cells printed in the plate is approximately 4.8 mm. 255 Figure 11.1 Experimental setup: S: massive copper cylinder with temperature-controlled heating element in direct thermal contact with the copper holder H supporting and surrounding the sample C. A thin pyrex glass cylinder W to minimize heat losses and thermal convection, around the sample, allows laser beams L to go through. A flat heat-isolating plate (not seen) covers the upper cylinder side. 260 Figure 11.2 Evolution of I Ω and I 2Ω during high temperature self-stabilized holographic recording (and compensation) for a typical experiment. 261 Figure 11.3 Diffraction efficiency of the overall grating during white-light development as a function of development time. Note that the time scale depends on the overall development light intensity on the sample. 261 Figure B.1

Diffraction efficiency as a function of out-of-Bragg angle 𝜃 in mrad for the measured data (•), theoretically computed for a = 0.35 mrad (continuous curve) and for a → 0 (dashed curve). From [242]. 273

Figure B.2

𝜈, computed from Eq. B.15, as a function of 𝜈 for in-Bragg condition and same parameters as in Fig. B.1. From [242]. 274

Figure B.3

Measurement of diffraction efficiency: The recording beams are not collimated and the sample adds focusing/defocusing effects. The output irradiance along each one of the incident directions is the coherent addition of the transmitted and the diffracted beams. The two different detectors, with different responses, should be centered on the same spot of the crystal. From [242]. 275

Figure C.1

Effective field coefficient: the figure shows a Gaussian cross-section irradiance I(x) illuminating a photoconductive material in steady-state regime with constant photocurrent j(x) = j, showing the resulting photoconductivity distribution 𝜎(x) and associated electric field E(x). The coordinate x (in arbitrary units) is along the two electrodes on the sample and all quantities represented (in ordinates) are also in arbitrary units. 280

Figure D.1

Volume A × dx with fixed ions of volume density ni of characteristic collision cross-section s, receiving a flux Γ of electrons of mass me and velocity 𝑣. 284

Figure E.1

np-junction showing the depletion layer and a diagram of the Schottky potential barrier. 288

Figure E.2

np-junction showing the depletion layer including the intrinsic layer and a diagram of the Schottky potential barrier. 288

Figure E.3

pn-junction showing the depletion layer including the intrinsic layer and a diagram of the Schottky potential barrier. The dashed curve shows the potential barrier under a direct bias potential V indicated by the dashed arrow. 288

List of Figures

Figure E.4

Figure E.5

Figure E.6

Photovoltaic mode operation for photodiodes. A shows its operation with a load RL , B shows the open-circuit operation and C shows the short circuit operation. 289 Photoconductive mode operation for photodiodes. A reverse bias voltage VB (usually VB ≫ V ) is applied as shown, to increase speed and improve linearity of the response. 289 Operational amplifier operated photodiode in the short-circuit photovoltaic regime. 290

xxxi

xxxiii

List of Tables Table 1.1 Table 2.1

Index of refraction of KDP. 17 Photovoltaic transport coefficient 𝜅phv for Fe- and Cu-doped LiNbO3 . 41

Table 2.2 Table 2.3

Photovoltaic transport coefficient 𝜅phv for BTeO and BSO. 41 Parameters for BTO and BSO from Figs. 2.5 and 2.6. 48

Table 6.1 Table 7.1

LiNbO3 :Fe samples. 144 Effective electro-optic coefficient for doped and undoped BTO.

Table 7.2 Table 7.3

Parameters: pure and doped sillenite crystals. 157 Absorption parameters for pure and doped BTO for 𝜆 = 532 nm. 160

Table 7.4 Table 7.5 Table 7.6

Saturated absorption for sillenites. 160 Dark conductivity 𝜎d measurement. 163 DOS for Bi12 TiO20 . From [29]. 177

Table 7.7 Table 8.1

Photoconductivity and derived parameters for BTO at 532 nm. 177 Properties of a KNSBN:Ti sample. 192

Table 8.2 Table 8.3

Debye length on illumination for Bi12 TiO20 . 193 Holographic sensitivity and gain for some materials. 194

Table 8.4 Table 8.5

Hole-electron competition in BTO:Pb – data from Fig. 8.7. 201 Sensitivity and relative photoconductivity: doped and undoped BTO at 𝜆 = 514.5 nm. 204

Table 8.6 Table 8.7

Running hologram: undoped BTO at 𝜆 = 514.5 nm. 219 Best fitting parameters from HPEMF experiments [153]. 227

Table 9.1 Table 9.2

Initial phase shift: for Bi12 TiO20 from data fitting in Fig. 9.2. 231 Parameters from experimental 𝜂 and K𝑣 data fitting as function of E0 ∕ED for undoped Bi12 TiO20 from Fig. 9.9. 238 Parameters for LiNbO3 :Fe samples. 241 LiNbO3 :Fe material parameters. 242

Table 9.3 Table 9.4

157

Table 9.5 Sensitivity and relative photoconductivity for doped and undoped BTO. Table 11.1 Fixed grating diffraction efficiency. 262 Table 12.1 Photoelectric conversion efficiency. 263

242

xxxv

Preface This book is a corrected and largely extended version of my former one (Photorefractives, John Wiley & Sons, 2007). The objective of this book is mainly focused on photorefractive materials, their properties and their technological possibilities. These materials are still the most interesting ones for dynamic optical recording, not only because their good photoconductivity and the good photovoltaic effects of some of them allow thinking about photoelectric conversion applications as well. The first part of this book is devoted to the analysis of the fundamental properties of this materials: electro-opticity and photoconductivity as well as other effects that some of them may exhibit and which should be taken into account while operating with them – photovoltaicity, light-induced absorption, luminescence and the Dember effect. Part II is focused on the dynamic recording of a spatial distribution of electric charge and the associated spatial electric field distribution leading to a corresponding index-of-refraction (and sometimes also light absorption coefficient) modulation in the material volume as a consequence of their electro-optic properties. Most of the recording is carried out using a spatially modulated interference (holographic) pattern of light, an index-of-refraction and sometimes associated absorption coefficient volume grating results. The real-time diffraction of the recording beams by the grating being built up results in complex wave coupling effects that should be taken into account to mathematically describe the dynamics of this recording process. Electrical coupling among charge carriers (electrons and/or holes) during recording allows the possibility that more than one photoactive type of defect (the Localized State in the material Band Gap) should be also taken into account. The recording of an interference pattern of light or hologram is usually subject to serious environmental perturbations that may undermine the recording quality, mainly for the rather long recording time processes that are usually the case with photorefractives. To cope with this problem, we describe here some dynamically stabilized setups that actively compensate the environmental phase perturbations on the interference pattern of light during recording. Some of these setups use, when possible, their own grating being recorded as a reference for the stabilization process, which is therefore labeled “self-stabilized recording”. Running holograms and self-stabilized running holograms are also discussed here. Part III is devoted to the characterization of photorefractives using holographic, nonholographic optical methods and electrical techniques, reporting a large number of actual experimental results on a variety of materials. Some practical applications including holographic real-time measurement of out-of-plane mechanical vibration modes in 2D and in-plane amplitude mechanical vibration using backscattered (“speckle” pattern) laser light are discussed in Part IV. Also, the possibility of using thin photorefractive crystal plate devices for photoelectric conversion is discussed in detail. As recording on photorefractive crystals is essentially reversible (recorded holograms may also be erased by the same light used for recording), we discuss here some fixing

xxxvi

Preface

techniques that may even allow the production of permanent micro- and sub-microscopic structures using different holographic techniques. Part V is an appendix where the physical meaning of some quantities closely related to photorefractives, such as Debye length, diffusion and mobility, as well as detailed practical techniques, such as how to measure diffraction efficiency of reversible holograms (which is a far from obvious matter), and even how to operate photodiodes and operational amplifiers for different light detection practical tasks, are discussed. Campinas-SP, April 2019

xxxvii

Acknowledgments This book is the result of direct and indirect cooperation of colleagues from Brazil and all over the world who have contributed with their experience, work and advice, as well as graduate students working on their theses under my direction, or just spending some time at my laboratory at the State University of Campinas, Campinas-SP, Brazil. My warm acknowledgments to all of them: Araújo, William R.

Arizmendi, Luis

Barbosa, Marcelo C.

Bassewitz, J.P.

Bian, Shaopin

Buse, Karsten

Carrascosa, Mercedes

Capovilla, Danilo

Carvalho, Jesiel F.

Cescato, Lucila H.

Freschi, Agnaldo A.

Garcia, Paulo Magno

Hernandes, Antonio C.

Inocente Junior, Nilson R.

Kamenov, V.P.

Kamshilin, Alexei A.

Kip, Detlef

Klein, Marvin

Krätzig, Eckhard

Kulikov, V.V.

Kumamoto, R.

Launay, Jean Claude†

Longeaud, Christophe

Lorduy G., Hector

Montegegro, Renata

Mosquera, Luis

Oliveira, Ivan de

Odoulov, S.G.

Prokofiev, Victor V.

Rasnik, Ivan

Ringhofer, Klaus H.†

Rupp, Romano A.

Salazar, A.

Santos, Paulo Acioly Marques dos

Santos, Pedro Valentim dos

Santos, Tatiane Oliveira dos

Schamonina, Ekaterina

Shcherbin, K.V

Shumelyuk, A.

Stepanov S.I.

Sugg, Bertrand

Sturman, B.I.

Telles, A.C.

Troncoso, L.S.

1

Part I Fundamentals

Figure 1 Naturally birefringent uniaxial lithium niobate crystal view under converging white light between crossed polarizers with its c-axis (optical axis) laying perpendicular to the plane (upper) and on the plane (lower).

3

Introduction

Photorefractive crystals are electro-optic and photoconductive materials. An electric field applied to an electro-optic material produces changes in its refractive index; a phenomenon also called the Pockels effect. On the other hand, photoconductivity means that light of adequate wavelength is able to produce electric charge carriers that are free to move by diffusion and also by drift under the action of an electric field. In the case of photorefractive materials, the light excites charge carriers from Localized States (LS-photoactive centers) in the forbidden Band Gap (BG) to Extended States (conduction or valence bands) where they move, are retrapped and excited again and so on. During this process, the charge carriers progressively accumulate in the darker regions of the sample. In this way, charges of one sign accumulate in the darker regions while leaving charges of the opposite sign in the brighter regions. This spatial modulation of charges produces an associated space-charge electric field. The combination of both effects gives rise to the so-called photorefractive effect: the light produces a photoconductive-based electric field spatial modulation that in turn produces an index-of-refraction modulation via the electro-optic effect. This change can be reversed by the action of light or by relaxation even in the dark. The action of light on a photosensitive material may produce changes in the electrical polarizability of the molecules and by this means a change in the complex index-of-refraction will result. This change may be sensible or not, depending on the wavelength spectral range analyzed. The imaginary part of the index (the extinction coefficient, related to absorption) or the real part (the so-called “index-of-refraction” itself ) may be more affected when observed in a certain wavelength spectral range. This is the case of dyes, some silver salts, chalcogenic glasses, photoresists and other materials. When sensible changes occur in the real part of the complex index of refraction, these materials are also called “photorefractives” because they actually show changes in the real refractive index under the action of light. These changes can be reversible or not. What is the essential difference between these processes and those we have mentioned before and we are dealing with in this book? The difference is that the latter always involve the establishment of a space-charge electric field and the production of index-of-refraction changes via an electro-optic (or Pockels) effect. We should therefore rather call them “photo-electro-refractive” materials instead of just using the “photorefractive” label. However, the latter generic name is so widespread nowadays in the scientific literature that it would be hard to change it now. In this book, we shall therefore only use the term “photorefractive”, but the reader should be aware that materials of a different nature are usually referred to under this same label. Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

4

Introduction

Chapter 1 contains a review of the electro-optic effect including a little bit of tensorial analysis. The effect of an applied electric field over the index ellipsoid of some usual electro-optic crystals is analyzed for the reader to get familiar with these procedures. We hope these examples will enable the reader to properly handle different materials and optical configurations. Chapter 2 deals with photoconductivity and light-induced absorption and their relation with the localized states (photoactive centers) in the forbidden band.

5

1 Electro-Optic Effect The electro-optic effect together with photoconductivity are the fundamental phenomena underlying the photorefractive effect. Most photorefractive crystals are anisotropic (their properties are different along different directions) and even those that are not become anisotropic under the action of an externally applied electric field. So, we shall start with a review of light propagation in anisotropic media. These materials usually exhibit piezoelectric effects too [1–3] but, for the sake of simplicity, we shall not consider them here. The electro-optic effect in photorefractive materials is of the highest importance because it is at the origin of the “imaging” of a space-charge field modulation into an index-of-refraction modulation; that is to say, a volume grating. In fact, the build up of a holographic grating in photorefractive materials consists of the spatial modulation of the index-of-refraction in the volume of the sample. In these materials, such a modulation arises from the build-up of a modulated space-charge field that on its turn modulates the index-of-refraction via an electro-optic effect.

1.1 Light Propagation in Crystals Crystals are, in general, anisotropic; that is to say, they have different properties for the light propagating along different directions. 1.1.1

Wave Propagation in Anisotropic Media

Let us start with the general vectorial relations ⃗ = 𝜀o E⃗ + P⃗ D P⃗ = 𝜖o 𝜒̂ E⃗

(1.1) (1.2)

⃗ E⃗ and D ⃗ are where 𝜀0 = 8.82 × 10−12 coul/(mV) is the permittivity of vacuum. The quantities P, the polarization, electric field and displacement fields, respectively, with 𝜒̂ (polarizability) being a tensor that, for isotropic media only, can be written as a scalar thus simplifying the relation in Eq. (1.2) P⃗ = 𝜖o 𝜒 E⃗

(1.3)

The relation in Eq. (1.2) can also be written as ⎡ P1 ⎤ ⎡ 𝜒11 𝜒12 𝜒13 ⎤ ⎡ E1 ⎤ ⎢ P2 ⎥ = 𝜀o ⎢ 𝜒21 𝜒22 𝜒23 ⎥ ⎢ E2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ P3 ⎦ ⎣ 𝜒31 𝜒32 𝜒33 ⎦ ⎣ E3 ⎦ Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

(1.4)

6

1 Electro-Optic Effect

and also ⃗ = 𝜀o (1̂ + 𝜒) ̂ E⃗ D

(1.5)

where 1̂ and 𝜒̂ are tensors that are written as: ⎡1 0 0⎤ 1̂ = ⎢ 0 1 0 ⎥ ⎢ ⎥ ⎣0 0 1⎦

⎡ 𝜒11 𝜒12 𝜒13 ⎤ 𝜒̂ = ⎢ 𝜒21 𝜒22 𝜒23 ⎥ ⎥ ⎢ ⎣ 𝜒31 𝜒32 𝜒33 ⎦

(1.6)

Let us recall that there is always a set of coordinate axes, called “principal axes” where 𝜒̂ assumes a diagonal form ⎡ 𝜒11 0 0 ⎤ 𝜒̂ = ⎢ 0 𝜒22 0 ⎥ ⎥ ⎢ ⎣ 0 0 𝜒33 ⎦ 1.1.2

(1.7)

General Wave Equation

The equation describing the electromagnetic wave in nonmagnetic and noncharged media can be deduced from Maxwell’s equations ⃗ 𝜕H ∇ × E⃗ = −𝜇o 𝜕t ⃗ ⃗ ⃗ = 𝜀o 𝜕 E + 𝜕 P + J⃗ with J⃗ = 𝜎 E⃗ ∇×H 𝜕t 𝜕t 1 ∇ . E⃗ = − ∇ . P⃗ 𝜀o

(1.8) (1.9) (1.10)

⃗ =0 ∇.H

(1.11)

In a system of principal coordinate axes, it is P1 = 𝜀o 𝜒11 E1 P2 = 𝜀o 𝜒22 E2 P3 = 𝜀o 𝜒33 E3 1.1.3

D1 = 𝜀11 E1 D2 = 𝜀22 E2 D3 = 𝜀33 E3

𝜀11 = 𝜀o (1 + 𝜒11 ) 𝜀22 = 𝜀o (1 + 𝜒22 ) 𝜀33 = 𝜀o (1 + 𝜒33 )

(1.12)

Index Ellipsoid

We shall write the expressions for the electric 𝑤e and magnetic 𝑤m energy densities in electromagnetic waves as [4] 1 ⃗ 1∑ 1 ⃗ = 1 𝜇H 2 E𝜖 E 𝑤m = B⃗ . H (1.13) = 𝑤e = E⃗ . D 2 2 kl k kl l 2 2 and write the Poynting formulation for the energy flux as ⃗ H ⃗ S⃗ = EX

(1.14)

After adequate substitutions and transformations taking into account Maxwell’s equations, for the principal coordinate axes we get D2x D2y D2z + + = 8𝜀o 𝜋𝑤e = constant 𝜖x 𝜖y 𝜖z

𝜖x ≡ 𝜖11 = 1 + 𝜒11 𝜖y ≡ 𝜖22 = 1 + 𝜒22 𝜖z ≡ 𝜖33 = 1 + 𝜒33

(1.15)

1.1 Light Propagation in Crystals

Following the definitions D x= √ x 𝑤e 𝜀 o Dy y= √ 𝑤e 𝜀 o Dz z= √ 𝑤e 𝜀 o with n2x = 𝜖x = 𝜀x ∕𝜀o n2y = 𝜖y = 𝜀y ∕𝜀o n2z = 𝜖z = 𝜀z ∕𝜀o we get the indicatrix formulation y2 x2 z2 + + =1 n2x n2y n2z

(1.16)

where nx , ny and nz are the index-of-refraction along coordinates x, y and z, respectively, as represented in Fig. 1.1. In order to use this ellipsoid to analyze the propagation of a plane wave ⃗ we just intersect the indicatrix with a plane orthogonal to the vector with propagation vector k, ⃗ k. An elliptic figure results where the extraordinary ne and ordinary no indexes for this wave are found from the intersection with the corresponding direction of vibration of the electric field, as shown in Fig. 1.2. In the next section, we shall analyze Eq. (1.16) in a more general form. Figure 1.1 Refractive index ellipsoid.

nz

z

x

y

ny

nx

z

Figure 1.2 Refractive indices for a plane wave propagating in an anisotropic medium.

nz k

ne

y x

nx

no

ny

7

8

1 Electro-Optic Effect

1.2 Tensorial Analysis Let us write the general equation [5] ∑

i=N,j=N

Sij xi xj = 1 or Si,j xi xj = 1

(1.17)

i=1,j=1

where xi and xj are variables and Sij are coefficients. If we assume that Sij = Sji , then Eq. (1.17) turns into the general ellipsoid representation: S11 x21 + S22 x22 + S33 x23 + 2S12 x1 x2 + 2S13 x1 x3 + 2S23 x2 x3 = 1 Equation (1.18) can be transformed into new coordinate axes transformation matrix, as follows x′1 = a11 x1 + a12 x2 + a13 x3 x′2 = a21 x1 + a22 x2 + a23 x3 x′3 = a31 x1 + a32 x2 + a33 x3

(1.18) x′i ,

by using the axes-

(1.19)

that can be written in a matricial form ⎡ x′1 ⎤ ⎡ a11 a12 a13 ⎤ ⎡ x1 ⎤ ⎢ x′ ⎥ = ⎢ a21 a22 a23 ⎥ ⎢ x2 ⎥ ⎢ 2′ ⎥ ⎢ ⎥⎢ ⎥ ⎣ x3 ⎦ ⎣ a31 a32 a33 ⎦ ⎣ x3 ⎦

(1.20)

From the matricial relation (1.20), we should deduce that it is also ′ ⎡ x1 ⎤ ⎡ a11 a21 a31 ⎤ ⎡ x1 ⎤ ⎢ ⎢ x2 ⎥ = ⎢ a12 a22 a32 ⎥ ⎢ x′ ⎥⎥ ⎥ 2 ⎢ ⎥ ⎢ ⎣ x3 ⎦ ⎣ a13 a23 a33 ⎦ ⎢⎣ x′ ⎥⎦ 3

(1.21)

The relation in Eq. (1.20) can be written in the form xi = aki x′k

xj = alj x′l

(1.22)

that substituted into Eq. (1.18) leads to Sij xi xj = Sij aki alj x′k x′l = Skl′ x′k x′l

(1.23)

where Skl′

are the coefficients in the new coordinate system. An ellipsoid can be used to describe any symmetric tensor (Sij = Sji ) of second order, and is especially useful to describe any property in a crystal that should be represented by a tensor. An important property of an ellipsoid is the presence of “principal axes”, in which case Eq. (1.18) can be simplified to S11 x21 + S22 x22 + S33 x23 = 1

⇒

⎡ S11 0 0 ⎤ Sij = ⎢ 0 S22 0 ⎥ ⎥ ⎢ ⎣ 0 0 S33 ⎦

(1.24)

1.3 Electro-Optic Effect The indicatrix in Eq. (1.16) is an ellipsoid in a principal coordinate axes system. Its general formulation is [5] 1 (1.25) Bij xi xj = 1 with Bij = 𝜖ij

1.3 Electro-Optic Effect

The slight variation in the refractive index produced by an electric field can be described by the third-order electro-optic tensor rijk (in the range of 10−12 m∕V for most materials) through the relation (1.26)

ΔBij = rijk Ek ⇒

From Bij = Bji

(1.27)

rijk = rjik

The B-tensor can be written as ⎡ B11 B12 B13 ⎤ ⎡ B1 B6 B5 ⎤ ⎢ B21 B22 B23 ⎥ = ⎢ B6 B2 B4 ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ B31 B32 B33 ⎦ ⎣ B5 B4 B3 ⎦

(1.28)

The electro-optic relation is therefore simplified to (i = 1, 2, 3, 4, 5, 6; j = 1, 2, 3) ΔBi = rij Ej

(1.29)

or explicitly written as ⎡ ΔB1 ⎤ ⎡ r11 ⎢ ΔB2 ⎥ ⎢ r21 ⎢ ΔB3 ⎥ = ⎢ r31 ⎢ ⎥ ⎢ ⎢ ... ⎥ ⎢ ... ⎣ ΔB6 ⎦ ⎣ r61

r12 r22 r32 ... r62

r13 ⎤ r23 ⎥ ⎡ ΔE1 ⎤ r33 ⎥ ⎢ ΔE2 ⎥ ⎥ ⎥⎢ ... ⎥ ⎣ ΔE3 ⎦ r63 ⎦

(1.30)

Let us assume that an electric field is applied, with components E1 , E2 , E3 as represented in Fig. 1.3 for a sillenite crystal so that Eq. (1.25) turns into: (B1 + r11 E1 + r12 E2 + r13 E3 )x21 + (B2 + r21 E1 + r22 E2 + r23 E3 )x22 + (B3 + r31 E1 + r32 E2 + r33 E3 )x23 + (B4 + 2r41 E1 + 2r42 E2 + 2r43 E3 )x2 x3 + (B5 + 2r51 E1 + 2r52 E2 + 2r53 E3 )x1 x3 + (B6 + 2r61 E1 + 2r62 E2 + 2r63 E3 )x1 x2 = 1

(1.31)

We are interested in the slow index-of-refraction build up produced by the slow accumulation of electric charges. Therefore, all the electro-optic coefficients referred to in this chapter are only the low-frequency ones only. In the following sections, we shall see what Eq. (1.31) looks like for some particular materials.

Figure 1.3 Crystallographic axes of a sillenite and an applied 3D electric field.

y z

(001) (110)

x

E3 x3

E2

E1 x1

x2

9

10

1 Electro-Optic Effect

Figure 1.4 Structure of an undistorted cubic perovskite structure with general chemical formula ABX3 . The differently shaded spheres represent X atoms (usually oxygens), B atoms (a smaller metal cation, such as Ti4+ ) and A atoms (a larger metal cation, such as Ca2+ ).

Figure 1.5 Three-dimensional sillenite structure: darker spheres represent Bi3+ ions and paler gray ones are O2− . Acknowledgments to Prof. Jesiel F. Carvalho, IF/UFG-Goiânia-GO, Brazil.

1.5 Sillenite Crystals

1.4 Perovskite Crystals Calcium titanium oxide (CaTiO3 ) is a typical representative of the Perovskite crystal structure. The general chemical formula is ABX3 , where “A” and “B” are two cations of very different sizes, and “X” is an anion (usually O) that bonds to both (see Fig. 1.4). The “A” atoms are larger than the “B” atoms and the latter is in six-fold coordination, surrounded by an octahedron of anions, and the “A” cation in 12-fold cuboctahedral coordination. Perovskites are the best-known and the largest family of ferroelectric and piezoelectric materials, such as single crystals of BaTiO3 , PbTiO3 , Pb(Zr, Ti)O3 and KNbO3 .

1.5 Sillenite Crystals Sillenites are cubic crystal structures with general chemical formula Bi12 MO20 where M=Si, Ti,Ge,Ga (see Fig. 1.5). The well-known crystals of this family are: Bi12 GeO20 (BGO),Bi12 GaO20 (BGaO), Bi12 SiO20 (BSO), and Bi12 TiO20 (BTO). They belong to the cubic noncentrosymmetric crystal point class 23 and are piezo-electric, electro- and elasto-optic, optically active and usually photoconductive. BTO is the crystal with the lowest optical activity (optical activity is undesirable for most applications) but is also the most difficult to grow because the chemical composition of the melt and the crystal are different, they are noncongruent. These crystals are usually grown using the so-called “top seed solution growth” (TSSG)20 . The axes in the sample are conveniently renamed, accounting for its cubic and isotropic nature in which case the axes [001], [010] and [100], for example, can be interchanged. In the slanted-sliced sample in Fig. 1.6, the striations are not visible through the polished (110)-face (see also Figs 1.7 and 1.8). The electro-optic tensor of this crystal family in the principal-axes coordinates [X1 , X2 , X3 ] has the following elements [7]: r41 = r52 = r63 ≈ 5 × 10−12 m∕V

(1.32)

all other elements being zero. 1.5.1

Index Ellipsoid

In the absence of electric field (E = 0), the index ellipsoid is x21 + x22 + x23 n2o

=1

Figure 1.6 Schematic representation of a raw BTO crystal boule with its striations, indicating the way it will be sliced (top left); already sliced crystal with striations not perpendicular to the (011)-face (top right) and ready-to-use crystal with renamed axes (bottom).

(1.33) [001]

[100]

(0 11 )

[001] (110)

11

12

1 Electro-Optic Effect

Figure 1.7 Bi12 TiO20 crystal boule as grown along its [001]-axis. ] 01

[0

Figure 1.8 Actual undoped sillenite crystals: raw Bi12 TiO20 crystal boule grown along its [001]-axis, showing striations on the lateral surfaces with both opposite (001)-faces cut and polished (left); Bi12 SiO20 crystal showing its (110)-surface cut and polished (center) and Bi12 TiO20 crystal with its larger (110)-face cut and polished with its [001]-axis direction along its longer dimension (right).

showing that we are dealing with an isotropic crystal. Applying an electric field along direction “x” as indicated in Fig. 1.10, we have the field components: √ 2 E1 = E2 = E (1.34) E3 = 0 2 so that the index ellipsoid is modified to: x21 n2o or

+

x22 n2o

+

x23 n2o

+ 2r41 E1 x2 x3 + 2r52 E2 x1 x3 = 1

√ 2 + 2 + 2 + 2r41 E (x x + x1 x3 ) = 1 2 2 2 3 no no no x21

x22

x23

(1.35)

(1.36)

1.5 Sillenite Crystals

Let us now rotate the system from coordinates X1 , X2 , X3 to coordinates X, Y , Z in Fig. 1.10: √ 2 x = (x1 + x2 ) (1.37) 2 √ 2 y = (x2 − x1 ) (1.38) 2 (1.39)

z = x3 which, when substituted into Eq. (1.36) and rearranged gives y2 z2 x2 + + 2 + 2r41 E x z = 1 2 2 no no no

(1.40)

To eliminate this term in “xz” it is necessary to carry out another rotation, now in the “x–z” plane as shown in Fig 1.11 √ 2 x = (𝜂 + 𝜁 ) (1.41) 2 √ 2 z = (𝜂 − 𝜁 ) (1.42) 2 which, when substituted into Eq. (1.40) gives the relation ) ) ( ( y2 1 1 2 𝜁2 − r E + 𝜂 + r E + 2 =1 (1.43) 41 41 2 2 no no no which means that the refractive indices along the new axes 𝜁 , 𝜂 and y are: 1 n𝜁 = no + n3o r41 E 2 1 3 n𝜂 = no − no r41 E 2 for no ≫ 1.5.1.1

(1.45) (1.46)

ny = no n3o r41 E∕2.

(1.44)

The wavelength dependence of no is reported in Fig. 1.9.

Index Ellipsoid with Applied Electric Field

Following the mathematical development here, it is possible to show that, for an electric field E⃗ along the axis [001], as shown in Fig. 1.12, the principal axes of the index ellipsoid are ( ( ) ) z2 1 1 2 2 + r63 E + y − r63 E + 2 = 1 (1.47) x 2 2 n0 n0 n0 with its principal axes directed along x, y and z and the corresponding indexes of refraction being: 1 nx = n0 − n30 r63 E 2 1 3 ny = n0 + n0 r63 E 2 nz = n0

(1.48) (1.49) (1.50)

13

1 Electro-Optic Effect

2.70

2.65

n

14

2.60

2.55

2.50 450

500

550

λ (nm)

600

650

700

Figure 1.9 Index-of-refraction of BTO that is formulated by n = 0.00863∕𝜆4 + 0.0199∕𝜆2 + 2.46 [6].

y z

Figure 1.10 Bi12 SiO20 -type cubic crystal and its crystallographic axes X1 , X2 and X3 with an externally electric field E applied along the “x”-direction.

(001) (110)

x2

x3 E

x

z

x1

η

η

η

n𝜁 x 45°

𝜁

nη

n𝜁

45° x

nη

no

no E

x

𝜁

E

𝜁

Figure 1.11 Principal coordinate axes system 𝜂 − 𝜁 arising by the effect of an electric field E applied along the “x”-axis, as shown in Fig. 1.10.

1.5 Sillenite Crystals

Figure 1.12 Sillenite crystal cut along its principal crystallographic axes, with an electric field along the [001]-axis.

[001]

Z

X3

X

X2 E

[010] Y X1 [100]

thus meaning that, in the input crystal plane (110) that is also the x–z plane, the index of refraction changes only along x and is constant along z. If the sillenite crystal is cut along its principal axes as shown in Fig. 1.12, with the (X1 , X3 ) being the input plane, and the electric field E⃗ always along axis [001], it will produce index-of-refraction variations only in the (X1 , X2 ) plane but nothing in the input plane (X1 , X3 ). That is the reason why, for practical applications, these crystals should be cut as in Fig. 1.10 and not along its crystallographic axes as in Fig. 1.12. 1.5.2

Other Cubic Noncentrosymmetric Crystals

GaAs, InP and CdTe are also cubic noncentrosymmetric crystals, though they belong to the point class 43m but have the same electro-optic tensor structure as sillenites, that is to say that all elements are zero except r41 = r52 = r63 = 1.72 pm V−1

for GaAs

(1.51)

r41 = r52 = r63 = 1.34 pm V−1

for InP

(1.52)

r41 = r52 = r63 = 5.5 pm V−1

for CdTe

(1.53)

The 43m symmetry, however, guarantees that there is no optical activity. The indexof-refraction of CdTe varies from 2.86 at 𝜆 = 1.06 μm to 2.73 at 𝜆 = 1.55 μm and follows the relation [8]: n2 = 4.744 + 1.5.3

𝜆2

2.424𝜆2 − 282181.61

(1.54)

Lithium Niobate

LiNbO3 is a ferroelectric material [9] the structure of which, in principle, can be described as a highly distorted perovskite structure, to which it can be related by a displacive transformation. The LiNbO3 structure is often considered as a distinct structure type from perovskites because small A cations have octahedral six-fold coordination instead of a 7–12-fold coordination in perovskites [10]. The electro-optic tensor, in the principal axes system [X1 , X2 , X3 ] for this material, has zero elements everywhere except the following ones [11]: r12 = −r22 = r61 ≈ 6.8 pm/V r13 = r23 = 10.0 pm/V

r33 = 32.2 pm/V

(1.55) r42 = r51 = 32 pm/V

(1.56)

15

16

1 Electro-Optic Effect

Figure 1.13 Lithium niobate crystal with an applied electric field along the photovoltaic c-axis.

x1

c x3 x2

E3

n0+Δn2

n0–Δn2

ne–Δn3

ne+Δn3 n0+Δn1

n0–Δn1

E3

E3

Figure 1.14 Lithium niobate crystal ellipsoid (black) and its modified (gray) size by the action of an applied field in opposite directions (left and right pictures) along the c-axis.

For an electric field E3 applied along axis x3 as shown in Fig. 1.13, the tensorial equation Eq. (1.31) becomes: ( ( ) ) ) ( 1 1 1 2 2 + r13 E3 x1 + + r13 E3 x2 + + r33 E3 x23 = 1 (1.57) n2o n2o n2e with no = 2.286 and ne = 2.200 at 𝜆 = 633 nm [1] and the following relations ( ) n3o 1 1 Δ(n ) = r E ⇒ Δ(n ) = − = −2 Δ r E 1 13 3 1 2 13 3 n2 n3o ( 1) n3o 1 1 Δ(n ) = r E ⇒ Δ(n ) = − = −2 r E Δ 2 13 3 2 2 13 3 n22 n3o ( ) n3e 1 1 Δ = −2 Δ(n ) = r E ⇒ Δ(n ) = − r E 3 33 3 3 2 33 3 n23 n3e

(1.58) (1.59) (1.60)

and the index-ellipsoid is modified as shown in Fig. 1.14. 1.5.4

KDP-(KH2 PO4 )

This crystal is actually not a photorefractive one but is here included as an example of electro-optic tensor somewhat similar to that of sillenites. It has the following electro-optic tensor: ⎡ 0 ⎢ 0 ⎢ 0 rij = ⎢ r ⎢ 41 ⎢ 0 ⎢ ⎣ 0

0 0 0 0 r52 0

0 ⎤ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ 0 ⎥ ⎥ r63 ⎦

r41 = r52 = 8.6 pm∕V

r63 = 10.6 pm∕V

The index-of-refraction for this material is reported in Table 1.1.

(1.61)

1.6 Concluding Remarks

Table 1.1 Index of refraction of KDP. 𝝀 (nm)

no

ne

546

1.5115

1.4698

633

1.5074

1.4669

The indicatrix equation formulated in the principal coordinate (crystallographic) axes X1 , X2 and X3 , as represented in Fig. 1.10, is x21 n2o

+

x22 n2o

+

x23 n2e

+ 2r41 E1 x2 x3 + 2r52 E2 x1 x3 + 2r63 E3 x1 x2 = 1

(1.62)

Le us assume an externally applied field E3 along the axis x3 only. In this case we should proceed as for the case of Bi12 SiO20 in Fig. 1.11 in order to get the following ellipsoid ( ) ( ) y2 1 1 2 2 𝜁 − r E + r E =1 (1.63) + 𝜂 + 63 3 63 3 n2o n2o n2e with 1 n𝜁 = no + n3o r63 E3 2 1 3 n𝜂 = no − no r63 E 2 ny = ne 1.5.5

(1.64) (1.65) (1.66)

Bismuth Tellurium Oxide-Bi2 TeO5 (BTeO)

This one is a photovoltaic crystal (see Section 2.4.1.2.1) with 𝛼 = 5 cm−1 at 𝜆 = 532 nm [12] and 𝜖 = 70 [13].

1.6 Concluding Remarks The aim of this chapter was just to recall some fundamental properties of optically anisotropic materials and the way an electric field is able to modify the index ellipsoid via an electro-optic effect. We have briefly shown how to calculate these effects in a few kinds of crystal having different electro-optic tensors. We hope these examples will enable the reader to understand how to operate on different materials, different crystals and different optical configurations.

17

19

2 Photoactive Centers and Photoconductivity Photorefractives are electro-optic and photoconductive [14] materials, which means that electrons and/or holes are excited, by the action of light, from photoactive centers (donors or acceptors) somewhere inside the forbidden energy Band Gap (BG) to the Conduction Band (CB) (electrons) or Valence Band (VB) (holes), where they accumulate and diffuse away under the action of the diffusion gradient or are drifted in the presence of an externally applied electric field. After moving along an average diffusion length LD (or drift length LE in the case of an applied field) they are retrapped somewhere else, excited again and retrapped again, and so on. Such a process leads, in the presence of a spatially modulated intensity of light onto the material, to charge carriers being progressively accumulated in the less illuminated regions whereas the more illuminated regions become oppositely charged. Such a spatial modulation of charged traps produces separation of electric charges and an associated electric (space-charge) field that is able to modify the index-of-refraction via electro-optic effect as explained in Section 1.3. The movement of charges under the action of the diffusion gradient is opposed by the growing space-charge field until an equilibrium is achieved. The presence of defects forming localized states in the Band Gap is therefore absolutely necessary to enable building up the space-charge field that is at the basis of the photorefractive effect. These defects may arise from doping (Fe in LiNbO3 , for example) and are called “extrinsic”. Or they may be the so-called “intrinsic” defects, produced during the growing process, that result from missing atoms or atoms occupying the position of other different atoms in the crystalline structure. To confirm the role of defective growing on the final crystal properties, some researchers have already reported [15, 16] that Bi12 SiO2 grown by hydrothermal methods produces almost perfect intrinsic crystals without photochromic and photorefractive properties while Czochralski and Bridgman–Stockbarger techniques, using the same raw chemicals, produce (defective enough) crystals with photorefractive properties. The interference of coherent laser beams is able to produce sinusoidally modulated patterns of light with small spatial periods of the order of the wavelength dimension of the recording light. Such small periods produce rather large diffusion gradients and consequently large opposing space charge fields can be obtained in this way. Space-charge fields of a few kV per cm are easily produced in this way and consequently rather large overall index-of-refraction changes can be observed. These effects may be produced by light with photonic energy h𝜈 high enough to excite charge carriers but lower than that of the Band Gap (Eg ) so that the material is rather transparent to this radiation. Therefore, the recording is carried out in the whole material volume and the recording beams are also able to detect the effect of their own action: They are refracted or diffracted by the index-of-refraction variation they are producing themselves in (almost) real time on the material volume. Of course, the whole process in the material volume depends on the distribution of light inside it, so that the bulk absorption and the light-induced absorption (if existing) effects need to be accounted for. The spatial modulation of charge is in Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

20

2 Photoactive Centers and Photoconductivity

fact represented by a spatial distribution of acceptors that have received an electron and donors that have lost one. The dielectric polarizability of such filled and emptied photoactive centers (traps1 ) is not necessarily the same as that one of their initial state. This means that the real (index-of-refraction itself ) and the imaginary (extinction coefficient or absorption) part of the complex index-of-refraction may be also modulated via trap modulation in the material’s volume. Such so-called trap-arising index-of-refraction and absorption modulation are related to the electro-optical-based effect but they are nevertheless additional effects of a different nature. We shall see further on that the index-of-refraction modulation arising from trap polarizability modulation and the one arising from space-charge modulation are mutually 𝜋∕2-phase-shifted and both are, in general, also shifted from the recording pattern of light. There is still the possibility to find an additional local index-of-refraction and absorption modulation effect arising from the direct action of light on the material without any relation with charge carriers’ excitation and trapping modulation. Both the trap-arising and the electric field-arising index-of refraction modulations are essentially originating from a spatial modulation of electric charges but have different sizes, properties and characteristics. The building up of such a space-charge modulation is determined by the dynamics of electric charges transport in the material and is characterized by a time constant that depends, among other parameters, on the Maxwell (or dielectric) relaxation time 𝜏M that on its turn is inversely proportional to the conductivity 𝜎. The relation between the holographic build-up time and the conductivity makes holography a particularly interesting technique for the measurement of conductivity. In practice, however, these relations are somewhat more complicated because the recording and erasure of holograms are influenced by self-diffraction effects. The conductivity may also vary along the inter-electrode distance because of the difficulty of avoiding nonuniform light distribution on the sample and may certainly also vary along the crystal thickness because of the nonuniform distribution of light produced by the bulk optical absorption effect [17] the size of which will depend upon the kind of material, the particular sample, and the light wavelength. The reader may foresee the difficult task that may be involved in the analysis of the experimental data depending on the characteristics of the sample under analysis. A sample exhibiting a behavior that can be understood using the so-called “one-center/two-valence/one charge carrier” model is simple to analyze. However, some materials may require a “two-center”, or a “one-center/three-valence” model and so on [18]. Shallow and deep traps may coexist, and even hole-electron competition may appear. The mathematical model may become so much complicated as to prevent a quantitative analysis unless considerable simplifications are accepted. This chapter starts with a brief description of photoactive centers in the Band Gap for some paradigmatic materials – CdTe, Bi12 TiO20 and LiNbO3 :Fe – in order to point out their complex nature and provide a more realistic background for better understanding the description of the theoretical models in the following sections.

2.1 Photoactive Centers: Deep and Shallow Traps In the following sections, we shall describe some well-known photorefractive materials to illustrate the physical model involved as well as to provide some information about these materials, which will be studied in Part III of this book. 1 Unless otherwise stated, we shall use the term “trap” in its more general sense meaning localized states in the Band Gap that are able to receive charge carriers.

2.1 Photoactive Centers: Deep and Shallow Traps

2.1.1

Cadmium Telluride

CdTe is a large Band Gap (1.6 eV at 4 K to 1.5 eV at 300 K [8]) semiconductor of the II–VI family with face-centered cubic structure, binary analog to diamond. The Cd—Te bonds are sp3 -type atomic hybrid orbitals. Each atom is surrounded by a tetrahedron of the other atom species [8]. The CdTe is a well-studied material and will be here analyzed as an example in order to understand the effect of dopants (deep and shallow traps) in the properties of materials. Pure intrinsic CdTe is theoretically very resistive with very low dark conductivity. It exhibits intrinsic defects which are believed to be a Cd vacancy (VCd ) at about 0.4 eV above the VB, acting as an acceptor and a Te occupying a Cd vacancy (Te in Cd antisite represented by the symbol TeCd ) at about 0.23 eV below the CB, acting as a donor [19]. There are also extrinsic defects like Fe, Mn and so on. Cd-vacancies give the p-type character to the dark conductivity. It is possible to increase the number of such vacancies by annealing under vacuum. It is also possible to reduce these Cd-vacancies by annealing under Cd-vapor atmosphere, but it is not possible to completely eliminate them. In principle it is possible to perfectly compensate the Cd vacancies near the VB with TeCd donors close to the CB: the electrons from the latter will fill in the Cd-vacancy acceptors and dark conductivity will be strongly reduced. Once more this is not practical because one or the other defect will be always in excess. The excess of Te-donors or Cd-vacancy acceptors, however, can be compensated by doping with vanadium. In the absence of other dopants the V2+ /V3+ level is near the middle of the Band Gap and is considered to be a deep trap. For a sufficiently large concentration of V2+ and V3+ the Fermi level is pinned to this V2+ /V3+ energy level. In the realistic example of Fig. 2.1 the Fermi level is thus located at EF = 0.68 eV above the VB with the V2+ slightly below (0.62 eV) and the V3+ slightly above EF . The Fermi level is here shown crossing the the V2+ distribution close to its upper end as well as the lower end of the V3+ distribution, thus indicating that the latter is a little bit filled with electrons whereas the former is a little bit emptied of electrons. The Fermi level is closer to the VB than to the CB so that dark conductivity is still predominantly by holes. In the presence of a small excess of either TeCd or Cd-vacancies, the electrons from donors or the holes from acceptors are fixed in the deep vanadium level and by this means the free charge carriers can be strongly reduced: that is to say that dark conductivity can be reduced by doping CdTe with vanadium. Figure 2.2 shows the Arrhenius [20] curve for a particular sample having a Fermi level almost exactly in the middle of the Band Gap. In this case, the dark conductivity is probably the lowest possible one for this material. Under illumination we should expect the density of holes and electrons photoexcited to be similar. It is, however, not the case because the mobility of electrons is roughly 10-fold higher (𝜇e ≈ 10μh ) than for holes. Furthermore, the density of V2+ is usually larger than of V3+ centers, thus increasing the influence of electrons in the process. CB

Figure 2.1 Energy diagram for a typical CdTe crystal doped with vanadium, with the Te in the Cd anti-sites at 0.23 eV below the CB and the Cd vacancies 0.4 eV above the VB [19].

0.23 eV

1.6 eV

Te+ Te2+ V3+

EF 0.68 ev 0.62 ev 0.4 ev

VCd

V2+ VB

21

2 Photoactive Centers and Photoconductivity –3.00E+00 1.80E+00

2.00E+00

2.20E+00

2.40E+00

2.60E+00

2.80E+00

3.00E+00

3.20E+00

3.40E+00

–4.00E+00

–5.00E+00

log sigma

22

–6.00E+00

–7.00E+00

–8.00E+00

–9.00E+00

–1.00E+01

1000/T

Figure 2.2 Dark conductivity measured at various temperatures for a CdTe:V crystal (labeled CdTeBR16B3) produced and measured by Dr. J.C. Launay, ICMCB-Bordeaux, France. From the Arrhenius plot, the energy of the Fermi level EA = 0.83 eV is computed.

2.1.2

Sillenite-Type Crystals

The sillenite crystal structure is believed [21] to be formed by linked distorted octahedra with five coordinated B—O bonds and the sixth corner occupied by a lone-electron pair as depicted in Fig. 2.3. The Bi—O octahedra share corners as in Fig. 2.4 to form the crystal framework within which regular tetrahedra sites, as represented in Fig. 2.5 are occupied by Bi ions. The occupied tetrahedra have three oxygen atoms and the fourth corner is occupied by a lone-electron pair or nonbonding pair of electrons. From the point of view of the chemical composition [22], the sillenite structure is represented by Bi12 (Bi3+ 0.8 ◽0.2 ) O19.2

(2.1)

with their tetrahedra centers being 80% occupied by Bi3+ and 20% vacant as represented by ◽ inside the parenthesis in Eq. (2.1). The vacant tetrahedra have four oxygen ions occupying their O

Figure 2.3 Representation of the sillenite octahedra unit with the lone-electron pair in one corner.

O O Bi3+

O

O

·· o

o

o Bi 3+

Figure 2.4 Octahedra sharing corners. o

o o

Bi 3+

··

o o

2.1 Photoactive Centers: Deep and Shallow Traps

Figure 2.5 Sillenite structure showing (dashed lines) the empty tetrahedra formed by four double-octahedra units.

four corners, but in the Bi3+ -occupied tetrahedra one of the oxygens is replaced by the lone electron pair from the central Bi3+ ion. The incorporation of Ti4+ ions in the sillenite structure occurs by partially filling the tetrahedral vacancies and substituting the Bi3+ in the tetrahedra according to the formula 4+ Bi12 (Bi3+ 0.8−4x Ti5x ◽0.2−x ) O19.2+4x

0 ≤ x ≤ 0.2

(2.2)

Besides Bi3+ donors, photoactive acceptors should be also present to allow for space charge spatial modulation, as required to enable optical recording in this material. Acceptors like Bi5+ cannot be included because they are believed [22] to be unstable at the high temperature the crystal is grown. Instead, there is Bi3+ plus a hole h+ in the form of the 4+ ion [Bi3+ + h+ ] that is simultaneously a donor [Bi3+ + h+ ] ⇒ Bi5+ + e−

(2.3)

an acceptor [Bi3+ + h+ ] + e− ⇒ Bi3+

(2.4)

+

is likely to be present, with h assumed to be resonantly distributed among the four oxygens in 3+ + the tetrahedra around the Bi3+ ion. Note that Bi3+ M and [BiM + h ] are absorbing centers whereas 5+ BiM is not. The introduction of this 4+ charged ion in the structure may follow the same pattern as with Ti4+ : one [Bi3+ + h+ ] filling a vacant tetrahedra site and four others substituting four Bi3+ with four oxygen ions completing the missing oxygens in the corners of the substituted tetrahedra and balancing the charges according to the following formula 3+ + h+ ]5z Ti4+ Bi12 (Bi3+ 0.8−4x−4z [Bi 5x ◽0.2−x−z ) O19.2+4x+4z

0 ≤ x + z ≤ 0.2

(2.5)

with the maximum x = 0.2 (and z = 0) leading to Bi12 TiO20 . It is interesting to point out that, when grown in hydrothermal equilibrium, rather perfect Bi12 MO20 structures are obtained that have no photorefractive properties. That is to say that out-of-equilibrium conditions are required to produce sillenite crystals with useful BiM anti-site defects to turn them photoconductors, hence actually photorefractives. The Band Gap energy for all Bi12 GeO20 (BGO), Bi12 SiO20 (BSO) and Bi12 TiO20 (BTO) was determined to be Eg = 3.2 eV (𝜆 ≈ 400 nm) at room temperature [23]. Same value was found for Bi12 GaO20 (BGaO). The fact that the absorption edge is the same for all four materials can be explained by assuming that an identical Bi—O lattice in all these crystals is responsible for the Band Gap. Their yellow color, on the other hand, is due to a broad absorption shoulder between 2.3 and 3.2 eV that may be due to the already mentioned incorrect occupation of an M (M≡Ge,Si,Ti) site in the oxygen tetraedron by a [Bi3+ + h+ ], a Bi3+ atom with a bound electron

23

24

2 Photoactive Centers and Photoconductivity

defect h+ resonantly distributed among the four oxygens in the tetrahedron known as “anti-site defect BiM ”. The density of these centers in Bi12 GeO20 is lower (2–3-fold) than in Bi12 SiO20 that on its turn is lower than for [Bi12 TiO20 ]. This is also assumed [23] to be a consequence of the Ge atoms being bonded 0.1 eV stronger than for Si and the latter on its turn being bonded 0.16 eV stronger than for Ti. Sillenites exhibit dark p-type conductivity that is assumed to arise from the fact that the [Bi3+ + h+ ] centers in its formulation of Eq. (2.5) are closer to the VB than to the CB. The activation energy of these electron-acceptor or hole-donor centers was measured using impedance spectroscopy at high temperature that led to 0.99 eV for BTO [24] and to 0.48 ± 0.02 eV for Bi12 GaO20 [25]. It was measured to be 1.1 eV for BGeO and BSO [26]. Direct measurement of dc dark conductivity in the range 50–130∘ C gave 1.06 eV for BTO [27]. Holograms were recorded on BTO with 𝜆 = 514.5 nm (h𝜈 ≈ 2.4 eV) light and their relaxation in the dark was measured [28] at different temperatures (from about 40 to 90∘ C) to construct an Arrhenius curve (as the one shown in Fig. 2.2 for CdTe) from which data an activation energy of 1.05 eV was obtained. This energy is close to the one measured by several researchers for direct plain dark p-type conductivity as reported previously. This means that holographic relaxation in the dark is probably due to p-type conductivity, from the Fermi level down to the VB, roughly about 1 eV below. However, the photoconductivity of these materials is largely n-type, proba+ bly arising from the same [Bi3+ M + h ] (see Eq. (2.5)) centers, 2.2 eV below the CB, which are now acting as electron donors, aside from the electrons also photoexcited from the Bi3+ centers represented in Eq. (2.5). The n-type nature of sillenites under the action of light may be due to a larger cross-section of [Bi3+ + h+ ] for photons, a higher mobility of electrons in the CB or a combination of both effects. Holograms based on photoexcitation of electrons can be recorded on undoped BTO (and on other undoped sillenites too) using light in the wavelength range at least from 𝜆 = 488 nm (h𝜈 ≈ 2.5 eV) to 𝜆 = 633 nm (h𝜈 ≈ 1.96 eV). Using 𝜆 = 780 nm (h𝜈 ≈ 1.6 eV) beams instead, holograms are based on hole-excitation. Even with 𝜆 = 670 (h𝜈 ≈ 1.85 eV) pre-exposure only p-type holograms are recorded as seen in Fig. 8.5. But at least from pre-exposure with 𝜆 = 634 nm (h𝜈 ≈ 1.96 eV) down, electrons excited from a different center appear together with holes as shown in Fig. 8.5. Recording with 𝜆 = 1064 nm (h𝜈 = 1.16 eV) was unsuccessful, whatever the pre-exposure. Direct 𝜆 = 780 nm p-type recording certainly proceeds by direct excitation of holes from [Bi3+ + h+ ] to the VB at an energy gap lower than 1.6 eV. Pre-exposure acting on n-donor centers may act by populating an intermediate center in between the CB and the Fermi level (2.2 eV below the CB) that should be closer than 1.6 eV from the bottom of the CB in order to n-based recording with 𝜆 = 780 nm be possible. The Wavelength-Resolved Photoconductivity (WRP) spectra [29] in Fig. 7.12 clearly show the presence of a filled photoactive center between 1.9 and 2 eV, probably being an electron donor, that explains the recording with 𝜆 = 633 nm (h𝜈 = 1.96 eV) on undoped BTO. The relaxed crystal in the same figure, however, does not show this center so that we should deduce that the latter is filled by previous illumination or by the same red 𝜆 = 633 nm light during recording itself. A large photoactive center is shown at 2.2 eV that is likely to be the Fermi level that is probably pinned to the [Bi3+ + h+ ] center described in Eq. (2.5). From the reported experimental facts, it is probable that the BTO Fermi level is pinned to the [Bi3+ + h+ ] donor-acceptor centers about 1 eV above the VB and 2.2 eV below the CB. These materials also exhibit a strong photochromic darkening effect upon illumination with light of wavelength at least in the 514.5–780 nm range, although the effect decreases with increasing wavelength. Photochromic darkening is a strong effect but a rather slow process that saturates at comparatively low light intensities, at least for the 532 and 514.5 nm

2.1 Photoactive Centers: Deep and Shallow Traps

wavelengths. This photochromic effect cannot be explained by the simple one-center model. In fact, the one-center model assumes that moderately low intensity light onto the sample will not significantly change the total-to-acceptor trap density ratio but will just produce a spatial modulation in its value where the spatial average will remain constant so that no photochromic effect could be detected under a uniform illumination. The two-center model instead may allow for a kind of light-induced absorption coefficient or photochromic effect as will be seen in Section 2.6. WRP, modulated photoconductivity, photochromic measurement and holographic recording [29], among other experiments, have indicated the presence of several localized states in the Band Gap of undoped Bi12 TiO20 , among which are a shallow empty level at 0.42 eV (probably below the CB) that is responsible for photochromism and an electron donor center at 2.2 eV below the CB. Dark p-type conductivity was associated with an activation energy of about 1 eV; the latter is probably referred to the VB and, according to the 3.2 eV Band Gap, is probably the same electron-donor level at 2.2 eV below the CB. This is probably the Fermi level associ+ ated with the position of the electron donor/acceptor (Bi3+ Ti + h ) center referred to in Eqs. (2.3) and (2.4). At least a couple (or more) empty levels (one certainly at 2.0 eV) should also be present between the 2.2 eV Fermi level and the CB to explain holographic recording using light with photonic energy as low as 1.6 eV. Other levels at 0.10, 0.14 and 0.29 eV, either located below the CB or above the VB, were detected by modulated photocurrent (MPC) techniques. Electron-donor levels farther than 2.2 eV from the CB were also detected by WRP. A possible representation [29] of some of the relevant states in the Band Gap of undoped Bi12 TiO20 is shown in Fig. 2.6, where the 0.42 eV level responsible for photochromism is shown as well as the 0.10, 0.14 and 0.29 eV that were arbitrarily placed close to the CB. In spite of the practical interest in sillenites and the large number of publications on these materials, their actual nature is still poorly known and is a subject of active research. Thus, the model of localized states in the Band Gap represented in Fig. 2.6, as well as the nature of most of the photoactive centers involved, should be considered to be tentative representations subject to revision. 2.1.2.1

Doped Sillenites

From the paper by Valant and Suvorov [22] we should get general rules about the way sillenites may be doped: • Doping with M2+ : M2+ occupy the tetrahedral site with three ions entering the vacant site while two of the ions substitute the Bi3+ . The removal of the latter with their lone-electron pairs from the tetrahedral site opens space to incorporate two oxygen ions for charge compensation as in the formula in Eq. (2.5) as follows: 3+ 2+ Bi12 (Bi3+ + h+ ]5z Ti4+ 0.8−4x−4z−2y [Bi 5x M5y ◽0.2−x−z−3y ) O19.2+4x+4z+2y

0 ≤ x + z + 3y ≤ 0.2

(2.6)

with the saturated substitution being 3+ 2+ Bi12 (Bi3+ + h+ ]5z Ti4+ 10y [Bi 5x M5y ) O20−10y

(2.7)

with x + z + 3y = 0.2

(2.8)

Most common dopants here are Cd, Pb, Co and Zn. For Pb-doped BTO, see Section 8.5.2.2.

25

2 Photoactive Centers and Photoconductivity

Conduction band 0.10 eV 0.14 eV 0.29 eV 0.42–0.44 eV

1.3 eV 1.4 eV

3.2 eV

1.7 eV 19 eV 2.0 eV

2.4 eV

2.5 eV 2.6 eV

1 eV

2.2 eV

26

Valence band

Figure 2.6 Localized states in the Band Gap of nominally undoped Bi12 TiO20 crystal, from [29]. Filled electron-donors are in gray and empty ones in white; the DOS (density of states) is qualitatively represented by the width of the full-line limited levels whereas the dashed-line ones are not. The succession of states close to the VB represents the almost continuous states except the few discrete ones at 2.4 and 2.5 eV. Reproduced from [29].

• Doping with M3+ : Two M3+ ions occupy the vacant tetrahedral sites and three substitute the Bi3+ allowing the incorporation of three oxygen ions with the formula 3+ 3+ + h+ ]5z Ti4+ Bi12 (Bi3+ 0.8−4x−4z−3y [Bi 5x M5y ◽0.2−x−z−2y ) O19.2+4x+4z+3y

(2.9)

with saturation 3+ 3+ Bi12 (Bi3+ + h+ ]5z Ti4+ 5y [Bi 5x M5y ) O20−5y

(2.10)

with x + z + 2y = 0.2

(2.11)

with dopants of this type already reported being: Ga3+ , Fe3+ , Cr3+ and Ti3+ .

2.1 Photoactive Centers: Deep and Shallow Traps

• Doping with M4+ : One M4+ fills a vacant tetrahedral site while four substitute the Bi3+ opening space for four oxygen ions as follows: 3+ 4+ Bi12 (Bi3+ + h+ ]5z Ti4+ 0.8−4x−4z−4y [Bi 5x M5y ◽0.2−x−z−y ) O19.2+4x+4z+2y

0 ≤ x + z + y ≤ 0.2

(2.12)

and the saturated formula being Bi12 M4+ O20

(2.13) 4+

4+

4+

4+

Most common examples here are M = Si , Ge and Ti . • Doping with M5+ : According to ref. [22], the addition of M5+ in the sillenite formulation of Eq. (2.5) occurs by substituting the Bi3+ ions by M5+ and incorporating an oxygen ion without modifying the tetrahedral vacancies, following the formula: 3+ 4+ Bi12 (Bi3+ + h+ ]5z M5+ y Ti5x ◽0.2−x−z ) O19.2+4x+y+4z 0.8−4x−y−4z [Bi

0 ≤ x + z ≤ 0.2

0 ≤ 4x + y + 4z ≤ 0.8

(2.14)

and the saturated formula is 4+ Bi12 ([Bi3+ + h+ ]1−5x−1.25y M5+ y Ti5x ◽0.25y ) O20 with 4x + y + 4z = 0.8

(2.15)

The most common dopant here is M5+ =V5+ . Note that by reducing the density of Bi3+ centers an indirect reduction in [Bi3+ + h+ ] also occurs because of the −1.25y term in the subindex of [Bi3+ + h+ ] in Eq. (2.15). – BTO:V exhibits a dramatic reduction in photoconductivity [30] compared to that of undoped BTO, certainly due to substitution of Bi3+ and [Bi3+ + h+ ] centers by V. As the exchange of charge carriers (electrons and holes) is essentially operated via these two centers, their lower concentration may explain its dramatic lowering in photoconductivity [30] compared to the undoped crystal. – Its lower optical absorption compared to that of the undoped material [30] also supports the hypotheses of Bi3+ and [Bi3+ + h+ ] being replaced by the nonphotoactive V3+ and V5+ ions. – EPR experiments showed [30] no signals for V, thus indicating it to be in paramagnetic states V3+ and V5+ , which may indicate that these 3+ and 5+ ions may be alternatively substituting the 4+ [Bi3+ + h+ ] centers. – EPR also showed a strong reduction of h+ ions in BTO:V, thus confirming that not only Bi3+ but also [Bi3+ + h+ ] centers are substituted by V. – It is also possible that the presence of the highly charged V5+ ion in BTO:V may inhibit, to some extent, the formation of Bi5+ from electron pumping by light, thus explaining the almost absent pre-exposure effect in WRP experiments [30] – The donor Bi3+ is directly affected by V3+ so that its relative concentration is lower, thus explaining the lower influence of electrons observed [30] in holographic experiments. – The effect of V-doping in reducing dark conductivity is briefly discussed in Section 2.3.2.3.

27

28

2 Photoactive Centers and Photoconductivity

2.1.3

Lithium Niobate

The deep trap centers in this material are known to be Fe2+ /Fe3+ at approximately 1 eV below the 2 CB. There are also shallow traps due to defect Nb4+ Li centers [31] producing polaronic electron conduction with an activation energy 0.1–0.4 eV. There is still ionic conductivity (in as-grown and in hydrogen-doped) predominantly due to H+ ions with characteristic activation energy of 1.2 eV. Hydrogen is located in the oxygen planes along the O–O bond and its relative contents can therefore be evaluated as the strength of the OH− stretching vibration absorption line near 2.87 μm [33]. Above 70–80∘ C, the ionic conductivity largely prevails over the Fe2+ -electron detrapping based dark conductivity. At temperatures below 60∘ C, however, dark conductivity is predominantly due to polaronic electrons from Nb4+ Li centers. For iron concentration larger than 0.05%wt Fe2 O3 , dark conductivity is predominantly due to tunneling of electrons between localized iron sites without significant influence of band transport [34]. 2.1.4

Bismuth Telluride Oxide: Bi2 TeO5

Holographic recording with 𝜆 = 633 nm laser light has put into evidence [35] the presence of a fast and a slow hologram of clear photorefractive nature, probably arising from electrons and holes respectively with, ΦF = 0.2 and ΦS = 0.01 being the quantum efficiency for photoelectron excitation from the fast and the slow centers, respectively. Accordingly, for a short recording time, only the fast grating is recorded. The presence of deep and shallow traps was also detected [36] for this material at 𝜆 = 532 nm, with an exponential thermal relaxation estimated 𝛽 ≈ 70 ± 5 s−1 .

2.2 Luminescence Figure 2.7 shows how a short wavelength (𝜆 = 405 nm in this case) having a very high absorption coefficient 𝛼 produces a larger wavelength (in this case 𝜆 = 570 nm) luminescence throughout the output; the higher photonic energy illumination excites electrons from the Fermi level (or localized states below it) to the CB, where they decay to empty localized states (see the BG diagram in Fig. 2.6) emitting lower photonic energy (h𝜈) light. Fig. 2.8 shows the spectrum of a quasi-monochromatic LED, centered at 𝜆 = 408 nm (left-side peak), incident onto a 2.8 mm thick undoped BTO crystal; the spectrum of the light behind the sample is represented by the much wider peak at the right-side and centered at about 570 nm. 570 nm 2.18 eV

Figure 2.7 Schematic representation of luminescence effect on a sillenite crystal.

405 nm 3.06 eV

BTO

2 The electron placed in a elastic or deformable lattice produces a strain in the lattice. The electron plus the associated strain field is called a polaron. The displacement of this associated field increases the effective mass of the electron: for the case of KCl, for example, the electron mass is increased by a factor of 2.5 with respect to the band theory mass in a rigid lattice [32].

2.3 Photoconductivity 8000

6000

Irradiance (au)

Figure 2.8 Photoluminescence in BTO-008. The dashed line is the spectrum of the light of an LED illuminating the BTO crystal sample. The continuous curve is the spectrum of the light measured at the crystal output, very closed to it. A luminescent peak appears at 570 nm (≈ 2.2 eV).

4000

2000

0

2000 300

500

700

900

1100

λ (nm)

2.3 Photoconductivity Electric conductivity depends on the concentration of free charge carriers (electrons or holes) in the Extended States (conduction or valence bands). In the presence of a relatively large Band Gap (BG), as is the case with most photorefractive materials, the density of free carriers in the Extended States largely depends on the number and quality of Localized (photoactive) States (LS) in the BG. We shall first analyze the effect of these LS and then discuss two simple models referred to in the literature: The one- and the two-center models. We shall then focus on the way the photoconductivity should be measured and the relation between the photocurrent and the photoconductivity. We shall also analyze the photochromic effects arising from the two-center model and relate the measured quantities (conductivity and absorption coefficients) with the theoretical parameters derived from the theory. 2.3.1

Localized States: Traps and Recombination Centers

It is worth recalling that, in an intrinsic semiconductor, the Fermi level is exactly in the middle of the BG as illustrated in Fig. 2.9, with roughly 100% electron occupied states below the Fermi level and zero above. The density of free electrons  in the conduction band (CB) and free holes  in the valence band (VB) are determined by the relations  = NC e−(EC − EF )∕kB T

 = NV e−(EF − EV )∕kB T

(2.16)

where NC and NV are the Density of States (DOS) at the bottom and at the top of the CB and VB, respectively, EC and EV are the corresponding energies and EF is the energy of the Fermi level. In the presence of a sufficiently large density of impurities, the Fermi level may be pinned by the position of these impurities, as illustrated in Fig. 2.10, where all donor levels above the Fermi level are empty in equilibrium (as expected) in the dark, as depicted by the occupation-of-states (from zero to one) diagram shown on the right-hand side. In the example of Fig. 2.10, the density of free holes is larger than of free electrons because EF is closer to the VB than to the CB and we have assumed that NC ≈ NV . This situation can be changed by the action of light. In fact, under the action of sufficiently energetic photonic light, charge carriers are excited so that initially empty LS become populated and the density of free carriers in the CB and/or VB also increases. In order to be able to account of these changes and still allow Eq. (2.16) to be verified, steady-state Fermi (or quasi-Fermi) levels for electrons Efn and for holes Efp are defined [14] as depicted in Fig. 2.11 with the occupation-of-states accordingly modified as represented by the right-hand

29

30

2 Photoactive Centers and Photoconductivity

CB

EC

EF = Eg/2 EF

Fermi

Eg EV 0

VB

1

Figure 2.9 Intrinsic semiconductor: Fermi level for an intrinsic semiconductor and its “energy vs. occupation-of-states diagram” (right side). Figure 2.10 Doped semiconductor: Fermi level pinned at the position of the dopant in the BG. On the right-hand side is the “energy vs. occupation-of-states” diagram.

CB

EF acceptors Fermi

Eg

donors 0

VB

1

Figure 2.11 Doped semiconductor: Fermi Ef and quasi-stationary Fermi levels upon illumination. The “energy vs. occupation-ofstates” graphics is shown on the right-hand side.

ILLUMINATION CB Efn EF Fermi Eg

Efp 0 VB

1

2.3 Photoconductivity

CB band-to-band recombination

recombination

EF

Efn

Fermi Efp

recombination

VB

Figure 2.12 Recombination centers.

diagram [37]. The density of free carriers is now as follows  = NC e−(EC − Efn )∕kB T

 = NV e

−(Efp − EV )∕k)BT

(2.17)

In this condition, charge carriers in LS in between Efn and Efp are stable and remain in these states for a long time until recombination with an oppositely charged carrier. These levels are therefore called “recombination centers” and are illustrated in Fig. 2.12. LSs outside the Efn -Efp energy band do easily relax their charge carriers to the nearest extended states and are called “traps” as illustrated in Fig. 2.13. In the case of sillenites, recombination centers produced by the action of light remain (at least partially) like that for hours, days or weeks in the dark.

CB trapping EF Efn Fermi Efp trapping VB

Figure 2.13 Traps.

31

2 Photoactive Centers and Photoconductivity

2.3.2

Theoretical Models

The behavior of all three examples (sillenites, CdTe and LiNbO3 :Fe) described before can be adequately generalized by assuming a single species (one ion) with two different valence states such as Fe2+ /Fe3+ for the case of lithium niobate in Section 2.1.3 or Bi3+ /Bi5+ for the case of sillenites in Section 2.1.2 and still V2+ /V3+ for the case of CdTe in Section 2.1.1. Donors and acceptors are incorporated and/or formed in the material during the growing of the crystal in an electrically neutral local environment. That is to say, in thermal equilibrium (in as-grown crystals) ions in their different valence states are therefore stabilized by an adequate environment to produce local electric neutrality so that they are certainly located at different energy positions in the Band Gap, as schematically illustrated in Fig. 2.14, with acceptors above and donors below the Fermi energy level. Depending on their density, these two levels may relevantly contribute to define the position of the material’s Fermi level. These donors and acceptors are intentionally shown in Fig. 2.14 to be distributed along a finite energy bandwidth in the Band Gap, thus emphasizing that they do not occupy one energy position but a narrow energy band. Under the action of light of adequate wavelength, electrons are shown in Fig. 2.15 to be excited from donors to the conduction band (CB), diffuse or are drifted (if there is an external electric field) and, after some time (photoelectron lifetime), they are likely to be retrapped somewhere else in available acceptors, be excited again and so on. On average, the density of electrons in the CB increases by the action of light so that the n-type photoconductivity increases too. A similar situation is described in Fig. 2.16 where, besides electrons, holes are also excited (but to the valence band VB) by the light. The photoconductivity is here produced by electrons and CB

Band Gap

32

+ –

+ – + –

+ – + –

+ –

+ –

+ –

+ –

+ –

+ –

+ –

+ –

+ + – –

VB

Figure 2.14 Schematic representation of a material with one center (one single species) with two valence states (electron donors and electron acceptors) on two correspondingly slightly different localized states in the Band Gap. Electron acceptors are here represented as positively charged so that a nonphotoactive negative ion should be close to it in order to produce electrical neutrality at equilibrium for the as-grown crystal.

2.3 Photoconductivity

CB

–

–

Band Gap

–

+ –

+

+ – + –

+ – +

+ –

+ + – – + +

+ –

+ – +

+ –

+ – +

+ –

+ –

VB

Figure 2.15 Under the action of light (of adequate wavelength) electrons are excited to the CB, thus increasing the electron density in the CB and therefore increasing the n-type (photo)conductivity. In the CB they diffuse (or are drifted if there is an externally applied electric field) and are retrapped (on the available acceptors) again and re-excited and so on.

holes, although electrons appear here to predominate. In other cases, holes could predominate or the photoconductivity could even be only due to holes, without electrons participating in the process. Under nonuniform illumination, as shown in Fig. 2.17, the charge carriers excited to the extended states (CB or VB) do diffuse and/or are drifted and are progressively accumulating in the darker (less illuminated) regions where the excitation rate by light is lower than in the brighter ones. It is important to realize that to produce an overall accumulation of electric charge in the illuminated volume it is necessary to have both donors and acceptors already available in adequately large concentrations. Otherwise, the excited charge carriers (electrons or holes) would have nowhere to be retrapped but to return back to emptied donors (for electrons) or filled acceptors (for holes) in the illuminated volume of the material where they were excited from. After switching the light off, the (deep) trapped electrons remain where they are because thermal excitation is very low to excite them back to the CB at a sensible rate. The result is that the regions that were illuminated become positively charged, whereas those that were less illuminated become negatively charged. If the charge carriers were holes, instead of electrons, the spatial distribution of charges would be obviously the opposite one. If charge carriers were both electrons and holes instead, without externally applied field, there should be a mutual partial compensation of the spatial charge distribution in the material and even no charge accumulation at all could occur in the hypothetical case of both electrons and holes be equally effective in the process. The participation of electrons and/or holes in this process is dependent on the presence of an externally applied electric field, on the respective density of donors

33

2 Photoactive Centers and Photoconductivity

CB

–

–

Band Gap

–

+ –

+

+ – + –

+ – +

–

+ –

+ + – – + +

–

+ – +

–

+ –

+ – +

+ –

+ –

+

+ VB

Figure 2.16 In this example, under the action of light, electrons and holes are excited to the CB and VB, respectively, so that the photoconductivity is due to electrons and holes. In this case, electrons do predominate but it could also be the opposite, or even be only holes being excited and the photoconductivity being of the p-type. CB

Band Gap

34

+ + + – + – + – – – + + negative positive –

–

–

+ –

+ –

+ – + +

negative

positive

+ –

–

+ –

+ –

negative

VB

Figure 2.17 Under nonuniform light, negative charges (in this case, we assume to be electrons only) accumulate in the darker (less illuminated) regions leaving behind, in the more illuminated regions, opposite (positive here) charges.

2.3 Photoconductivity

and acceptors, their respective cross-section coefficients for the illumination wavelength and the mobilities of holes and electrons in their respective extended states (CB or VB). It is important to point out that the overall electric charge density variation, resulting from a local illumination, is due to the local (spatial) variation of donor/acceptor densities independent of this variation being produced by electrons, holes or both in different proportions. That is to say, for one single system of donor/acceptor level, one single structure results even if donors and acceptors are placed on different energy levels. For the sake of simplicity, we may sometimes show such donors and acceptors on the same energy level in the Band Gap just to emphasize the fact that we are handling with one single species, leading to one single structure of spatial trap modulation. For the case of sillenites and at least for undoped Bi12 TiO20 , it is known that at least two distinct gratings are recorded under usual conditions. The single species (or single center) with two valence states cannot explain such a behavior and more than one species should, therefore, be involved. In this case, two independent modulated photoactive (two centers) systems can be produced by the action of light and two gratings can be recorded, each one of them involving electrons and/or holes as is for the case of one single species discussed before. The particular expressions for the density of free electrons in the conduction band and for the photo- and dark conductivity depend on the theoretical model used to describe the material behavior. Here, we shall analyze the two simplest models: one-center and two-center, always with one single charge carrier. Section 3.4.1 in Chapter 3 will discuss the case of two different species (photoactive centers), one based on electron transport and the other based on hole transport. Much more complicated structures can be analyzed following the procedures used to handle these few rather simple examples. 2.3.2.1

One-Center Model

For the simplest one-center/one charge (electrons) carrier model, one single type photoactive + ) is assumed, in which case the charges and space-charge electric field are detercenter (ND1 , ND1 mined by the rate, continuity and Poisson equations: 𝜕 (x, t) 1 = G − R + ∇.⃗j 𝜕t e 𝜕ND+ (x, t) 𝜕t

(2.18) (2.19)

=G−R

G = (ND − ND+ (x, t))

(

sI +𝛽 h𝜈

) (2.20)

R = rND+ (x, t) (x, t)

(2.21)

⃗j = e (x, t)𝜇E(x, ⃗ t) + e∇ (x, t)

(2.22)

⃗ t)) = e(N + (x, t) −  (x, t) − N − ) ∇.(𝜖𝜀o E(x, A D

(2.23)

where e = 1.6 × 10−19 Coul is the absolute electric charge of an electron,  is the free electrons density in the conduction band, ND+ (x, t), ND are the density of ionized empty (electron-acceptors) and total photoactive electron donor centers, respectively, and NA− is the concentration of nonphotoactive negative ions that compensate for the initial (“as grown”) positively ND+ charged traps. Equations (2.18) and (2.19) are charge conservation equations, Eq. (2.22) describes the charge current in terms of drift (the first term) and diffusion (the last term), Eqs. (2.20) and (2.21) describe photoelectron generation and recombination,

35

36

2 Photoactive Centers and Photoconductivity

respectively, with r being the recombination constant. Equation (2.23) is the Poisson’s relation between charge density and electric field. The mobility and diffusion constant for electrons are 𝜇 and , respectively, s is the effective cross-section for photoelectron generation, 𝛽 is the thermal photoelectron generation coefficient and 𝜖 is the dielectric constant of the crystal. For the case of holes (instead of electrons) being the charge carriers, Eqs. (2.18)–(2.23) should be substituted with Eqs. (3.97)–(3.101). Steady State Under Uniform Illumination: Low Irradiance For steady-state equilibrium under uniform illumination, all time and spatial derivatives are zero, so it is G = R, ∇.⃗j = 0 and ⃗ t) = 0. From Eqs. (2.20) and (2.21) we get ∇.E(x, ( ) sI (ND − ND+ ) (2.24) + 𝛽 = rND+  h𝜈 with

2.3.2.1.1

𝜏 ≡ (rND+ )−1

(2.25)

being the photoelectron lifetime that is substituted into Eq. (2.24) and leads to ( ) sI  = (ND − ND+ )𝜏 +𝛽 (2.26) h𝜈 We may describe the photoelectron generation, in terms of the absorbed light, as follows (ND − ND+ )s = 𝛼Φ

(2.27)

which substituted into Eq. (2.26) leads to dIabs Φ𝛼I dI + (ND − ND+ )𝛽𝜏 = − = 𝛼I (2.28) h𝜈 dz dz with 𝛼 being the overall intensity absorption coefficient and z (from z = 0 to z = d) the coordinate along the crystal thickness. In this case, 𝛼I is the effectively absorbed irradiance per unit volume at z and Φ is the quantum efficiency for photoelectron generation. The parameter 𝜏 in Eq. (2.25) is a constant if we assume that the effect of the light on ND+ and ND − ND+ is weak enough not to significantly affect their values. The concentrations of free electrons in the conduction band in the dark and under the action of light are, respectively  =𝜏

d = (ND − ND+ )𝛽𝜏 ph = (ND − ND+ )

sI 𝜏 h𝜈

(2.29) (2.30)

Note that for a spatially uniform and constant illumination I 0 it is sI 0 h𝜈 The general expression for the conductivity is 0 ph = (ND − ND+ )𝜏

(2.31)

𝜎 = e𝜇

(2.32)

and the corresponding expressions for the photo- and dark conductivity are 𝜎ph = e(ND − ND+ )

sI 𝜇𝜏 h𝜈

𝜎d = e(ND − ND+ )𝛽𝜇𝜏

(2.33) (2.34)

2.3 Photoconductivity

Steady State Under Uniform Illumination: High Irradiance The development here assumes that the light irradiance is low enough so that the density of free charges  is negligible compared to the density of acceptors and donors in the material. If this is not the case, Eqs. (2.18)–(2.23) should include the density of electrons  and related equations should be reformulated accordingly and, for the particular case of the density of free electrons, Eq. (2.26) should turn into [38]: ] [ √ N− (2.35)  = A −(1 + f ) + (1 + f )2 + 4f (ND − NA− )∕NA− 2 2.3.2.1.2

f ≡

sI∕(h𝜈) + 𝛽 𝛾NA−

(2.36)

For low irradiances it is f ≪ 1, which substituted into Eq. (2.35) turns into  = f (ND − NA− ) = (ND − NA− )

sI∕(h𝜈) + 𝛽 𝛾NA−

(2.37)

where we have substituted NA− with ND+ and found the expression in Eq. (2.26). For the limiting condition of high irradiances instead, it is f ≫ 1 and the density of free electrons becomes saturated at:  = (ND − NA− )

(2.38)

So far we have been dealing with the “one-center/one-charge carrier” model only. 2.3.2.2

Two-Center/One-Charge Carrier Model

This model is essentially related to the presence of shallow traps ND2 , as represented in Fig. 2.18, which are certainly influencing the electrical conductivity in these materials. We assume [18] + represent the total density of deep centers and the density of the empty deep that ND1 and ND1 + represent the same but for the shallow centers, centers, respectively, whereas ND2 and ND2 + + ∕(ND2 − ND2 ) may be strongly affected by with ND2 being small enough so that the ratio ND2 ILLUMINATION

conduction band CB + –

+ –

+ –

+ –

conduction band CB

+ –

+ –

+ –

+ –

+ –

+ –

+ –

+ –

+ –

+ –

+ –

+ –

–

–

+ –

– –

+ + – + + – + + – + + – +

+ –

– + –

–

+ + – + + – + + –

+ –

E valence band VB – –

nonphotoactive

valence band VB + N+D1 acceptor + N+D2 acceptor

ND1–N+D1 donor

– –

nonphotoactive

+ N+D1 acceptor

+ N+D2 acceptor

N+D1–N+D1 donor

Figure 2.18 Photochromic effect and the band-transport model. On the left side, we represent deep photoactive centers (acceptors and donors) and shallower centers close to the CB, with empty donors + (acceptors) only, labeled ND2 . In this figure electron acceptors, both for deep and for shallow centers, are represented as positively charged so that a nonphotoactive negative ion should be close to these charged acceptors to ensure local electric neutrality. On the right side, we see that under the action of light (represented by the arrows), the electrons are excited into the conduction band. Some of the electrons are retrapped to the + ND+ and some others to the ND2 centers. The latter ones, that slowly relax to the deeper ND+ centers in the dark, have a higher light absorption coefficient and are therefore responsible for the photochromic darkening effect.

37

38

2 Photoactive Centers and Photoconductivity

the action of light, which is not the case for the deep traps. We shall also assume that only one single type (electrons) of charge carrier is involved here. In equilibrium conditions, shallow traps are empty and under the action of light they start to be filled with electrons pumped from the deeper donor centers ND1 . The filled shallow traps ND2 -N+D2 may have a much larger absorption cross-section s2 than the s1 one of the filled deep traps ND1 -N+D1 so that their filling may produce a considerable increase in the overall absorption coefficient. This is the origin of the light-induced photochromic darkening. As the irradiance of the light increases, the density of filled shallow traps increases too until nearly all of them are filled and the light-dependent absorption coefficient saturates. Steady State Under Uniform Illumination In this case, the rate equations in Section 2.3.2.1 should also include those for the shallow traps, always excluding spatial-derivatives, as follows

2.3.2.2.1

+ 𝜕ND1

𝜕t + 𝜕ND2

𝜕t

= G1 − R1

(2.39)

= G2 − R2

(2.40)

𝜕 = G1 + G2 − R1 − R2 𝜕t ( ) sI + + with Gi = (NDi − NDi ) i + 𝛽i and Ri = ri NDi  h𝜈

(2.41) i = 1, 2

(2.42)

+ + or ND2 − ND2 are zero or close to zero, it is probably For the limit conditions where either ND2 not possible to find the equilibrium by zeroing Eq. (2.40). Instead, the equilibrium value for  may be found out by writing the rate equation for  as + + 𝜕ND1 𝜕ND2 𝜕 = + 𝜕t 𝜕t 𝜕t

and assuming the quasi-equilibrium condition 𝜕 = 0, so that we get 𝜕t ) ) ( ( s I s I + + (ND1 − ND1 ) h𝜈1 + 𝛽1 + (ND2 − ND2 ) h𝜈2 + 𝛽2  = + 1∕𝜏1 + r2 ND2

(2.43)

(2.44)

The corresponding expressions for Eqs. (2.29) and (2.30) for this model should be, respectively, written as + + (ND1 − ND1 )𝛽1 + (ND2 − ND2 )𝛽2 d == (2.45) + 1∕𝜏1 + r2 ND2 ph =

+ + (ND1 − ND1 )s1 + (ND2 − ND2 )s2 I + h𝜈 1∕𝜏1 + r2 ND2

(2.46)

We have two limit situations for Eq. (2.44), always assuming that we are far from saturation for the deep traps (ND1 ): • The case where the irradiance is large enough to reach shallow trap saturation ) ( ) ( s I s I + ) h𝜈1 + 𝛽1 + ND2 h𝜈2 + 𝛽2 (ND1 − ND1 + ⇒ 0 so that  ⇒ ND2 1∕𝜏1

(2.47)

2.3 Photoconductivity

• The case where the irradiance is weak enough for the shallow traps to be empty ) ( s I + ) h𝜈1 + 𝛽1 (ND1 − ND1 + ND2 ⇒ ND2 so that  ⇒ 1∕𝜏1 + r2 ND2

(2.48)

As a consequence, the conductivity does also vary between two levels, with a lower value for low irradiances in Eq. (2.48) and a higher value for a larger irradiance in Eq. (2.47). In the general case, however, the photo- and dark conductivity can be calculated from Eq. (2.44), respectively, as: I + + 𝜎ph = e𝜏𝜇[(ND1 − ND1 )s1 + (ND2 − ND2 )s2 ] (2.49) h𝜈 + + 𝜎d = e𝜏𝜇[(ND1 − ND1 )𝛽1 + (ND2 − ND2 )𝛽2 ]

(2.50)

+ 1∕𝜏 ≡ 1∕𝜏1 + r2 ND2

(2.51)

It is still possible to be in the presence of hole-electron competition, in which case the formulation here should be modified. It is interesting to analyze the meaning of Eq. (2.50): it states that after having been strongly illuminated, dark conductivity is higher than its steady state in the dark. That is to say that illumination affects the dark conductivity too, which evolves until its lower steady-state value is reached. Unlike Eq. (2.27) for the One-Center model, here we should substitute Φ𝛼 with Φ𝛼0 + 𝛼li

(2.52)

+ Φ𝛼0 ≡ (ND1 − ND1 )s1

(2.53)

+ )s2 𝛼li ≡ (ND2 − ND2

(2.54)

where

for the Two-Center model, where 𝛼li stays for the action of light, producing a variation Δ𝛼 on the absorption coefficient: + Δ𝛼 = (s2 − s1 )Δ(ND2 − ND2 )

(2.55)

Note that such a variation is due to a number of filled deep traps that lose their electrons and + + become emptied to fill-in an equal number of shallow traps (−Δ(ND1 − ND1 ) = Δ(ND2 − ND2 )) that have a different cross-section (s2 ) from that (s1 ) one of the former deeper ones. + ∕𝜕t = 0) for ND2 in In order to compute 𝛼li we assume the steady-state equilibrium (𝜕ND2 Eq. (2.40) and substitute the expression for  in Eq. (2.44) into the expression for R2 in Eq. (2.42) to compute [39]: ) ( s I + ) h𝜈1 + 𝛽1 r2 ND2 (ND1 − ND1 + = (2.56) ND2 − ND2 [ ] + + 𝛽2 ∕𝜏1 + r2 𝛽1 (ND1 − ND1 ) + r2 s1 (ND1 − ND1 ) + s2 ∕𝜏1 h𝜈I that substituted into Eq. (2.54) leads to 𝛼li =

aI + d bI + c

+ )∕(h𝜈) a ≡ 𝜏1 r2 s1 (ND1 − ND1

(2.57) (2.58)

39

40

2 Photoactive Centers and Photoconductivity + b ≡ 𝜏1 r2 s1 s2 ND2 (ND1 − ND1 )∕(h𝜈) +

s2 h𝜈

(2.59)

+ ) ≈ 𝛽2 c ≡ 𝛽2 + 𝜏1 r2 𝛽1 (ND1 − ND1

(2.60)

+ d ≡ 𝜏1 r2 ND2 (ND1 − ND1 )s2 𝛽1 ≈ 0

(2.61)

We may find out simple expressions for 𝛼li for the limiting conditions: lim 𝛼li = 0 I→0

lim 𝛼li =

I→∞

+ (ND1 − ND1 )𝜏1 r2 s1 a ≈ s2 ND2 = s2 ND2 + b (ND1 − ND1 )𝜏1 r2 s1 + s2

(2.62) (2.63)

2.3.2.3 Dark Conductivity and Dopants

As analyzed in Section 2.3.2.2, shallow photoactive centers are responsible for a higher dark conductivity immediately after the recording illumination is switched off and in this way dark stability of recorded information is rapidly degradated. Such an effect may be compensated by the action of dopants in a deep level in the energy Band Gap, as illustrated for the case of CdTe crystals [19]. Such crystals exhibit shallow centers at approximately 0.2 eV below the CB and also at approximately 0.4 eV above the VB that are responsible for an enhancement of dark conductivity. The introduction of V3+ -V2+ impurities at a deep level, roughly in the middle of in the energy Band Gap, considerably reduces the influence of the shallow centers effect by acting as a sink for the electron-donors and filling up the hole-donors and, by this means, considerably reducing shallow traps-arising free charge carriers (electrons in the CB and holes in the VB) in the dark. This is also probably the case for Bi12 TiO20 doped with Ru [27], where 𝜎d decreases more than three-fold from undoped to [Ru]=1019 cm−3 Ru-doped samples.

2.4 Photovoltaic Effect Photovoltaic is a bulk effect that is experimentally put into evidence by the generation of an electric current under the action of light of adequate wavelength without any externally applied electric field on the sample. This effect is observed in some ferroelectric crystals and is supposed to be produced by the photoexcitation of electrons from asymmetric impurity potentials [40, 41]. This effect appears in poled uniform single crystals with noncentrosymmetry. It is different from the P-N junction observed in semiconductors or metal-semiconductor interfaces. The photovoltaic effect has interesting practical applications because it results in a higher space-charge modulation, and therefore leads to enhanced diffraction efficiency for the recorded holograms and may have potential applications (although not yet practical) for photoelectric conversion. At the origin of photovoltaic effect there seems to be a nonsymmetric distribution of donors and acceptors so that the electron photoexcited from a trap is closer to an acceptor in a certain sense rather than in the opposite one. Therefore, electrons do move preferentially along the same sense, the so-called “photovoltaic C axis”, when excited by the light. The resulting photovoltaic current density is jphv (z) = 𝜅phv I(z)𝛼

(2.64)

𝜕I(z) with I(z) = I(0) e−𝛼z (2.65) 𝜕z with 𝜅phv being called the photovoltaic Glass constant, which depends on the nature of the absorbing center and the illumination wavelength, as reported in Tables 2.1 and 2.2, with I(z)𝛼 I(z)𝛼 = −

2.4 Photovoltaic Effect

Table 2.1 Photovoltaic transport coefficient 𝜅phv for Fe- and Cu-doped LiNbO3 . Material

LiNbO3

Dopant

Fe

𝜆 (nm)

532

514.5

472.7

514.5

472.7

980[12]

3000[42]

4800[42]

550[42]

960[42]

Cu

𝜅phv (pA cm/W)

Table 2.2 Photovoltaic transport coefficient 𝜅phv for BTeO and BSO. Material

Bi12 SiO20

Bi2 TeO5

𝝀 (nm)

488

532

2000[44]

𝜅010 ≈ 390†

𝜅phv (pA cm/W)

†external value 𝜅010 = 319 pA cm/W from [46] corrected for air-crystal interphase reflection using n ≈ 2.44[47].

representing the absorbed intensity per unit thickness of crystal sample (total thickness of d) along the coordinate z. The photovoltaic current differential is diphv (z) = jphv (z)Hdz = −H𝜅phv d

iphv = H𝜅phv

∫0

𝜕I(z) dz 𝜕z

𝜕I(z) dz = 𝜅phv HI(0)(1 − e−𝛼d ) 𝜕z

(2.66) (2.67)

with H being the height of the sample and H × d being the area of each one of the electrodes. The average photovoltaic current density is jphv ≡ 2.4.1

iphv Hd

= 𝜅phv I(0)(1 − e−𝛼d )∕d

(2.68)

Photovoltaic Crystals

Some of the photovoltaic samples mentioned in this section are referred to in Figs. 2.19–2.21. 2.4.1.1

Lithium Niobate and Other Ferroelectric Crystals

Photocurrents were first reported [42] at least for ferroelectric photorefractive crystals such as BaTiO3 and Fe- and Cu-doped LiNbO3 (see Table 2.1) under light of 𝜆 = 514.5 nm and found to be due to a bulk photovoltaic effect and not to the presence of an internal field in the crystal as was formerly proposed by Chen [43]. 2.4.1.2

Some Photovoltaic Nonferroelectric Materials

A photovoltaic effect was recently reported [12] for Bi2 TeO5 and also for sillenites [44, 45] such as Bi12 SiO20 and Bi12 TiO20 , with some of their experimentally measured photovoltaic constants 𝜅phv reported in Table 2.2.

41

42

2 Photoactive Centers and Photoconductivity

[001]

Figure 2.19 Schema for the crystal samples: undoped Bi12 TiO20 (labeled BTO-J40), lead-doped Bi12 TiO20 (labeled BTO-Pb), undoped Bi12 SiO20 (labeled BSO) and photovoltaic iron-doped LiNbO3 (labeled LNb) with the photovoltaic “c” axis parallel to the [110] crystal axis. The light is always incident on the (110) crystal plane. Dimensions for all samples are reported in Fig. 2.20.

(001)

[ˉ110] (110) H

t l

Sample BTO-J40 BTO-Pb BSO LNb

l (mm) 6.0 4.4 4.2 5.2

H (mm) 5.4 4.5 5.8 4.5

[001]

[001]

t (mm) 2.0 1.8 1.7 0.85

Figure 2.20 Crystal samples.

Figure 2.21 Bi2 TeO5 (left) and LiNbO3 :Fe (right) crystal samples showing the [010] and c-axis that are their photovoltaic axes, respectively.

t

H C⃗

[010] a [100]

[100]

Bismuth Tellurium Oxide and Sillenites Bismuth telluride oxide (Bi2 TeO5 ) has been recently reported [12] to be photovoltaic, with its [010]-axis being the “c” photovoltaic axis, a Glass photovoltaic coefficient of 𝜅phv ≈ 400 pA cm W−1 along axis < 010 > under 𝜆 = 532 nm light as reported in Table 2.2. Its photovoltaic current was shown to be linear as for Fe-doped LinbO3 , although much weaker than for the latter as shown in Fig. 2.22. The origin of photovoltaic effect in this crystal is still to be investigated but may be related to the local asymmetry produced by stereochemically active electron “lone-pairs” (pairs of valence electrons that are not shared with another atom and are also called “nonbonding” pairs) of both cations Bi3+ and Te4+ . These lone-pairs (as for the case of sillenites) are responsible for the distorted anionic coordination around them and greatly influence the physical properties of this material [48, 49]. The dipoles originating from these local asymmetric structures combined with photoactive defects (like oxygen vacancies) can be responsible for the photovoltaic effect. 2.4.1.2.1

2.4 Photovoltaic Effect

30

1,0 LiNbO3:Fe

25

Bi2TeO5

20

0,6

15 0,4

10

0,2 0,0

Jph(pA/mm2)

Jph (pA/mm2)

0,8

5 0

100

200

300

400

0 600

500

Intensity (mW/cm2)

Figure 2.22 Average photovoltaic current density measured along axes [010] and “c”, respectively, on the BTeO and LNbO:Fe crystal samples (depicted on the left side) illuminated with spatially uniform 𝜆 = 532 nm laser light normally incident on their (100) faces, as a function of the intensity I(0) as computed at the input plane inside the material. Reproduced from [12]. Fitting data to Eq. (2.67) with 𝛼 = 5 cm−1 for BTeO [50] and 𝛼 = 7.3 cm−1 for LNbO:Fe [12] it is possible to compute their corresponding 𝜅ph𝑣 , which are reported in Tables 2.1 and 2.2. Reproduced from [12].

2.4.2

Light Polarization-Dependent Photovoltaic Effect

Experimental results have shown that the photovoltaic effect may depend on the polarization direction of the incident light, as demonstrated for Fe-doped LiNbO3 and undoped Bi2 TeO5 in Fig. 2.23 as well as for undoped sillenites in Fig. 2.24. Such a dependence was already reported before for single ferroelectric crystals like BaTiO3 [51], Pb(Zn1∕3 Nb2∕3 )O3 [52] and formalized as a third-order tensor [41] but the physical mechanism involved is still unclear. Because the electric field of the laser beams usually employed (assuming an average intensity of 20

1.0 0.9 0.8

16

0.7 14

0.6

12 10

LiNbO3:Fe

0.5

Bi2TeO5 0

50

100

150

200

Jph(pA/mm2)

Jph (pA/mm2)

18

250

0.4 300

θ (degree)

Figure 2.23 Polarization-dependent photovoltaic photocurrent for both BTeO and LNbO:Fe crystal samples, as a function of the polarization direction of the 𝜆 = 532 nm laser light, with the angular position referred to the axes [010] and “c”, respectively, for the incident (onto the (100) crystal faces) intensity (outside the material) I0 = 480 mW/cm2 . Reproduced from [12].

43

2 Photoactive Centers and Photoconductivity

4

3 Iph (pA)

44

2

1

0

30

60

90

120

150

180

θ (degree)

Figure 2.24 Photocurrent (•) Iph , for undoped Bi12 TiO20 as a function of the angle 𝜃. The photocurrent was measured along the [110]-crystal axis using 𝜆 = 532 nm and incident light intensity I0 = 102 mW/cm2 measured outside the crystal. The initial point, 𝜃 = 0o , corresponds to the polarization parallel to the [110]-axis (see Fig. 2.19).

I ≈ 10 mW/cm2 represents an electric field of E ≈ 28 V/m) is orders of magnitude lower than typical effective photovoltaic fields (Ephv ≈ 10 kV/m as reported in Section 2.4.1.2, it is difficult to believe that the electric field from the incident light may be directly affecting the photovoltaic current, as well as because the latter is DC and the light-electric field is alternating with a very high frequency. It is possible, however, that the direction of polarization of light may affect the photoexcitation process itself and by this bias affect the overall photovoltaic current too. This matter deserves further experimental research.

2.5 Nonlinear Photovoltaic Effect As for the case of light-induced absorption (see Section 2.3.2.2), the presence of deep and shallow photovoltaic photoactive centers in the crystal Band Gap may lead to a nonlinear photovoltaic effect. Taking into account two such types of center with different photovoltaic coefficients 𝜅1 and 𝜅2 , respectively, Eq. (2.64) should be modified to [53]: jphv (z) = (𝜅1 𝛼1 + 𝜅2 𝛼2 )I(z)

(2.69)

and from Eq. (2.27) we should write Eq. (2.69) as ] I(z) [ + + jphv (z) = 𝜅1 (ND1 − ND1 )s1 + 𝜅2 (ND2 − ND2 )s2 h𝜈

(2.70)

+ and sj (j = 1, 2) are described in Section 2.3.2.2. Assuming that  in the CB is where NDj , NDj much smaller than the density of deep and shallow traps so as to be able to assume that + + + ND2 ≈ NA− ND1

(2.71)

where NA− (see Eq. (2.23)) is the density of nonphotosensitive negative ions necessary to electri+ + cally equilibrate for the “as-grown” positive (ND1 and ND2 ) ions in the crystal. Once steady-state

2.5 Nonlinear Photovoltaic Effect

equilibrium (which means zeroing the time derivatives in Eqs. (2.39) and (2.40)) is achieved, that is to say for + 𝜕ND1

+ 𝜕ND2

=0 𝜕t 𝜕t + with Eq. (2.71) we obtain an equation for ND1 =

(2.72)

+ 2 + (ND1 ) + ND1 A(I) + B(I) = 0

with

[( 1+ A(I) ≡ NA−

ND1 NA−

)

( s1 ∕(h𝜈) − 1 −

(2.73) ND2 NA−

) ] ( s2 I − 1 −

ND2 NA−

)

𝛽2

𝛽2 + (s2 − s1 )I∕(h𝜈)

B(I) ≡ −NA−

ND1 s1 I∕(h𝜈) 𝛽2 + (s2 − s1 )I∕(h𝜈)

(2.74) (2.75)

with the solution

√ A(I)2 − 4B(I) = 2 + and substituting ND1 into Eq. (2.71) we get √ A(I) − A(I)2 − 4B(I) + − ND2 (I) = NA + 2 Substituting Eqs. (2.76) and (2.77) into Eq. (2.70) we get a nonlinear expression in I(z) [ ] I(z) + + jphv (z) = 𝜅1 (ND1 − ND1 (I(z))s1 ∕Φ1 + 𝜅2 (ND2 − ND2 (I(z)))s2 ∕Φ2 h𝜈 −𝛼z I(z) = I(0) e + (I) ND1

−A(I) +

(2.76)

(2.77)

(2.78) (2.79)

with I(0) being the incident light intensity computed inside the material. Thus, the photovoltaic current density jphv can be written as a function of light intensity I ≡ I(z): + + jphv (I) = 𝜅1 S1 [ND1 − ND1 (I)]I + 𝜅2 S2 [ND2 − ND2 (I)]I

(2.80)

Note that jphv (I) in Eq. (2.80) is no longer linear with respect to the light intensity I and this behavior is a direct mathematical consequence of 𝛽2 ≠ 0 in Eqs. (2.74) and (2.75), which characterizes ND2 as being a shallow center. Aside from this purely mathematical explanation for the requirement of ND2 to be a shallow trap, it is easy to understand that as the light intensity increases, the initially empty shallow centers become progressively filled with electrons from the deeper centers, thus increasing their participation in the photovoltaic process, changing the nature of the photovoltaic centers involved and, by this means, becoming a nonlinear process. If only deep centers would be considered instead, with similar recombination constants, the photovoltaic process would be based on a linear combination of the properties of roughly linearly varying deep centers’ concentration, and a linear process would result. The photovoltaic current is approximately written as d

iphv (I(0)) ≈ H

∫0

+ + (𝜅1 S1 [ND1 − ND1 (I(0))] + 𝜅2 S2 [ND2 − ND2 (I(0))])

× I(0) e−𝛼z dz

(2.81)

where H is the height of the crystal (electrodes surface being Hd each) and the traps densities assumed to be constant (preliminary calculations showing the deep traps varying by only

45

2 Photoactive Centers and Photoconductivity

0.001% and the shallow ones by 0.05% throughout the whole sample thickness z at the highest illumination of I(0) ≈ 600 W/m2 , at least for BSO) and the integration being carried out just on the irradiance I(z), which varies exponentially according to Eq. (2.81). Nonlinear photovoltaic effect has been already reported for Fe-doped [54] and undoped LiNbO3 [55] for rather high light irradiances. For undoped Bi12 TiO20 and Bi12 SiO20 , however, such nonlinear behavior was observed [56] but for much lower irradiances as reproduced in Figs. 2.25 and 2.26, respectively. For roughly the same irradiance range, however, Fe-doped LiNbO3 in Fig. 2.28 exhibits a perfect linear behavior and Pb-doped Bi12 TiO20 in Fig. 2.27 can hardly be considered to be nonlinear [36].

2.5.1

Light-Induced Absorption and Nonlinear Photovoltaic Effects

Although nonlinear photovoltaic and light-induced absorption effects are of different nature, both depend on the presence of deep and shallow centers and also rely on the filling of shallow centers from the deeper ones via the Conduction Band by the action of light. Let us recall that nonlinear absorption requires both centers to have wide different photoelectric cross-sections, whereas the nonlinear photovoltaic effect requires them to be photovoltaic (that is to say, to be placed in a crystal structure with spatially asymmetric donor and acceptor centers) having a different (or even opposite sign) photovoltaic constant 𝜅. Sillenite crystals and particularly undoped BTO and BSO, as well as BTO-Pb, all exhibit strong light-induced absorption [39, 60, 61] and undoped BTO and BSO also show nonlinear photovoltaic effects, so that one may wonder if the latter effect on BTO and BSO could simply arise from the former one so that, by increasing the absorption coefficient with light intensity, we may reduce the illumination inside the crystal and thus produce an apparent nonlinear photovoltaic current response. Such a possibility was straightforwardly ruled out by showing [56] that light-induced absorption is already saturated (that means a constant absorption coefficient) in the light intensity range where nonlinear photovoltaic effects take place. 12

9 Iph (pA)

46

6

3

0

0

50

100

150

200

250

300

Intensity (mW/cm2)

Figure 2.25 Photovoltaic current versus light intensity I(0) (uniform 𝜆 = 532 nm laser incident on the (110) crystal plane with light polarization direction along [110]) for undoped Bi12 TiO20 (BTO-J40) sample. The ◽ and • represent the photovoltaic current measured along the [001] and [110]-axis, respectively. The continuous line is the best fitting with Eq. (2.78) and the parameters computed from fitting are reported in Table 2.3.

2.5 Nonlinear Photovoltaic Effect

6 5

Iph (pA)

4 3 2 1 0

60

0

180

120

240

2)

Intensity (mW/cm

Figure 2.26 Photovoltaic current versus light intensity (uniform 𝜆 = 532 nm laser incident on the (110) crystal plane with light polarization direction along [001]) for an undoped Bi12 SiO20 (BSO) sample. The continuous line is the best fitting with Eq. (2.80) and the parameters computed from fitting are reported in Table 2.3. 2.0

Iph (pA)

1.6 1.2 0.8 0.4 0.0

0

80

160

240

320

400

Intensity (mW/cm2)

Figure 2.27 Photovoltaic current versus light intensity (uniform 𝜆 = 532 nm laser incident on the (110) crystal plane with light polarization direction along [001]) for a lead-doped Bi12 TiO20 (BTO-Pb) sample. The dashed line is only a guide for the eyes.

2.5.2

Deep and Shallow Centers

Nonlinear effects (absorption and photovoltaic) both appear on some materials and both such nonlinearities are based on the presence of deep and shallow centers, with the latter being filled from the deep ones by the action of light. Are both nonlinearities connected to the same deep and shallow centers? We have already discussed this in Section 2.5.1 and concluded that the nature of centers involved in both phenomena differ widely. However, it is not impossible for these centers simultaneously to have the necessary properties to produce both nonlinear effects. As regards the sillenite samples here (undoped BTO and BSO), however, this possibility is again ruled out for the same reason discussed in Section 2.5.1, where we pointed out that both effects occur in different light intensity ranges.

47

2 Photoactive Centers and Photoconductivity

2.5 2.0 Jph (nA/cm2)

48

1.5 1.0 0.5 0.0

0

70

140

210

280

350

Intensity (mW/cm2)

Figure 2.28 Average photovoltaic current density data, measured along the c-axis, versus light (𝜆 = 532 nm) intensity (light polarization direction along crystal c-axis) for an iron-doped LiNbO3 (LNb) sample show a strict linear behavior with the continuous line being the best fitting with Eq. (2.67). Table 2.3 Parameters for BTO and BSO from Figs. 2.25 and 2.26. Parameters

Samples BTO-J40

BSO

along [001]

along [110]

along [001]

ND1 (1024 m−3 )*

10

10

10

ND2 (1024 m−3 )*

0.1

0.1

0.1

NA− (1024 m−3 )*

4

4

4

𝜅1 (10−16 Am/W)

6.83

7.47

7.83

𝜅2 (10−16 Am/W)

−259

−299

−315 4.58

s1 (10

−24

2

m)

0.32

1.84

s2 (10−24 m2 )

380

2690

210

𝛽2 (s−1 )

0.327

0.327

0.00431

𝛼 (m−1 )

700

700

470

*From [57–59]

2.6 Light-Induced Absorption or Photochromic Effect Light-induced absorption, also known as the photochromic effect, occurs in materials with shallow photoactive centers close to the bottom of the Conduction Band. As already mentioned in Section 2.3.2.2, photochromic effects are not expected for the one-center model. Let us

2.6 Light-Induced Absorption or Photochromic Effect

therefore refer to the two-center model, including shallow traps. In this case, the light-induced absorption 𝛼li has already been formulated in Eq. (2.54) and is a function of the irradiance I. Photochromic effects are easy to measure and may give valuable information to be compared with that obtained from photoconductivity. In order to compute 𝛼li , we substitute Eq. (2.44) + , into Eq. (2.40) and find the stationary equilibrium (that is, the null time-derivative) for ND2 + + which is only possible if we are far from the extremes ND2 = ND2 or ND2 = 0. In this case, we get an expression

+ ND2 − ND2

) ( s I + ) h𝜈1 + 𝛽1 ND2 r2 (ND1 − ND1 = [ + + 𝛽2 ∕𝜏1 + r2 (ND1 − ND1 )𝛽1 + r2 (ND1 − ND1 )s1 ∕(h𝜈) +

1 s ∕(h𝜈) 𝜏1 2

] I (2.82)

that, substituted into Eq. (2.54), results in an expression for the light-induced absorption 𝛼li =

aI + d bI + c

(2.83)

s1 h𝜈 1 s2 + s1 ) + b ≡ r2 (ND1 − ND1 h𝜈 𝜏1 h𝜈 𝛽 𝛽2 + + r2 𝛽1 (ND1 − ND1 )≈ 2 c≡ 𝜏1 𝜏1

(2.85)

+ d ≡ r2 ND2 (ND1 − ND1 )s2 𝛽1 ≈ 0

(2.87)

+ )s2 a ≡ r2 ND2 (ND1 − ND1

(2.84)

(2.86)

the limit values of which are + )𝜏1 𝛽1 r2 ND2 s2 (ND1 − ND1 𝛽 d + ≈ r2 ND2 s2 (ND1 − ND1 lim 𝛼li = = )𝜏1 1 ≈ 0 + I→0 c 𝛽2 𝛽2 + r2 𝛽1 𝜏1 (ND1 − ND1 ) (2.88) lim 𝛼li =

I→∞

+ (ND1 − ND1 )𝜏1 r2 s1 a = ND2 s2 + b (ND1 − ND1 )𝜏1 r2 s1 + s2

(2.89)

where the approximated values here indicate that we assume that photoelectrons are mainly generated by the action of light on the deep traps and that thermally excited electrons are only produced from the shallow centers. In this case, Eq. (2.83) is simplified to aI (2.90) bI + c The typical darkening light-induced absorption in undoped Bi12 TiO2 (BTO) is observed in Fig. 2.29 and the activation energy Ea of these photochromic centers was measured, in the case of BTO (sample labeled BTO-8), by saturating the sample at 514.5 nm and then measuring the photochromic effect relaxation in the dark, using the Arrhenius [20] law as shown in Fig. 2.30, from which data it was found to be [28] Ea = 0.42 eV. 𝛼li =

49

2 Photoactive Centers and Photoconductivity

Figure 2.29 Light-induced absorption spots produced in the center of an undoped Bi12 TiO20 crystal by the action of a thin 𝜆 = 532 nm laser line beam; the second spot is due to the beam reflected from the rear crystal face.

10

5

τ (min)

50

2

1

0.5 2.70

2.85

3.00

3.15

3.30

3.45

1000/T (K–1)

Figure 2.30 Photochromic relaxation time for Bi12 TiO20 as a function of inverse absolute temperature. Arrhenius data fitting leads to an activation energy of 0.42 ± 0.02 eV.

2.7 Dember or Light-Induced Schottky Effect

0.6 y = + 0.00142x1 + 0.00345, max dev:2.69E–4, r2 = 1.00

Pt (μW)

0.4

0.2 y = + 5.70E–4x1 + 0.0899, max dev:0.00, r2 = 1.00 0

200

0

400 Po (μW)

600

800

Figure 2.31 Transmitted versus incident power (both measured in the air) for a 8.1 mm thick photorefractive Bi12 TiO20 crystal slab labeled BTO-010 using a 𝜆 = 532 nm Gaussian cross-section intensity laser beam (1.3 mm radius, P = 800 μW corresponding to I ≈ 150 mW/m2 ). Data in the graphics are fitted by a linear equation for the limits Po → 0 (black line) and Po → ∞ (gray line) as shown in the graphics.

2.6.1

Transmittance with Light-Induced Absorption

The formulation of the transmitted light in the presence of light-induced absorption follows the usual pattern (see Fig. 2.31) dI = −(𝛼0 + 𝛼li )I dz

𝛼0 + 𝛼li =

(𝛼0 b + a)I + 𝛼0 c bI + c

(2.91)

(bI + c) dI = −dz I[(𝛼0 b + a)I + 𝛼0 c] I(d)

b

∫I(0)

I(d)

dI dI +c = −d ∫I(0) I[(𝛼0 b + a)I + 𝛼0 c] (𝛼0 b + a)I + 𝛼0 c (𝛼 + a∕b)I(0) + 𝛼0 c∕b a∕b I(d) ln 0 + ln = −𝛼0 d 𝛼0 + a∕b (𝛼0 + a∕b)I(d) + 𝛼0 c∕b I(0)

(2.92)

with some simple expressions for the limit conditions: I(d) = I(0) e−𝛼0 d for I(0) ⇒ 0 I(d) = I(0) e

a −(𝛼0 + )d b for I(0) ⇒ ∞

(2.93) (2.94)

2.7 Dember or Light-Induced Schottky Effect The Schottky effect is the build-up of an electric potential barrier occurring at the interface between semiconductors with different dopants (n-Si and p-Si, for example), or between a metallic film on a n-type semiconductor: electrons from the n-type semiconductor flow into the p-type one or into the metallic electrode, leaving behind a positively charged depletion

51

52

2 Photoactive Centers and Photoconductivity ITO – + – electrode – + – – –– – – +++ – – – – + – –– + – – – – + + –– + – – + – – – – –+ + + – +– – – – + + –

Figure 2.32 Light-induced Schottky barrier at the illuminated transparent conductive ITO electrode-photorefractive crystal interface. –

– – – –

potential barrier

[010]

d

H

Figure 2.33 Schema of a photorefractive BTO crystal plate between two conductive transparent ITO electrodes including crystal axes and the illuminated front (001) plane. +

(001)

[100] –

ITO

l

Figure 2.34 Cross-section schema of the ITO-sandwiched BTO plate indicating the photocurrent flow under illumination.

iph

Light

ITO electrode front face – – + + – – + + – – + + + – – + – + – + + – – + + – – + + – – + + – + – + + – – + – – + + + – –

ITO electrode rear face + – – + + – + – +

+ +

+

–

– + – +

– –

+ – – + –

Voltage

Depth

2.7 Dember or Light-Induced Schottky Effect

Figure 2.35 ITO sandwiched 0.81 mm thick BTO crystal plate with electrodes wired to a lock-in amplifier.

front ITO electrode

back ITO electrode

(00 1) illumination

to lock-in amplifier

10

0

–10 –2000

photocurrent (pA)

photocurrent (pA)

–1000

–20

–3000

0

100

200

–30 300

f (Hz)

Figure 2.36 Measured photocurrent data referred to Fig. 2.35 with •, ◾ and ▾ indicating the front illuminated sample, whereas ○, ◽ and ∇ refer to rear plane illumination, with ○ and • data refer to the left-side ordinate axis.

layer, until the resulting electric potential barrier becomes strong enough to counteract such a flow. Because of the potential barrier, this junction becomes a rectifying one. In 1931, Dember [62] reported that a local electric potential barrier may also arise in the volume of a semiconductive crystal under nonuniform illumination of adequate wavelength. This so-called “Dember Effect” was mathematically formulated later on in 1977 [63]. We have already shown [64] that in the interface between a photorefractive (that is a photoconductive material) crystal (undoped BTO, for example) and a transparent conductive electrode (an ITO film, for example) a potential barrier may be also build up by the action of light, as illustrated in the schema of Fig. 2.32. In the absence of light the ITO-crystal interface remains of ohmic nature so that it can be switched from ohmic (without illumination) to rectifying (Schottky) and back to ohmic by switching on and off light of adequate wavelength onto the photorefractive crystal-electrode junction. A photorefractive (in this case BTO) crystal plate covered on the front and rear planes with conductive transparent electrodes as depicted in Fig. 2.33 develops a light-induced Schottky barrier at the front (strongly illuminated) crystal-electrode junction and at the rear one, the

53

2 Photoactive Centers and Photoconductivity

1000

1000

100

100

10

10

1

1

0.1

0

200

100

Dember current (pA)

10 000

10 000

photovoltaic current (pA)

54

0.1 300

f (Hz)

Figure 2.37 Photovoltaic-based current data (•, ◾ and ▴) computed from curves in Fig. 2.36 are plotted on the left-side ordinate axis, whereas computed Dember-based currents (○, ◽ and △) are plotted on the right-side ordinate axis. Because of logarithmic scales, all current are plotted as positive, although Dember and photovoltaic based ones have opposite signs. Data for I0 ≈ 1276 mW/cm2 are represented by ○ and • whereas ◾ and ◽ are for I0 ≈ 12.8 mW/cm2 . Data for I0 ≈ 1.02 mW/cm2 are represented by ▴ and △.

latter barrier being sensibly weaker than the front one because less illuminated, as illustrated in Fig. 2.34, where this ITO-sandwiched crystal is shown to operate as a photoelectric conversion device. 2.7.1

Dember and Photovoltaic Effects

Some photorefractive crystals including sillenites [36] exhibit photovoltaic effects. Dember and photovoltaic effects may be separately measured as the latter effect is always producing a photoelectric current flowing in sense and direction determined by the crystal c-axis whereas that arising from Dember effect is determined by the way light illuminates the transparent electrode-sandwiched crystal plate: by reversing illumination from the front to the rear electrode, the Dember-based photocurrent flowing sense is also reversed without changing the sense of the photovoltaic-based one. The overall ac photocurrent on the d = 0.81 mm thick ITO-sandwiched Bi12 TiO20 crystal plate of Figs. 2.33 and 2.35, under a spatially uniform chopped expanded 𝜆 = 532 nm laser light was measured using a lock-in amplifier and the result is plotted in Fig. 2.36 as a function of chopper frequency and for different light intensities I0 = 1.02, 12.8 and 1275 mW/cm2 . Data from reverse illumination is also plotted in Fig. 2.36 and from these data it is possible to separately compute the photovoltaic-based and the Dember-based photocurrents that are separately plotted in Fig. 2.37.

55

Part II Holographic Recording

56

Introduction

This second part of the book is devoted to holographic recording in photorefractive materials. These materials are particularly interesting for holographic recording and many applications in this field and related ones have been and are currently being developed. Some of their advantages over other photosensitive recording materials are: almost real-time optical recording, reversibility, indefinite number of recording-erasure cycles and very high spatial resolution. Also, the final recording state does not depend on the irradiance and on the total energy (time-integrated irradiance) but on the pattern-of-light modulation, and this is particularly interesting for recording with low levels of irradiance, as is usually the case for image processing applications. Chapter 3 describes the recording of a space-charge electric field without caring about the associated index-of-refraction modulation, whereas Chapter 4 is devoted to the build-up of an index-of-refraction modulation in the material’s volume; that is to say, a phase volume hologram. Because of the real-time nature of the recording process, the hologram does diffract the recording beams during recording, thus modifying their relative amplitudes and their mutual phase-shift, which also modifies the hologram being recorded and in turn further modifies the recording beams and so on. This kind of feedback process is called wave-mixing or self-diffraction, is characteristic of real-time reversible recording materials and is also dealt with in Chapter 4. Holograms recorded in some materials show diffracted light having a polarization direction different from that of the transmitted light, and this subject will be treated in Chapter 5. Chapter 6 is the last one in this part and will describe a practical feedback-controlled stabilized holographic recording procedure that requires no external reference for stabilization and reduces environmental perturbations during recording, thus strongly improving the recording process. The process and its application to a couple of very representative materials are described in detail.

Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

57

3 Recording a Space-Charge Electric Field This chapter focuses on the mechanisms responsible for the build-up of a modulated space-charge electric field under the action of a modulated pattern of light projected onto the sample, without considering wave-mixing effects; that is to say, without caring about the diffraction of the recording beams by the index-of-refraction modulation associated with the space-charge field hologram being built up. The theoretical model is based on the Band Transport Theory and rate equations proposed by Kukhtarev and co-workers [65, 66]. Before starting with the mathematical development, let us qualitatively describe the processes involved. The material is usually characterized by a relatively large energy Band Gap compared to the recording light so that the latter can go through the whole sample volume. Inside the Band Gap there are one or more localized states (photoactive centers) from where electrons and/or holes can be excited to the conduction band (CB) or to the valence band (VB) as illustrated by Fig. 3.1. To ensure electrical neutrality in equilibrium, charged donors or acceptors should be in close proximity to an oppositely charged nonphotoactive ion. Under the action of a modulated pattern of light onto the crystal, electrons (for the sake of simplicity, we shall assume that only electrons are involved) are excited to the CB where they diffuse along the direction of their concentration gradient, with a characteristic diffusion length distance, are retrapped again, are re-excited and so on. After some time, electrons are accumulated preferentially in the less illuminated regions because there they are less efficiently excited than anywhere else. A spatial distribution of electric charge is therefore built up with exceeding positive charges being left in the illuminated regions and negative ones in the less illuminated regions, as illustrated in Fig. 3.2. The spatial modulation of charge produces an associated space-charge electric field modulation, as illustrated in Fig. 3.3, which is π∕2-phase shifted to the spatial modulation of charge because of the well-known Poisson equation relating charge and electric field. If the material is electro-optic, besides being photoconductive, then the space-charge field modulation produces a corresponding modulation in the index-of-refraction, in phase with the field, as already described in Section 1.3 and illustrated in Figs. 3.4 and 3.5. Under the action of an externally applied field the electrons move because of the electric drift apart from the action of diffusion concentration gradient. Because of the nonsymmetric action of the drift, the resulting spatial modulation of charge is not any more in phase with the pattern of light modulation. A sinusoidal pattern of light that can be used for holographic recording is produced by the simple interferometric (or holographic) setup, schematically illustrated in Fig. 3.6. It is worth pointing out a general property of any holographic setup: the angular deviation α of the input laser beam produces a linear deviation of the pattern of fringes at the recording plane that is proportional to 𝛼 2 × ΔL [67] where ΔL is the optical path difference between the two interfering beams. To reduce such an instability, it is therefore highly recommended to reduce ΔL as much as possible. The choice of ΔL does also depend on the coherence length of the laser in the setup: Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

3 Recording a Space-Charge Electric Field

light

light

light

BAND GAP

CONDUCTION BAND

– +

– +

– +

– +

– +

– +

– +

– +

– +

– +

– +

– +

– +

– +

– +

– +

VALENCE BAND +

– empty trap ACCEPTOR

filled trap DONOR

nonphotoactive ion

Figure 3.1 Photoactive centers inside the Band Gap. There are filled traps ND − ND+ (electron-donors), empty traps ND+ (electron-acceptors) and nonphotoactive ions (+) to provide local charge neutrality. light

light

light

Conduction band

BAND GAP

58

– – – + + overall negative charge

+ + overall positive charge

– – – – – + + + overall negative charge

+ + overall positive charge

– – – – – + + + overall negative charge

– – – + + + + overall positive charge

VALENCE BAND

– empty trap ACCEPTOR

filled trap DONOR

+ nonphotoactive ion

Figure 3.2 Under the action of light the electrons are excited from the traps into the conduction band where they diffuse and are retrapped in the darker regions. A space modulation of electric charge results, with overall positive charge in the illuminated and negative charge in the less illuminated regions.

3 Recording a Space-Charge Electric Field light

light

light

CONDUCTION BAND

BAND GAP

E – – – + + overall negative charge

E

E

E

– – – – – + + + + + overall overall positive negative charge charge

E

– – – – – + + + + + overall overall positive negative charge charge

E – – – + + + + overall overall positive negative charge charge

VALENCE BAND

Figure 3.3 The charge distribution produces a space-charge electric field modulation. light

light

E

E

overall negative charge

overall positive charge

E

overall negative charge

light

E

E

overall positive charge

overall negative charge

E

overall positive charge

overall negative charge

crystal lattice deformation

Figure 3.4 The electric field modulation may produce deformations in the crystal lattice. light

E

E

overall negative charge

light

overall positive charge

E

overall negative charge

light

E

overall positive charge

E

overall negative charge

E

overall positive charge

index of refraction modulation

Figure 3.5 If the photoconductive material is also electro-optic, that is to say it is photorefractive, the space-charge field may produce an index-of-refraction modulation in the crystal volume that is in-phase (or counterphase) with the space-charge field modulation and is π∕2-shifted to the recording pattern of light.

59

60

3 Recording a Space-Charge Electric Field

Sh1 laser

Sh3

BS

M1

Sh2

M2

C D1

D2

Figure 3.6 Holographic setup: A laser beam is divided by the beamsplitter BS, reflected by mirrors M1 and M2 and interfering with an angle 2𝜃. A sinusoidal pattern of light, as described in the text, is produced in the volume where these two beams interfere. A photorefractive crystal C is placed in the place where this pattern of light is produced. The irradiance of the two interfering beams is measured behind the crystal using photodetectors D1 and D2. Shutters Sh1, Sh2 and Sh3 are used to cut off the main beam and each one of interfering beams, if necessary.

The latter should be much longer than ΔL, otherwise a poor pattern-of-fringes contrast or no fringes at all may be produced. The whole recording and reading process can be schematically described by Figs. 3.7–3.14. These qualitatively described processes will be developed in the remainder of this chapter on a quantitative mathematical basis.

3.1 Index-of-Refraction Modulation Let us think about the way the space-charge field modulation may act on the index ellipsoid in order to produce the index-of-refraction modulation that is necessary to produce a volume grating in a photorefractive material. Let us take the example of sillenites, as represented in Fig. 1.11 and described in Eqs. (1.44)–(1.47). It is easy to understand that for any polarization Figure 3.7 Generation of an interference pattern of fringes. LASER BEAM

LASER BEAM ∆

INTERFERENCE PATTERN OF FRINGES OF PERIOD ∆

3.1 Index-of-Refraction Modulation

Figure 3.8 Light excitation of electrons to the CB in the crystal. ∆ –– –– –

–– –– –

–– –– –

–– –– –

–– –– –

–– –– –

Figure 3.9 Generation of an electric charge spatial modulation in the material.

– – –

– – – – –

+ + + + +

– – – – –

+ + + + +

– – – – –

+ + + + +

– – – – –

+ + + + +

– – – – –

+ + + + +

– – – – –

Figure 3.10 Generation of a space-charge electric field modulation.

Figure 3.11 The electric field modulation produces a index-of-refraction modulation (volume grating) via electro-optic effect.

direction of the reading beam (the incident beam that is diffracted by the grating in the crystal’s volume) the index-of-refraction will be changing as the space-charge field (E⃗ in the Fig. 1.11) will be changing too in value and sense. However, it is not as obvious to understand why the index-of-refraction modulation is invariant for any polarization direction of the reading beam, as far as the electro-optic configuration represented in Figs. 1.3, 1.10 and 1.11 is concerned.

61

62

3 Recording a Space-Charge Electric Field

Figure 3.12 The recorded grating can be read using one of the recording beams that is transmitted and diffracted.

Figure 3.13 The grating is erased during reading.

Figure 3.14 Until all recording is erased.

incident beam

transmitted beam

3.2 General Formulation

In fact, any linearly polarized reading beam may be decomposed in two eigen waves propagating along each one of the two principal axes, 𝜂 and 𝜁 , in Fig. 1.11. In one grating period the space-charge field will change from the maximum value along x (Fig. 1.11) to the maximum along the other direction. Therefore, the index-of-refraction variation along axis 𝜂 and along axis 𝜁 in one grating spatial period will be, in absolute values, the same: |Δn𝜁 | = |Δn𝜂 | =

1 3 1 n r E − (−) n30 r41 E = n30 r41 E 2 0 41 2

(3.1)

In conclusion, as the index-of-refraction modulation along any of the two principal axes is the same, the proportion of the incident reading wave that is decomposed and propagated along each one of the principal axes does not affect at all the overall phase modulation of the reading wave. Therefore, the diffraction efficiency measured with the reading beam, as far as the index-of-refraction modulation is concerned, will be invariant with the direction of polarization of the reading wave. Note that this conclusion is independent of optical activity or any other effect of another nature on the diffraction efficiency. This result has been experimentally reported and also theoretically demonstrated in a more quantitative basis by several authors such as those in References [58, 68, 69].

3.2 General Formulation We shall now analyze the charge transport and associated equations for the particular case of an interference pattern of light being projected onto the sample, as schematically illustrated in Fig. 3.15. The interference of two plane waves of complex amplitudes of the form ⃗ = S⃗0 e𝚤(k⃗S . x⃗ − 𝜔t) S(0)

with S0 = |S0 | e𝚤𝜙

(3.2)

⃗ = R⃗0 e𝚤(k⃗R . x⃗ − 𝜔t) R(0)

with R0 = |R0 |

(3.3)

produces a pattern of light onto the sample that is represented in Fig. 3.15 and is described by ⃗ + R(0)∣ ⃗ 2 I = ∣ S(0)

[

|S⃗0 . R⃗0 | I = (∣ S⃗0 ∣2 + ∣ R⃗0 ∣2 ) 1 + 2 cos(K⃗ . x⃗ − 𝜙) ∣ S0 ∣2 + ∣ R0 ∣2

] (3.4)

I = I0 [1+ ∣ m ∣ cos(K⃗ . x⃗ − 𝜙)] [ ] I = I0 + I0 ∕2 m e𝚤Kx + m∗ e−𝚤Kx Figure 3.15 Space-charge electric field grating being recorded by the 𝜙-shifted sinusoidal pattern of fringes.

(3.5) (3.6)

⃗ S(0) 2θ

⃗ R(0)

space-charge field grating and associated hologram

X

R⃗

ϕ pattern of light

Z ∆ d

S⃗ K = 2Π/∆

63

64

3 Recording a Space-Charge Electric Field

where 𝜙 is the phase shift between the pattern of fringes and the space-charge electric field grating, with the following definitions: IS0 ≡ |S0 |2

IS ≡ |S|2

IR0 ≡ |R0 |2

IR ≡ |R|2

I0 = IR0 + IS0

(3.7)

with ∗

m≡

∗

2S⃗0 . R⃗0 |S⃗0 |2 + |R⃗0 |2

= |m| e−𝚤𝜙

|m| ≡ 2

|S⃗0 . R⃗0 | |S⃗0 |2 + |R⃗0 |2

(3.8)

and K⃗ ≡ k⃗R − k⃗S

⃗ ≡ 2π∕Δ = 2k sin 𝜃 |K|

with k =∣ k⃗S ∣=∣ k⃗R ∣

(3.9)

where 𝜔 is the angular frequency of the light, k⃗S and k⃗R are the corresponding (symmetric) propagating vectors, 2𝜃 is the angle between the interfering beams and Δ is the spatial period of the sinusoidal pattern of fringes. This pattern of light is projected on the material in order to record an elementary hologram (grating), where S, IS and R, IR are the complex amplitudes and corresponding irradiances of each one of the two interfering beams. The index “0” means their values at the input plane. The quantity m is the so-called complex pattern-of-light fringe modulation. 3.2.1

Rate Equations

Unless otherwise stated it the simplest “one center, two valence, one charge-carrier” model will be assumed, with electrons being the charge carriers, as depicted in Figs. 3.1–3.5. The equations for this model were already formulated in Eqs. (2.18)–(2.23) as follows 𝜕 (x, t) 1 = G − R + (∇ . ⃗j) 𝜕t e 𝜕ND+ (x, t) =G−R 𝜕t ( G = (ND − ND+ (x, t))

sI +𝛽 h𝜈

)

R = rND+ (x, t) (x, t) ⃗j = e (x, t)𝜇E(x, ⃗ t) + e∇ (x, t) ⃗ t)) = e(N + (x, t) −  (x, t) − N − ) ∇ . (𝜖𝜀0 E(x, A D 3.2.2

Solution for Steady-State

We should now find a solution to the rate equations in Section 3.2.1 for the steady-state. In this case all variables are time-independent and the time derivatives are zero, so that we can rewrite the rate equations as G−R=0

(3.10)

∇ . ⃗j = 0 ⇒ j = j0 = constant

(3.11)

in which case we get  (x) ≈ 0 (1 + m cos K x)

(3.12)

3.2 General Formulation

0 ≡

ND − ND+ sI0 rND+

(3.13)

h𝜈

where the dark excitation (𝛽) was neglected and the spatial modulation of the trap ratio (ND − ND+ )∕ND+ was also neglected compared to the pattern of fringes modulation. In this way, the density of free electrons  in the CB as derived in Eq. (3.12) follows exactly the spatial pattern of the light. Therefore, for steady state and from Eq. (2.22) with j = j0 we get 𝜕 (x) (3.14) 𝜕x and substituting  (x) and its spatial derivative by their expressions from Eq. (3.12), we get an expression for the space-charge electric field j0 = e (x)𝜇E(x) + e

E(x) =

j0 e𝜇0 (1 + m cos Kx)

+m

K sin Kx 𝜇 1 + m cos Kx

(3.15)

where E(x) and j0 are the values of the respective x-component vectors, which are the only ones in our unidirectional geometry. The integration of Eq. (3.15) may help simplifying the previous relations. In fact, the applied external voltage V0 is L

V0 =

∫0

(3.16)

E(x)dx

and the integrated terms in Eq. (3.15) are L

L 1 dx = √ ∫0 1 + m cos Kx 1 − m2 L sin Kx dx = 0 for L ≫ 2π∕K ∫0 1 + m cos Kx

(3.17) (3.18)

Accordingly, we should write E0 ≡ V0 ∕L =

j0 e𝜇0

1 √ 1 − m2

(3.19)

which, substituted into Eq. (3.15), together with Eq. (3.19), gives the electric field expression √ 1 − m2 sin Kx + mED (3.20) E(x) = E0 1 + m cos Kx 1 + m cos Kx k T K =K B (3.21) with ED ≡ 𝜇 e where ED is the diffusion-arising space-charge field and E0 is the externally applied electric field. The result here shows that the sinusoidal pattern of fringes does not lead, in general, to a sinusoidal space-charge field. For the particular case of small pattern-of-fringes modulation (|m| ≪ 1), however, Eq. (3.20) can be approximated to √ E E ≈ E0 1 − m2 (1 − m cos Kx) + mED sin Kx − m2 D sin 2Kx (3.22) 2 which contains the first and the second harmonic terms in Kx. For a sufficiently small m, however, the second harmonic in m2 can be neglected. Figure 3.16 shows the theoretically computed shape of the space-charge field for different pattern-of-fringes visibility m: It is obvious that the field is completely asymmetric for m = 0.99 and is rather sinusoidal for m = 0.30. Accordingly, we should rather consider a “first-spatial approximation” only for m ≤ 0.3.

65

3 Recording a Space-Charge Electric Field 10

10

10

0.4

0.4

5

5

0.5

0.5

0.2

0.2

0

0

–5

–5

–10

0

0.5

1.0

1.5

–10 2.0

x (au)

0

0

–0.5 –1.0

0

0.5

1.0

1.5

x (au)

E (au)

10

E (au)

E (au)

66

0

–0.5

–0.2

–1.0 2.0

–0.4

0 –0.2

0

0.5

1.0

1.5

–0.4 2.0

x (au)

Figure 3.16 Space-charge electric field without an externally applied field for a pattern of fringes with modulation m = 0.99 (left), 0.60 (center) and 0.30 (right).

3.3 First Spatial Harmonic Approximation The procedure in Section 3.2.2 allows one to compute the space-charge field for an arbitrarily large pattern-of-fringes contrast m but the calculation is limited to find out the final stationary state only. In this section, we shall limit ourselves to m ≪ 1 but shall be able to develop an expression for the temporal evolution too. If the light modulation onto the crystal, as described by Eq. (3.8), is sufficiently small (|m| ≪ 1), we may assume that  (x, t), ND+ (x, t) and the space-charge electric field E(x, t), are all periodic real functions of coordinate x and may be described by their first Fourier series development term, the so-called “first spatial harmonic approximation”, as follows: ] [ (3.23)  (x, t) = 0 + 0 ∕2 a(t) eiKx + a∗ (t) e−iKx [ ] ND+ (x, t) = ND+ + ND+ ∕2 A(t) eiKx + A∗ (t) e−iKx (3.24) ] [ ∗ (3.25) E(x, t) = E0 + (1∕2) Esc (t) eiKx + Esc (t) e−iKx ND+ = NA− + 0 ≈ NA− Substituting Eqs. (3.6) and (3.24) into Eq. (2.20) one can write the generation term as: ] G′ [ G(x, t) = G0 + 0 g(t) eiKx + g ∗ (t) e−iKx 2 where ( ] )[ + sIo sIo ∕(h𝜈) 1 ND ∗ ∗ G0 ≡ (ND − ND+ ) + A(t) m) +𝛽 1− × (A(t)m h𝜈 4 ND − ND+ sIo ∕(h𝜈) + 𝛽 ( ) sIo  G0′ ≡ (ND − ND+ ) +𝛽 = 0 h𝜈 𝜏 ND+ A(t) g(t) ≡ meff − ND − ND+ sIo ∕(h𝜈) meff ≡ m sIo ∕(h𝜈) + 𝛽

(3.26)

(3.27)

(3.28) (3.29) (3.30) (3.31)

For the case of small fringes visibility ∣ A(t) m ∣≪ 1 these expressions are simplified to G0 ≈ G0′ = 0 ∕𝜏

(3.32)

By substituting Eqs. (3.23) and (3.24) into Eq. (2.21) an expression for the retrapping is also obtained: ] R [ (3.33) R(x, t) = R0 + 0 (a(t) + A(t)) eiKx + (a∗ (t) + A∗ (t)) e−iKx 2

3.3 First Spatial Harmonic Approximation

o 𝜏 ∣ A(t)a∗ (t) ∣≪ 1

where R0 = rND+ 0 =

(3.34)

assuming

(3.35)

For quasistationary conditions defined as 𝜕 (x, t) =0 𝜕t and substituted into Eq. (2.18), we deduce 1 G − R = − ∇ . ⃗j e

⃗j = e (x, t)𝜇E(x, ⃗ t) + e∇ (x, t)

(3.36)

Substituting the corresponding terms in eiKx from Eq. (3.27) and Eq. (3.33) into the expression in Eq. (3.36), we get G0 E (t) R   g(t) − 0 (a(t) + A(t)) = −𝜇0 iK sc − 𝜇E0 0 a(t)iK −  0 (iK)2 a(t) 2 2 2 2 2 that is rearranged to get a(t) explicitly as a(t) =

meff − A(t)ND ∕(ND − ND+ ) + 𝜇𝜏iKEsc (t)

(3.37)

1 − 𝜇𝜏iKE0 + K 2 𝜏

Following the same procedure for the Eq. (2.19) we get ND+

  ND 𝜕A(t) 0 sIo m∕(h𝜈) − 0 a(t) = − A(t) o + 𝜕t 𝜏 sIo ∕(h𝜈) + 𝛽 𝜏 ND − ND 𝜏

(3.38)

Also substituting the expressions in Eq. (3.24) and Eq. (3.25) into Eq. (2.23) with the assumption o ≪ ND+ − NA− , and solving for the terms in eiKx only, we get 𝚤K𝜖𝜀0 Esc (t) ≈ eND+ A(t)

(3.39)

Combining Eq. (3.37) and Eq. (3.38) we get an equation in A(t) 𝜕A(t) ND+ 𝜕t

N

D 0 0 −𝚤K𝜇𝜏Esc (t) − meff + A(t) ND −ND+ 0 ND + = m − A(t) 𝜏 eff 𝜏 𝜏 1 − 𝚤K𝜇𝜏E0 + K 2 𝜏 ND − ND+

Substituting A(t) in (3.39) by its expression in Eq. (3.39) we get an explicit expression in Esc (t) 𝚤K𝜖𝜀0 𝜕Esc (t) = e 𝜕t

K𝜖𝜀

0 2 0 meff (−𝚤K𝜇𝜏E0 + K 2 𝜏) 0 −𝚤K𝜇𝜏 + 𝚤 e(ND )eff (𝚤K𝜇𝜏E0 − K 𝜏) = + Esc (t) 𝜏 1 − 𝚤K𝜇E0 + K 2 𝜏 𝜏 1 − 𝚤K𝜇E0 + K 2 𝜏

with the effective trap concentration (ND )eff ≡ ND+ (ND − ND+ )∕ND

(3.40)

After rearranging terms, the resulting expression for the space-charge electric field becomes 𝜏M

𝜕Esc (t) −1 + 𝚤KlE − K 2 ls2 E0 + 𝚤ED Esc (t) − meff = 2 𝜕t 1 − 𝚤KLE + K 2 LD 1 − 𝚤KLE + K 2 L2D

(3.41)

67

3 Recording a Space-Charge Electric Field

1.0

Figure 3.17 Simulated recording (from 0 to 20 au) and erasure (from 20 to 50 au) of a space-charge field with E0 = 0 and 𝜏sc = 10 au.

recording

0.8 ESC (t) (au)

68

0.6

erasure

0.4 τSC = 10 (au)

0.2 0

0

10

20 30 Time (au)

40

50

which can be formulated in a more compact form as 𝜕Esc (t) + Esc (t) = −meff Eeff 𝜕t 1 + K 2 L2D − 𝚤KLE 𝜏sc ≡ 𝜏M 1 + K 2 ls2 − 𝚤KlE E0 + 𝚤ED Eeff ≡ 1 + K 2 ls2 − 𝚤KlE

𝜏sc

where

(3.42) (3.43) (3.44)

√ 𝜏

(3.45)

LE ≡ 𝜇𝜏E0

(3.46)

LD ≡

are the diffusion and drift lengths, respectively, and 𝜏 ≡ (rND+ )−1

(3.47)

𝜏M ≡ 𝜖𝜀0 ∕(eμ0 )

(3.48)

are the free electron lifetime and Maxwell (or dielectric) relaxation time, respectively, with kB being the Boltzmann constant, with K 2 𝜖𝜀0 kB T (ND )eff e2 K𝜖𝜀0 E0 KlE ≡ E0 ∕Eq = (ND )eff e e(ND )eff Eq ≡ K𝜖𝜀0

K 2 ls2 ≡ ED ∕Eq =

(3.49) (3.50) (3.51)

where Eq represents the saturation space-charge field and ls is the Debye screening length. Figure 3.17 shows the evolution of Esc during recording and erasure, in arbitrary units with E0 = 0 and 𝜏sc = 10 au, as computed from Eq. (3.42). 3.3.1

Steady-State Stationary Process

For stationary steady-state conditions, it is 𝜕Esc (t)∕𝜕t = 0, which substituted into Eq. (3.42) gives the stationary space-charge field: Esc (t → ∞) = Esc = −meff Eeff

(3.52)

3.3 First Spatial Harmonic Approximation

Unless otherwise stated, we shall hereafter always assume that meff = m. We shall also understand that “stationary” means that it is fixed in space, whereas “steady-state” means that it has reached an equilibrium with a time-invariant formulation, even if it includes a function of time, such as a wave function. 3.3.1.1

Diffraction Efficiency

Figure 3.18 represents a pattern of fringes producing a spatial modulation of charges and an associated space-charge field of amplitude Esc , which produces, via a linear electro-optic effect, an index-of-refraction modulation of amplitude n1 , as defined, for example, in Eqs. (1.45) and (1.46). The index-of-refraction modulation is always in-phase or counter phase with the space-charge field and represents a volume phase grating or hologram. The latter diffraction efficiency (𝜂) is computed, as described in detail in Chapter 4, from the well-known Kogelnik [70] formula: ( ) πn1 d 2 𝜂 = sin (3.53) 𝜆 cos 𝜃 with n1 = −(n3 ∕2)reff |Esc |

(3.54)

where reff is the effective electro-optic coefficient for the given crystal configuration, |Esc | is the amplitude of the space-charge electric field modulation, n is the average refractive index, 𝜆 is the illumination wavelength, 2𝜃 is the angle between the incident beams inside the crystal, and d is the crystal thickness. Equation (3.53) assumes the simplifying approximation of a uniform index-of-refraction modulation along the sample’s thickness. 1.0

IRRADIANCE

0.5 0 –0.5 –1.0 0

0.5

1.0

ρ

DENSITY OF CHARGES +

+

1.5 1.0 0.5 0

–

0

–0.5 0.5

1.0

–1.0 1.5

90°

1.0

SPACE-CHARGE FIELD ∆n

0.5 0 –0.5 –1.0

0

0.5

1.0

1.5

Figure 3.18 Index-of-refraction modulation arising in the crystal volume. The upper figure shows the pattern of light fringes projected onto the crystal, the middle figure shows the resulting charge density and the lower figure shows the spatial-charge field and index-of-refraction modulation in the absence of any externally applied electric field (E0 = 0). All vertical coordinates are in “arbitrary units”.

69

70

3 Recording a Space-Charge Electric Field

3.3.1.2 Hologram Phase Shift

The phase position (𝜙) of the recording pattern of fringes referred to an arbitrarily selected fixed reference (as illustrated in Fig. 3.16) is given by the phase of the complex modulation m in Eq. (3.8) or of the effective modulation meff in Eq. (3.31), whereas that of the resulting hologram (that is to say, the index-of-refraction modulation pattern, for a pure refractive index hologram) is given by the phase of the complex Esc as the latter two are necessarily always in-phase as indicated in Eq. (3.54). The so-called photorefractive hologram phase shift 𝜙P (which is the actual interesting parameter) is then the phase difference between the recorded hologram and the corresponding pattern of light fringes and is the phase of the complex quantities Esc ∕m or Esc ∕meff . For steady-state conditions, as in Eq. (3.52), where it is Esc = −meff Eeff , the phase-shift 𝜙P is computed as tan{𝜙P } =

ℑ{Eeff } ℜ{Eeff }

(3.55)

where “ℑ” and “ℜ” mean the “imaginary” and “real” parts, respectively, and from the expression for Eeff in Eq. (3.44) we get tan{𝜙P } =

E0 KlE + ED (1 + K 2 ls2 )

(3.56)

E0 (1 + K 2 ls2 ) − ED KlE

For E0 = 0, and consequently KlE = 0, it is straightforward to show that 𝜙P = ±π∕2. 3.3.2

Time-Evolution Process: Constant Modulation

For the general case, the hologram being recorded (because of the growing of a space-charge electric field modulation Esc ) does modify the pattern of light throughout the crystal volume so that the light modulation m is not at all constant but varies along the crystal thickness. In this case, Eq. (3.42) can be solved with the help of the coupled-wave theory, as we shall show in Chapter 4. We shall assume here, however, that meff in Eq. (3.42) is constant. This may be approximately true for low diffraction efficiency and for no sensible energy exchange between the interfering beams as they propagate through the sample thickness. In this case, Eq. (3.42) is easily solved to give: Esc (t) = −mEeff (1 − e−t∕𝜏sc )

(3.57)

We shall take into consideration here the fact that 𝜏sc , Eeff , m and, consequently Esc , are all complex quantities, so we should explicitly write { } { } Esc (t) Esc (t) Esc (t) =ℜ + iℑ (3.58) m m m { } [ ] 2 Esc (t) ℜ = −ℜ{Eeff } 1 − e−tℜ{𝜏sc }∕|𝜏sc | cos (tℑ{𝜏sc }∕|𝜏sc |2 ) + m 2 (3.59) −ℑ{Eeff } e−tℜ{𝜏sc }∕|𝜏sc | sin (tℑ{𝜏sc }∕|𝜏sc |2 ) { } [ ] 2 Esc (t) ℑ = −ℑ{Eeff } 1 − e−tℜ{𝜏sc }∕|𝜏sc | cos (tℑ{𝜏sc }∕|𝜏sc |2 ) + m 2 (3.60) +ℜ{E } e−tℜ{𝜏sc }∕|𝜏sc | sin (tℑ{𝜏 }∕|𝜏 |2 ) eff

sc

sc

where from Eq. (3.43): ℜ{𝜏sc } = 𝜏M

(1 + K 2 L2D )(1 + K 2 ls2 ) + KLE KlE (1 + K 2 ls2 )2 + K 2 lE2

(3.61)

3.3 First Spatial Harmonic Approximation

ℑ{𝜏sc } = 𝜏M

KlE (1 + K 2 L2D ) − KLE (1 + K 2 ls2 )

(3.62)

(1 + K 2 ls2 )2 + K 2 lE2

and from Eq. (3.44): ℜ{Eeff } = ℑ{Eeff } =

E0 (1 + K 2 ls2 ) − ED KlE

(3.63)

(1 + K 2 ls2 )2 + K 2 lE2 E0 KlE + ED (1 + K 2 ls2 )

(3.64)

(1 + K 2 ls2 )2 + K 2 lE2

The evolution of Esc in modulus (|Esc |2 ∝ 𝜂) and phase are described by: √ 2 2 |Esc (t)| = |m||Eeff | 1 + e−2tℜ{𝜏sc }∕|𝜏sc | − 2 e−tℜ{𝜏sc }∕|𝜏sc | cos(t∕ℑ{𝜏sc }∕|𝜏sc |2 ) (3.65) and tan 𝜙P (t) =

ℑ{Esc (t)∕m} ℜ{Esc (t)∕m}

ℑ{Eeff } − [ℜ{Eeff } sin (tℑ{𝜏sc }∕|𝜏sc |2 ) + ℑ{Eeff } cos (tℑ{𝜏sc }∕|𝜏sc |2 )] e−tℜ{𝜏sc }∕|𝜏sc |

2

=

ℜ{Eeff } − [ℜ{Eeff } cos (tℑ{𝜏sc }∕|𝜏sc |2 ) − ℑ{Eeff } sin (tℑ{𝜏sc }∕|𝜏sc |2 )] e−tℜ{𝜏sc }∕|𝜏sc | (3.66) 2

Note that the terms ℜ{𝜏sc }∕|𝜏sc |2 and ℑ{𝜏sc }∕|𝜏sc |2 in Eqs. (3.65) and (3.66) are, respectively, 𝜔R and 𝜔I : 𝜔R ≡ ℜ{1∕𝜏sc }

𝜔I ≡ ℑ{1∕𝜏sc }

(3.67)

which in turn are represented from Eq. (3.43) as: 1 = 𝜔R + 𝚤𝜔I 𝜏sc

(3.68)

and explicitly formulated as 2 2 2 2 1 (1 + K ls )(1 + K LD ) + KlE KLE 𝜏M (1 + K 2 L2D )2 + K 2 L2E KLE − KlE 1 𝜔I = 𝜏M (1 + K 2 L2D )2 + K 2 L2E

𝜔R =

(3.69) (3.70)

We can compute the expressions for the initial and for the stationary conditions for Esc (t) from Eq. (3.65): lim |Esc (t)| = 0

(3.71)

t→0

lim |Esc (t)| = |m||Eeff |

t→∞

√ = |m| ED2 + E02

√ ED2 E E 1 + K 2 lE2 + 2 E2D+E02 KlE + 3 E2 +E K 2 ls2 2 D

(1 +

0

K 2 ls2 )2

0

+

K 2 lE2

D

(3.72)

71

72

3 Recording a Space-Charge Electric Field

Substituting Eqs. (3.68) and (3.69) into Eq. (3.66), we can compute the initial (𝜙I ) and the stationary values for 𝜙P , where the former one is 𝜔 ℜ{Eeff } + 𝜔R ℑ{Eeff } tan 𝜙I = lim tan{𝜙P (t)} = I t→0 𝜔R ℜ{Eeff } − 𝜔I ℑ{Eeff } [(1 + K 2 ls2 )ED + KlE E0 ]𝜔R + [(1 + K 2 ls2 )E0 − KlE ED ]𝜔I (3.73) = [(1 + K 2 ls2 )E0 − KlE ED ]𝜔R − [KlE E0 + (1 + K 2 ls2 )ED ]𝜔I and the stationary value is ℑ{Eeff } E Kl + ED (1 + K 2 ls2 ) (3.74) = 0 E t→∞ ℜ{Eeff } E0 (1 + K 2 ls2 ) − ED KlE Equation (3.74) is, of course, the same as the one reported for a Steady-State Stationary process in Eq. (3.55). The expressions in Eqs. (3.71)–(3.73) for the initial and for the steady state can be used to determine some of the materials’ parameters, as discussed in Chapter 8 and reported elsewhere [71, 72]. lim tan{𝜙P (t)} =

3.4 Steady-State Nonstationary Process: Running Holograms Running holograms in photorefractives (Fig. 3.19) were first reported in Bi12 SiO20 crystals by Huignard et al. [73, 74], who pointed out the resonant behavior of the two-wave mixing amplitude gain and demonstrated its interest for coherent beam amplification and vibration measurement. Stepanov et al. [75] further developed the subject by reporting illustrative experimental results and by establishing a sound theoretical basis to explain the main features. Refregier et al. [76] analyzed these holograms with special attention to amplitude gain for sillenites and showed that the resonance velocity condition was particularly suitable for amplitude gain, not only because of its characteristic large diffraction value but also because it exhibits a nearly 90∘ hologram phase-shift that optimizes amplitude coupling. The research extended RUNNING HOLOGRAM

Figure 3.19 Schematic description of running hologram generation in photorefractives. A moving pattern-of-fringes onto the sample produces a synchronously moving volume hologram that reaches a maximum amplitude at a resonance speed.

V I n X

V

RESONANCE SPEED VO I n

X VO

3.4 Steady-State Nonstationary Process: Running Holograms

to semi-insulating semiconductor photorefractives such as GaAs and InP [77–80]. The whole subject of moving holograms in photorefractives has been analyzed under the general approach of the so-called space-charge wave formalism [81–84]. Running holograms were experimentally detected and measured in photorefractive materials under the action of a moving pattern of fringes in the presence of an externally applied electric field [85]. Just to start understanding the matter before going to equations, let us think of an already recorded space-charge field with charge carriers being continuously excited to the CB and preferentially drifted (because of the external field) and retrapped along one sense of the applied field direction. This is just like inducing the grating to move along that sense. But the pattern of fringes is stationary so that the hologram does not move. If we allow the pattern of fringes to move along with the charge carriers, we shall allow the grating to move. An optimum speed does exist that depends on the drifting force of the external field and the response time of the material. By moving the pattern of fringes with such a speed a resonance is achieved where the moving grating amplitude is maximum. If the pattern of fringes moves faster or slower than this resonance speed, the recorded hologram will certainly follow the pattern-of-fringes speed but a weaker grating will result. Let us now put these ideas into equations. We can theoretically find a solution for the moving holograms by just solving out the same fundamental set of Eqs. (2.18)–(2.23), where the stationary pattern of fringes in Eq. (3.5) is substituted by a moving pattern of fringes of the form I = Io (1+ ∣ m ∣ cos(Kx − K𝑣t + 𝜙)) = Io + (Io ∕2)[m exp(iKx − iK𝑣t) + m∗ exp(−iKx + iK𝑣t)]

(3.75)

where 𝑣 is the speed of the fringes moving along the x-axis. In this case the development in Section 3.3 leads to a general equation of the form 𝜕E (t) 𝜏sc sc + Esc (t) = −mEeff e−𝚤K𝑣t 𝜕t 1 + K 2 L2D − 𝚤KLE E0 + 𝚤ED with Eeff = 𝜏 = 𝜏 (3.76) sc M 1 + K 2 ls2 − 𝚤KlE 1 + K 2 ls2 − 𝚤KlE Note that the expressions for Eeff and 𝜏sc here are the same as those for the stationary hologram in Eqs. (3.44) and (3.43), respectively. The difference here is that an exponential term e−iK𝑣t is factoring the expression for Eeff in the differential equation Eq. (3.76). The general solution of the differential equation in Eq. (3.76) is E (t) = −mEst e−iK𝑣t + [Etrans e−𝜔R t ] e−𝚤𝜔I t (3.77) sc

sc

sc

Eeff (𝜔R + 𝚤𝜔I ) 𝜔R + 𝚤(𝜔I − K𝑣) where the complete formulation of Eq. (3.77) according to Eq. (3.25) is: E(x, t) = E (t) e𝚤Kx st with Esc =

(3.78)

sc

st 𝚤(Kx − K𝑣t) trans −𝜔R t 𝚤(Kx − 𝜔I t) e + Esc e e = −mEsc

(3.79)

The first term in Eq. (3.79) is the steady-state solution, representing a hologram moving along with the pattern of fringes with velocity 𝑣; the second term is a resonantly excited transient hologram where the amplitude decays with a time constant 𝜔R and moves along with a linear speed 𝑣trans = 𝜔I ∕K. Note that 𝜔I

(3.80)

is the natural oscillation frequency of the system; that is to say, the running hologram resonance frequency.

73

3 Recording a Space-Charge Electric Field

5

ωI

3 2 ωR

sc

2

(au)

4

Est

74

2

1

0 –5

0

5

10

15

Kv (rad/s) st 2 Figure 3.20 Plot of |Esc | ∝ 𝜂 for the assumed parameters: LD = 0.20 μm, lS = 0.02 μm, Φ = 0.5, 𝜔I = 5.1 rad/s, Q ≈ 2 and 𝛼 = 11.5 cm−1 for 𝜆 = 514.5 nm; with the experimental conditions being K = 10 μm−1 , E0 = 106 V/m, and an intensity inside the front crystal plane I(0) = 100 W/m2 . From Eq. (3.21) and K we compute ED = 2.59 × 105 V/m at T = 300 K, from K 2 ls2 and ED in Eq. (3.49) we get Eq = 6.5 × 106 V/m, from E0 and Eq in Eq. (3.50) we compute KlE = 0.15, from Q and 𝜔I in Eq. (3.87) we get 𝜔R = 2.55 rad/s.

st The unit-modulation steady-state amplitude Esc in Eq. (3.78) can be written as: st Esc =

[(𝜔R E0 − 𝜔I ED ) + 𝚤(𝜔R ED + 𝜔I E0 )] [𝜔R (1 + K 2 ls2 ) + (𝜔I − K𝑣)KlE ] + 𝚤[(𝜔I − K𝑣)(1 + K 2 ls2 ) − 𝜔R KlE ]

(3.81)

Assuming some typical material and usual experimental parameters, and taking into account st 2 | is computed that from Eqs. (3.49) and (3.50) it is KlE = K 2 ls2 E0 ∕ED , the square modulus |Esc st 2 and plotted in Fig. 3.20, which represents the diffraction efficiency (|Esc | ∝ 𝜂), for 𝜂 ≪ 1 conditions, being maximum at the resonance speed 𝑣res = 𝜔I ∕K

(3.82)

The real and imaginary parts of

st Esc

are, respectively:

(𝜔R E0 − 𝜔I ED )(𝜔R A + B KlE ) + (𝜔I E0 + 𝜔R ED )(BA − 𝜔R KlE ) (𝜔R A + B KlE )2 + (B A − 𝜔R KlE )2 (𝜔 E + 𝜔R ED )(𝜔R A + B KlE ) − (𝜔R E0 − 𝜔I ED )(B A − 𝜔R KlE ) st ℑ{Esc }= I 0 (𝜔R A + B KlE )2 + (B A − 𝜔R KlE )2

st }= ℜ{Esc

with: B ≡ (𝜔I − K𝑣)

(3.83) (3.84)

A ≡ (1 + K 2 ls2 )

which are particularly interesting because they determine, from Eq. (3.55), the hologram phase shift 𝜙P , which can therefore be now written as st ℑ{Esc } st ℜ{Esc } 𝜔 [E M + ED N] + B[ED M − E0 N] = R 0 𝜔R [E0 N − ED M] + B[E0 M + ED N]

tan 𝜙P =

with: M ≡ 𝜔I A + 𝜔R KlE

N ≡ 𝜔R A − 𝜔I KlE

(3.85)

3.4 Steady-State Nonstationary Process: Running Holograms

2

ϕP (rad)

1

0

–1

–2

0

2

4

6

8

10

Kv (rad/s)

Figure 3.21 Plot of 𝜙P from Eq. (3.85) for the same parameters referred to in Fig. 3.20.

At the resonance condition K𝑣 = 𝜔I , we should write B = 0 in Eq. (3.85) and the latter simplifies to: tan 𝜙P ]K𝑣=𝜔I =

E0 M + ED N E0 N − ED M

(3.86)

It is easy to mathematically verify that the phase shift for a resonantly running hologram in Eq. (3.86) is the same as that one in Eq. (3.73) for initial holographic recording conditions. This similarity is not entirely surprising because a running hologram operates by continuously recording and erasing to record again a little bit farther on and, for an adequate speed (not too slow to develop a large hologram and not too fast to avoid recording even a small one) the necessary conditions to reproduce an “initial hologram phase-shift” value may (as it actually seems to) be achieved even if a rather large travelling nonstationary hologram is finally produced. The hologram phase shift 𝜙P is plotted in Fig. 3.21 as a function of K𝑣. Verify that for K𝑣 = 0, Eq. (3.85) reduces to the simple case of Eq. (3.74) characterizing a steady-state stationary hologram. Note that Fig. 3.20 actually shows a resonant behavior with a characteristic resonance frequency 𝜔I and a dissipative term 𝜔R with a quality factor Q that is defined in the usual way as 𝜔 Q≡ I (3.87) 𝜔R Substituting 𝜔R and 𝜔I , respectively, from Eqs. (3.70) and (3.69) into Eq. (3.87) we get: Q=

KLE − KlE (1 +

K 2 ls2 )(1

+ K 2 L2D ) + KlE KLE

(3.88)

75

76

3 Recording a Space-Charge Electric Field

and still substituting KlE from Eqs. (3.50) and (3.49) and KLE from Eqs. (3.46) and (3.45) and (3.21) KlE = K 2 ls2 E0 ∕ED KLE = K 2 L2D E0 ∕ED

(3.89) (3.90)

into Eq. (3.88) we get the final expression Q=

K 2 L2D − K 2 ls2 (1 +

K 2 ls2 )(1

+

K 2 L2D )

+

K 2 ls2

K 2 L2D

(E0 ∕ED

)2

E0 ED

(3.91)

which is represented in the 3D graphics of Fig. 3.22 as a function of LD (in μm) and ls (in μm), whereas Fig. 3.23 shows its dependence on K (in μm−1 ), for some typical parameter values. st 2 st 2 | (with |Esc | ∝ 𝜂, always for 𝜂 ≪ 1) is maximum at resonance Figure 3.24 shows that |Esc st }, where the latter determines the extent of amplitude coupling (𝜔I = K𝑣) as well as ℑ{Esc (energy exchange) in two-wave mixing experiments, as discussed in Section 4.2.1.1. 3.4.1

Running Holograms with Hole-Electron Competition

It is possible to have electrons and holes excited simultaneously, by the action of light, in order to produce their correspondingly associated space-charge field gratings. If both electrons and holes are excited from the same photoactive species in the Band Gap, as illustrated in Fig. 3.25 there is one single spatial trap (charges) modulation and the recording process follows a one-exponential law [86] where the characteristic exponential time depends on the properties of traps and of both type of carriers [86]. In the absence of an external field, the movement of these charge carriers is controlled by diffusion. In this case, there should be one charge carrier predominating over the other for an effective space-charge modulation to be built up. Figure 3.22 Plotting of Q as a function of LD (LD-axis) and ls (LS-axis) for E0 = 106 V/m, K = 10 μm−1 , 𝜆 = 514.5 nm with 𝛼 = 11.5 cm−1 , Φ = 0.5 and an intensity inside the front crystal plane I(0) = 100 W/m2 . 4 3 Q

0.05

2

0.04

1 0

0.03

0

0.02

0.2 0.4 LD (µm)

0.01

0.6 0.8 1

0

Quality factor Q

Figure 3.23 Plotting of Q as a function of K, from Eq. (3.91), for typical values LD = 0.15 μm, ls = 0.03 μm and different applied electric fields from 5 × 105 , 7 × 105 , 10 × 105 to 15 × 105 V/m, represented by the progressively increasing size of the dashed lines, respectively.

2 1.5 1 0.5 2

L3 (µm)

4

6

8

10

K (μm–1)

3.4 Steady-State Nonstationary Process: Running Holograms

lm[Est] Re[Est]

sc

∣Est ∣2sc sc sc

4

1

3

0

2

–1

1

sc

sc

Re[Est ], |m[Est ] (au)

2

5

│Est│2 (au)

3

–2 –5

0

0 15

10

5 Kv (rad/s)

st 2 st st Figure 3.24 Plotting of |Esc | (continuous curve), ℜ{Esc } (long dashing curve) and ℑ{Esc } (short dashing curve) versus K𝑣, for the same parameters referred to in Fig. 3.20.

ILLUMINATION

CB

–

–

+

+ –

+

–

+ –

+ –

–

–

+ –

+ +

+ –

–

+

VB

– N– A

–

nonphotoactive

+ + ND

acceptor

+

+ –

+

–

+

ND ND

donor

Figure 3.25 One-species/two-valence/two-charge carrier model contributing to charge transport: one single spatial trap modulation structure is produced.

Otherwise, both carriers will compensate each other and no actual charge separation will occur. Instead, if electrons and holes are excited from different species (photoactive centers or localized states), as illustrated in Fig. 3.26, there may be an effective build-up of two physically distinct (each one on different centers) and opposite sign gratings, and the overall recording dynamics may follow a two-exponential law [86], one for each one of the species, which in this case also corresponds to two different charge carriers.

77

78

3 Recording a Space-Charge Electric Field

ILLUMINATION

CB

–

– +

+ – + –

–

–

– + – +

+

+ –

+

+

– +

+ –

acceptor – + ND2 ND2 acceptor

+ NA 2 nonphotoactive

–+

+

+ –

+

+ + ND

nonphotoactive

– – +

+ + –

+

VB

– N– A

+– + –

+

–

–

– – +

+

+ –

–

+

ND– ND –

donor

+ ND2 donor

Figure 3.26 Two-species/two-valence/two-charge carrier model contributing to charge transport: two distinct spatial trap modulation structures are produced.

CB

– +

– – – + +

+++ – –

+

+ –

+

–

+

–

+

–

– – +

+ –

– +

+ –

– +

+ –

VB

+ –

Figure 3.27 Hole-electron competition on different photoactive centers under the action of low energetic photon recording light: only charge carriers close to the CB (electrons) and to the VB (holes) can be excited, but electrons cannot be excited from the hole-donor level or holes from the electron-donor level, because of energy considerations. In this case, an electron-based hologram is recorded in the level closer to CB, and the same for holes in the level close to VB. However, electrons progressively accumulate in the (deeper) level closer to the VB and holes accumulate in the level close to the CB, where they cannot be re-excited again because the recording light is not energetic enough. The recording is progressively decreasing, because of the decrease in the corresponding charge carriers, until a steady state is achieved because of the exhaustion of any one of the two levels.

There is still the very realistic possibility that both electrons and holes might be excited from both photoactive centers with holes dominating one but electrons dominating the other. It is still possible that the two centers might be able to allow holes and electrons to be excited but that these centers might be placed differently in the Band Gap, with the photonic energy being small enough to allow only excited electrons from one species and only holes from the other, as schematically depicted in Fig. 3.27. 3.4.1.1 Mathematical Model

Here, we shall handle the case where holograms involve holes in one LS and electrons in another LS, with holes and electrons originating, of course, from different centers and for the case where

3.4 Steady-State Nonstationary Process: Running Holograms

there is a mechanism for providing reversibility. The simultaneous presence of electron- and hole-photoactive centers produces an electrically coupled system of equations that, under certain conditions, can be analytically solved [83]. Considering the formulation in Eq. (3.75) for the pattern of fringes being I = Io (1+ ∣ m ∣ cos(Kx − K𝑣t + 𝜙)) = Io + (Io ∕2)[m exp(iKx − iK𝑣t) + m∗ exp(−iKx + iK𝑣t)] and from the previous assumption that electrons and holes are originated from different and independent photoactive centers, we should consider the system of fundamental Eqs. (2.18)–(2.21) for electrons, just including a sub-index “1”: 𝜕 (x, t) 1 = G1 − R1 − ∇ . j⃗1 𝜕t e + 𝜕ND1 (x, t)

𝜕t

(3.93)

= G1 − R1

+ G1 = (ND1 − ND1 (x, t))(

(3.92)

sI + 𝛽1 ) h𝜈

(3.94)

+ R1 = r1 ND1 (x, t) (x, t)

(3.95)

⃗ t) + e1 ∇ (x, t) j⃗1 = e (x, t)𝜇1 E(x,

(3.96)

and a similar independent set of equations for holes with sub-index “2”: 𝜕(x, t) 1 = G2 − R2 − ∇ . j⃗2 𝜕t e − (x, t) 𝜕ND2 = G2 − R2 𝜕t ⃗ t) − e2 ∇(x, t) j⃗2 = e(x, t)𝜇2 E(x, − G2 = (ND2 − ND2 (x, t))(

s2 I + 𝛽2 ) h𝜈

− R2 = r2 ND2 (x, t)(x, t)

(3.97) (3.98) (3.99) (3.100) (3.101)

with  being the density of free holes in the VB. The coupling between holes and electrons is mathematically formulated by the Poisson equation + − 𝜀0 𝜖∇ . E⃗ = e(ND1 +  − NA− − ND2 −  + NB+ )

(3.102)

with NB+ having the same meaning that NA− but for holes. We should neglect thermal excitation and assume a linearized set of equations for all parameters involved, of the same form as those in Eqs. (3.23)–(3.25), now also including similar expressions for the holes and their photoactive centers, which are represented by the sub-index “2”, whereas “1” is for the electrons. The solution of the previous system of coupled equations starts with the usual assumption of quasi-stationary condition: 𝜕 ∕𝜕t ≈ 𝜕∕𝜕t ≈ 0

(3.103)

79

80

3 Recording a Space-Charge Electric Field

In this case, we are able to write sI sI + + ) 1 o m e−iK𝑣t − 1 o ND1 A1 (t) + G1 − R1 = (ND1 − ND1 2h𝜈 2h𝜈 + + −o ND1 A1 (t)r1 ∕2 − 0 a1 (t)ND1 r1 ∕2 (3.104) sI sI − − G2 − R2 = (ND2 − ND2 ) 2 o m e−iK𝑣t − 2 o ND2 A2 (t) + 2h𝜈 2h𝜈 − − −o ND2 A2 (t)r2 ∕2 − o a2 (t)ND2 r2 ∕2 (3.105) ∇ . ⃗j1 (3.106) = ∇ . (0 𝜇1 Esc (t)∕2 + o a1 (t)𝜇1 E0 ∕2) − 1 K 2 0 a1 (t)∕2 e ∇ . ⃗j2 (3.107) = ∇ . (o 𝜇2 Esc (t)∕2 + o a2 (t)𝜇2 E0 ∕2) + 2 K 2 o a2 (t)∕2 e Substituting Eqs. (3.105) and (3.107) into Eq. (3.97) and proceeding in a similar way with the electrons, we get sIm sI + + + + (ND1 − ND1 ) 1 o e−iK𝑣t − 1 o ND1 A1 (t) − 0 ND1 A1 (t)r1 − 0 a1 (t)ND1 r1 = h𝜈 h𝜈 e𝜇  + e𝜇  − = 1 K 2 0 a1 (t) − 1 o ND1 A1 (t) + 1 o ND2 A2 (t) − iK0 a1 (t)𝜇1 E0 𝜖𝜀0 𝜖𝜀0 sIm sI − − − − (ND2 − ND2 ) 2 o e−iK𝑣t − 2 o ND2 A2 (t) − o ND2 A2 (t)r2 − o a2 (t)ND2 r2 = h𝜈 h𝜈 e𝜇  − e𝜇  + A2 (t) + 2 o ND1 A1 (t) + iKo a2 (t)𝜇2 E0 = 2 K 2 o a2 (t) − 2 o ND2 𝜖𝜀0 𝜖𝜀0 From the previous two equations we get the relations between ai (t) and Ai (t): [ sI + + 2 + + o ND1 r1 + a1 (t)[−0 ND1 r1 − 1 K 0 + iKo 𝜇1 E0 ] = A1 (t) 1 o ND1 h𝜈 ] e𝜇  + sI e𝜇  − + − 1 o ND1 A2 (t) − 1 o m(ND1 − ND1 ) e−iK𝑣t + 1 o ND2 𝜖𝜀0 𝜖𝜀0 h𝜈 sI − − − a2 (t)[−0 ND2 r2 − 2 K 2 o − iKo 𝜇2 E0 ] = A2 (t)[ 2 o ND2 + o ND2 r2 + h𝜈 e𝜇  + sI e𝜇  − − ] + 2 o ND1 A1 (t) − 2 o m(ND2 − ND2 ) e−iK𝑣t − 2 o ND2 𝜖𝜀0 𝜖𝜀0 h𝜈

(3.108)

(3.109)

Substituting Eq. (3.104) into Eq. (2.9) and proceeding similarly with Eq. (3.97) for holes, we get the two equations [ ] sI s1 Io + + 𝜕A1 (t) + + ND1 ) 1 o m e−iK𝑣t + = −A1 (t) ND1 + o ND1 r1 + (ND1 − ND1 𝜕t h𝜈 h𝜈 + −o ND1 r1 a1 (t) [ ] sI − sI − 𝜕A2 (t) − − + o ND2 r2 + (ND2 − ND2 ) 2 o m e−iK𝑣t + = −A2 (t) 2 o ND2 ND2 𝜕t h𝜈 h𝜈 − r2 a2 (t) −o ND2 Coupled Equations Substituting the values of a1 (t) and a2 (t) in the previous equations by their expressions computed from Eqs. (3.108) and (3.109) and also using the linearized relation

3.4.1.1.1

+ − A1 (t) − ND2 A2 (t)) iK𝜖𝜀0 Esc (t) ≈ e(ND1

3.4 Steady-State Nonstationary Process: Running Holograms

from Eq. (3.102), we get the two coupled differential equations 𝜕Esc1 (t) + Esc1 (t) = −mEeff1 𝜕t 𝜕E (t) 𝜏sc2 sc2 + Esc2 (t) = −mEeff2 𝜕t with the usual parameter definitions

e−iK𝑣t − 𝜅12 Esc2 (t)

(3.110)

e−iK𝑣t − 𝜅21 Esc1 (t)

(3.111)

= 𝜔(1) + i𝜔(1) I R

(3.112)

1 = = 𝜔(2) + i𝜔(2) I R 𝜏sc2 𝜏M2 1 + K 2 L2D2 − iKLE2

(3.113)

𝜏sc1

1 𝜏sc1

=

Eeff1 = Eeff2 =

1

2 1 + K 2 ls1 − iKlE1

𝜏M1 1 + K 2 L2D1 − iKLE1 2 2 1 1 + K ls2 − iKlE2

E0 + iED1 1+

2 K 2 ls1

− iKlE1 E0 + iED2

2 1 + K 2 ls2 − iKlE2

kB T e k T = −K B e

ED1 = K

(3.114)

ED2

(3.115)

KLE1 = K 2 L2D1 E0 ∕ED1

(3.116)

KLE2 = K 2 L2D2 E0 ∕ED2

(3.117)

2 E0 ∕ED1 KlE1 = K 2 ls1

(3.118)

2 E0 ∕ED2 KlE2 = K 2 ls2

(3.119)

and the coupling coefficients definition: 2 − iKlE1 1∕𝜅12 ≡ 1 + K 2 ls1

(3.120)

2 − iKlE2 1∕𝜅21 ≡ 1 + K 2 ls2

(3.121)

Esc (t) = Esc1 (t) + Esc2 (t)

(3.122)

with

Let us search for a steady-state solution of Eq. (3.110) and (3.111) of the same form as in Eq. (3.77): st st e−iK𝑣t and Esc2 (t) = −mEsc2 e−iK𝑣t Esc1 (t) = −mEsc1

that substituted into the coupled equations gives st st st = Esc1 + Esc2 Esc

=

Eeff1 (𝜔(1) + i𝜔(1) )[(𝜔(2) + i𝜔(2) )(1 − 𝜅21 ) − iK𝑣] I I R R (1) (1) (2) (2) (1) (1) (𝜔(2) + i𝜔(2) I − iK𝑣)(𝜔R + i𝜔I − iK𝑣) − (𝜔R + i𝜔I )(𝜔R + i𝜔I )𝜅12 𝜅21 R

+

(1) (1) Eeff2 (𝜔(2) + i𝜔(2) I )[(𝜔R + i𝜔I )(1 − 𝜅12 ) − iK𝑣] R

(𝜔(2) + i𝜔(2) − iK𝑣)(𝜔(1) + i𝜔(1) − iK𝑣) − (𝜔(2) + i𝜔(2) )(𝜔(1) + i𝜔(1) )𝜅12 𝜅21 I I I I R R R R

(3.123)

where the effect of holes and electrons on the space-charge field grating are coupled. Note that Eqs. (3.120) and (3.121) define the electrical coupling constants for charges that are roughly at the same position in space. This is not necessarily the case with actual materials

81

82

3 Recording a Space-Charge Electric Field

where charges may be located at different photoactive centers and may therefore be somewhat separated in space. The electric coupling between charges that are spatially separated depends on their corresponding Debye lengths of the photoactive centers involved: The larger the Debye length, the lower the effect of charge separation. This means that the electrical coupling should be adjusted for different photoactive centers using a phenomenological parameter 𝜁 [87], in which case the coupling constants in Eqs. (3.120) and (3.121) should be written as: 𝜁1 𝜅12 = (3.124) 2 2 1 + K ls1 − iKlE1 𝜁2 (3.125) 𝜅21 = 2 1 + K 2 ls2 − iKlE2 where 𝜁1 and 𝜁2 vary with the spatial separation between the interacting charges and their respective Debye lengths, with their values being determined by the experiment itself. Neglecting Coupling If we assume that the coupling constants are sufficiently small (that may happen for 𝜁1 ≈ 𝜁2 ≪ 1 or even for the condition E0 ∕ED ≫ 1) so that

3.4.1.1.2

𝜅12 ≈ 𝜅21 ≈ 0, Equation (3.123) can be written as a pair of independent terms, each one depending on one type of charge carrier only st Esc = Eeff1

(𝜔(1) + i𝜔(1) I ) R 𝜔(1) + i𝜔(1) I − iK𝑣 R

+ Eeff2

(𝜔(2) + i𝜔(2) I ) R 𝜔(2) + i𝜔(2) I − iK𝑣 R

(3.126)

Steady-State Stationary Limit It is interesting to find out what the expression in Eq. (3.123) would look like for the steady state stationary limit condition where K𝑣 = 0. In this case, we have Eq. (3.123) simplified to 1 − 𝜅21 1 − 𝜅12 st Esc = Eeff1 + Eeff2 (3.127) 1 − 𝜅12 𝜅21 1 − 𝜅12 𝜅21 In the absence of an externally applied field, Eq. (3.127) becomes further simplified to 3.4.1.1.3

st Esc =K

2 2 − K 2 ls1 K 2 ls2 kB T 2 2 2 2 q K 2 ls1 + K 2 ls2 + K 2 ls1 K 2 ls2

(3.128)

2 2 and for nonsaturated conditions (K 2 ls1 ≪ 1 and K 2 ls2 ≪ 1) we get st Esc =K

2 2 − K 2 ls1 kB T K 2 ls2 2 2 q K 2 ls1 + K 2 ls2

(3.129)

which leads to zero effective space charge field if the Debye lengths are similar for electrons and holes, which in fact is not usual. If the concentration of one of the centers (for ex. holes) is 2 2 much lower than the other (K 2 ls2 ≫ K 2 ls1 ) instead, the overall space-charge field is dominated by the nondepleted center, as expected k T 1 st Esc =K B (3.130) 2 q 1 + K 2 ls1 which in this example are the electron-based centers. More complicated situations, including wave mixing and bulk light absorption effects, which usually occur in photorefractive materials, do not usually lead to analytical solutions. The solution for a particular simple case will be treated in Section 8.6.5.

3.4 Steady-State Nonstationary Process: Running Holograms

Hologram Erasure The erasure of a hologram with hole-electron competition on different localized states as developed previously appears to have a rather simple mathematical solution. In fact, let us go back to Eqs. (3.110) and (3.111), where we should write m = 0 for erasure so that the independent terms disappear and the coupled equations can be written as two uncoupled equations

3.4.1.1.4

𝜕 2 Esc1 (t) 𝜏sc1 + 𝜏sc2 𝜕Esc1 (t) 1 − 𝜅12 𝜅21 + E (t) = 0 + 𝜕t 2 𝜏sc1 𝜏sc2 𝜕t 𝜏sc1 𝜏sc2 sc1 𝜕 2 Esc2 (t) 𝜏sc1 + 𝜏sc2 𝜕Esc2 (t) 1 − 𝜅12 𝜅21 + E (t) = 0 + 𝜕t 2 𝜏sc1 𝜏sc2 𝜕t 𝜏sc1 𝜏sc2 sc2

(3.131) (3.132)

The solution of the equations here is Esc1 = A1 er1 t + B1 er2 t

(3.133)

Esc2 = A2 er1 t + B2 er2 t

(3.134)

with the total space charge field being EscT = Esc1 + Esc2 = (A1 + A2 ) er1 t + (B1 + B2 ) er2 t with

√ 1 − b) √ r2 ≡ −a(1 + 1 − b) r1 ≡ −a(1 −

1 𝜏sc1 + 𝜏sc2 2 𝜏sc1 𝜏sc2 1 − 𝜅12 𝜅21 b ≡ 4𝜏sc1 𝜏sc2 (𝜏sc1 + 𝜏sc2 )2

a≡

(3.135) (3.136)

(3.137) (3.138) (3.139) (3.140)

For the condition E0 = 0 and b ≪ 1, Eq. (3.136) simplifies to EscT = (A1 + A2 ) e−𝛽t + (B1 + B2 ) e−𝛼t 1 − 𝜅12 𝜅21 𝜏sc1 + 𝜏sc2 𝜏sc1 + 𝜏sc2 𝛼≈ 𝜏sc1 𝜏sc2 𝛽≈

(3.141) (3.142) (3.143)

where 𝛽 and 𝛼 are real positive values. If the exponential time constant (𝜏sc2 ) for the hole-based space charge field is much larger than that (𝜏sc1 ) for the electron-based one, then we may further simplify Eqs. (3.142) and (3.143) to 𝛼 ≈ 1∕𝜏sc1 𝛽≈

1 − 𝜅12 𝜅21 𝜏sc2

(3.144) (3.145)

Experimental erasure of a hologram with hole-electron competition in Pb-doped Bi12 TiO20 is described in Section 8.4.2.2.

83

84

3 Recording a Space-Charge Electric Field

3.5 Photovoltaic Materials Space charge electric field build-up in photovoltaic crystals exhibits specific features that need special attention. Let us write the continuity and the Gauss equation ∇ . j(x,⃗ t) +

𝜕𝜌(x, t) =0 𝜕t

(3.146)

⃗ t) = 𝜌(x, t) ∇ . D(x,

(3.147)

𝜕 (x, t) with j(x,⃗ t) = q + x̂ 𝜎Ephv + x̂ 𝜎E(x, t) 𝜕 x⃗

(3.148)

with 𝜎Ephv ≡ 𝜅phv I𝛼 𝜎 = 𝜎ph + 𝜎d

(3.149)

where the first, second and third terms on the right side of Eq. (3.148) are the diffusion, photovoltaic and ohmic components, respectively, 𝜎d the dark conductivity, 𝜎ph the photoconductivity and I𝛼 = dI(z)∕dz

(3.150)

is the light intensity absorbed per unit sample thickness (z) or light power absorbed per unit crystal volume. The x̂ is the unit vector along coordinate axis x. The 𝜅phv is the photovoltaic transport coefficient that was found [42] to depend on the nature of the absorbing center and the wavelength as reported in Table 2.1. From Eqs. (3.146)–(3.148) we get: ⃗ t) 𝜕 D(x, ∇ . (j(x,⃗ t) + )=0 (3.151) 𝜕t ⃗ t) 𝜕 D(x, and: ⃗j(x, t) + (3.152) = j⃗0 𝜕t For the following, we shall assume all vectors along the x-coordinate only. Substituting the expression for j(x,⃗ t) in Eq. (3.148) into Eq. (3.152) we get a differential equation: 𝜀𝜖0 3.5.1

𝜕 (x, t) 𝜕E(x, t) + 𝜎Ephv = j0 + 𝜎E(x, t) + q 𝜕t 𝜕 x⃗

(3.153)

Uniform Illumination: 𝝏 ∕𝝏x = 0

We shall analyze the electric field build up from Eq. (3.153) for the simple case of uniform illumination for a short-circuited as well as for an open-circuited sample: • Short circuit: E = D = 0 j0 = 𝜎Ephv

(3.154)

• Open circuit: j0 = 0 𝜕E(t) + 𝜎E(t) + 𝜎Ephv = 0 𝜕t

(3.155)

E(t) = −Ephv (1 − exp(−t∕𝜏sc ))

(3.156)

𝜀𝜖0 so that:

3.5 Photovoltaic Materials

where: 𝜏sc = 𝜀𝜖0 ∕𝜎

𝜎 = 𝜎d + 𝜎ph

and from Eqs. (3.149): Ephv = 𝜅phv

rND+ I0 𝛼d 𝜇e(ND − ND+ )(sI0 ∕(h𝜈) + 𝛽)

(3.157)

]𝛼d≪1

Note that open circuit leads to a progressive electric polarization under uniform light illumination that is opposite to the photovoltaic field Ephv and may therefore even compensate the latter and prevent any further optical recording [88]. 3.5.2

Interference Pattern of Light

The electric field build-up in the presence of a modulated pattern of light I(x) = Io (1 + m cos(Kx)) and the resulting self-diffraction effects make it difficult to find an analytical solution. In order to get such a solution, however, we shall neglect self-diffraction and assume the photovoltaic effect to be much more relevant than diffusion so as to neglect the latter term (e𝜕 ∕𝜕x ≈ 0) in Eq. (3.153) to be able to simplify it down to: 𝜕E(x, t) + 𝜎E(x, t) + 𝜎Ephv (1 + m cos(Kx)) = j0 𝜕t • Open circuit: j0 = 0 𝜖𝜀0

𝜖𝜀0

(3.158)

𝜕E(x, t) + 𝜎E(x, t) + 𝜎Ephv (1 + m cos(Kx)) = 0 𝜕t

(3.159)

so that E(x, t) = −Ephv (1 + m cos(Kx))(1 − exp(−t∕𝜏sc ))

(3.160)

𝓁 ∫0

E(x, t) . dx = 0 • Short circuit: In this case, the crystal is usually short-circuited using conductive silver glue as represented in Fig. 3.28. Integrating the expression in Eq. (3.158) from x = 0 to x = 𝓁, where 𝓁 is the interelectrode distance, we get the expression in Eq. (3.154): j0 = 𝜎Ephv Equating Eq. (3.158) and Eq. (3.154) we get an expression for the electric field: E(x, t) = −mEphv cos(Kx)(1 − exp(−t∕𝜏sc ))

C

(3.161)

C

Figure 3.28 Short circuit schema using conductive silver glue to electrically connect the opposite faces along the photovoltaic axis c⃗ (left) and open circuit schema, without any electrical connection (right).

85

86

3 Recording a Space-Charge Electric Field

3.5.2.1 Influence of Donor Density

The formulation of the photovoltaic effect as stated in Eq. (3.148) does not show evidence of the influence of the electron-donor density involved here. Considering: sI 𝜏 h𝜈 to be substituted into Eq. (3.148) it turns out to be 𝜎 ≈ e𝜇ph

ph = (ND − ND+ )

(3.162)

⃗j = e 𝜕 + x̂ eL⃗phv (ND − N + ) sI + x̂ e 𝜇E D h𝜈 𝜕 x⃗

(3.163)

Lphv = 𝜇𝜏Ephv

(3.164)

where Lphv is formulated in a way similar to LE = 𝜇𝜏E0 in Eq. (3.45). All calculations can be repeated now with Eq. (3.163) instead of Eq. (3.148), all other hypothesis being maintained. The space-charge field time-derivative expression in Eq. (3.158) now becomes: N+

meff (E0 + Ephv + iED ) + Esc (t)(1 + K 2 ls2 − 𝚤KlE − 𝚤Klphv ND 𝜕Esc (t) D =− 𝜕t 𝜏M (1 + K 2 L2D − iKLE − 𝚤KLphv )

sIo ∕(h𝜈) ) sIo ∕(h𝜈)+𝛽

(3.165) with Klphv = Ephv ∕Eq

meff = m

sIo ∕(h𝜈) sIo ∕(h𝜈) + 𝛽

(3.166)

or written in a more compact form 𝜕Esc (t) + Esc (t) = −meff Eeff 𝜕t which solution is: E (t) = −m E (1 − e−t∕𝜏sc ) 𝜏sc

sc

(3.167)

(3.168)

eff eff

with Eeff ≡ meff

E0 + Ephv + iED N+

1 + K 2 ls2 − iKlE − iKlphv ND D sIo ∕(h𝜈) =m (sIo ∕(h𝜈) + 𝛽

sIo ∕(h𝜈) sIo ∕(h𝜈)+𝛽

(3.169) (3.170)

and 𝜏sc = 𝜏M

1 + K 2 L2D − iKLE − iKLphv N+

1 + K 2 ls2 − iKlE − iKlphv ND

D

that we should also write as 1 = 𝜔R + i𝜔I 𝜏sc

sIo ∕(h𝜈) sIo ∕(h𝜈)+𝛽

(3.171)

(3.172)

3.5 Photovoltaic Materials

with

[ 1 𝜔R ≡ 𝜏M

(1 + K 2 ls2 )(1 + K 2 L2D ) + KlE (KLE + KLphv ) (1 + K 2 L2D )2 + (KLE + KLphv )2 N+

⎤ ⎥ (1 + K 2 L2D )2 + (KLE + KLphv )2 ⎥ ⎦ [ 2 2 2 2 1 (1 + K ls )(KLE + KLphv ) − KlE (1 + K LD ) (KLE + KLphv )Klphv ND

D

+

𝜔I ≡

𝜏M

N+

−

sI0 sI0 +𝛽

(1 + K 2 L2D )2 + (KLE + KLphv )2

⎤ ⎥ (1 + K 2 L2D )2 + (KLE + KLphv )2 ⎥ ⎦ (1 + K 2 L2D )Klphv ND

D

+

sI0 sI0 +𝛽

(3.173)

+

(3.174)

Note that 𝜔I represents the phase shift speed of the grating under uniform illumination so that the total phase shift during the characteristic time 𝜏M is, from Eq. (3.70): 𝜏M 𝜔I =

(1 + K 2 ls2 )(KLE + KLphv ) − KlE (1 + K 2 L2D ) (1 + K 2 L2D )2 + (KLE + KLphv )2 N+

−

(1 + K 2 L2D )Klphv ND

D

sI0 sI0 +𝛽

(1 + K 2 L2D )2 + (KLE + KLphv )2

+

(3.175)

87

89

4 Volume Hologram with Wave Mixing The space-charge electric field modulation that is produced by the action of a spatially modulated pattern of light as described in Chapter 3 also produces a phase-shifted associated real-time index-of-refraction spatial modulation due to the electro-optic effect analyzed in Chapter 1. As a consequence, a phase grating, because of its real-time build-up, diffracts the light during the recording process itself, thus modifying the recording pattern of light that on its turn further affects the recorded grating and so on in a continuous mutual feedback process. This feedback phenomenon is known as “self-diffraction” or “wave mixing” and is the subject of this chapter.

4.1 Coupled Wave Theory: Fixed Grating We shall first make a short review of the coupled wave theory that deals with light diffraction by a fixed volume hologram and then shall extend this theory for the case of a dynamic reversible recording material where diffraction and recording occurs simultaneously and therefore feedback is established between both processes. Following Kogelnik’s theory [70], let us represent a fixed volume grating of period Δ, wavevec⃗ thickness d and a reading beam of amplitude R incident at the angle 𝜃 (always measured tor K, inside the grating volume), as described in Fig. 4.1. The reading beam is one of the two beams previously used to record this same hologram, as shown in Fig. 4.2 where the index-of-refraction modulation pattern is shifted by 𝜙P , referring to the recording pattern of light as represented in Fig. 4.2. We also assume that the average index-of-refraction of the grating is the same as that of the surrounding medium. Let us also assume the presence of a 𝜙A -shifted fixed amplitude grating (not represented in the figure). Let us write the wave equation inside the material [89] as with ∇2 Ψ + k 2 Ψ = 0 ( ) 2 𝜔 𝜎 k2 = 2 1 + 𝜒 + i c 𝜔𝜀0

(4.1) n2 =

c 𝜎 =1+𝜒 +i 𝑣2 𝜔R 𝜀 0 2

𝜖 ≡1+𝜒

(4.2)

Kogelnik writes these relations in the form k2 =

𝜔2 𝜖 + i𝜔𝜇𝜎 c2

𝛽2 =

𝜔2 𝜖 c2 o

𝜇c𝜎 𝛼= √o 2 𝜖o

(4.3)

The index-of-refraction and amplitude gratings are represented by the modulations 𝜖1 and 𝜎1 in the dielectric constant 𝜖 = n2 (n being the refraction index) and in the conductivity 𝜎, respectively, as follows ⃗ r + 𝜙P ) ⃗ i(K.⃗ + e−i(K.⃗r + 𝜙P ) ⃗ r + 𝜙P ) = 𝜖o + 𝜖1 e 𝜖 = 𝜖o + 𝜖1 cos(K.⃗ (4.4) 2 Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

90

4 Volume Hologram with Wave Mixing

HOLOGRAM x

R⃗(0)

R⃗

z

θ

θ θ S⃗

Δ K = 2Π

d

Figure 4.1 Reading the recorded hologram with one of the recording beams.

/Δ

hologram S⃗(0)

x

2θ

R⃗

z

ϕ pattern of light

R⃗(0)

Figure 4.2 Recording a fixed volume index-of-refraction hologram that is phase-shifted by 𝜙 = 𝜙P referred to the recording pattern of fringes with 2𝜃 being the angle inside the material.

S⃗

Δ

K = 2Π

d

/Δ

⃗ r + 𝜙A ) ⃗ i(K.⃗ + e−i(K.⃗r + 𝜙A ) ⃗ r + 𝜙A ) = 𝜎o + 𝜎1 e 𝜎 = 𝜎o + 𝜎1 cos(K.⃗ 2

(4.5)

Substituting Eqs. (4.4) and (4.5) into the formulation for k 2 in Eq. (4.3) we get the expression ⃗ ⃗ k 2 = 𝛽 2 + i2𝛽𝛼 + 2𝛽(𝜅+ eiK.⃗r + 𝜅− e−iK.⃗r )

(4.6)

where 𝜅+ and 𝜅− are defined next. Searching a solution for Eq. (4.1), with the form ⃗r 𝜌.⃗r + S(z) ei𝛿.⃗ Ψ = R(z) ei⃗

(4.7)

with the Bragg condition (see Appendix B.1 and, for example, page 388 in [1]), K⃗ + 𝛿⃗ = 𝜌⃗

(4.8)

with ∣ 𝜌⃗ ∣=∣ 𝛿⃗ ∣= 2𝜋∕𝜆 and K = 2𝜋∕Δ represented in Fig. 4.3 with the assumption of a weak coupling 𝜕2R 𝜕2S ≈ 2 ≈0 (4.9) 𝜕z2 𝜕z and following the development of Kogelnik, for TE-polarization, we get the coupled equations cos 𝜃

𝜕R + 𝛼R = i𝜅+ S 𝜕z

ρ⃗

K⃗

δ⃗

(4.10) Figure 4.3 Bragg condition where 𝜌⃗ and 𝛿⃗ are the incident beam and the diffracted beam wavevectors, respectively (or vice versa), and K⃗ is the grating wavevector.

4.1 Coupled Wave Theory: Fixed Grating

cos 𝜃

𝜕S + 𝛼S = i𝜅− R 𝜕z

(4.11)

𝜅+ =

𝜇𝜎 c 1 𝜔𝜖1 i𝜙P 1 𝜔Δn i𝜙P + i √ 1 ei𝜙A ) = ( + iΔ𝛼 ei𝜙A ) ( √ e e 4 c 𝜖o 2 c 𝜖o

(4.12)

with

𝜇𝜎 c 1 𝜔𝜖 1 𝜔Δn −i𝜙P + iΔ𝛼 e−i𝜙A ) e 𝜅− = ( √ 1 e−i𝜙P + i √ 1 e−i𝜙A ) = ( 4 c 𝜖o 2 c 𝜖o (4.13) 𝜖 where Δn = √1 2 𝜖o 4.1.1

𝜇c𝜎 Δ𝛼 = √ 1 2 𝜖o

(4.14)

Diffraction Efficiency

From the equations before, Kogelnik showed that the diffraction efficiency of an unslanted grating of purely index-of-refraction nature is ( ) 𝜋Δn d 2 𝜂 = sin (4.15) 𝜆 cos 𝜃 whereas for a purely absorption grating it is ( ) Δ𝛼 d 𝜂 = sinh2 2 cos 𝜃

(4.16)

Both Eqs. (4.15) and (4.16) do not consider the effect of average bulk absorption that is consistent if we assume the phenomenological definition 𝜂 = I d ∕(I d + I t ) in terms of the beams (transmitted I t and diffracted I d ) behind the crystal. 4.1.2

Out of Bragg Condition

These calculations assume that the incident reading beam exactly matches the Bragg condition represented by Eq. (4.8), which can be also written as 2k sin 𝜃 = K

(4.17)

However, the incident beam can be shifted away from this condition because of a mismatch Δ𝜃 of the incidence angle or because of a mismatch in the wavelength Δ𝜆 or both. Both parameters are related by the Eq. (4.17) and such a relation can be explicitly formulated by derivation of this equation K d𝜃 = (4.18) d𝜆 4𝜋 cos 𝜃 It is possible to show [70] that the diffraction efficiency for slightly out-of-Bragg conditions can be accounted for by introducing the mismatch parameter 𝜉 = Δ𝜃Kd∕2

(4.19)

2

Δ𝜆K d 8𝜋 cos 𝜃 in the modified formulation for 𝜂 √ sin2 𝜈 2 + 𝜉 2 𝜂= 1 + 𝜉 2 ∕𝜈 2 =−

(4.20)

(4.21)

91

92

4 Volume Hologram with Wave Mixing

𝜋Δn d (4.22) 𝜆 cos 𝜃 Photorefractive crystals allow recording gratings with rather large Kd that result in a sensibly high angular and wavelength selectivity as deduced from Eqs. (4.19) and (4.20). Their high Bragg selectivity together with their adaptability (because of their real-time and reversible recording properties) make these materials particularly suitable as, for example, efficient filters in extended-cavity semiconductor lasers for improving single-mode laser operation [90]. 𝜈≡

Exercise Which is the Bragg angular mismatch that reduces the diffraction efficiency to half its original 100% diffraction efficiency value for a 2-mm-thick purely index-of-refraction grating with 0.5 μm spatial period grating?

4.1.2.0.1

4.2 Dynamic Coupled Wave Theory In the case of a dynamic recording media such as photorefractive crystals are, the coupling between the interfering beams is not characterized by constant parameters such as 𝜅+ and 𝜅− . Instead, a feedback mechanism is present in this case, relating the holograms being recorded and the diffraction of beams being used for recording, a phenomenon called “self-diffraction” (or wave mixing) that does not exist for fixed gratings and is therefore not accounted for in the original Kogelnik’s formulation. In order to understand these differences, let us recall the expressions of the amplitude of the index-of-refraction modulation as expressed in Eqs. (1.45) and (1.46) and in Eqs. (1.59)–(1.61), which can be generalized as n3 reff ∣ Eeff ∣ (4.23) 2 where reff and Eeff are the effective values for these parameters. Let us also recall that the steady-state amplitude of the electric field generated by holographic recording in a photorefractive material is given by mEeff as reported in Eq. (3.52) where Eeff is constant but m is the visibility of the recording pattern of light fringes that may vary along the sample thickness. It is therefore necessary to write Eq. (4.23) as Δn = −

n3 reff ∣ Eeff ∣ (4.24) 2 in order to explicitly show the dependence of the index-of-refraction modulation Δn on the fringes visibility m. Δn = mn1

4.2.1

n1 ≡ −

Combined Phase-Amplitude Stationary Gratings

Therefore, for the general case of a dynamic grating exhibiting at the same time a phase modulation (with index-of-refraction amplitude modulation n1 ) 𝜙P -shifted and an amplitude modulation (with amplitude 𝛼1 ) 𝜙A -shifted to the interference pattern of light onto the crystal, the Kogelnik’s formulation in Eqs. (4.10) and (4.11) is easily modified into 𝜕R + 𝛼R = i𝜅+ mS 𝜕z 𝜕S + 𝛼S = i𝜅− m∗ R cos 𝜃 𝜕z

cos 𝜃

with: 𝜅+ =

(4.25) (4.26) (4.27)

𝜋n1 i𝜙 𝛼 e P + i 1 ei𝜙A 𝜆 2

(4.28)

4.2 Dynamic Coupled Wave Theory

and: 𝜅− =

(4.29) 𝜋n1 −i𝜙 𝛼 P + 𝚤 1 e−𝚤𝜙A e 𝜆 2

(4.30)

and:

(4.31)

n1 = −n3 reff |Eeff |∕2

(4.32)

where 𝜙P is the phase of the complex parameter Eeff and is therefore the phase shift between the recording pattern of fringes and the recorded grating. We have here assumed that 𝛼 is the average bulk light absorption and Δ𝛼 = m𝛼1 is the absorption modulation, which we assume to be proportional to m as for the case of Δn. Such an assumption is rather reasonable if we consider that the modulation Δ𝛼 also arises from the spatial modulation of traps in the sample. Both n1 and 𝛼1 represent the maximum possible respective modulation amplitudes that are achieved for m = 1. For the case of a uniform light background (Ib ) the pattern of light onto the crystal may be written as: I = Ib + I0 [1+ ∣ m ∣ cos(Kx + 𝜙)] I = (Ib + I0 )[1+ ∣ m ∣

(4.33)

I0 cos(Kx + 𝜙)] Ib + I0

(4.34) I

so that in this case ∣ m ∣ should be converted to ∣ m ∣ I +Io everywhere. With the simple transb 0 formation S → S e−𝛼z∕cos𝜃 , the bulk absorption term 𝛼S may be eliminated in Eq. (4.26), and √ √ similarly for R. Substituting S = I e−i𝜓S and R = I e−i𝜓R into the previous simplified S

R

equations and comparing the imaginary and real terms, the following set of results: 𝜅+I 4IR IS 𝜕IR =− 𝜕z cos𝜃 IR + IS

(4.35)

𝜕IS 𝜅 I 4IR IS =− − 𝜕z cos𝜃 IR + IS

(4.36)

𝜅+R 2IS 𝜕𝜓R =− 𝜕z cos 𝜃 IS + IR

(4.37)

𝜕𝜓S 𝜅 R 2IR =− − 𝜕z cos 𝜃 IS + IR

(4.38)

where 𝜅+I , 𝜅−I and 𝜅+R and 𝜅−R are the imaginary and the real terms, respectively, of 𝜅+ and 𝜅− . Eqs. (4.35)–(4.38) may be rearranged to: 𝜕(IR + IS ) 4𝛼 cos 𝜙A IR IS =− 1 𝜕z cos 𝜃 IR + IS

(4.39)

𝜕(IS − IR ) 8𝜋n1 sin 𝜙P IR IS = 𝜕z 𝜆 cos 𝜃 IR + IS

(4.40)

𝜕(𝜓S − 𝜓R ) 𝜋n cos 𝜙P IS − IR 𝛼1 sin 𝜙A − =2 1 𝜕z 𝜆 cos 𝜃 IR + IS cos 𝜃

(4.41)

𝜕(𝜓S + 𝜓R ) 𝜋n cos 𝜙P 𝛼1 sin 𝜙A IS − IR = −2 1 + 𝜕z 𝜆 cos 𝜃 cos 𝜃 IS + IR

(4.42)

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4 Volume Hologram with Wave Mixing

4.2.1.1 Fundamental Properties

The analysis of Eqs. (4.39)–(4.42) allows one to formulate the fundamental properties of combined amplitude-phase gratings. Let us discuss them in detail: • Energy conservation: Energy conservation means that, not considering bulk absorption, IR + IS is constant along z in the sample thickness. Energy conservation does hold for any condition making zero the right side in Eq. (4.39): energy is conserved for a 90∘ -shifted amplitude grating (cos 𝜙A = 0) or in the absence of any amplitude grating (𝛼1 = 0). • Energy exchange or amplitude coupling: Energy exchange as described by Eq. (4.40) is dependent on the imaginary part of the phase grating only and is illustrated in Fig. 4.4. The amplitude grating has no effect at all. If there is no phase grating, there is no possibility for energy to be exchanged from one beam to the other. • Phase shifting or phase coupling: The shifting of the interference pattern phase planes is described by the evolution of 𝜓S − 𝜓R (see Eq. (4.41)) and is illustrated in Fig. 4.5. The phase difference (hologram phase shift) between the recording pattern of light and the hologram being recorded is determined by the material and experimental parameters as described in Eq. (3.55). This means that the hologram will follow the shifting of the recording pattern of light in order to keep constant the holographic phase shift. Accordingly, both the pattern of light and the recorded hologram will be synchronously shifted. Equation (4.41) shows that there is no hologram phase-shifting in the following cases: 1. in-phase (or counterphase) amplitude and ±90∘ -shifted phase grating, 2. in-phase (or counterphase) amplitude grating and in-phase phase grating (meaning no beam coupling) with equal input irradiance beams (ISo = IRo ). Any one of these conditions will make the right-hand side in Eq. (4.41) equal to zero, thus meaning that the phase difference remains constant through the crystal thickness so that there is no phase-coupling and therefore no hologram “bending” due to self-diffraction. Figure 4.4 Amplitude coupling in two-wave mixing: in this example, the weaker beam receives energy from the stronger, but could also be the other way round.

Figure 4.5 Phase coupling in two-wave mixing: the pattern of fringes and associated grating are progressively shifted by the same amount. The picture shows some degree of amplitude coupling too.

4.2 Dynamic Coupled Wave Theory

4.2.1.2

Irradiance

The general case of mixed phase/amplitude gratings as described by Eqs. (4.35)–(4.38) does not verify either energy conservation, or phase uncoupling. Multiplying Eq. (4.35) by 𝜅−I and Eq. (4.36) by −𝜅+I , and adding both equations, we get: 𝜕(𝜅+I IS − 𝜅−I IR ) 𝜕IR 𝜕I + 𝜅+I S = =0 (4.43) 𝜕z 𝜕z 𝜕z Equation (4.43) shows that the quantity  = 𝜅+I IS − 𝜅−I IR is a constant and may play the same role as energy conservation in the solution of coupled Eqs. (4.35)–(4.38). Unfortunately, the general solution in this case does not provide an explicit analytic formulation for IS and for IR . Therefore, substituting  into Eq. (4.36) and rearranging terms, we get −𝜅−I

cos 𝜃

𝜅+I IS −  𝜕IS = −4𝜅−I IS 𝜕z IS (𝜅+I + 𝜅−I ) − 

(4.44)

rearranging terms and integrating from the input (d = 0 and IS0 ) to the output (d and IS ) we get the following expression: IS d IS (𝜅+I + 𝜅−I ) −  cos 𝜃 dI = −4 dz S ∫0 𝜅−I ∫IS0 IS2 𝜅+I −  IS

(4.45)

where cos 𝜃(𝜅+I + 𝜅−I )∕𝜅−I

IS

dIS

∫I 0 IS 𝜅+I −  S

−

IS dIS  cos 𝜃 = −4d I I 2 ∫ 𝜅+ IS0 IS 𝜅+ −  IS

(4.46)

Knowing that dIS 1 = I ln(IS 𝜅+I −  ) ∫ IS 𝜅+I −  𝜅+ 𝜅+I IS dIS −1 = and ln ∫ − IS + I 2 𝜅+I  𝜅+I IS − 

(4.47) (4.48)

S

and substituting Eqs. (4.47) and (4.48) into Eq. (4.46) and rearranging terms, we get the final solution: ) ]r [( IS IS r −1 + 1 = exp(−4d𝜅+I ∕ cos 𝜃) (4.49) ISo ISo 𝛽2 A similar solution is found for the beam in the other direction: ) 2 [( ]r IR IR 𝛽 −1 r ≡ 𝜅+I ∕𝜅−I + 1 = exp(−4d𝜅−I ∕ cos 𝜃) IRo IRo r

(4.50)

where 𝛽 2 ≡ IR0 ∕IS0

(4.51)

For the particular case of 𝛼1 = 0, Eqs. (4.49) and (4.50) become: 1 + 𝛽2 1 + 𝛽 2 exp(−Γd) 1 + 𝛽2 and IR = IR0 2 𝛽 + exp(Γd) IS = IS0

(4.52) (4.53)

which are the same expressions that will be found later for the case of pure photorefractive holograms in Eqs. (4.84) and (4.94).

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4 Volume Hologram with Wave Mixing

4.2.2

Pure Phase Grating

We shall discuss the particular case where 𝛼1 = 0, in which case we deduce from Eq. (4.39) that energy conservation holds for a pure phase grating, so that we may write IR + IS = IR0 + IS0 = I0

(4.54)

4.2.2.1 Time Evolution

Let us consider the build-up of a space-charge electric field and the associated index-ofrefraction modulation and corresponding phase grating. For a purely phase-modulated (𝛼1 = 0) grating and substituting the expression of n1 in Eq. (4.24) into Eqs. (4.25)–(4.32) we should write 𝜋n3 reff Esc (t) m 𝜅(t) = m 𝜅+ (t) = m∗ 𝜅−∗ (t) = − (4.55) 2𝜆 Now 𝜅+ (t) and 𝜅− (t) are associated with the index-of-refraction modulation amplitude for |m| = 1. Assuming no absorption at all, neither bulk nor modulation effects, the Kogelnik coupled-wave equations are now written as: 𝜕R (4.56) = im𝜅(t)S 𝜅(t) ≡ [𝜅+ (t)]𝛼1 =0 cos 𝜃 𝜕z 𝜕S 𝜅 ∗ (t) ≡ [𝜅− (t)]𝛼1 =0 (4.57) = im∗ 𝜅 ∗ (t)R cos 𝜃 𝜕z Time-deriving Eq. (4.57), substituting Eq. (4.55) into the latter and considering Eq. (3.42) with Eq. (3.8), we get the following expression for S: ∗ ∗ −𝜋n3 reff Eeff −𝜋n3 reff Esc (t) 𝜕R 2 ∣ R ∣2 S cos 𝜃 𝜕S 𝜕2S + ∗ +i ∗ = i cos 𝜃 𝜕z𝜕t 𝜏sc 𝜕z 𝜏sc (∣ R ∣2 + ∣ S ∣2 ) 2𝜆 2𝜆 𝜕t

(4.58) Following a similar procedure we get an expression for R too: cos 𝜃

−𝜋n3 reff Eeff −𝜋n3 reff Esc (t) 𝜕S 2 ∣ S ∣2 R 𝜕2R cos 𝜃 𝜕R + +i = i 𝜕z𝜕t 𝜏sc 𝜕z 𝜏sc (∣ R ∣2 + ∣ S ∣2 ) 2𝜆 2𝜆 𝜕t (4.59)

4.2.2.1.1

Undepleted Pump Approximation If we assume energy conservation

Io = I =∣ R∣2 + ∣ S∣2 that is reasonable for a pure index-of-refraction grating, and state the so-called “undepleted pump approximation” (∣ R∣2 ≈ I, I constant) condition, we deduce the following relations: 2RS∗ ≈ 2S∗ ∕R∗ I I 𝜕m 𝜕S∗ ≈ 𝜕z 2R 𝜕z 1 𝜕R 1 𝜕R |≪1 | |≪1 | R 𝜕t R 𝜕z

(4.61)

|Esc (t)| ≤ |mEeff |

(4.63)

m=

(4.60)

(4.62)

that, substituted into Eq. (4.58), dividing the whole by R and computing the conjugate, results in 3 1 𝜕m m 𝜋n reff Eeff 𝜕2m + +i =0 𝜕z𝜕t 𝜏sc 𝜕z 𝜏sc 𝜆 cos 𝜃

(4.64)

4.2 Dynamic Coupled Wave Theory

because |

(4.65)

mE 𝜕R∗ Esc (t) 𝜕R∗ | ≤ | ∗eff | ≪ |mEeff | ∗ R 𝜕t R 𝜕t A similar expression is found for S

(4.66)

3 𝜕 2 S∗ 1 𝜕S∗ 1 𝜋n reff Eeff ∗ + +i S =0 𝜕t𝜕z 𝜏sc 𝜕z 𝜏sc 𝜆 cos 𝜃

(4.67)

There is no similar formulation for the pump beam R because the right side term cannot be neglected in Eq. (4.59). However, in the case where R is the weak beam and S is the pump, another couple of differential equations can be found 3 𝜕 2 mR 1 𝜕mR m 𝜋n reff Eeff + −i =0 𝜕z𝜕t 𝜏sc 𝜕z 𝜏sc 𝜆 cos 𝜃 3 1 𝜕R 1 𝜋n reff Eeff 𝜕2R + −i R=0 𝜕t𝜕z 𝜏sc 𝜕z 𝜏sc 𝜆 cos 𝜃

(4.68) (4.69)

where mR means the modulation for the case where R is the weak beam. The differential Eqs. (4.64) and (4.67) may be written in the general form [91–93]: 𝜕 2 A(z, t) 1 𝜕A(z, t) + − bA(z, t) = 0 𝜕z𝜕t 𝜏sc 𝜕z

b = −i

𝜋n3 reff Eeff 𝜆 𝜏sc cos 𝜃

(4.70)

with a solution of the form: A(z, t) = Ao (to )A1 (z, t) exp(−t∕𝜏sc )

(4.71)

that substituted into Eq. (4.70) gives 𝜕 2 A1 (z, t) (4.72) − bA1 (z, t) = 0 𝜕z𝜕t With the change of variable 𝛼 = bzt we get the second-order differential equation in one single variable: 𝜕A1 𝜕2 A (4.73) − A1 = 0 𝛼 21 + 𝜕𝛼 𝜕𝛼 Equation (4.73) may be√transformed into a Bessel one just by making the following variable √ change 𝜁 = i2 𝛼 = i2 bzt: 𝜕 2 A1 1 𝜕A1 + + A1 = 0 𝜕𝜁 2 𝜁 𝜕𝜁 whose solution is the zero-order Bessel function: √ A1 = J0 (i2 bzt) From Eqs. (4.74) and (4.75) we get the solution for Eqs. (4.64) and (4.67): (√ ) 8𝜅 z t m(or S∗ ) = Ao (to )J0 i exp(−t∕𝜏sc ) 𝜏sc cos 𝜃 (√ ) 𝜅 zt −8i exp(−t∕𝜏sc ) mR (or R) = Ao (to )J0 𝜏sc cos 𝜃 where we have defined 𝜋n3 reff Eeff 𝜅 = lim 𝜅(t) = t→∞ 2𝜆

(4.74)

(4.75)

(4.76) (4.77)

(4.78)

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4 Volume Hologram with Wave Mixing

from Eqs. (4.55) and (3.52). Note that the holographic phase-shift in Eq. (3.55) can be now written as 4ℑ{𝜅} 4ℜ{𝜅} Γ and 𝛾 ≡ (4.79) tan 𝜙P = with Γ ≡ 𝛾 cos 𝜃 cos 𝜃 Response Time with Feedback The coupled wave theory shows that diffraction of beams along the directions S and R, due to the grating being recorded, results in energy transfer from one of these beams into the other, a process called “two-wave mixing”. If a grating is erased using one of these beams (let us say R), which transfers energy to the other beam (S), the erasure can be considered a positive feedback process. In fact, the signal beam S is increased (“amplified”) due to such a transfer of energy and by this means the erasure is slowed down. On the opposite case, if R is used to erase but energy transfer occurs from S to R, there is negative feedback because the signal beam S is not amplified but reduced and the hologram erasure is speeds up. The transfer of energy from one beam to the other is determined by the direction of propagation of the beam and the crystal parameters; that is to say, is determined by the sign of 𝜅 or 𝜅 ∗ in the corresponding coupled wave equation. As for the case of amplifiers in electronic circuits, negative feedback results in an increase of the frequency bandwidth; that is, it results in a faster response. The opposite occurs for positive feedback. This similarity between photorefractive two-wave mixing and electronic amplification has been already pointed out elsewhere [93, 94]. The matter can be mathematically analyzed by assuming an erasure process where the weak beam is S whose evolution is described by Eq. (4.76) with adequately selected constants (A0 (t0 ) =∣ S∗ (0) ∣) given by the corresponding boundary conditions. The corresponding intensity evolution is therefore given by (√ ) 𝜅 zt 8i ∣ S ∣2 = ∣ A0 (to )J0 e−t∕𝜏sc ∣2 𝜏sc cos 𝜃 ℜ{𝜏sc } (√ ) −t2 𝜅 zt ∣ 𝜏sc ∣2 = ∣ A0 (to )∣2 ∣ J0 8i (4.80) ∣2 e 𝜏sc cos 𝜃

4.2.2.1.2

Figures (4.6–4.9) show the numerical plotting of Eq. (4.80) for |A0 (0)|2 = 1 and for some usual values for a BTO crystal with E0 ∕ED = 2 and K = 12 μm−1 : ℜ{𝜏sc } = 0.4s and ℑ{𝜏sc } = −0.65s where the abscissa are always the normalized time t∕|𝜏sc |. It is clear from the four figures that 1 0.8 0.6

Time t ⁄

sc

0.4 0.2

2

4

S2

6

8

10

Figure 4.6 Numerical plotting of |S|2 versus the normalized time t∕|𝜏sc |, from Eq. (4.80) for Γd = 1, ℜ{𝜏sc } = 0.4s and ℑ{𝜏sc } = −0.65s with 𝛾z = −0.5, −0.25, 0.25 and 0.5 from the smaller to the larger dashed lines, respectively.

4.2 Dynamic Coupled Wave Theory

1 0.8 0.6

Time t ⁄

sc

0.4 0.2

2

4

6

S

8

10

2

Figure 4.7 Numerical plotting of |S|2 versus the normalized time t∕|𝜏sc |, from Eq. (4.80) for Γd = −1, ℜ{𝜏sc } = 0.4s and ℑ{𝜏sc } = −0.65s with 𝛾z = −0.5, −0.25, 0.25 and 0.5 from the smaller to the larger dashed lines, respectively. 1 0.8 0.6

Time t ⁄

sc

0.4 0.2

2

4

6

8

10

S2

Figure 4.8 Numerical plotting of |S|2 versus the normalized time t∕|𝜏sc |, from Eq. (4.80) for 𝛾d = 1, ℜ{𝜏sc } = 0.4s and ℑ{𝜏sc } = −0.65s with Γz = −0.5, −0.25, 0.25 and 0.5 from the smaller to the larger dashed lines, respectively.

for any value of the 𝛾z parameter, the erasure is always faster for lower values of Γd; that is, for a condition where the transfer of energy from the pump to the signal beam is reduced, where z = d is the sample’s thickness. The transient effects theoretically discussed here are experimentally illustrated in Fig. 4.10 for the case of running holograms [94]. In this case, a perturbation is automatically established by the starting of a ramp-shaped voltage (thick curve) applied to a PZT-supported mirror (in order to produce the necessary detuning K𝑣 for the running hologram generation) in the setup. We see how fast the running hologram diffraction efficiency (thin oscillating curve) evolves to equilibrium after the setup is perturbed by the starting of the mirror movement: The negative-gain experiment (lower graphics) shows a much faster and less oscillatory evolution to equilibrium than the positive-gain experiment in the upper graphics, in agreement with the theoretical predictions.

99

4 Volume Hologram with Wave Mixing

1 0.8 0.6

Time t ⁄

sc

0.4 0.2

1

2

3

S

4

5

2

600

1.2 1.1

200 0

1.0 0

2

4

6

8

η (%)

400

0.9 10 0.55 0.50

400

0.45

η (%)

Ramp Voltage (V)

Figure 4.9 Numerical plotting of |S|2 versus the normalized time t∕|𝜏sc |, from Eq. (4.80) for 𝛾d = −1, ℜ{𝜏sc } = 0.4s and ℑ{𝜏sc } = −0.65s with Γz = −0.5, −0.25, 0.25 and 0.5 from the smaller to the larger dashed lines, respectively.

Ramp Voltage (V)

100

0.40 200

0

2

4

6

8

0.35 10

Time (s)

Figure 4.10 Transient effect of a perturbation, in the form of a ramp voltage (thick curve) applied to the PZT-supported mirror in the holographic setup, on the diffraction efficiency (thin curve) of a running hologram recorded in a photorefractive BTO-crystal using the 514.4-nm wavelength. The diffraction efficiency evolution to equilibrium is faster for the negative-gain (lower graphics, with K = 2.55 μm−1 ) than for the positive-gain (upper graphics with K = 4.87 μm−1 ) experiment. In both cases, the applied external field is E0 ≈ 7.5 kV/cm, the total incident irradiance is Io ≈ 22.5 mW/cm2 and the beam ratio is 𝛽 2 ≈ 40. Reproduced from [94].

4.2.2.2 Stationary Hologram

Substituting Eq. (4.54) into Eq. (4.36), where a pure phase grating is considered, we get: 𝜕IS 4𝜅 I I (I − IS ) =− − S 0 𝜕z cos 𝜃 I0

(4.81)

with the variable substitution t = 1∕IS , we get 4𝜅−I 4𝜅−I 𝜕t − t+ =0 𝜕z cos 𝜃 I0 cos 𝜃

(4.82)

4.2 Dynamic Coupled Wave Theory

where the solution is I t = t0 e4𝜅− z∕ cos 𝜃 + 1∕I0 (4.83) From the boundary conditions we get IS (z) = IS0 where

1 + 𝛽2 1 + 𝛽 2 e−Γz

Γ = 4𝜅0 sin 𝜙P ∕ cos 𝜃 = ℑ{

(4.84) 4𝜅 } cos 𝜃

(4.85)

𝜋n3 reff Eeff with 𝜅 = 𝜅0 ei𝜙P = 𝜅+ = (𝜅− )∗ = 2𝜆 𝜋n1 2 and 𝜅0 =∣ 𝜅 ∣= (4.86) , 𝛽 = IR0 ∕IS0 𝜆 Eeff here is the complex parameter described in Eq. (3.170) and 𝜙P is the phase-shift of the pure phase hologram. We shall only consider an index-of-refraction modulation that, in photorefractive crystals, arises from the electro-optic properties of these materials. In fact, a modulated pattern of space-charge electric field is produced under the action of a corresponding pattern of fringes of light, as already described in Chapter 3. As a consequence of the electro-optic effect a corresponding modulated pattern of index-of-refraction is produced, the amplitude Δn of which is, according to Eq. (4.23) for the stationary case (t → ∞): n3 r E (4.87) Δn ei𝜙P = − eff sc 2 that substituted into Eqs. (4.28) and (4.30), for a purely phase-modulated (Δ𝛼 = 0) grating may be written as 𝜋n3 reff Esc m 𝜅 = m 𝜅+ = m∗ 𝜅−∗ = − (4.88) 2𝜆 Now 𝜅+ and 𝜅− are associated with the index-of-refraction modulation amplitude for |m| = 1. Substituting Eq. (4.54) into Eq. (4.38), we find an equation for 𝜓S : 𝜕𝜓S 𝛾 I − IS (4.89) =− 0 𝜕z 2 I0 𝜅+R 𝜅R 𝜅 = 4 − = 4ℜ{ } (4.90) where 𝛾 = 4 cos 𝜃 cos 𝜃 cos 𝜃 Note that the hologram phase shift defined in Eq. (3.55) and written as Eq. (4.79) can now also be written, from Eqs. (4.85) and (4.90), as tan 𝜙P = Γ∕𝛾

(4.91)

Substituting Eq. (4.84) into Eq. (4.89) and rearranging terms, we get the differential equation: ) ( 𝜕𝜓S 𝛾 1 −1 (4.92) = 𝜕z 2 1 + 𝛽 2 e−Γz 𝛾z 1 + 𝛽 2 e−Γz 1 whose solution is: 𝜓S (z) = 𝜓S (0) − ln (4.93) + 4 2 tan 𝜙p 1 + 𝛽2 We find a similar set of equations for IR and 𝜓R with their corresponding solutions being: IR (z) = IRo

1 + 𝛽2 𝛽 2 + eΓz

(4.94)

101

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4 Volume Hologram with Wave Mixing

𝜓R (z) = 𝜓R (0) −

𝛾z 𝛽 2 + eΓz 1 ln − 4 2 tan 𝜙p 1 + 𝛽2

(4.95)

From Eqs. (4.93) and (4.95), an expression is found for 𝜓S (z) − 𝜓R (z) = 𝜓S (0) − 𝜓R (0) +

(𝛽 2 + eΓz )2 1 ln tan 𝜙p (1 + 𝛽 2 )2 eΓz

(4.96)

which describes the pattern-of-light phase planes and consequently the hologram phase planes too. Equation (4.96) represents a pattern-of-fringes that is being continuously shifted from the input 𝜓S (0) − 𝜓R (0) to the output 𝜓S (d) − 𝜓R (d) and is therefore leading to a “bent” hologram. This is a direct consequence of phase coupling represented by Eq. (4.41). For the case of materials exhibiting optical activity, we can show that the coefficients 𝜅+I and 𝜅−I in the right side of Eqs. (4.35–4.38) are not constants but are functions of z [95]. In this case, the differential equations in Eq. (4.81) and Eq. (4.89) should be represented as: 𝜕IS (z) I (z)I (z) = Γ(z) R S 𝜕z IR (z) + IS (z)

(4.97)

𝜕𝜓S (z) IR (z) 𝛾(z) =− 𝜕z 2 IR (z) + IS (z)

(4.98)

where the explicit dependence of the real 𝛾(z) and imaginary Γ(z) parts of the coupling constant on the crystal thickness (z) are indicated. Equation (4.97) can be written as: Γ(z) 𝜕t + Γ(z)t = 𝜕z I0

t≡

1 IS (z)

I0 = IS (0) + IR (0)

(4.99)

whose general solution is [96] ⎞ ⎛ ⎜ Γ(z) ∫ Γ(z)dz ⎟ − ∫ Γ(z)dz e dz⎟ e t = ⎜t0 + ∫ I0 ⎟ ⎜ ⎠ ⎝

(4.100)

Rearranging terms, we get the general formulation IS (z) = IS0

1+

𝛽2

1 + 𝛽2 exp(− ∫ Γ(z)dz)

(4.101)

where the amplitude coupling is shown to depend on the integral of Γ(z). For the case of the phase, and for the undepleted pump approximation (IR (z) ≫ IS (z)), we can write Eq. (4.98) as 𝜕𝜓S (z) 𝛾(z) ≈− (4.102) 𝜕z 2 where the dependence of phase coupling upon the integral of 𝛾(z) is obvious. The results in Eqs. (4.101) and (4.102) show that for the case of Γ and 𝛾 varying along the crystal thickness, their influence upon amplitude and phase coupling are represented by their corresponding integrals. That is to say that the simple Γz and 𝛾z products should be substituted by their integrals. This conclusion is in fact a general one that may be applied whenever Γ and 𝛾 are dependent on the sample thickness.

4.2 Dynamic Coupled Wave Theory

4.2.2.2.1

Diffraction Let us recall the formulation of a coupled wave in Eqs. (4.25) and (4.26)

𝜕R(z) 𝜕S(z) 𝜅 𝜅∗ (4.103) =i m(z) S(z) =i m(z)∗ R(z) 𝜕z cos 𝜃 𝜕z cos 𝜃 S∗ (z)R(z) m(z) = 2 (4.104) I where absorption has been neglected and therefore I ≡ |S(z)|2 + |R(z)|2 = |S(0)|2 + |R(0)|2 is constant along z. The solution of the corresponding intensities and phases were already computed in Section 4.2.2.2. We shall now investigate the situation when the already written (that is, when m(z) is fixed) grating is read by another couple of waves (z) and (z) that are identical to the corresponding writing ones. Such a formulation is necessary to allow one to compute the diffracted beam (and therefore the diffraction efficiency), which is to be measured when a probe beam, different but in principle identical to the recording one, is diffracted by the grating without erasing it so that Eq. (4.103) should be now written as: 𝜕(z) 𝜅 =i m(z) (z) (4.105) 𝜕z cos 𝜃 𝜕(z) 𝜅∗ (4.106) =i m(z)∗ (z) 𝜕z cos 𝜃 We shall be then reading the “reference” beam (z) with the boundary conditions (0) = 1 and (0) = 0. Likewise, when reading out the same grating with the “signal beam” (z), we need the solutions of the same equations fulfilling the boundary conditions (0) = 0 and (0) = 1. How to find this fundamental system of solutions will be shown later. But, as soon as these solutions are available, the “reference”  and the “signal”  beams can be written, respectively, in the form (z) = R(0) R + S(0) S

(4.107)

(z) = R(0) R + S(0) S

(4.108)

where R is the transmittance and R is the diffraction complex coefficients, respectively, for beam  and similarly, with S and S , for beam . Unshifted holograms Here, we shall show how to evaluate the expressions in Eqs. (4.107) and

(4.108) for the simple case of an unshifted (local) grating. Let us assume an unshifted grating, 𝜙 = 0 (or Γ = 0), which is just a homogeneous (may be also tilted) grating. From Eq. (4.96) we compute, for Γ = 0, the phases 𝜓= 𝜓R = 𝜓R (0) −

1 Δk z 2

𝛾 IS (0) z 2 I

Δk =

𝛾 IS (0) − IR (0) 2 I

𝜓S = 𝜓S (0) −

𝛾 IR (0) z 2 I

(4.109)

and therefore  = R(0) e

−i

𝛾 IS (0) z 2 I

 = S(0) e

−i

𝛾 IR (0) z 2 I

(4.110)

so that the intensities are fixed: IR = IR (0) and IS = IS (0) but the modulation depends on the crystal depth: m(z) = m(0) e−i Δk z

(4.111)

103

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4 Volume Hologram with Wave Mixing

We insert the expression of the modulation in Eq. (4.111) into Eqs. (4.105) and (4.106), and taking into account that 𝜙 = 0 so that 𝜅∕ cos 𝜃 = 𝛾∕4, we obtain 𝛾 𝜕 = i m(0) e−i Δk z  (4.112) 𝜕z 4 𝛾 𝜕 = i m(0)∗ ei Δk z . 𝜕z 4 Differential equations with constant coefficients are obtained by the transformation ̂ e−iΔkz∕2 =

(4.113)

 = ̂ eiΔkz∕2

that result in the coupled equations ̂ 𝛾 𝜕 = i m(0) ̂ 𝜕z 4

(4.114)

𝛾 𝜕 ̂ ̂ (4.115) = i m(0)∗ . 𝜕z 4 which are easily solved. The result for diffraction of the reference wave from the dynamic grating is the solution of Eqs. (4.112) and (4.113) with the boundary conditions R (0) = 1 and R (0) = 0: 𝛾 IS (0) 𝛾 IR (0) z IS (0) i z IR (0) −i R = e 2 I e 2 I + I I 𝛾 IS (0) ⎤ ⎡ −i 𝛾 IR (0) z i z 1 ∗⎢ ⎥ R = m(0) e 2 I −e 2 I ⎥ ⎢ 2 ⎦ ⎣

R (0) = 1 (4.116) R (0) = 0.

From this, we find the diffraction efficiency 𝛾z 𝜂 = |R |2 = |m(0)|2 sin2 4 This is Kogelnik’s formula [70] with 𝛾z 1 𝜉 = 𝜓 = Δk z. 𝜈 = |m(0)| 4 2

(4.117)

(4.118)

For modulation m = 1 we have 𝜉 = 0. There is no tilting and we obtain a simple sin2 -function for small modulation ∣ 𝜉 ∣≫∣ 𝜈 ∣. To obtain the corresponding formulae for diffraction of the signal wave, it is enough to observe that Eqs. (4.112) and (4.113) are invariant under  ↔ , R ↔ S and Δk ↔ −Δk; 𝛾 hereby remains the same. Therefore, diffraction efficiency is not changed and 𝛾 IR (0) ⎤ ⎡ −i 𝛾 IS (0) z i z 1 ⎥ ⎢ 2 I S = m(0) e −e 2 I ⎥ ⎢ 2 ⎦ ⎣ 𝛾 IR (0) 𝛾 IS (0) z IR (0) i z I (0) −i S = S e 2 I e 2 I + I I

S (0) = 0 (4.119) S (0) = 1.

Phase-shifted holograms We shall now deal with an arbitrarily phase-shifted grating. We shall

start again from Eq. (4.103) and consider again a fixed (m(z)) grating. Note that R and S still solve this system. The method for how to obtain a second, linearly independent solution of

4.2 Dynamic Coupled Wave Theory

Eq. (4.103) can be found in any textbook on ordinary differential equations. Here, however, it is enough to take the complex conjugate of Eq. (4.103) to find that  = −S∗

 = R∗

(4.120)

is a second solution of Eqs. (4.105) and (4.106). Because the determinant of these two solutions is equal to the intensity I, they are indeed linearly independent. Now R(0)∗ ∕I times the old solution vector (R, S) minus S(0)∕I times the new solution vector (, ) gives the solution corresponding to the diffraction of the reference wave: R = [R(0)∗ R + S(0) S∗ ]∕I R = [−S(0) R∗ + R(0)∗ S]∕I

R (0) = 1 R (0) = 0.

(4.121)

The solution corresponding to the diffraction of the signal wave is obtained by this symmetry: S = −R∗ = [S(0)∗ R − R(0) S∗ ]∕I S = ∗R = [R(0) R∗ + S(0)∗ S]∕I

S (0) = 0 S (0) = 1.

(4.122)

The diffraction efficiency is hereby given by 𝜂(z) = |R |2 = |S |2 = 2

𝛽 2 cosh Γz∕2 − cos 𝛾z∕2 . 1 + 𝛽 2 𝛽 2 e−Γz∕2 + eΓz∕2

(4.123)

It is interesting to compute ( )2 [( )2 ( )2 ] 𝜋n3 reff 2 2𝛽 𝛾d Γd 2 2 2 = |m| + |E | d ( ) lim 𝜂 = eff d→0 1 + 𝛽2 4 4 2𝜆 cos 𝜃 with ∣ m ∣2 = (

(4.124)

2𝛽 2 ) 1 + 𝛽2

which is the well-known Kogelnik formula [70] for 𝜂 ≪ 1. It is possible to verify that R = R(0) R + S(0) S

(4.125)

S = R(0) R + S(0) S

(4.126)

and also that |R |2 + |R |2 = |S |2 + |S |2 = 1. The phase shift 𝜑 between the transmitted and the diffracted beams along the same direction at the hologram output can be computed from Eqs. (4.125) or (4.126). From the latter, for example, we get tan 𝜑 =

ℑ{R(0)R S(0)∗ S∗ } ℜ{R(0)R S(0)∗ S∗ }

(4.127)

Verifying that, substituting the parameters here by their expressions in Eq. (4.122) and using the expressions in Eqs. (4.94) and (4.84), we obtain 𝛾 sin z 2 tan 𝜑 = − . (4.128) 𝛾 ) 1 − 𝛽2 ( Γ Γ cosh z − cos z + sinh z 1 + 𝛽2 2 2 2 which relates the output phase shift 𝜑 with the material parameters Γ and 𝛾 and the experimental parameter 𝛽 2 .

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4.2.2.3 Steady-State Nonstationary Hologram with Wave-Mixing and Bulk Absorption

The mathematical model describing the generation of running (nonstationary) holograms with wave mixing is the same as in Section 3.4, except that in this case diffraction efficiency should be formulated in terms of Γd and 𝛾d as described in Eq. (4.123), where Γ and 𝛾 are computed st in Eq. (3.78). from the expression of Esc Bulk light absorption does not affect the way a stationary (not moving) hologram is recorded except for the fact that its build-up is slower. In fact, the average light intensity decreases along the crystal thickness z and the Maxwell (or dielectric) relaxation time 𝜏M (which determines the recording response time) in Eq. (3.48) is no longer a constant, but varies along z as: 𝜖𝜀0 h𝜈 𝜏M (z) = 𝜏M (0) e𝛼z 𝜏M (0) = (4.129) q𝜇𝜏ΦI0 𝛼 Consequently, the deeper layers in the sample are slower because progressively less light is left for recording, due to absorption. st st For stationary holograms, the formulations of Γ ∝ ℑ{Esc } and 𝛾 ∝ ℜ{Esc } do not depend on bulk absorption, so diffraction efficiency and the hologram phase shift 𝜙 (both depending on Γd and 𝛾d) in this case are also not affected and are the same as in the absence of bulk absorption as formulated in Section 3.4. For the case of nonstationary (moving) holograms, however, bulk absorption has a more complicated effect and does actually affect both Γ and 𝛾. It is straightforward to show that Eqs. (3.83) and (3.84) are, because of absorption, dependent on z and should be written as ar e𝛼z K𝑣 + cr st ℜ{Esc (4.130) }= a e2𝛼z (K𝑣)2 + b e𝛼z K𝑣 + c ai e𝛼z K𝑣 + ci st (4.131) ℑ{Esc }= a e2𝛼z (K𝑣)2 + b e𝛼z K𝑣 + c a = [K 2 L2E + (1 + K 2 L2D )2 ]𝜏M (0)2 b = 2𝜏M (0)[K 2 ls2 − K 2 L2D ]

E0 ED

(4.132) (4.133)

c = (1 + K 2 ls2 )2 + K 2 lE2

(4.134)

ar = −[(1 + K 2 L2D )ED + KLE E0 ]𝜏M (0)

(4.135)

cr = E0

(4.136)

ai = E0 𝜏M (0)

(4.137)

ci = E0 KlE + ED (1 + K 2 ls2 )

(4.138)

𝜏M (0) =

𝜖𝜀0 (kB T∕e)h𝜈 eL2D 𝛼I0 (0)Φ

(4.139)

The parameters Γ and 𝛾 in Eqs. (4.85) and (4.90) are accordingly written as st Γ(z) = 4𝑤ℑ{Esc }

(4.140)

st } 𝛾(z) = 4𝑤ℜ{Esc

(4.141)

with 𝑤 =

𝜋n3 reff 2𝜆

(4.142)

4.2 Dynamic Coupled Wave Theory

and vary along the crystal thickness z. It is possible to argue that, if Γ(z) and 𝛾(z) are not constants, their products Γd and 𝛾d should be substituted everywhere (particularly in the expression for 𝜂 in Eq. (4.123)) by their integrals, as already explained in Section 4.2.2.2 for the case of optical activity. In the present case, the integrals are [ ]z=d z=d bci 2aK𝑣 e𝛼z + b 2 arctan √ Γ(K𝑣, z)dz = Γd = 4𝑤(ai − + ) √ ∫z=0 2c 𝛼 4ac − b2 4ac − b2 z=0 [ ]z=d c e2𝛼z +4𝑤 i ln (4.143) 2𝛼c a(K𝑣)2 e2𝛼z + bK𝑣 e𝛼z + c z=0 [ ]z=d z=d bcr 2aK𝑣 e𝛼z + b 2 arctan √ 𝛾(K𝑣, z)dz = 𝛾d = 4𝑤(ar − + ) √ ∫z=0 2c 𝛼 4ac − b2 4ac − b2 z=0 [ ]z=d cr e2𝛼z (4.144) ln +4𝑤 2𝛼c a(K𝑣)2 e2𝛼z + bK𝑣 e𝛼z + c z=0 with the condition 4ac ≥ b2 . These results indicate that the formulations of 𝜂 and 𝜙 should be correspondingly revised, which is the subject of the following sections. Diffraction Efficiency If bulk absorption is considered, the usual expression for diffraction efficiency, reported in Eq. (4.123), under self-diffraction effects should be further modified to

4.2.2.3.1

𝜂=

2𝛽 2 cosh(Γd∕2) − cos(𝛾d∕2) 1 + 𝛽2 𝛽 2 e−Γd∕2 + eΓd∕2

(4.145)

with Γd and 𝛾d as already defined in Eq. (4.144), respectively. It is interesting to see how the different material parameters affect 𝜂. For a hypothetical experimental condition using typical values: E0 = 500 kV/m, I0 = 200 W/m2 , d = 2.05 mm, 𝛽 2 = 50 and 𝜆 = 514.5 nm, and typical material parameters for an undoped BTO crystal: 𝛼 = 1165/m and reff = 5.6 pm/V, 𝜂 was computed (ordinates) for different K (ranging from K = 20 to 0.5 μm−1 ) as a function of the detuning K𝑣 (horizontal axis in rad/s) and plotted in Figs. 4.11–4.14. From these figures we draw the following conclusions: • Quantum efficiency Φ has no influence on the 𝜂 peak value. It just acts on the position of the peak and the shape of the curve, the lower the K, the larger its influence. • ls that is always appearing as K 2 ls2 has no influence on either the peak position or on the shape of the curve. It just acts on the peak size, but has no influence for K 2 ls2 ≪ 1; that is to say, for far-from-saturation conditions, as expected. • LD that is always appearing as K 2 L2D acts on the peak position and the shape of the curve, increasing the K𝑣-value for the peak and widening the curve for increasing LD . For K 2 L2D ≫ 1, LD has no influence at all. The physical meaning of the features referred to here is usually not easy to grasp. The increase of the abscissae of the 𝜂 peak with increasing LD , for instance, is easy to understand because the latter is related to the carrier’s mobility L2D = 𝜏 = (kB T∕q)𝜇𝜏. The fact that LD has no effect on the hologram movement for K 2 L2D ≫ 1 is probably due to the fact that LD is large compared to the grating period, the distribution of photoelectrons is somewhat randomized and the actual value of LD no longer has importance in the dynamics of the process. The increase in the resonance speed (the position of the peak) as Φ increases is also reasonable because the latter

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4 Volume Hologram with Wave Mixing

–5

0.009

0.009

0.0085

0.0085

0.008

0.008

0.0075

0.0075

0.007

0.007

0.0065

0.0065 5

0.0055

10

15

20

–5

ls = 0.03 μm, LD = 0.2 μm and Φ = 0.3, 0.4, 0.5, 0.6 increasing with the size of the dashing of the lines.

5

0.0055

10

15

20

Φ = 0.3, LD = 0.2 μm and ls = 0.03, 0.04, 0.05, 0.06 μm increasing with the size of the dashing of the lines.

0.0175 0.015 0.0125 0.01 0.0075 0.005 0.0025 10

20

30

40

Φ = 0.3, ls = 0.03 μm and LD = 0.1, 0.2, 0.3, 0.4 μm increasing with the size of the dashing of the lines.

Figure 4.11 Computed running hologram 𝜂 as a function of K𝑣 (rad/s) for K = 0.5 μm−1 and different material parameters.

–5

0.016

0.016

0.014

0.014

0.012

0.012

0.008

5

10

15

20

–5

0.006

0.008

5

10

15

0.006 0.004

0.004 ls = 0.03 μm, LD = 0.2 μm and Φ = 0.3, 0.4, 0.5, 0.6 increasing with the size of the dashing of the lines.

Φ = 0.3, LD = 0.2 μm and ls = 0.03, 0.04, 0.05, 0.06 μm increasing with the size of the dashing of the lines.

0.03 0.025 0.02 0.015 0.01 0.005 –5

5

10

15

20

Φ = 0.3, ls = 0.03 μm and LD = 0.1, 0.2, 0.3, 0.4 μm increasing with the size of the dashing of the lines.

Figure 4.12 Computed running hologram 𝜂 as a function of K𝑣 (rad/s) for K = 2 μm−1 and different material parameters.

4.2 Dynamic Coupled Wave Theory

0.02

0.025 0.02

0.015

0.015

0.010

0.010

0.005 –5

0.005 5

10

15

–4

20

ls = 0.03 μm, LD = 0.2 μm and Φ = 0.3, 0.4, 0.5, 0.6 increasing with the size of the dashing of the lines.

–2

2

4

6

8

10

Φ = 0.3, LD = 0.2 μm and ls = 0.03, 0.04, 0.05, 0.06 μm increasing with the size of the dashing of the lines.

0.02 0.015 0.010 0.005 –5

5

10

15

20

Φ = 0.3, ls = 0.03 μm and LD = 0.1, 0.2, 0.3, 0.4 μm increasing with the size of the dashing of the lines.

Figure 4.13 Computed running hologram 𝜂 as a function of K𝑣 (rad/s) for K = 10 μm−1 and different material parameters.

increases the light effectively involved in the process and in this way 𝜏M (0) decreases and the material becomes faster. The large influence of ls is easy to understand too: an increase of ls means a reduction in the effective photoactive center’s concentration and the peak of 𝜂 therefore also becomes limited by the lack of charge carriers to build up the charge modulation in the crystal. Output Beams Phase-Shift In the presence of bulk light absorption, there is no one single value for 𝜙 but it varies along the sample’s thickness because of the variation of Γ and 𝛾, as described in Eqs. (4.140) and (4.141). The phase-shift 𝜑 between the transmitted and diffracted beams behind the sample is affected too. Its formulation in the presence of self-diffraction is discussed in Section 4.3.1.2 and in Reference [97]. As for the case of 𝜂, the formulation of 𝜑 in Eq. (4.128) should be also modified accordingly in the presence of bulk absorption by substituting Γ and 𝛾 by Γ and 𝛾, respectively, leading to

4.2.2.3.2

tan 𝜑 = − 1−𝛽 2 1+𝛽 2

sin(𝛾d∕2) (cosh(Γd∕2) − cos(𝛾d∕2)) + sinh(Γd∕2)

(4.146)

In order to understand the influence of the different material parameters and experimental conditions on 𝜑, Eq. (4.146) was numerically computed for a typical BTO crystal, for the 514.5-nm wavelength, for the same parameters as for 𝜂 (E0 = 500 kV/m, I(0) = 200 W/m2 and 𝛽 2 = 50), always for a negative two-wave mixing amplitude gain. Comparing Fig. 4.15 with Fig. 4.16, we see that the effect of LD is lower for K 2 L2D ≫ 1, as already pointed out in Section 4.2.2.3.1 for 𝜂. There is apparently some optimum value of K that enhances the effect

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4 Volume Hologram with Wave Mixing

0.015

0.025

0.0125

0.02

0.01

0.015

0.007

0.010

0.005

0.005

0.0025 –5

5

10

15

20

ls = 0.03 μm, LD = 0.2 μm and Φ = 0.3, 0.4, 0.5, 0.6 increasing with the size of the dashing of the lines.

–4

–2

2

4

6

8

10

Φ = 0.3, LD = 0.2 μm and ls = 0.03, 0.04, 0.05, 0.06 μm increasing with the size of the dashing of the lines.

0.015 0.0125 0.01 0.007 0.005 0.0025 –4

–2

2

4

6

8

10

Φ = 0.3, ls = 0.03 μm and LD = 0.1, 0.2, 0.3, 0.4 μm increasing with the size of the dashing of the lines.

Figure 4.14 Computed running hologram 𝜂 as a function of K𝑣 (rad/s) for K = 20 μm−1 and different material parameters.

of LD for some value of K𝑣, as seen in Fig. 4.15. No definite conclusions can be drawn instead about the effects of ls and Φ on 𝜑. Figure 4.17 shows a simulated result for a hypothetical very thin crystal, where self-diffraction effects can be neglected: in this case positive (not shown) and negative gain gives the same result. Figure 4.18 is also for a hypothetical thin but low absorption 𝛼 = 1 m−1 sample so as to neglect the effect of absorption on the movement of the hologram: These figures show that in these conditions tan 𝜑 becomes constant, for K𝑣 values sufficiently different from zero, with its value only depending on LD , whatever the value of K. The influence of ls and Φ are clearly negligible here. It seems that, in the absence of self-diffraction and absorption, the movement of the hologram (and consequently the value of the hologram phase-shift) is only dependent on the charge carriers’ diffusion length LD : in this case, there is no such “randomization” of the charge carriers’ distribution in the CB (see comments in Section 4.2.2.3.1) because the whole grating is moving along with the charge carriers. The whole set of figures 4.15–4.18 shows that for K𝑣 = 0 (that is, stationary holograms) the only parameter of relevance is ls , in agreement with Eq. (3.56). 4.2.2.4 Gain and Stability in Two-Wave Mixing

As already mentioned before, photorefractive two-wave mixing (neglecting absorption to simplify) represents a feedback process where energy is transferred from one to the other recording beam and in this sense it behaves as an electronic amplifier: if the weaker beam (signal) receives energy from the stronger pump beam, we may think of positive feedback and of negative feedback if energy goes the other way round. The gain G of an amplifier with feedback  and intrinsic

4.2 Dynamic Coupled Wave Theory

tan φ 6 5 4 3 2 1 –5

5

10

15

Kv (rad/s)

ls = 0.03 μm, Φ = 0.3 and LD = 0.1, 0.2, 0.3, 0.4 μm increasing with the size of the dashing of the lines. tan φ 4 3 2 1

–5

5

10

15

Kv (rad/s)

Φ = 0.3, LD = 0.15 μm and ls = 0.03, 0.04, 0.05, 0.06 μm increasing with the size of the dashing of the lines. tan φ 8

6

4

2

–5

5

10

15

Kv (rad/s)

LD = 0.15 μm, ls = 0.3 μm and Φ = 0.3, 0.4, 0.5, 0.6 increasing with the size of the dashing of the lines.

Figure 4.15 Tan 𝜑 versus K𝑣 (rad/s), computed for K = 2 μm−1 and different material parameters, for a typical BTO crystal 2.05 mm thick and 𝛼 = 1165 m−1 .

111

112

4 Volume Hologram with Wave Mixing

tan φ 2 1.5 1 0.5 –5

5

10

15

Kv (rad/s)

–0.5

ls = 0.03 μm, Φ = 0.3 and LD = 0.1, 0.2, 0.3, 0.4 μm increasing with the size of the dashing of the lines. tan φ 1.5 1 0.5

–5

5

10

15

Kv (rad/s)

–0.5

Φ = 0.3, LD = 0.15 μm and ls = 0.03, 0.04, 0.05, 0.06 μm increasing with the size of the dashing of the lines. tan φ 2.5 2 1.5

0.5 –5

5

10

15

Kv (rad/s)

–0.5 –1

LD = 0.15 μm, ls = 0.3 μm and Φ = 0.3, 0.4, 0.5, 0.6 increasing with the size of the dashing of the lines.

Figure 4.16 Tan 𝜑 versus K𝑣 (rad/s), computed for K = 11 μm−1 and different material parameters, for a typical BTO crystal 2.05-mm thick and 𝛼 = 1165 m−1 .

4.2 Dynamic Coupled Wave Theory

tan φ

1.5

1

0.5

–5

5

10

Kv (rad/s)

15

ls = 0.03 μm, Φ = 0.3 and LD = 0.1, 0.2, 0.3, 0.4 μm increasing with the size of the dashing of the lines. tan φ 1.75 1.5 1.25 1 0.75 0.5 0.25 –5

5

10

Kv (rad/s)

15

Φ = 0.3, LD = 0.15 μm and ls = 0.03, 0.04, 0.05, 0.06 μm increasing with the size of the dashing of the lines. tan φ 2 1.5 1 0.5 –5

5

10

15

Kv (rad/s)

–0.5

LD = 0.15 μm, ls = 0.3 μm and Φ = 0.3, 0.4, 0.5, 0.6 increasing with the size of the dashing of the lines.

Figure 4.17 Tan 𝜑 versus K𝑣 (rad/s), computed for K = 11 μm−1 and different material parameters, for a typical BTO crystal 2.05-mm thick and 𝛼 = 1165 m−1 .

113

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4 Volume Hologram with Wave Mixing tan φ

tan φ

4

0.25

2 –5

5

–2

10

15

Kv (rad/s)

–5 –0.25 –0.5

5

10

15

Kv (rad/s)

–0.75 –1 –1.25

–4 –6 ls = 0.03 μm, Φ = 0.3 and LD = 0.1, 0.2, 0.3, 0.4 μm increasing with the size of the dashing of the lines.

Φ = 0.3, LD = 0.15 μm and ls = 0.03, 0.04, 0.05, 0.06 μm increasing with the size of the dashing of the lines.

tan φ 0.25 –5

–0.25

5

10

15

Kv (rad/s)

–0.5 –0.75 –1 –1.25 LD = 0.15 μm, ls = 0.3 μm and Φ = 0.3, 0.4, 0.5, 0.6 increasing with the size of the dashing of the lines.

Figure 4.18 Tan 𝜑 versus K𝑣 (rad/s), computed for K = 1 μm−1 and different material parameters, for a typical BTO crystal 2.05 mm thick and a hypothetically low 𝛼 = 1m−1 .

large amplification A is: A 1 + A and becomes

(4.147)

G=

G ≈ 1∕

A ≫ 1

(4.148)

where the latter expression shows that gain is no longer depends on the amplifier characteristics represented by A, and therefore the amplification process becomes more stable. Also, amplifiers with negative feedback have a larger frequency bandwidth; that is to say, they exhibit a faster response. All of the experiments and mathematical simulations in this section confirm that photorefractive two-wave with energy flowing from the stronger pump to the weaker signal beam does behave like a negative feedback amplifier. Additional photorefractive two-wave mixing experiments reported elsewhere [94] still showed that diffraction efficiency measured under negative feedback are considerably more stable than under positive feedback conditions, at least in experiments where an external electric field is applied to the photorefractive crystal. If no electric field is applied instead, whether the feedback is positive or negative has no sensible effect. To understand such a different behavior with or without an external field, let us recall that the latter field produces an intrinsic resonant excitation (see Eq. (3.79)) giving rise to transient running holograms that are certainly the source of instability; the negative feedback eliminating perturbations of intrinsic nature, it is straightforward to understand the improving effect under applied electric field. In the absence of such a field, however, perturbations come from the setup itself and are processed along with the signal we are interested in without being attenuated by the feedback being negative or not.

4.3 Phase Modulation

Figure 4.19 Phase modulation setup: BS: beamsplitter, PZT piezoelectric-supported mirror, D: photodetector, LA-Ω and LA-2Ω: lock-in amplifiers tuned to Ω and 2Ω respectively, HV high voltage source for the PZT, OSC oscillator to produce the dithering signal.

+Vo

M BS

IR0

IR

I0S BTO

PZT OSC Ω

IS

+

D

HV

LAΩ V

LA2Ω

Ω

V2Ω

4.3 Phase Modulation Phase modulation in two-wave mixing is produced by phase-modulating (with amplitude 𝜓d ) one of the interfering beams with (angular) frequency Ω x + 𝜙 − 𝜔t + 𝜓d sin Ωt) ⃗ = S⃗0 ei(k⃗S .⃗ S(0)

(4.149)

whereas the other beam remains unchanged x − 𝜔t) ⃗ = R⃗0 ei(k⃗R .⃗ R(0)

(4.150)

In this case, the interference pattern of light in Eq. (3.5) becomes (assuming S⃗0 .R⃗0 = S0 R0 ) I(x, t) = I0 + I0 ∣ m ∣ cos(Kx + 𝜙 + 𝜓d sin Ωt)

(4.151)

which represents a sinusoidal pattern of light vibrating with frequency Ω and phase amplitude 𝜓d along K⃗ that is also parallel to the coordinate axis x⃗. If Ω𝜏sc ≪ 1, then the hologram is faster than the movement of the pattern of light so that it follows the pattern and is recorded (and erased) continuously. Both hologram and pattern are moving simultaneously and, because it is moving, the recorded grating is the same as if it were recorded from a standing pattern of light. If Ω𝜏sc ≫ 1 instead, the recording is much slower and cannot follow the movement of the pattern. The result is a hologram produced by a pattern of light that is the time-average of the actual moving one. For intermediate cases, the corresponding differential equation must be taken into account and the strength of the resulting hologram will depend on the relation between Ω and 1∕𝜏sc . This case is analyzed in Section 8.6.3 and is used to measure the response time 𝜏sc of the recording material. We shall here focus on the second case when Ω𝜏sc ≫ 1. In this case, it is necessary to compute the time-average of the pattern of light t

0 1 I(x, t)dt t0 →∞ t ∫0 0 < I(x, t) > = I0 + I0 ∣ m ∣ cos(Kx + 𝜙) < cos(𝜓d sin Ωt) > +

< I(x, t) > ≡ lim

−I0 ∣ m ∣ sin(Kx + 𝜙) < sin(𝜓d sin Ωt) > where t0 ≫ 1∕Ω. Developing the time-dependent terms in Bessel series and making the time-average < cos(𝜓d sin Ωt) > = J0 (𝜓d ) + 2J2 (𝜓d ) < cos 2Ωt > +2J4 (𝜓d ) < cos 4Ωt > +... = J0 (𝜓d )

(4.152)

115

116

4 Volume Hologram with Wave Mixing

and < sin(𝜓d sin Ωt) > = 2J1 (𝜓d ) < sin Ωt > +2J3 (𝜓d ) < sin 3Ωt > +... =0

(4.153)

and substituting these results into the expression for < I(x, t) > results in < I(x, t) >= I0 + I0 ∣ m ∣ J0 (𝜓d ) cos(Kx + 𝜙)

(4.154)

where Jn (x) represents the ordinary Bessel function of order n (n, integer). Equation (4.154) means that the pattern of light behaves as if it were standing with an effective fringes modulation of ∣ m ∣ J0 (𝜓d ) instead of just ∣ m ∣. The phase modulation experimental setup is schematically depicted in Fig. 4.19. In these conditions, the phase-modulated beam “S” is not modified by the recorded hologram because the latter is too slow for that. Accordingly, the transmitted and diffracted “S” beams behind the crystal exhibits the same 𝜓d sin Ωt phase modulation imposed at the crystal input. The overall irradiances behind the sample are formed by the coherent addition of the phase modulated transmitted “S” plus the diffracted (nonmodulated) “R” beam, for IS , and the phase modulated diffracted “S” plus the nonmodulated transmitted “R” beams for IR . Because of the nonlinear relation between irradiance and phase, such a dither signal of frequency Ω gives rise to multiple harmonics in Ω that may be detected in IS and IR at the sample’s output using phase-sensitive frequency-tuned lock-in amplifiers. Phase modulation can be used for operating a stabilized holographic recording setup as developed in Chapter 6. It can be also used in different ways to characterize material parameters, which is the subject of Part III of this book. 4.3.1

Phase Modulation in Dynamically Recorded Gratings

Diffraction from a dynamically recorded space-charge grating cannot be modeled easily, because in general it will not be homogeneous, and it will be tilted and bent. Kogelnik’s formula [70] in Eq. (4.15) is only valid for the diffraction from a homogeneous nontilted grating and the question arises of how to generalize his results to the case of a phase-modulated dynamic grating. We shall here describe in detail the accurate formulation of temporal harmonic components in phase modulated photorefractive two-wave mixing (TWM). We shall show how to obtain the general solution of the problem of diffraction from a fixed dynamic grating (described by a system of linear ordinary differential equations) by exploiting the solution obtained from solving the nonlinear two-wave mixing equations. From these results we shall derive analytical expressions for the first and second temporal harmonics of the signal output beam. According to the assumptions here, we shall assume that the pattern of light and the corresponding hologram is not affected by the oscillating pattern of light (with Ω𝜏sc ≫ 1), except for the fact that the fringe visibility is now J0 (𝜓d ) ∣ m ∣ instead of ∣ m ∣. We shall focus first on the simple particular case of 𝜙 = 0, and then on the more complex general case of arbitrarily 𝜙-shifted holograms. Approximate expressions relating the first and second temporal harmonic terms to the hologram phase shift 𝜙 [71], and accurate formulations for particular conditions, such as equal incident beams [98] or undepleted pump approximation [99], have already been published. We shall describe next an accurate general formulation relating the fundamental photorefractive material parameters to the temporal harmonics in a two-wave mixing phase modulation experiment. 4.3.1.1 Phase Modulation in the Signal Beam

We shall now investigate the development of the expressions for the dynamic coupled wave in Section 4.2.2.2 when the amplitude of the signal beam oscillates in the form ei 𝜓d sin Ωt

4.3 Phase Modulation

with an angular frequency that is large relative to the reciprocal holographic relaxation time of the crystal Ω𝜏sc ≫ 1. In this case, Eqs. (4.107) and (4.108) should be substituted by the corresponding  = R(0) R + S(0) ei 𝜓d sin Ωt S

(4.155)

 = R(0) R + S(0) ei 𝜓d sin Ωt S

(4.156)

= S + S(0) S ( ei𝜓d sin Ωt − 1).

(4.157)

Note that we have neglected the twice-diffracted modulated beam at the output along the direction of the directly transmitted modulated beam. Such an approximation may not be possible for sufficiently highly diffractive gratings and the exact handling of this case has been reported by Ringhofer and co-workers [100]. Expanding the expression in Eq. (4.156), to compute the intensity I = ||2 of the signal beam in terms of 𝜓d2 , allows one to find harmonic terms in Ωt I = IS + IΩ sin Ωt + I2Ω cos 2Ωt + ...

(4.158)

IΩ = − 4J1 (𝜓d )ℑ{S∗ S(0)S }

(4.159)

I2Ω = 4J2 (𝜓d )(ℜ{S∗ S(0)S } − IS (0) |S |2 ).

(4.160)

with

Unshifted Hologram To evaluate Eqs. (4.159) and (4.160) for the special case of 𝜙 = 0 (unshifted) and according to the results from Section 4.2.2.2.1.1, we need to calculate [ 𝛾 ] i z IS (0) ∗ S S(0)S = IS (0) + IR (0) e 2 I

4.3.1.1.1

and |S |2 = 1 − 𝜂 = 1 − |m(0)|2 sin2

𝛾z . 4

From that, we obtain IR (0)IS (0) 𝛾 sin z 2 I 2 I (0)I (0) I (0) − IS (0) 2 𝛾 R S R sin z. IS2Ω ∕I = −2J2 (𝜓d ) I2 I 4 For the ratio of the intensities, we obtain ISΩ ∕I = 4J1 (𝜓d )

IS2Ω ISΩ

=

J2 (𝜓d ) IR (0) − IS (0) 𝛾 tan z. J1 (𝜓d ) I 4

(4.161) (4.162)

(4.163)

Shifted Hologram Again, assuming that the signal beam amplitude oscillates in the form ei 𝜓d sin Ωt , with Ω𝜏sc ≫ 1, we first need the results in Section 4.2.2.2.1.2 to compute the expressions 𝛾 Γ z −i z 2 I (0) e + IS (0) e 2 S∗ S(0)S = IS (0) R Γ Γ z − z IR (0) e 2 + IS (0) e 2

4.3.1.1.2

117

118

4 Volume Hologram with Wave Mixing

and |S |2 = 1 − 𝜂 =

Γ Γ − z z IR (0)2 e 2 + IS (0)2 e 2 + 2IR (0)IS (0) cos 𝛾2 z Γ Γ ⎡ − z z ⎢IR (0) e 2 + IS (0) e 2 ⎢ ⎣

⎤ ⎥I ⎥ ⎦

.

From that we find 𝛽 2 sin 𝛾2 z

IΩ (z) = 4J1 (𝜓d )IS (0) 𝛽2

(4.164)

Γ Γ z − z e 2 + e2

and

I2Ω = −4J2 (𝜓d )IS (0)

𝛽2 𝛽 1 + 𝛽2

2

Γ Γ − z z 2 e − e 2 + (1 − 𝛽 2 ) cos 𝛾 z 2

𝛽2

Γ Γ − z z e 2 + e2

The ratio of these intensities is Γ Γ − z z 𝛾 2 2 2Ω I J2 (𝜓d ) 𝛽 e 2 − e 2 + (1 − 𝛽 ) cos 2 z = − J1 (𝜓d ) (1 + 𝛽 2 ) sin 𝛾2 z IΩ

(4.165)

(4.166)

Note that equation (7b) in reference [98] is just a particular case of Eq. (4.166) for 𝛽 2 = 1. 4.3.1.2 Output Phase Shift

We may also directly operate on Eq. (4.156) to describe the total irradiance at the output along the S-beam direction I = ∣  ∣2 =∣ R(0) R + S(0) ei 𝜓d sin Ωt S ∣2 √ √ = ∣ R(0) 𝜂 + S(0) 1 − 𝜂 ei𝜑 + i𝜓d sin Ωt ∣2 (4.167) where I is the irradiance along the incident beam S, measured behind the crystal, with

and

S(0)S R(0)R =∣ S(0)S R(0)R ∣ ei𝜑

(4.168)

IR0 =∣ R(0)∣2

IS0 =∣ S(0)∣2

(4.169)

√ 𝜂 =∣ R ∣

√ 1 − 𝜂 =∣ S ∣

(4.170)

as already defined in Section 4.2.2.2, 𝜑 represents the phase shift between the transmitted and diffracted beams behind the crystal, as shown in Fig. 4.20 where the hologram phase shift 𝜙 is also shown. Equation (4.167) is formulated in terms of parameters directly measured at the input and output of the sample. Developing Eq. (4.167), one gets a phenomenological formulation √ √ 𝛽 ISdc =∣ S0 ∣2 (1 − 𝜂)+ ∣ RO ∣2 𝜂 + 2(IS0 + IR0 ) 2 (4.171) J0 (𝜓d ) 1 − 𝜂 𝜂 cos 𝜑 𝛽 +1 √ √ ISΩ = −4J1 (𝜓d ) IS0 IR0 𝜂(1 − 𝜂) sin 𝜑 (4.172)

4.4 Four-Wave Mixing

Figure 4.20 Wave-mixing schema showing the hologram phase shift 𝜙 and the phase shift 𝜑 between the transmitted and diffracted beams at the crystal output.

0 S

ϕ

tran

0 R pattern of fringes

IS2Ω = 4J2 (𝜓d )

√

R

diff ra

sm itte

cte

d

d S shifted by φ

hologram

√ IS0 IR0 𝜂(1 − 𝜂) cos 𝜑

ISΩ J2 (𝜓d ) tan 𝜑 = − 2Ω IS J1 (𝜓d )

(4.173) (4.174)

Substituting Eq. (4.166) into Eq. (4.174), we get the following expression for the output beam phase shift 𝛾 sin z 2 tan 𝜑 = − . 2 ( 1−𝛽 𝛾 ) Γ Γ cosh z − cos z + sinh z 1 + 𝛽2 2 2 2 that is, of course, the same expression already reported in Eq. (4.128), although computed following a different procedure. This formula specializes to 𝛾 1 − 𝛽2 tan 𝜑 tan z = −1 1 + 𝛽2 4

(4.175)

for 𝜙P = 0 (so-called local materials) and to tan 𝜑 tan 𝜙P = −1

(4.176)

for 𝛽 = 1 with Γz ≪ 1 and 𝛾z ≪ 1; that is to say, in the absence of phase coupling, but for any 𝜙P . Another interesting special case is the one of thin crystals, which corresponds to the exclusion of self-diffraction effects, in which case it also simplifies to 2

tan 𝜑 tan 𝜙P = −1

for

z → 0.

(4.177)

Exercise Verify that the negative sign in Eq. (4.172) is in agreement with the negative sign in Eq. (4.159). Note that the sign does depend on the way 𝜑 is defined, so be sure that it is defined in the same way in both formulations.

4.3.1.2.1

4.4 Four-Wave Mixing So far, we have been dealing with the interference of two waves only, either with the same (degenerate) or with slightly different (nearly degenerate) frequencies. These two waves produce a hologram in a nonlinear (in our case, a photorefractive) material and the resultant hologram

119

120

4 Volume Hologram with Wave Mixing

R

P

R

S

S*

S

Figure 4.21 Degenerate four-wave mixing showing the signal S and reference R beams interfering to produce a real-time hologram in the nonlinear material (left); then a pump beam P, identical to R but much stronger and counter propagating, is diffracted by the already recorded hologram and the diffracted beam is the conjugate S* of the signal S beam, reflecting back along the same incidence direction.

diffracts the recording waves. There is feedback between the recording waves and the nonlinear media, and by means of this interaction the amplitudes and/or phases of the recording waves are changed: This is “two-wave mixing” or TWM. We can similarly mix four waves instead of two and this is called “four-wave mixing” or FWM. If all the waves involved have the same frequency, this is known as “degenerate” FWM or DFWM. The mathematics is somewhat more involved than for TWM but the phenomena are essentially the same. In this case, waves S and R, mutually coherent and having the same temporal frequency, do interfere in the material and a real-time (or almost) hologram arises as represented in Fig. 4.21. A pump beam P with same wavefront shape as R and same temporal frequency (not necessarily coherent with R) but usually much stronger and propagating in the opposite direction is diffracted by the real-time hologram as shown in the right-hand side of Fig. 4.21. The diffracted P beam is S* , which represents the conjugate of S. Our simplified picture shows a hologram arising only from the interference of S and R, which is true if the pump P is not coherent with the former two beams. The whole is behaving as a so-called “phase-conjugate” mirror and Fig. 4.21 schematically describes its behavior: The incident wave S is phase conjugated and reflected back exactly along its incidence direction. FWM or DFWM is extremely interesting for a number of applications, but it is rarely used for material characterization, so we shall not extend further on this subject. More details about this can be found in reference [101], among many other books.

4.5 Conclusions Before closing this chapter, we would like to draw the reader’s attention to the fact that photorefractives exhibit unique interesting features: • the adaptive and multiplicative characteristic of real-time recording, • the low pass filtering arising from the finite material response time, • the amplitude coupling or energy transfer derived from the phase-shifted nature of photorefractive recording that are at the root of many applications focused on image and signal processing (see, for example, references [102–104] among many others) and make photorefractive materials an extraordinarily fertile field for applied research.

121

5 Anisotropic Diffraction Some crystals exhibit anisotropic diffraction, that is to say that the polarization of the incident and the diffracted light have different directions. This is the case, among others, of sillenite-type crystals and this feature derives from the structure of their electro-optic tensor. In fact, let us recall the expressions for the index-of-refraction along axes 𝜁 , 𝜂 and y in Section 1.5 that we now designate, in a more conventional way, as x, y and z, respectively. 1 nx = no + n3o r41 E 2 1 3 ny = no − no r41 E 2 nz = no

5.1 Coupled-Wave with Anisotropic Diffraction Let us now analyze the expression of the coupled wave equations for a pure phase grating in Eqs. (4.56) and (4.57) where the coupling constant 𝜅, as formulated in Eq. (4.86), should now be written in tensorial form as 𝜅̂ =

n3o r41 𝜋E 2𝜆

⎡1 0 0⎤ ⎢ 0 −1 0 ⎥ ⎢ ⎥ ⎣0 0 0⎦

(5.1)

In this case, the coupled equations (4.56) and (4.57), neglecting absorption (𝛼 = 0) and simplifying to |m| = 1, look like: ⃗ 𝜕 R(z) = i𝜅̂ S⃗ 𝜕z ∗ 𝜕 S⃗∗ (z) = −i𝜅̂ R⃗ cos 𝜃 𝜕z cos 𝜃

(5.2) (5.3)

⃗ respecIf we define r⃗ and s⃗ as the unit vectors indicating the polarization direction of R⃗ and S, tively, we should write the Eqs. (5.2) and (5.3) as: 𝜕R = i(⃗r.𝜅⃗ ̂ s)S 𝜕z 𝜕S∗ = −i(⃗s.𝜅̂ r⃗)R∗ cos 𝜃 𝜕z cos 𝜃

Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

(5.4) (5.5)

122

5 Anisotropic Diffraction

y

R(z) ∧ r

S(z) ∧ S y

S(z) 𝛼 ∧ S

y

Figure 5.1 Input and output light polarization. 𝛼

X

𝛼 = –γ

Figure 5.2 Input and output polarization referred to actual principal axes coordinates.

X γ 𝛼 = –γ

R(z) ∧ r

where r⃗.𝜅⃗ ̂s =

n3o r41 𝜋E ⎡⎢ cos 𝛾 ⎤⎥ ⎡⎢ 1 0 0 ⎤⎥ ⎡⎢ cos 𝛼 ⎤⎥ sin 𝛾 . 0 −1 0 sin 𝛼 ⎢ ⎥ ⎢ ⎥⎢ ⎥ 2𝜆 0 0 0 0 ⎣ ⎦ ⎣ ⎦⎣ 0 ⎦

(5.6)

with s⃗(cos 𝛼, sin 𝛼, 0) and r⃗(cos 𝛾, sin 𝛾, 0). This means that the value (modulus) of the effective coupling constant n3 r 𝜋E n3o r41 𝜋E (cos 𝛾 cos 𝛼 − sin 𝛾 sin 𝛼) = o 41 cos(𝛾 + 𝛼) (5.7) 2𝜆 2𝜆 is maximum for: cos(𝛾 + 𝛼) = 1 n3 r 𝜋E for 𝛼 = −𝛾 (5.8) [⃗r.𝜅⃗ ̂ s]max = 𝜅 = o 41 2𝜆 The diffracted light, with a polarization direction verifying the conditions here, is optimized and will develop over all other possibilities. Equation (5.8) means that the polarization directions of the incident and the diffracted beams are symmetric relative to the coordinate axis (x or y) in the crystal incidence plane, as illustrated in Fig. 5.1. Note that in the case of a sillenite-type crystal with an electric field applied as illustrated in Fig. 1.10, the principal coordinate axes of the index ellipsoid are rotated 45∘ , as illustrated in Fig. 1.11, so that in this case the actual picture stands as represented in Fig. 5.2 that is rotated 45∘ to that of Fig. 5.1 but does not change the fact that the output polarization directions of the incident and diffracted beams are symmetric along (the new) axes x and y. r⃗.𝜅⃗ ̂s =

5.2 Anisotropic Diffraction and Optical Activity A possible solution for the coupled equations Eqs. (5.4) and (5.5) is R(z) = R cos(𝜅z) + iS ei𝜙 sin(𝜅z) 0

0

S(z) = iR0 sin(𝜅z) + S0 ei𝜙 cos(𝜅z) Assuming 𝜅z ≪ 1, the components polarized along 𝛾 and 𝛼 = −𝛾 are, respectively, R (z) = R cos(𝜅z) S (z) = S ei𝜙 cos(𝜅z) 𝛾

0

𝛾

R−𝛾 (z) = iS0 ei𝜙 sin(𝜅z)

0

S−𝛾 = iR0 sin(𝜅z)

(5.9) (5.10)

(5.11) (5.12)

5.2 Anisotropic Diffraction and Optical Activity

5.2.1

Diffraction Efficiency with Optical Activity, 𝝆

From Eqs. (5.11) and (5.12), we can write dS−𝛾 = iR𝛾 𝜅dz

(5.13)

R𝛾 = R0 cos(𝜅z)

(5.14)

with the x- and y-components at the crystal output z = d being dS−𝛾 ]x = iR0 𝜅 cos(𝜅z) cos[−𝛾 + 𝜌(d − z)]dz

(5.15)

dS−𝛾 ]y = iR0 𝜅 cos(𝜅z) sin[−𝛾 + 𝜌(d − z)]dz

(5.16) (5.17)

After factoring the trigonometric functions here and integrating, we get [ ] iR0 𝜅 sin(𝜅z + 2𝜌z + 𝛾0 − 𝜌d) sin(𝜅z − 2𝜌z − 𝛾0 + 𝜌d) z=d S(z)−𝛾 ]x = + 2 𝜅 + 2𝜌 𝜅 − 2𝜌 z=0 [ ] iR0 𝜅 cos(𝜌d − 𝛾0 − 𝜅z − 2𝜌z) cos(𝜌d − 𝛾0 + 𝜅z − 2𝜌z) z=d S(z)−𝛾 ]y = + 2 𝜅 + 2𝜌 𝜅 − 2𝜌 z=0 𝛾 = 𝛾0 + 𝜌z

𝛾0 = 𝛾(0)

(5.18) (5.19) (5.20)

Assuming 𝜅 ≪ 2𝜌, we have iR0 𝜅 sin(𝜅d + 𝜌d + 𝛾0 ) − sin(𝛾0 − 𝜌d) − sin(𝜅d − 2𝜌d − 𝛾0 ) 2 2𝜌 iR0 𝜅 cos(𝜅d + 𝜌d + 𝛾0 ) − cos(𝜌d − 𝛾0 ) + cos(𝛾0 + 𝜌d − 𝜅d) S(d)−𝛾 ]y = 2 2𝜌 that can be written as iR 𝜅 2 cos(𝜅d) sin(𝜌d + 𝛾0 ) + 2 sin(𝜌d − 𝛾0 ) S(d)−𝛾 ]x = 0 2 2𝜌 iR0 𝜅 2 cos(𝜅d) cos(𝜌d + 𝛾0 ) − 2 cos(𝜌d − 𝛾0 ) S(d)−𝛾 ]y = 2 2𝜌 where R2 𝜅 2 ISdiff = |S(d)−𝛾 ]x |2 + |S(d)−𝛾 ]y |2 = 0 2 [sin(𝜌d)]2 𝜌 [ ]2 diff I sin(𝜌d) 𝜂 = S 2 ≈ (𝜅d)2 |R0 | 𝜌d S(d)−𝛾 ]x =

5.2.2

(5.21) (5.22)

(5.23) (5.24)

(5.25) (5.26)

Output Polarization Direction

⃗ at the In Section 5.2.1 it was stated that 𝛾0 is the angle of the polarization direction of wave R(z) ⃗ at the output. input. Let us then assume 𝛼s to be the corresponding angle (see Fig. 5.2) for S(z) From Eqs. (5.23) and (5.24) we may compute 𝛼s as follows tan 𝛼s =

S(d)−𝛾 ]y S(d)−𝛾 ]x

= − tan 𝛾0

(5.27)

always for 𝜅d ≪ 1. Figures 5.3–5.5 do illustrate some typical results for a crystal with 𝜌d = 20∘ and interfering beams with the same input polarization direction.

123

124

5 Anisotropic Diffraction

d

[001]

70°

d 70°

t

t

ρd = 20°

Figure 5.3 General illustration of the polarization direction of the transmitted and diffracted beams through a crystal with optical activity and anisotropic diffraction. At midcrystal thickness, the polarization directions of the transmitted and diffracted beams are 10∘ shifted from the [110] and [001] axes, respectively.

[010] t

[110]

t

[100]

d

[001]

90°

d 90°

t

t

ρd = 20°

Figure 5.4 Transmitted and diffracted beams orthogonally polarized at the output through a crystal with optical activity and anisotropic diffraction. Assuming ρd = 20∘ , the incident beam’s polarization direction at the input plane should be −10∘ with reference to the [110]-axis.

[010] t

[110]

t

[100]

d

[001] d

ρd = 20°

t t

[010] t

[110]

t

[100]

Figure 5.5 Transmitted and diffracted beams parallel-polarized at the output through a crystal with optical activity and anisotropic diffraction. Assuming 𝜌d = 20∘ , the incident beam’s polarization direction at the input plane should be 35∘ with reference to the [110]-axis.

125

6 Stabilized Holographic Recording 6.1 Introduction Holographic setups are extremely sensitive to environmental perturbations (thermal drifts, air currents, mechanical vibrations etc.) and this fact makes it difficult to obtain reproducible holographic recordings, unless the recording time is much smaller than the period of the perturbations. The characteristic recording time, in photorefractive materials at least, is roughly inversely proportional to the average irradiance onto the crystal, so that stable holograms may only be produced with high intensity laser beams. High intensity, however, is rarely achieved in many applications like image processing experiments, for example, where the light scattered back from the target is usually weak. For a Bi12 TiO20 sample illuminated by 200–300 μW/cm2 in the 514 nm wavelength, the recording time is of the order of a few seconds, and it is still larger for the 633 nm wavelength. The simplest way to overcome holographic instability is to perform a stabilized holographic recording using a feedback setup as described further on. Such a stabilization can be performed using a reference (for example, a previously recorded hologram placed as close as possible to the hologram to be recorded) to fix the pattern of fringes. It is still possible to stabilize the pattern of fringes using the hologram being recorded as the reference itself. The latter procedure is the so-called “self-stabilized” recording. Both procedures will be described and applied to different materials. In any case and taking into account the complex nature of holographic recording in rather thick (compared to the pattern of fringes period) samples and high diffraction efficiencies that may result, the behavior of the material involved should be carefully studied in order to choose an adequate holographic recording procedure. Several examples using different materials in this chapter will certainly illustrate this subject to the reader. Figures 6.1 and 6.2 clearly show the possibilities of self-stabilized recording in microelectronics grade photoresists: The large height-to-width ratio of the structures shown in these figures could never have been obtained without using self-stabilized holographic recording [106]. In fact, this technique allows one to firmly fix the recording pattern of light in order to obtain the very sharp spatial light contrast that is necessary to produce these structures. Another feature of stabilized recording is shown in Fig. 6.3 where the two first spatial harmonics (with adequate spatial frequency, relative amplitude and mutual phase shift) of a “sawtooth” profile were recorded on a photoresist film. The profile does not look much like a sawtooth but its behavior under diffraction definitely does, because the diffraction properties rely mainly on the first few harmonics [105]. Stabilization here has a double purpose: avoiding perturbations on the recording setup and also adequately fixing the mutual phase shift between the two spatial harmonics during recording. A pioneer research leading to the present self-stabilized holographic recording was proposed by Neumann and Rose [107] by 1967. They first proposed to amplify the recording interference Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

126

6 Stabilized Holographic Recording

Figure 6.1 Scanning electronic microscopy image of a 1D hollow sleeve structure first recorded on photoresist film, then metallic vacuum deposited and finally washed away from all remaining photoresist to produce hollow metallic structures. Produced and photographed by Lucila Cescato, Laboratório de Óptica, Instituto de Física, Universidade Estadual de Campinas, Brazil.

Figure 6.2 Scanning electronic microscopy image of a 2D-array holographically recorded and chemically developed on photoresist film. Produced and photographed by Lucila Cescato, Laboratório e Óptica, Instituto de Física, Universidade Estadual de Campinas, Brazil.

pattern of fringes using a microscope objective and project this amplified pattern onto a photodetector to operate an electronic feedback loop to stabilize this recording pattern of fringes. MacQuigg [108] further improved this technique by using an auxiliary (previously recorded) hologram, instead of a microscope objective, for amplifying the pattern of fringes on the photodetector. In this way, a much brighter amplified pattern of fringes was obtained. Additionally, he phase-modulated one of the interfering beams to produce a temporal harmonic term, then the mixed beams were sent along both directions behind the reference grating, one of them being projected onto a photodetector connected to a phase-selective frequency tuned amplifier known as a “lock-in” amplifier for demodulating the temporal harmonic to be used as error signal for the feedback stabilization loop.

6.2 Mathematical Formulation

Figure 6.3 Scanning electronic microscopy image of a blazed grating made by the holographic recording of the first and the second spatial harmonic components of a sawtooth-shape profile on photoresist film. Produced and photographed at Laboratório de Óptica, Instituto de Física, Universidade Estadual de Campinas, Brazil. Reproduced from [105].

6.2 Mathematical Formulation The plain stabilized recording is a particular simple case where one uses an external reference (a fixed hologram or a glass plate adequately placed by the side of the sample) that is able to produce an interference pattern of light that can be used to operate the feedback loop. In this chapter, we shall focus on the self-stabilized recording because it is the more complex situation and is also the most interesting procedure for holographic recording. This technique can also be used with nonreversible materials like photoresists [109, 110], for example. The self-stabilization procedure does not use an external reference. This technique is based on phase modulation, as described in Section 4.3, where a modulation of amplitude 𝜓d and angular frequency Ω (Ω much larger than the frequency response of the hologram) is produced in the phase of one of the two interfering beams (of irradiances IR and IS ) in the holographic setup. By this means, the phase-shift 𝜑 between the transmitted and diffracted beams behind the sample is correspondingly modulated, so that the expression of the overall irradiance along the direction IS behind the sample can be written as √ √ IS = IS0 (1 − 𝜂) + IR0 𝜂 + 2 𝜂(1 − 𝜂) IS0 IR0 cos(𝜑 + 𝜓d sin Ωt) (6.1) where IR0 and IS0 are the values at the input. Because of the nonlinear relation between 𝜑 and IS , harmonic terms in Ω do appear where the amplitude of the first and second ones were already formulated in Eqs. (4.172) and (4.173): √ √ ISΩ = −4J1 (𝜓d ) 𝜂(1 − 𝜂) IS0 IR0 sin 𝜑 √ √ IS2Ω = 4J2 (𝜓d ) 𝜂(1 − 𝜂) IS0 IR0 cos 𝜑 with 𝜑 described in Eq. (4.128) as tan 𝜑 = −

𝛾 sin z 2

1 − 𝛽2 ( 𝛾 ) Γ Γ cosh z − cos z + sinh z 2 1+𝛽 2 2 2

For nonphotovoltaic photorefractive materials, in the absence of an external electric field the hologram phase shift (phase difference between the recording pattern of fringes and the resulting hologram) is 𝜙P = 𝜋∕2 (with tan 𝜙P ≡ Γ∕𝛾), which, substituted into the previous expression for tan 𝜑, leads to 𝜑 = 0 (or 𝜋). This means that, in equilibrium, it should be ISΩ = 0 and we can

127

128

6 Stabilized Holographic Recording

therefore use ISΩ as error signal to operate a stabilization system to keep the holographic setup actively fixed to this 𝜑 = 0 condition. Unless otherwise stated, we shall therefore assume that ISΩ is always used as error signal in self-stabilization experiments. The setup is schematically represented in the block-diagram of Fig. 6.4 and the schema of the actual setup in Fig. 6.5. The effect of a phase perturbation (noise) 𝜑N on the two-wave mixed output is illustrated in Fig. 6.6. The photodetector D transforms the overall irradiance IS at the crystal output into an electric signal, the harmonic term amplitudes of which in Ω and 2Ω are, respectively, VSΩ = KdΩ ISΩ

(6.2)

VS2Ω = Kd2Ω IS2Ω ,

(6.3) φN

PM OSC

v(t)

φf

φ

HOLOGRAPHIC SETUP

Is

+

Vf HV

D

LA–Ω M

r se La

IS

C I0S

OSC

D

IR0

BS

Ω

Figure 6.4 Block-diagram of a self-stabilized setup: D photodetector, LA-Ω phase sensitive lock-in amplifiers tuned to Ω, HV voltage source for the phase modulation device PM, OSC oscillator at frequency Ω. The output phase shift, feedback and noise phases are 𝜑, 𝜑f and 𝜑N , respectively.

PZT Vd

+

Vf

IR

VC

HV

LAΩ

LA2Ω VS2Ω

Figure 6.5 Schematic description of the actual self-stabilized holographic recording setup: C photorefractive crystal, D photodetector, LAΩ and LA2Ω phase sensitive lock-in amplifiers tuned to Ω and 2Ω, respectively, HV high voltage source for the piezo-electric supported mirror PZT acting as phase modulator, OSC oscillator at frequency Ω. φN NOISE 0 S

Figure 6.6 Schematic description of the effect of noise on the two-wave mixing in the holographic setup.

ϕ+φN

R

tran

sm

0 R pattern of fringes

diff

rac

itte d

ted

S

mutually shifted φ+φN by hologram

6.2 Mathematical Formulation

with KdΩ and Kd2Ω being the photodetector responses to signals with frequencies Ω and 2Ω, respectively. A Ω-tuned lock-in amplifier LA-Ω selects the first harmonic term and produces a demodulated and amplified signal VC = AΩ VSΩ ,

(6.4)

that is, the correction signal in the feedback loop where AΩ is the amplification. This signal is fed to the voltage source HV, which produces an electric feedback signal Vf = K0 VC ,

(6.5)

where K0 is the HV amplification. The signal Vf acts on the phase modulator device PM (in this case, a piezoelectric supported mirror, PZT), which produces a correction feedback phase 𝜑f on the holographic setup 0 𝜑f = KPM Vf = A sin 𝜑 0 A = KPM K0 AΩ KdΩ 4J1 (𝜓d )

(6.6) √

√ IS0 IR0 𝜂(1 − 𝜂),

(6.7)

0 where KPM is the voltage-to-phase response at the PM for Ω ≈ 0. At the same time, an oscillator OSC produces a small ac voltage of frequency Ω of the form

𝑣(t) = Vd sin Ωt

(6.8)

which is added to Vf and is fed to the PM to produce the phase modulation of frequency Ω and amplitude Ω 𝜓d = KPM Vd ,

(6.9)

which is necessary to produce the phase modulation that is represented in Eq. (6.1). The quanΩ tity KPM is the voltage-to-phase response of the PM at Ω. 6.2.1

Stabilized Stationary Recording

In the absence of feedback, the output phase in the setup can be written as 0 𝜑 = 𝜓0 + 𝜓H − KPM V0

(6.10)

0 V0 , 𝜙P = 𝜓H − KPM

(6.11)

where 𝜓H is the (phase) position of the recorded hologram, V0 is the dc bias voltage applied to 0 the PM, KPM V0 is the pattern-of-fringes position and their difference is the so-called hologram phase-shift 𝜙P . The 𝜓0 is a correction term that depends on the nature of the hologram, the value of 𝜙P and the degree of phase coupling [65]. In steady-state conditions, 𝜙P and 𝜓0 are constants and the corresponding steady-state value of 𝜑 is also a constant 𝜑0 𝜑0 = 𝜓0 + 𝜙P .

(6.12)

The term 𝜓0 is implicitly contained in the Eq. (4.128) and the expression for 𝜙P in Eq. (4.91) 𝛾 sin z Γ 2 (6.13) tan 𝜑0 = − tan 𝜙P = ) 𝛾 𝛾 1 − 𝛽2 ( Γ Γ cosh z − cos z + sinh z 1 + 𝛽 2I 2 2 2

129

130

6 Stabilized Holographic Recording

where 𝜑 was substituted by its nonfeedback constrained value 𝜑0 . Note that 𝛾 in Eq. (4.90) and Γ in Eq. (4.85) are proportional to the components of the space-charge electric field amplitude, which are in-phase and 𝜋∕2-shifted, respectively, to the pattern of fringes. For the particular case when 𝜙P = 𝜋∕2, it is 𝛾 = 0 and consequently 𝜑0 = 0 (or 𝜋) and implicitly 𝜓0 = ±𝜋∕2. For other values of 𝜙P , the corresponding values for 𝜑0 can be computed from Eq. (6.13). In the steady state under feedback conditions, the expression of 𝜑f in Eq. (6.6) is subtracted (negative feedback) from the Eq. (6.10) to give the steady-state feedback equilibrium value 𝜑eq 0 𝜑eq = 𝜓0 + 𝜓H − KPM V0 − 𝜑f

with 𝜑f = A sin 𝜑eq

(6.14)

where 𝜑eq is, in general, different from the non feedback constrained 𝜑0 one (𝜑eq ≠ 𝜑0 ), except for the case 𝜑0 = 0 (that is, 𝜙P = ±𝜋∕2) in which case it is also 𝜑eq = 0 in Eq. (6.14). This means that under feedback constraint, the system will be in stationary equilibrium only for 𝜙P = 𝜋∕2. Otherwise, the feedback will force the pattern of fringes and associated hologram to move because of the mismatch between 𝜑eq and 𝜑0 . 6.2.1.1 Stable Equilibrium Condition

For the stationary (nonmoving hologram) case, where 𝜑0 = 0, it is still necessary to analyze the stability of the equilibrium condition 𝜑eq = 0. In the presence of a phase perturbation 𝜑N in the setup, Eq. (6.14) becomes 0 V0 − 𝜑f 𝜑 = 𝜑N + 𝜓0 + 𝜓H − KPM

(6.15)

A stable equilibrium condition requires that d𝜑∕d𝜑N ≈ 0, which, substituted into Eq. (6.15), results in d𝜑 1 1 ] = ] = ≈0 (6.16) d𝜑N 𝜑eq 1 + A cos 𝜑 𝜑eq 1 + A where 𝜑 = 𝜑eq ≈ 0 was assumed to enable the use of ISΩ as error signal. The condition in Eq. (6.16) is actually verified for a large negative (A ≫ 1) feedback. 6.2.2

Stabilized Recording of Running (Nonstationary) Holograms

Moving holograms or space-charge waves have been described in Section 3.4. If the pattern of light is moving with the resonance speed characteristic for the hologram in the crystal (that is K𝑣 = 𝜔I ) a maximum in diffraction efficiency is reached [85] as shown in Fig. 3.20. The operation of the feedback in order to produce running holograms in a self-stabilized way is the central point here. Such holograms are also known as “fringe-locked” running holograms. As discussed previously, a nonstationary (running) hologram is automatically established when 𝜑0 ≠ 𝜑eq . The latter relation is verified when 𝜑0 ≠ 0. In this condition, the hologram is forced to be erased and re-written continuously somewhere ahead and by this means continuous movement occurs. The block diagram of this new experimental setup is depicted in Fig. 6.7, whereas φN

PM OSC

v(t)

φf

HOLOGRAPHIC SETUP

Figure 6.7 Block-diagram of fringe-locked running hologram setup: same as for Fig. 6.4 with the addition of an integrator INT at the output of the lock-in amplifier.

φ

Is

+

Vf HV

INT

LA–Ω

D

6.2 Mathematical Formulation

M

r se La

IS

C

BS I0S

Ω OSC

D

IR0

PZT Vd

+

Vf

IR

HV

VC

LA Ω

INT

LA 2Ω VS2Ω

Figure 6.8 Schematic actual setup for self-stabilized running hologram recording: same as for Fig. 6.4 with the addition of an integrator INT at the output of the lock-in amplifier.

the actual setup schema is depicted in Fig. 6.8. An integrator was included here between the lock-in amplifier output and the voltage source HV so that the correction feedback phase 𝜑f is no longer described by Eq. (6.6) but by the integral t

𝜑f =

A sin 𝜑 dt 𝜏i ∫0

(6.17)

where 𝜏i is a factor arising from the integrating circuit. This 𝜑f is required to produce the mismatch between 𝜑eq and 𝜑0 that is necessary for running hologram generation and at the same time to fulfill the feedback loop condition 𝜑eq ≈ 0 that is determined by the use of ISΩ as error signal in the feedback. The expression for 𝜑eq under these new feedback constraints can be formulated as t

0 V0 − 𝜑eq = 𝜓0 + 𝜓H − KPM

A sin 𝜑eq dt 𝜏i ∫0

(6.18)

The steady-state condition is represented by d𝜑eq

d𝜓0 d𝜓H (6.19) = + − 𝜅f sin 𝜑eq = 0 dt dt dt where 𝜅f ≡ A∕𝜏i . The equilibrium condition d𝜓0 ∕dt = 0 should be considered to get the expression for the hologram speed d𝜓H (6.20) = 𝜅f sin 𝜑eq dt But the hologram speed 𝜔H is a function of the mismatch between 𝜑eq and 𝜑0 , so that a relation can be stated in the form 𝜔H =

𝜔H = f (𝜑eq − 𝜑0 ) with

f (0) = 0

(6.21)

where f (𝜑eq − 𝜑0 ) is a function depending on material and experimental parameters. From Eqs. (6.20) and (6.21), we should write 𝜔H = f (𝜑eq − 𝜑0 ) = 𝜅f sin 𝜑eq

(6.22)

showing that 𝜅f → ∞ leads to 𝜑eq → 0 in which case, Eq. (6.22) becomes lim 𝜔H = f (−𝜑0 ) = 𝜅f sin 𝜑eq

𝜅f →∞

(6.23)

131

132

6 Stabilized Holographic Recording

which means that for a sufficiently large amplification 𝜅f , the hologram speed 𝜔H is independent of 𝜅f and dependent on the unconstrained equilibrium value 𝜑0 through the functional relation f (−𝜑0 ). The last function will be discussed further on. 6.2.2.1 Stable Equilibrium Condition

As for the case of stationary holograms, we need to analyze the stability of the equilibrium; that is to say, the way the feedback loop does reduce the effects of both a noise (see Fig. 6.6) 𝜑N on the phase and a noise 𝜔N on the speed, near the equilibrium position. For this purpose let us write the differential equation (6.19), describing the output phase shift 𝜑eq under feedback at equilibrium as d𝜑eq

+ 𝜅f 𝜑eq = 𝜔H dt where we have assumed sin 𝜑eq ≈ 𝜑eq ≪ 1, in which case the general solution is [96]: − 𝜅f dt [𝜑N + 𝜑eq = e ∫

∫

𝜔H e ∫

𝜅f dt

dt]

(6.24)

(6.25)

If we assume 𝜅f to be independent of time, it is ∫ 𝜅f dt = 𝜅f t. Also, using the theorem of integration by parts, we can write: 𝜔 𝜔̇ 𝜔̈ 𝜔H e𝜅f t dt = H e𝜅f t − H2 e𝜅f t − H3 e𝜅f t − · · · + C ∫ 𝜅f 𝜅f 𝜅f which simplifies to ∫

𝜔 𝜔H e𝜅f t dt ≈ H e𝜅f t + C 𝜅f

for

𝜔̇H ≪ 𝜅f2

and the expression for Eq. (6.25) becomes: 𝜔 𝜔 𝜑eq ≈ H + [𝜑i + N ] e−𝜅f t 𝜅f 𝜅f

(6.26)

(6.27)

where we have written C ≡ 𝜔N with 𝜑N and 𝜔N being constants arising from the solution of the homogeneous differential equation and consequently representing the transient solutions. The quantities 𝜑N and 𝜔N can be thought to be perturbations or noises on the phase and on the speed, respectively. For a large negative amplification (𝜅f ≫ 1) the term in square brackets (where noises 𝜑N and 𝜔N are included) is rapidly vanishing. The remaining stationary term 𝜔H ∕𝜅f represents the steady-state (inhomogeneous differential equation) solution of the nonperturbed system. For a sufficiently large 𝜅f it is 𝜔H ∕𝜅f ≈ 0 and consequently 𝜑eq ≈ 0 in Eq. (6.27) as required for the feedback operation. 6.2.2.2 Speed of the Fringe-Locked Running Hologram

Equation (6.23) states that the speed of the hologram is a function of f (−𝜑0 ) where 𝜑0 is the phase shift between the transmitted and diffracted beams along the same direction behind the crystal, in equilibrium, without feedback. On the other hand, the actual expression for the self-stabilized (also known as fringe-locked) running hologram speed can be computed from ISΩ = 0

(6.28)

A condition that is inherent to the use of ISΩ as error signal in the feedback stabilization loop. This condition means, from Eqs. (4.164), (4.90) and (4.86), that 𝛾 ∝ ℜ{Eeff } = 0

(6.29)

6.2 Mathematical Formulation

For the case of a running hologram, however, the expression Eeff in Eq. (6.29) should be substist as formulated in Eq. (3.78), where the bulk light absorption effect on the hologram tuted by Esc speed has been neglected. Then, from st 𝛾 ∝ ℜ{Esc }=0

(6.30)

and its explicit expression in Eq. (3.83), we get K𝑣 =

E0 ∕ED 1 2 𝜏M (1 + E0 ∕ED2 )K 2 L2D + 1

(6.31)

From Eq. (6.31) we see that K𝑣 depends on both ED ∝ Γ and E0 ∝ 𝛾, which in turn determine 𝜑0 in Eq. (6.13). So, K𝑣 itself is implicitly determined by 𝜑0 and the theoretical statement in Eq. (6.23) is therefore justified. The effect of bulk absorption on the hologram speed is more difficult to analyze and will be studied in Section 9.2.1. In order to experimentally verify the independence of K𝑣 on the settings of the feedback loop, K𝑣 was measured for different values of 𝜅f on a BTO crystal using the 514.5-nm wavelength and a nominally applied field of 4.7 kV/cm as seen in Fig. 6.9, where it is shown that for a 50-fold variation in 𝜅f (in arbitrary units), K𝑣 does vary by ±1.5% only, a variation that is roughly of the order of magnitude of the data dispersion in the experiment. The increase of 𝜅f in Eq. (6.23) just makes 𝜑eq approach zero without sensibly affecting 𝜔H , as predicted by the theory and experimentally confirmed here. Nevertheless, the adequate choice of 𝜅f is of the highest practical relevance in the sense that the operation of the feedback is very sensitive to the amplification in the loop: a low amplification may be not enough, whereas a too-large amplification may produce instabilities and even drive the setup to an oscillatory behavior. 6.2.3

Self-Stabilized Recording with Arbitrarily Selected Phase Shift

The self-stabilized holographic recording setup described previously suffers from a serious limitation: the phase-shift 𝜑 between the transmitted and diffracted beam behind the sample is fixed either to 𝜑 = 0, 𝜋 if the first harmonic term V Ω is used as error signal, or to 𝜑 = ±𝜋∕2 if the second harmonic term V 2Ω is used instead. 0.100

Kv (rad/s)

0.095

0.090

0.085

0.080

0

20

40

60

𝜅f (au)

Figure 6.9 Fringe-locked running hologram speed: Kv (rad/s) versus feedback amplification 𝜅f (arbitrary units) in a fringe-locked running hologram experiment carried out on an undoped Bi12 TiO20 crystal using the 514.5 nm wavelength, with E0 = 4.7 kV∕cm, IRo = 533 μW/cm2 , ISo = 20 μW/cm2 , Ω∕(2𝜋) = 2.1 kHz, K = 7.55 μm−1 and 𝜓d ≈ 0.011 rad.

133

134

6 Stabilized Holographic Recording

M

er

s

La

BS HV

Vf

IS

D

IR0 C PM

VD (t)

IS0

+

IR

M

Vd sin Ωt VC

OSC Ω

BP Ω Vd sin Ωt

×

INT

VX

VY

A

PS

LA2Ω

𝜃

BP 2Ω

V1(t)

+

V2(t)

V+(t)

Figure 6.10 Schema of the self-stabilized setup in Fig. 6.8 modified to operate with arbitrarily selected 𝜑: PM is a generic phase ⨂ modulation that could also be the PZT, BPΩ and BP 2Ω are bandpass filters tuned to Ω and 2Ω, respectively, is a function multiplier, PS is a phase shifter, LA 2Ω is a dual-phase lock-in amplifier tuned to 2Ω with orthogonally shifted outputs X and Y, with all other components as already described in Fig. 6.8.

However, holographic recording in photorefractive crystals produces, in general, amplitude and phase coupling of the incident interfering beams [65]. Phase coupling occurs whenever the hologram phase shift 𝜙P does not verify the condition 𝜙P = ±𝜋∕2, as is the case for photovoltaic crystals (like LiNbO3 where 𝜙P = 𝜋 [111]) or when an external electric field is applied on a nonphotovoltaic crystal, in which case bending of the hologram phase planes (hologram bending) results in a consequent reduction in diffraction efficiency. In order to avoid hologram bending, Freschi et al. [112, 113] proposed a modification in the electronics of the feedback stabilization loop that allows 𝜑 to be actively fixed to any value at will and in this way to enable operating under the desirable 𝜙P = ±𝜋∕2 condition. The modified setup is schematically represented in Fig. 6.10 and this new procedure was theoretically analyzed by Sturman and co-workers [114]. It is worth calling readers’ attention to the fact that any stabilization procedure (such as this one) producing a running hologram has the additional advantage of reducing random scattered light recording, as experimentally demonstrated by Garcia et al. [115]. Let us recall the expressions of the first and second harmonic terms, in the output voltage 𝑣D (t) from the photodetector D, as derived from Eqs. (4.172) and (4.173) and Eqs. (6.2) and (6.3): √ √ V0Ω ≡ KdΩ 4J1 (𝜓d ) 𝜂(1 − 𝜂) IS0 IR0 (6.32) 𝑣Ω (t) = V0Ω sin 𝜑 sin(Ωt + 𝜖1 ) 𝑣2Ω (t) = V02Ω cos 𝜑 cos(2Ωt + 𝜖2 )

V02Ω ≡ Kd2Ω 4J2 (𝜓d )

√ √ 𝜂(1 − 𝜂) IS0 IR0

(6.33)

where KdΩ and Kd2Ω are the irradiance-to-voltage conversion at the photodetector D for frequencies Ω and 2Ω, respectively, and 𝜖1 and 𝜖2 are the phase shifts due to the electronics. The signal 𝑣D (t) is, on one side, filtered to extract the 2Ω component (bandpass filter BP 2Ω), which is amplified (by the amplifier A) and the result is the signal 𝑣2 (t) = V 2Ω a cos(2Ωt + 𝜖) cos 𝜑

(6.34)

6.3 Self-Stabilized Recording in Actual Materials

The same signal 𝑣D (t) is also filtered to extract the Ω component (bandpass filter BP Ω), which is phase-shifted with PS and then multiplied with the signal from the oscillator 𝑣Ω (t) = 𝑣d sin Ωt

(6.35)

A second harmonic term 𝑣1 (t) arises from this operation 𝑣1 (t) = V Ω2 sin(2Ωt + 𝜖) sin 𝜑 + dc

(6.36)

where the low frequency dc terms can be neglected because they will be filtered out in the next step by the lock-in amplifier LA-2Ω tuned to 2Ω. The phase shift 𝜖 in 𝑣1 (t) and 𝑣2 (t) is the same because it is possible to adjust it by means of the PS on the 𝑣1 (t) signal from before. The signals 𝑣1 (t) and 𝑣2 (t) are also adjusted to have the same amplitude (by means of the amplifier A) and then added to get the signal 𝑣+ (t) = V0 sin(2Ωt + 𝜖) sin 𝜑 + V0 cos(2Ωt + 𝜖) cos 𝜑 = V0 cos(2Ωt + 𝜖 − 𝜑)

(6.37)

This signal is fed to the double phase lock-in amplifier LA-2Ω, which allows one to measure the two components VX = V0 sin(𝜑 − 𝜑S )

(6.38)

VY = V0 cos(𝜑 − 𝜑S )

(6.39)

with 𝜑S ≡ 𝜖 + 𝜃

(6.40)

where 𝜃 is the phase shift selected for the reference signal in the lock-in amplifier. The advantage of this new signal processing is the fact that we are able to include the phase shift 𝜑 between the transmitted and diffracted beams behind the sample into the temporal argument of the second harmonic term so are now able to operate with the phase 𝜑 − 𝜑S instead of simply 𝜑, where 𝜑S is adjusted at will by acting on the reference phase shift 𝜃 in the lock-in amplifier. Now it is possible to select the signal VX as error signal, for example, in which case the system will automatically set the argument of the sin to zero so that the phase shift will be set to 𝜑 = 𝜑S = 𝜖 + 𝜃

(6.41)

This abitrary phase-shift stabilized setup is interesting not only for photorefractives but also for classical optical interferometry in general. For the particular case of photorefractives, as we shall see in the following sections, the output phase shift 𝜑 is not constant throughout the recording process except for the case of 𝜋∕2-shifted holographic phase shift (𝜙P = ±𝜋∕2). It is therefore not possible, in general, to adjust the operating 𝜑eq in the setup to the unconstrained 𝜑 in order to avoid phase mismatching and keep the hologram self-stabilized without moving. It is, however, possible to measure 𝜑 during the recording process, as already proposed elsewhere [116], and continuously feed this information to the feedback stabilization system to keep the system stationary self-stabilized.

6.3 Self-Stabilized Recording in Actual Materials We shall apply the self-stabilization referred to in Section 6.2.3 to holographic recording on two widely differing materials: Bi12 TiO20 and LiNbO3 :Fe. In both cases, we shall see

135

136

6 Stabilized Holographic Recording

that self-stabilized recording not only reduced external perturbations on the setup but also sometimes modifies the recording process itself. In the case of LiNbO3 :Fe, for example, this procedure allows one to produce a 100% diffraction efficiency volume grating for wide different conditions for the sample and for any recording beams ratio 𝛽 2 , which is not always possible in nonself-stabilized regime. In order to understand this feature, one should figure out that the feedback-driven pattern of fringes movement is automatically adjusted to produce the required hologram phase shift to achieve 𝜂 = 1, whatever the material and experimental conditions. The present conclusions may certainly be extended to other photorefractive materials besides lithium niobate, provided their holographic phase shift and coupling effects are adequately considered. Self-stabilized recording in sillenites will focus on the ability of the setup to cope with environmental perturbations, whereas the section on LiNbO3 aims to illustrate the way selfstabilization may interfere in the recording process itself, regardless of the always underlying ability to reduce external perturbations, which is extremely important for long-term duration recording materials as this one certainly is. 6.3.1

Self-Stabilized Recording in Sillenites

The presently analyzed self-stabilization holographic recording may be used to record holograms in undoped BTO crystals using the setup schematically shown in Fig. 6.5 with the crystal in the transverse optical configuration as shown in Fig. 6.11. For a 90∘ -shifted photorefractive hologram (which is the present case for a nonphotovoltaic crystal without an externally applied field), we deduce from Eq. (4.91) that it should √ be 𝛾 = 0 and from Eq. (4.164) we get I Ω = 0 whereas from Eq. (4.165) we know that I 2Ω ∝ 𝜂. This means that the I Ω signal may be used as an “error-signal” in our negative feedback stabilization loop. In this case, the feedback loop actively keeps the hologram and the pattern of light in the stable 90∘ -shifted position, and the recording proceeds in a self-stabilized mode. Because the freeand feedback-constrained conditions are the same (𝜑0 = 𝜑eq = 0 as deduced from Eq. (4.128)), a stationary (nonmoving) hologram is recorded. We have already demonstrated (see Section 6.2.1.1) that such a system is in stable equilibrium so that, each time a perturbation shifts the system away from 90∘ , a correction signal acting on the piezo-mirror drives the system back to the I Ω = 0 stable position. A lock-in amplifier tuned to 2Ω may be √ used to follow the evolution of the diffraction efficiency profiting from the relation I 2Ω ∝ 𝜂. The good performance of self-stabilized holographic recording on BTO and the usefulness of using the VS2Ω ∝ IS2Ω harmonic to follow the recording are evident from the results reported in Figs. 6.12 and 6.13. 6.3.2

Self-Stabilized Recording in LiNbO3

Self-stabilized recording in a strongly photovoltaic material such as LiNbO3 is also possible. In this case, however, it is 𝜙P ≈ 180∘ (that is, Γ ≈ 0) instead of ±𝜋∕2. In the absence of self-diffraction, Eq. (4.178) shows that 𝜙P ≈ 𝜋 leads to 𝜑 ≈ ±𝜋∕2 and to IS2Ω ≈ 0 in Eq. (4.173). Figure 6.11 Transverse optical configuration for holographic recording on BTO: the incident beams, incidence plane and pattern-of-fringes onto the input crystal face are shown, with the holographic vector K⃗ being perpendicular to the [001]-axis and parallel to the [110]-axis.

[001] K

[110]

Signal (Volts)

Figure 6.12 Self-stabilized recording in a Bi12 TiO20 crystal: The upper figure shows the evolution of the VSΩ (thin black line) and the VS2Ω (thick gray line) when the stabilization is off. The lower figure shows the evolution of both signals when VSΩ is used as the error √ signal, in which case VS2Ω ∝ 𝜂. The recording was o with 𝜆 = 633 nm with IR = 0.52 mW/cm2 and ISo = 11 μm/cm2 , interfering with an angle 2𝜃 = 60∘ on a 10-mm-thick crystal with the pattern-of-fringes on the (110) plane and the hologram vector K⃗ perpendicular to the [001]-axis and parallel to [110].

Signal (Volts)

6.3 Self-Stabilized Recording in Actual Materials

1.2 0.8 0.4 0 –0.4 1.2 0.8 0.4

0

1

0

1

2

3

4

5

2 3 4 Recording time (min)

5

0 –0.4

Figure 6.13 Second harmonic evolution during holographic recording in a nominally undoped photorefractive BTO crystal with the self-stabilization off (left) and on (right), for IR0 + IS0 = 12 mmW/cm2 , using the 𝜆 = 514.5 nm laser line and K ≈ 4.5 μm−1 .

Therefore, in this case IS2Ω should be used as the error signal in the feedback stabilization loop, as shown in Fig. 6.14, instead of ISΩ as was the case for sillenites. 6.3.2.1

Holographic Recording without Constraints

Holographic recording under an externally applied electric field or on a photovoltaic crystal produces phase coupling (see Section 4.2.1) with tilted holograms [117] resulting in expressions for IS2Ω and ISΩ (see Section 4.3.1.1.2) that show they are not suitable as feedback error signals. In this case, self-stabilization is obviously not possible. This is true, in general, for photovoltaic lithium niobate crystals [118]. In reduced LiNbO3 :Fe crystals, phase coupling can be avoided using recording beams of equal irradiances, as explained in Section 4.2.1, but for oxidized samples it is not possible to avoid phase coupling. It is nevertheless always possible to stabilize on an external reference, with the hologram being freely recorded without constraints using a stabilized pattern of fringes. Holographic recording using a closely placed glassplate is described in Section 6.3.2.2.2. Tilted holograms are automatically out of Bragg (always refering to the direction of any of the recording beams). Tilting arising from phase-coupling prevents achieving 100% diffraction efficiency [97] unless the crystal is rotated, after recording, to adjust to Bragg condition [119].

137

138

6 Stabilized Holographic Recording

M

r se La

IS

C

BS I0S

Ω OSC

D

IR0

PZT Vd

+

Vf

IR

HV

VC

LA 2Ω

INT

LA Ω VSΩ

Figure 6.14 Experimental setup: BS beamsplitter, C: LiNbO3 :Fe crystal, M mirror, PZT pzt-driven mirror, OSC signal generator, HV high voltage source, INT integrator, D1,2 detectors, LA-Ω and LA-2Ω lock-in amplifiers tuned to Ω and 2Ω, respectively.

Space-Charge Electric Field Build-up Because the self-stabilized setup produces a running hologram, in general, here we shall refer to a moving (with speed 𝑣 along the grating ⃗ space-charge electric field grating that is, neglecting self-diffraction effects, ruled by vector K) a differential equation [120–122] 𝜕E (t) (6.42) 𝜏sc sc + Esc (t) = −mEeff e−iK𝑣t 𝜕t Ephv + iED Ephv ≈ (6.43) with Eeff = + N N+ 1 + K 2 ls2 − iKlphv ND 1 − iKlphv ND

6.3.2.1.1

D

and: 1∕𝜏sc = 𝜔R + i𝜔I

D

(6.44)

2 2 ND − ND+ 1 1 + K ls 1 ≈ ∝ 𝜏M 1 + K 2 L2D 𝜏M ND+ + Klphv Klphv ND+ ND 1 𝜔I = − ≈ − 𝜏M (1 + K 2 L2D )2 ND 𝜏M ND

𝜔R =

(6.45) (6.46)

with the definition in Eq. (3.167) Ephv ND Klphv ≡ ∝ Eq ND − ND+ which includes the effect of a moving grating [76, 85, 94]. The differential equation here is essentially the same as in Eq. (3.168) with an additional moving-pattern term and excluding the external field E0 . ND+ and ND are the concentration of the empty (electron-acceptors Fe3+ ) and the total (acceptors Fe3+ plus donors Fe2+ ) photoactive centers in the sample, respectively. The approximate relations in Eqs. (6.43), (6.45) and (6.46) derive from the usual assumptions for LiNbO3 :Fe: (1) far from photoactive centers saturation, that is to say, K 2 ls2 ≪ 1, (2) diffusion length short compared to the grating period as stated by K 2 L2D ≪ 1 and (3) photovoltaic field much larger than the diffusion field, say, Ephv ≫ ED . The proportionality relation in Eq. (6.45) is derived from Eqs. (3.48). The solution of Eq. (6.42) for recording is (6.47) E (t) = −mEst e−iK𝑣t + mEst e−(𝜔R + i𝜔I )t sc

st Esc ≡

sc

Eeff (𝜔R + i𝜔I ) 𝜔R + i(𝜔I − K𝑣)

sc

(6.48)

6.3 Self-Stabilized Recording in Actual Materials

where the first right-hand term represents the stationary space-charge wave moving synchronously along with the pattern-of-fringes, whereas the second term represents the transient effect fading away with a time constant 1∕𝜔R . 6.3.2.1.2

Hologram Phase Shift The hologram phase shift is always computed from

tan 𝜙 = Γ∕𝛾 with 2𝜋n3 reff (6.49) ℑ{Esc (t)∕m} = ℑ{4𝜅∕ cos 𝜃} 𝜆 cos 𝜃 2𝜋n3 reff (6.50) 𝛾= ℜ{Esc (t)∕m} = ℜ{4𝜅∕ cos 𝜃} 𝜆 cos 𝜃 as described in Eqs. (4.85) and (4.90). In the present case, for a quasi-stationary (a slowly st was substituted by the slowly varying time time-dependent) steady-state recorded grating, Esc function Esc (t)∕m for the calculation of Γ and 𝛾. ℜ{Esc (t)∕m} and ℑ{Esc (t)∕m} represent the real and imaginary parts of Esc (t)∕m, respectively, and 𝜃 the incidence angle inside the crystal. In the absence of self-stabilization, the pattern of fringes onto the sample is a stationary one and therefore only a stationary (𝑣 = 0) hologram arises. In this condition and for the case of reduced LiNbO3 :Fe crystals, it is possible to assume that Klphv ND+ ∕ND ≪ 1 in which case 𝜔I ≪ 𝜔R that substituted into Eq. (6.47) leads to Γ=

Esc (t)∕m ≈ Ephv (1 − e−t∕𝜏M )

(6.51)

In this case, (see Eqs. (6.49) and (6.50)) Γ ≈ 0 and therefore it is 𝜙 ≈ 0, 𝜋 that characterizes a local hologram. For the case of oxidized samples instead, and still for stationary (𝑣 = 0) holograms, it is Esc (t)∕m ≈ Ephv

1 + iKlph (ND+ ∕ND )

2 1 + K 2 lph (ND+ ∕ND )2 ( ) 𝚤Klph (ND+ ∕ND )t∕𝜏M −t∕𝜏 M × 1−e e

(6.52)

which, substituted into the expression for the hologram phase shift in Eq. (4.91) with Eqs. (6.49) and (6.50), leads, in general, to 𝜙 ≠ 0, 𝜋. In fact, the holographic phase 𝜙 here depends (among other parameters) on the degree of oxidation ND+ ∕ND and may therefore considerably differ from 0 and 𝜋. Diffraction Efficiency with Wave Mixing In the presence of self-diffraction (wave-mixing) there is, in general, amplitude and phase coupling between the interfering beams, in which case the expression for the diffraction efficiency (measured along the direction of any one of the recording beams) of a quasi-steady state (that is to say, slowly time-dependent) recorded grating may be assumed to be formulated as for the steady-state case in Eq. (4.123)

6.3.2.1.3

𝜂(d) = 2

𝛽 2 cosh Γd∕2 − cos 𝛾d∕2 1 + 𝛽 2 𝛽 2 e−Γd∕2 + eΓd∕2

(6.53)

with Γ and 𝛾 as defined in Eqs. (6.49) and (6.50), with 𝛽 2 = IR0 ∕IS0 , IR0 and IS0 being the irradiances of the incident beams. The computed value of 𝜂 is shown in Fig. 6.15 as a function of 2𝜅d for samples with a different degree of oxidation: a reduced sample (leading to a grating with 𝜙 = 𝜋) and two others with an

139

6 Stabilized Holographic Recording

1 0.8

η

0.6 0.4 0.2 0

0

2

4

6

8

10

12

0

2

4

6

8

10

12

0

2

4

6 2𝜅d

8

10

1 0.8

η

0.6 0.4 0.2 0

1 0.8 0.6

η

140

0.4 0.2 0

12

Figure 6.15 Computed 𝜂 as a function of 2𝜅d from Eq. (6.53) for nonstabilized recording in LiNbO3 :Fe with a different degree of oxidation: a reduced sample with 𝜙 = 𝜋 (top), an oxidized sample with 𝜙 = 2.8 rad (middle) and a still more oxidized sample with 𝜙 = 2.5 rad (bottom). The figures were computed for 𝛽 2 = 1 (thick curve), 2 (thin curve) and 10 (dashed curve). Reproduced from [123].

increasing degree of oxidation, each one for different values of 𝛽 2 . Here the coupling constant 𝜅 in Eq. (4.86) is √ 𝜅 ≡ Γ2 + 𝛾 2 ∕4 (6.54) as deduced from Eqs. (6.49) and (6.50) and is used here to represent the size of the index-ofrefraction modulation in a sample of thickness d. Figure 6.15 shows that, for 𝜙 = 𝜋 (which is the case of a reduced sample), 𝜂 is oscillating and reaches 𝜂 = 1 only for 𝛽 2 = 1. This result can

6.3 Self-Stabilized Recording in Actual Materials

Figure 6.16 Computed 𝜂 as a function of 2𝜅d and 𝜙, for 𝛽 2 = 1. Reproduced from [123].

30

2κd

20

10 0 1 0.6 0.3 3

η

0

2 1

ϕ

0

Figure 6.17 Computed 𝜂 as a function of 2𝜅d and 𝜙, for 𝛽 2 = 10. The plane 𝜂 = 0.98 superimposed in the lower picture is a guide for the eyes only. Reproduced from [123].

30

20

2κd 10 0 1 0.6 0.3 3

η

0

2 1 0

30

ϕ

20

2κd 10 0 1 0.6 0.3 3

η

0

2 1

ϕ

0

be straightforwardly obtained by substituting Γ = 0 into the expression for 𝜂 previously, which leads to 𝜂(d) =

4𝛽 2 sin2 (𝜅d) (1 + 𝛽 2 )2

(6.55)

which is the general result [65, 70] for a tilted (out-of-Bragg) uniform stationary grating where, for 𝛽 2 ≠ 1, it is always 𝜂 < 1. For the special case of 𝛽 2 = 1 an in-Bragg grating results instead and 𝜂 = 1 may be achieved. A more general view is shown in Figs. 6.16 and 6.17, where 𝜂 is plotted as a function of 2𝜅d and 𝜙 for 𝛽 2 = 1 and for 𝛽 2 = 10, respectively.

141

142

6 Stabilized Holographic Recording

These figures confirm that it is possible to reach 𝜂 = 1 at 𝜙 = 0 or 𝜋, only for 𝛽 2 = 1, and also at some discrete values of 𝜙 for 𝛽 2 = 10. From these figures, one may induce that, for any 𝛽 2 , it is always possible to achieve 𝜂 = 1 but for discrete values of 𝜙 only. 6.3.2.2 Self-Stabilized Recording

In a self-stabilized regime, 𝜂 = 1 may be achieved for any 𝛽 2 and for any value of 𝜙. This is possible because in this case 𝜙 is automatically adjusted to the required value, by the feedback operation, as will be shown next. Self-stabilized holographic recording arises from the action of a negative feedback optoelectronic loop that forces the phase between the transmitted and the diffracted beams behind the sample to be fixed at a particular value. The latter value is, in general, different from the open loop (nonself-stabilized regime) value so that the hologram is erased to be written at a different position to comply with the feedback constraint. The result is the establishment of a continuously moving hologram, the speed of which depends on the material and the feedback conditions. The phase modulating setup produces the harmonic terms already reported in Eqs. (4.172) and (4.173) √ √ ISΩ = −4J1 (𝜓d ) 𝜂(1 − 𝜂) ISo IRo sin 𝜑 √ √ IS2Ω = 4J2 (𝜓d ) 𝜂(1 − 𝜂) ISo IRo cos 𝜑 with the formulation for 𝜑 in Eq. (4.128) 𝛾 sin z 2 tan 𝜑 = − 1 − 𝛽2 ( 𝛾 ) Γ Γ cosh z − cos z + sinh z 1 + 𝛽2 2 2 2 Substituting the expression for 𝜂 in Eq. (4.123) and for 𝜑 in Eq. (4.128) into the previous expressions for the first and second harmonics in Ω, and rearranging terms, we get the formulations already reported in Eqs. (4.164) and (4.165) IΩ = 4J1 (𝜓d )IS (0)

𝛽2

I2Ω = −4J2 (𝜓d )IS (0)

𝛽 2 sin 𝛾d∕2 e−Γd∕2 + eΓd∕2

𝛽 2 𝛽 2 e−Γd∕2 − eΓd∕2 + (1 − 𝛽 2 ) cos 𝛾d∕2 1 + 𝛽2 𝛽 2 e−Γd∕2 + eΓd∕2

The signal IS2Ω (instead of ISΩ as was the case for sillenites) is selected out from the overall irradiance IS behind the crystal, amplified using a phase-selective 2Ω-tuned lock-in amplifier and used as error signal in the feedback as represented in Fig. 6.14. For the particular case of a stationary grating (that is, the case of 𝑣 = 0, without feedback) in a reduced sample, it is 𝜔I ≈ Klph ND+ ∕ND ≈ 0. In this case, Eq. (6.52) shows that Esc (t)∕m is a real quantity so that it is 𝜙 = 0, 𝜋 and therefore Γ ≈ 0. In this case, IS2Ω is plotted in Fig. 6.18 (with Γ = 0) as a function of 2𝜅d for 𝛽 2 = 1, 2 and 10 where we see that, only for 𝛽 2 = 1, it is always IS2Ω = 0 and consequently, from the expression for IS2Ω before, we see that it is 𝜑 = ±𝜋∕2. This means that for a reduced crystal and 𝛽 2 = 1 one can use IS2Ω as error signal to operate the active stabilization setup. Additionally, because in this case the open loop and closed loop values of 𝜑 are the same (𝜑 = ±𝜋∕2), a stationary (non moving) self-stabilized hologram results, but only for 𝛽 2 = 1. For oxidized crystals and/or for any crystal with 𝛽 2 ≠ 1, it is 𝜑 ≠ ±𝜋∕2, in general, and in these cases it is I 2Ω ≠ 0. However, even in this case it is also possible to use IS2Ω as an error

6.3 Self-Stabilized Recording in Actual Materials

1.5

IS2Ω (au)

Figure 6.18 Computed IS2Ω (in arbitrary units), with Γ = 0 (that is, 𝜙 = 0,𝜋) as a function of 2𝜅d for 𝛽 2 = 1 (dashed curve), 2 (thin curve) and 10 (thick curve). Reproduced from [123].

1

0.5

0

0

1

2 3 2𝜅d (rad)

4

5

signal to operate the self-stabilized setup. To understand this possibility, we shall realize that the condition IS2Ω = 0 in its expression previously means that 𝜑 = ±𝜋∕2. Substituting the latter value into the expression for tan 𝜑 previously, we get the feedback-constrained relation 1 − 𝛽2 (cosh Γd∕2 − cos 𝛾d∕2) + sinh Γd∕2 = 0 1 + 𝛽2

(6.56)

that substituted on its turn into Eq. (4.123) leads to 𝜂(d) =

𝛽2 eΓd − 1 for 𝛽 2 ≠ 1 − 1 𝛽 2 + eΓd

(6.57)

𝛽2

For 𝛽 2 = 1 instead, the 𝜑 = ±𝜋∕2 condition in the expression for tan 𝜑 leads to Γd∕2 = 0 and the corresponding 𝜂 in Eq. (6.55) becomes 𝜂(d) = sin2 𝛾d∕4

for

𝛽2 = 1

(6.58)

From Eqs. (6.57) and (6.58) it is clear that under self-stabilized conditions it is always possible to achieve 𝜂 = 1 when eΓd∕2 = 𝛽 2 for 𝛽 2 ≠ 1 and when 𝛾d∕4 = 𝜋∕2 for 𝛽 2 = 1. In order to illustrate this important result, we simulate the evolution of 𝜂, 𝜙 and ISΩ during self-stabilized recording for 𝛽 2 = 1.1 ≈ 1 in Fig. 6.19 and for 𝛽 2 = 10 in Fig. 6.20. Both figures show a result that can be straightforwardly deduced from Eq. (4.172): as far as 𝜑 is actively fixed to 𝜋∕2, ISΩ = 0 means that 𝜂 = 0 or 1 and its maximum (in absolute value) occurs for 𝜂 = 0.5. Another important feature is shown in Fig. 6.19: for 𝛽 2 ≈ 1 it is 𝜙 ≈ 𝜋 (in self-stabilized regime), which is also the (open-loop, that is to say, without feedback) 𝜙 (rad) 3.2

1.0 0

𝜙

3.0 2.8 2.6 2.4

0.8

𝜂

–0.2

0.6

η

Figure 6.19 Computed evolution of 𝜙 (⚬), ISΩ (◽) in arbitrary units and 𝜂 (∇) as functions of 2𝜅d for self-stabilized conditions (IS2Ω = 0) and 𝛽 2 = 1.1. Note that 𝜙 ≈ 𝜋 throughout. Reproduced from [123].

IΩS

0.4

–0.4 –0.6

0.2 0

1

2 2𝜅d

3

0

143

6 Stabilized Holographic Recording

Figure 6.20 Computed evolution of 𝜙 (⚬), ISΩ (◽) in arbitrary units, and 𝜂 (∇) as functions of 2𝜅d for self-stabilized conditions (IS2Ω = 0) and 𝛽 2 = 10. Note that 𝜙 rapidly shifts away from 𝜋 during recording. Reproduced from [123].

𝜙 (rad) 0.5

3.2

1.0

𝜙

0

3.0

–0.5 2.8 2.6 2.4

𝜂

0.8 0.6

I ΩS

η

144

–1.0

0.4

–1.5

0.2

–2.0

0

1

2 2𝜅d

3

4

0

Table 6.1 LiNbO3 :Fe samples. Sample

Thickness

[Fe2+ ]/[Fe3+ ]

(mm)

[Fe]

[H+]

1019 /cm3

1018 /cm3

–

LNB5

0.85

0.03

2

LNB4

0.35

0.013

20

6.4

LNB3

1.39

0.013

2

0.32

LNB2

0.96

0.0037

2

22

LNB1

1.5

0.0021

2

0.34

equilibrium value for a stationary grating in a reduced sample. Because of this fact the recording pattern of fringes remains stationary (at a fixed position in space) during recording on a reduced crystal for 𝛽 2 = 1. For 𝛽 2 ≠ 1 instead, 𝜙 is neither 𝜋 nor even constant throughout the recording process, as shown in Fig. 6.20. In this case, for 𝛽 2 ≠ 1 and/or oxidized crystals, 𝜙 is certainly not equal to its open-loop equilibrium value and because of such a phase mismatch, a running hologram is established as already reported before. To illustrate the theoretical development from before, some holograms were recorded in LiNbO3 :Fe crystals with different degree of oxidation as described in Table 6.1. All crystals were short-circuited using conductive silver glue and illuminated on all their volume. The recording was always carried out with K ≈ 10 μm using the extraordinarily polarized 514.5 nm wavelength line of an argon laser and self-stabilization was operated as already described in previous sections but now using IS2Ω as the error signal. The diffraction efficiency here is defined as 𝜂 = I d ∕(I d + I t ) where I d and I t are the diffracted and the transmitted irradiances, respectively, and are always measured using the in-Bragg recording beam as described in Appendix B. Such a definition allows one to get rid of bulk absorption and interface losses. It is also possible to compute 𝜂 from the ISΩ -signal, during self-stabilized recording because, as already mentioned before, in this condition ISΩ (corrected from scattering) is maximum at 𝜂 = 0.5 and is ISΩ = 0 at 𝜂 = 1. Any (or both) of these conditions are used to adjust Eq. (4.172) and from the latter any intermediate value for 𝜂 may be computed. Figure 6.21 reports the results using the less oxidized sample LNB5. In this case, 𝜂 ≈ 1 was directly measured, from the diffraction of the recording beam and from the evolution of ISΩ , at the end of the recording cycle. The difference in behavior while using 𝛽 2 > 1 or 𝛽 2 < 1 arises from the nonsymmetric dependence of Eq. (6.57) on the sense of

6.3 Self-Stabilized Recording in Actual Materials

8 IsΩ (au)

6

4

2 Is2Ω

0 0

50

100

150

200

Time (s)

Figure 6.21 Self-stabilized recording in the less-oxidized crystal (sample LNB5) with 𝛽 2 ≈ 1 (IR0 = 141.1 W∕m2 and IS0 = 116 W∕m2 ). The evolution of ISΩ during the self-stabilized holographic recording experiment and the error signal I2Ω are shown both in arbitrary units. At the end of the cycle, 𝜂 = 1 was measured. Reproduced from [123].

Figure 6.22 Self-stabilized recording in an oxidized crystal (sample LNB1) with 𝛽 2 ≈ 1 (IR0 = 113.5 W∕m2 and IS0 = 108.1 W∕m2 ) showing the evolution of the ISΩ (in arbitrary units). The 𝜂 = 1 value by the time ISΩ reached zero was qualitatively verified. Reproduced from [123].

10

IΩ s 5

0

Is2 Ω 0

4000

2000

6000

Time (s)

Figure 6.23 Self-stabilized recording in an oxidized crystal (sample LNB1) with 𝛽 2 = 12 (IR0 = 243.2 and IS0 = 20.3 W∕m2 ) showing the evolution of the ISΩ (in arbitrary units). The 𝜂 = 1 value by the time ISΩ reached zero was qualitatively verified. Reproduced from [123].

4

Ω

IS

2

0

0

5000

10 000 Time (s)

15 000

20 000

145

6 Stabilized Holographic Recording

energy transfer: in fact, 𝜂 = 1 is only possible for 𝛽 2 > 1 if Γd > 0 and for 𝛽 2 < 1 if Γd < 0 where the former was the actual case in the experiment described here. Self-stabilized recordings on the most oxidized sample (LNB1), for 𝛽 2 ≈ 1 and for 𝛽 2 ≈ 12, are reported in Figs. 6.22 and 6.23, respectively. The small secondary peak in Fig. 6.23 is probably due to some oscillatory kinetics and was only observed in these conditions. In both cases, 𝜂 = 1 is achieved, in agreement with theory. The recording time needed to achieve 𝜂 = 1 increases as 𝛽 2 shifts away from 1 and also increases with the degree of oxidation as observed from Figs. 6.21–6.23. The movement of the recorded hologram (and corresponding recording pattern-of-fringes), by the effect of self-stabilization, can be measured using the interference pattern IG produced by the reflection and transmission of some of the recording beams on a small glass plate fixed by the side of the crystal as schematically represented in Fig. 6.24. The output beams through the crystal are used in the usual way for operating the self-stabilization loop. The time-evolution of IG and the K𝑣 computed from these data for a typical self-stabilized experiment are shown in Fig. 6.25. The same experiment was carried out on a reduced sample (labelled 758-1) in similar conditions with the speed being K𝑣 < 0.03 rad/min, thus experimentally confirming our conclusions about the different behavior of oxidized and reduced samples. Effect of Light Polarization LiNbO3 is a naturally birefringent crystal with wide different ordinary and extraordinary index-of-refractions and electro-optic coefficients as already discussed in Section 1.5.3. These parameters are involved in the expression of 𝜂 as for example in Eq. (6.58)

6.3.2.2.1

𝜂 = sin2 𝛾d∕4 where 1 3 𝜋n r |E | for ordinarily polarized light 2 o 13 sc 1 𝛾d∕4 = 𝜋n3e r33 |Esc | for extraordinarily polarized light 2

𝛾d∕4 =

(6.59) (6.60)

Figure 6.24 Overall beam IG produced by the interference of the recording beams transmitted and reflected by a thin glassplate G adequately placed close to the photorefractive crystal C being studied.

C

G

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3 0

1000

2000 Time (s)

3000

4000

Kv (rad/min)

IG

IG (au)

146

Figure 6.25 Measurement of the running hologram speed for the sample LNB1, 𝛽 2 ≈ 1, IS0 + IR0 ≈ 17 mW∕cm2 and K = 10 per μm. The oscillating shape curve is the interference of the transmitted plus reflected beams in a glassplate fixed close to the sample. Its decreasing amplitude is due to scattering of light in the sample. The filled circles represent the computed pattern-of-fringe speed, corrected from scattering, and the dashed curve is only a guide for the eyes.

6.3 Self-Stabilized Recording in Actual Materials 2.5

Figure 6.26 Self-stabilized recording on the same LiNbO3 :Fe sample (LNB3) with ordinarily and extraordinarily polarized 𝜆 = 514.5 nm light simultaneously and 𝛽 2 ≈ 1, all other experimental conditions being similar. Reproduced from [124].

Extraordinary

I Ω (au)

2.0 Ordinary

1.5 1.0 0.5 0 0

500

Time (s)

1000

1500

We already know that self-stabilized recording is limited to 𝜂 = 1, that is to say to 𝛾d∕4 = 𝜋∕2. On the other hand, it is (n3e r33 )∕(n3o r13 ) ≈ 3 in the visible spectral range. This means that self-stabilized recording with ordinarily polarized light allows the achievement of roughly a three-fold higher space-charge modulation |Esc | than operating with extraordinarily polarized light. Figure 6.26 reports two experiments carried out with the same LiNbO3 :Fe crystal (sample LNB3) in similar conditions, except that one was with ordinary and the other with extraordinary recording beams, where the latter was roughly four-fold faster than the former. More details about the use of light polarization to improve the index-of-refraction modulation recorded in LiNbO3 has been published elsewhere [124]. The larger time (four-fold) compared to the (three-fold) space-charge ratio is probably due to the exponential relation between both parameters during recording. It is interesting to point out that, while it was very easy to record holograms in this sample, it was almost impossible to record a hologram in sample LNB4, which has the same oxidation degree but a 10-fold larger Fe concentration. Only a weak hologram could be recorded in LNB4 that was erased in a few minutes, even in the dark. Some researchers have already reported [34] this particular behavior of highly Fe-doped crystals: they believe that the short distance between highly concentrated Fe photoactive center traps allows electrons to tunnel among these centers and in this way the electric charge distribution could not be produced or at least could not be kept in place for a sufficiently long time. Glassplate-Stabilized Recording The holographic recording can be also stabilized using the small glassplate of Fig. 6.24 in the setup schematically depicted in Fig. 6.27: the transmitted R-beam and the phase-modulated S-beam reflected from this glassplate, both propagating along R do mutually interfere producing harmonic terms in Ω. In this case, either the first IGΩ or the second harmonic IG2Ω terms in IG can be used as an error signal in a feedback opto-electronic loop to keep the pattern of fringes stabilized in relation to the glassplate. At the same time, we may use the harmonic terms independently measured through the sample (e.g. ISΩ ) in order to obtain information about the evolution of the holographic recording itself. In this case, stabilization does not refer to the hologram itself: the recording pattern of fringes is fixed in space because of the glassplate operated feedback but the recording occurs without constraints (the recording process itself is not affected by stabilization) because of the absence of self-stabilization. Such a recording was carried out on the oxidized sample LNB1 for 𝛽 2 ≈ 1 and the result is reported in Fig. 6.28. Different from self-stabilization, the ISΩ = 0 condition here does not necessarily mean that 𝜂 = 0 or 𝜂 = 1, because in this case 𝜑 is not actively fixed so it is free to vary: it might be ISΩ = 0

6.3.2.2.2

147

6 Stabilized Holographic Recording

just because sin 𝜑 = 0. In fact, it was measured 𝜂 = 0.85 at I Ω = 0 in Fig. 6.28 and, although not shown in this figure, 𝜂 = 1 was never achieved in this experiment. Such a result for oxidized crystals is in agreement with information from Figs. 6.16 and 6.17 where it is obvious that it is not possible to get 𝜂 = 1 for 𝛽 2 = 1 unless 𝜙 = 0 or 𝜋, which is not the case for oxidized samples. To further explain these facts, a mathematical simulation is shown in Fig. 6.29, where we plot the expressions for 𝜂 in Eqs. (4.123), for 𝜑 from Eq. (4.128) and for ISΩ in Eq. (4.172), using tan 𝜙 = 2.8 that corresponds to one of the examples in Fig. 6.15. This simulation does apparently qualitatively explain the main features in Fig. 6.28. For the same sample LNB1 and same experimental conditions but for 𝛽 2 = 12, the glassplate-stabilized experiment didn’t work: the recording was so noisy that stabilization (and recording) in these conditions was impossible. ISΩ

LA-Ω

M 4

1 r5

nm

D1 IS

IR0

se

La

LA-2 Ω C

BS

G

IS0 IS0 Ω OSC

IS2Ω

IG D2

LA-Ω

PZT +

IGΩ HV

INT

Figure 6.27 Recording setup stabilized on a nearby placed glassplate G, all other elements being the same as described in Fig. 6.14. Reproduced from [123]. 1.2 ISΩ 0.9

(a.u)

148

0.6

0.3 IΩG

𝜂 = 85%

0 0

20

40 60 Time (min)

80

100

Figure 6.28 Glassplate-stabilized experimental data for the recording on an oxidized sample (LNB1) with 𝛽 2 ≈ 1 and ISΩ in arbitrary units. The error signal IGΩ through the glassplate is also shown. At the end of the cycle when ISΩ = 0 it was measured 𝜂 = 0.85. Reproduced from [123].

6.3 Self-Stabilized Recording in Actual Materials

1.25 1 0.75 0.5 0.25 0 –0.25 0

2

4

6

8

10

2𝜅d

Figure 6.29 Mathematical simulation of non self-stabilized recording with 𝛽 2 = 1. The thick curve is 𝜂, the thin curve is ISΩ and the dashed is 𝜑, for tan 𝜙 = 2.8 that seems to qualitatively fit data for LNB1 in Fig. 6.28. Reproduced from [123].

Self-stabilized recording in highly diffractive materials exhibiting phase coupling (hologram bending), as is the case for photovoltaic LiNbO3 crystals, has additional advantages besides the main one (reducing environmental perturbations) as already reported elsewhere [115]: • Reduces hologram bending: in fact, the use of I 2Ω as error signal leads to 𝜑 = 𝜋∕2 as reported in Eq. (4.173) and in this case it is always possible to achieve 𝜂 = 1 as reported in Section 6.3.2.2. However, if the hologram is bended (out of Bragg) it is not possible to achieve 𝜂 = 1, 0.20

1.0

0.8

0.15

0.20

1.0

0.8

0.15

𝜂 0.6

0.6

𝜂

0.10

0.10 0.4

0.4 0.05

0.2

PSL

0

0

10

20 Time (min) (a)

30

0

0.05

0

PSL

0

10

20 Time (min) (b)

30

0.2

0

Figure 6.30 Evolution of 𝜂 and scattering PSL during stabilized holographic recording with (figure A) and without (figure B) self-stabilization in LiNbO3 :Fe using 𝜆 = 514.5 nm with IR0 ∕IS0 ≈ 16 and IR0 + IS0 ≈ 4 mW/cm2 . The diffraction efficiency 𝜂 do not consider bulk light absorption and PSL is the scattered light (in %). Reproduced from [115].

149

150

6 Stabilized Holographic Recording

as is the case in Fig. 6.28 and also in the simulation of graphics A in Fig. 6.29. The latter figure shows the evolution of 𝜂 during holographic recording where the setup is stabilized on a glassplate by the side of the sample: this recording is therefore stabilized but nonself-stabilized and 𝜂 starts growing, but after some time it starts decreasing because it progressively comes out of Bragg condition. The graphics B instead show a self-stabilized experiment where 𝜂 steadily grows up to the limit (because of self-stabilization constraints) of 𝜂 = 1. We should conclude that, by keeping 𝜑 = 𝜋∕2, self-stabilization somehow keeps the hologram in-Bragg during recording. • Reduces scattering: light scattering arises from the diffraction of random gratings produced by the interference of light scattered in defects inside the crystal. The movement of the recording pattern of light produced by self-stabilization maximizes the recording of this pattern of light and sensibly reduces the recording of other holograms that are not related to this one, as is the case for those producing scattering. This effect is clearly verified comparing graphics A and B in Fig. 6.30.

151

Part III Materials Characterization

152

Introduction

Properties of practical interest, such as sensitivity, diffraction efficiency, energy transfer (amplitude coupling), phase coupling and so on, depend on material parameters such as the diffusion length LD , the Debye screening length ls , the quantum efficiency Φ for photoelectron excitation for the recording wavelength and other parameters such as light absorption coefficient and optical activity (if any). It becomes, therefore, a matter of paramount importance to get information about these parameters. The research on the presence and the nature of the localized (photoactive) levels in the Band Gap of photorefractive materials is an important subject too because these photoactive levels are the ones where the spatial modulation of charges occurs under the action of light. Such a modulation is the starting point of optical recording in photorefractives. The adequate characterization of these localized states in the Band Gap will certainly allow one to better understand the optical recording process in a particular material. The characterization of materials, including the research on their photoactive centers, is the objective of this third part of the book. We have chosen optical methods, mainly holographic ones, for materials characterization because photorefractives are photosensitive materials and, on the one hand, their interaction with the light is at the basis of the processes we want to study while, on the other hand, holography underlies most of their practical applications. Nonholographic optical methods with emphasis on photoconductivity are dealt with in Chapter 7. Holographic techniques are the subject of Chapters 8 and 9, where particular attention is devoted to phase modulation and self-stabilized recording techniques, which are rarely described in a comprehensive way in the scientific literature. Although these techniques are of general interest, their application will be limited here to a few paradigmatic materials – those about which we can provide reliable first-hand experimental results. Self-stabilization is separately described in Chapter 9 because of the extent and the complex nature of this subject. The objective of this part of the book is to give an overview of the possibilities of Optics in general, and Holography in particular, for characterization of photorefractive materials, but not necessarily restricted to them. We should point out that most of these techniques may be applied to other photosensitive materials as well.

Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

Introduction

The reader should bear in mind that the actual values here reported for some material parameters should be handled with caution because they may vary from one sample to an other, since they usually depend on the fabrication technique and raw chemicals used to produce a particular sample.

153

155

7 General Electrical and Optical Techniques This chapter will report some few useful optical and electrical methods (among the large amount of them) for materials characterization, with the exclusion of Holography, which will be dealt with in the following chapter.

7.1 Electro-Optic Coefficient The electro-optic or Pockels coefficient is one of the most important parameters of photorefractive materials. Although an effective value of the electro-optic coefficient can be obtained from the measurement of diffraction efficiency of the recorded holograms, direct optical nonholographic methods are rather simple to carry out. They are based on the measurement of the ellipticity of a linearly polarized light going through a slab of the material under analysis with an applied transverse electric field on it as published by Henry et al. [127], Bayvel et al. [128], De Oliveira et al. [125] and Papazoglou et al. [129]. Materials exhibiting optical activity (like sillenites) are somewhat more difficult to measure because optical activity may also act on the ellipticity of the light and should therefore be separately evaluated. An additional difficulty arises from the the photoconductive nature of photorefractive crystals in general that leads to a light-induced space-charge field opposing the externally applied field during electro-optic coefficient measurement, thus resulting in an apparently lower value for the coefficient as experimentally reported in [125]. As photoconductivity is strongly dependent on illumination wavelength, it may also induce to an erroneous apparent wavelength dependence of this coefficient too. Any experimental setup for measurement of the electro-optic coefficient should therefore take into account optical activity and should use the lowest possible illumination intensity and/or be fast enough to avoid the building-up of a light-induced electric field that may jeopardize measurements. A practical setup the measurement of the unclumped electro-optic coefficient reff = r41 = r52 = r63 in sillenite crystals published by de Oliveira et al. [125] is schematically described in Fig. 7.1. Almost monochromatic light-emitting diodes (LED) are used as low intensity source of light here, followed by a thin grounded glass plate to improve illumination uniformity and a lens to guide the light through the crystal to be measured up to the detector at the output. A mechanical chopper operating at 2900 Hz is used to modulate the light and allow detection (even under ambience light) using a tuned lock-in amplifier. A fixed polarizer is placed before the crystal under analysis and a low frequency (13 Hz) rotating one is placed behind it in order to measure the ellipticity of light at the crystal output that is displayed using an oscilloscope. The crystal is used in a transverse configuration with the [001] crystal axis upward (out of the plane Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

156

7 General Electrical and Optical Techniques

lens

fixed polarizer

[110]

rotating polarizer detector

(001) LED grounded grass

chopper

crystal

lock-in amplifier

oscilloscope

Figure 7.1 Schema of the experimental setup for electro-optic coefficient measurement in sillenite crystals as described in [125]: almost monochromatic led (LED), grounded glass plate to improve light uniformity, a lens to collect the light through polarizers and the crystal sample and guide it to the output detector (DET) feeding a lock-in amplifier tuned to the chopper frequency and connected to an oscilloscope for displaying and measurement of the elliptically polarized light at the output. From [125].

of the paper) and the dc electric field being applied along the [110]-axis, light going through the sample thickness as indicated in the figure. A wavelength (from an appropriate LED) is selected and a fixed dc electric field, E0 , is applied transverse to the light path. The linearly polarized light shines the sample with an arbitrarily selected angular position 𝜃 for the linear polarization at the input crystal plane. The whole setup is adjusted and, prior to proceeding with measurements, the sample is short-circuited under illumination for a while, for space-charge zeroing. Then, E0 is applied and the signal in the oscilloscope is displayed and saved for measurement of the maximum (IM) and minimum (Im) of the low frequency sinusoidally modulated signal on the oscilloscope, in the shortest possible time to reduce space-charge building up. The parameter IM − Im (7.1) V. = IM + Im is computed. The input fixed polarizer is then rotated by 𝜋∕4 so that the angular position of the input polarization is now changed to 𝜃 + 𝜋∕4 and the whole procedure previously (including sample short-circuiting) is repeated to measure the new ellipticity as in Eq. (7.1). The following equation is then computed: [ ( )2 ]2 𝛿 2 sin 𝜙∕2 2 2 (7.2) V𝜃 + V𝜃+𝜋∕4 = 1 + 1 − 2 𝜙∕2 2𝜋 3 𝛿≡ 𝜙2 = 𝜌 2 + 𝛿 2 (7.3) n reff E0 d 𝜆 with d being the crystal thickness along the light path, n the average refractive index at 𝜆 (from [6] or any other available source) and 𝜌 is twice the rotation angle of the light polarization due to optical activity through the sample. The procedure is repeated for 𝜃 varying from about 10 to roughly scan a semicircle so that an average value for reff is obtained for this wavelength. Some results reproduced from [125] and other sources are displayed in Table 7.1. Table 7.1 shows that reff remains rather constant (less that 4.5% variation) for undoped BTO and for some doped ones like BTO:V, BTO:Ce, at least in the 𝜆 = 510–650 nm range. Also BTO:Nb and BTO:Tb do not show sensible wavelength-dependent electro-optic coefficients. We may conclude then, together with other researchers [129, 130] that sillenite crystals do not

7.2 Light-Induced Absorption

Table 7.1 Effective electro-optic coefficient for doped and undoped BTO. reff (pm/V) wavelength (nm) Crystal

515

520

543.5

580

590

632.8

645

BTO

5.4†

5.4†

5.4

5.3†

5.3†

5.45†

5.6†

BTO:Ce

5.7†

5.8†

–

5.5†

5.7†

5.4†

5.45†

BTO:V

5.55†

5.5†

–

5.6†

5.7†

5.35†

5.45†

BTO:Nb

–

–

5.9‡

–

5.9‡

–

BTO:Tb

–

–

5.5‡

–

5.2‡

–

†: from [125] ‡: from [126]

Table 7.2 Parameters: pure and doped sillenite crystals. 𝝆 (deg/mm)

Sample

reff (pm/V)

𝝐 low. freq.

BTO

5.5 ± 0.2† [125, 129]

47

at 𝝀 (nm) 633

514.5

6.7 ± 0.3

12

6.4 ± 0.3 [25]

–

BSO

4.1†† [127]

–

–

–

Bi12 Ti0.9 Ga0.1 O20

5.5

–

7.5 ± 0.3

–

Bi12 Ti0.7 Ga0.3 O20

5.6

–

9.7 ± 0.3 [25]

–

Bi12 GaO20

4.8 ± 0.1

–

18 ± 0.2

–

BTO:Ce

5.6 ± 0.2† [125]

–

5.9

–

BTO:Pb

4.1−4.2

–

5.5

11.5

BTO:V

5.5 ± 0.1† [125]

–

4.5

–

† 515–645 nm ††: 630–700 nm ‡: reff = r41 = r52 = r63

exhibit sensible wavelength-dependent effects on their electro-optic coefficients, and if ever such a dependence appears, attention should be paid to the jeopardizing effect of light-induced space-charge field due to photoconductivity that is effectively strongly wavelength-dependent [125]. Dopants, instead, may have a somewhat sensible effect on reff , which is not surprising because they may affect crystal structures.

7.2 Light-Induced Absorption This subject was already theoretically developed in Chapter 2, where Eq. (2.57) describes the light-induced absorption coefficient 𝛼li , the limits of which were shown to be lim 𝛼li = 0 I→0

lim 𝛼li = ND2 s2

I→∞

+ (ND1 − ND1 )𝜏1 r2 s1 + (ND1 − ND1 )𝜏1 r2 s1 + s2

157

158

7 General Electrical and Optical Techniques

as described in Eqs. (2.62) and (2.63), respectively. The relation between output I(d) and input I(0) irradiances (always defined inside) the sample of thickness d in the presence of light-induced absorption was solved in Eq. (2.92) as: (𝛼 + a∕b)I(0) + 𝛼0 c∕b a∕b I(d) ln 0 + ln = −𝛼0 d 𝛼0 + a∕b (𝛼0 + a∕b)I(d) + 𝛼0 c∕b I(0) with the limit initial and saturated conditions as reproduced from Eqs. (2.93) and (2.94), respectively: I(d) = I(0) e−𝛼0 d for I(0) ⇒ 0 a −(𝛼0 + )d b for I(0) ⇒ ∞ I(d) = I(0) e We recall that I(0) and I(d) are not directly available but are related to the corresponding quantities I0 and I t that are measured at the sample’s interface but in the air, just outside the sample: cos 𝜃 cos 𝜃 ′ I t cos 𝜃 I(d) ≈ 1 − R cos 𝜃 ′

I(0) ≈ I0 (1 − R)

(7.4) (7.5)

where R is the interface reflectance, 𝜃 and 𝜃 ′ are the incidence angles outside and inside the sample. In order to measure 𝛼0 and 𝛼li , a light beam is projected perpendicularly onto the sample and the transmitted I t irradiance is measured as a function of the incident I0 ; substituting these values into Eqs. (7.4) and (7.5) with the approximations cos 𝜃 ≈ cos 𝜃 ′ ≈ 1 because of the normal incident beam, and from Eqs. (2.93) and (2.94) or the full expression in Eq. (2.92), 𝛼0 and 𝛼li may be computed. The whole light, mainly at the crystal output, should be collected with a lens (Pt and P0 , respectively) and focused on a photodetector behind the crystal to be free from the possible lenslike effect produced by thick, high index-of-refraction samples. Figure 2.31 shows transmittance data (circles) for an undoped BTO crystal sample (labeled BTO-010, 8.1 mm thick) using the 532 nm laser wavelength light. The reflectance R is R≡

(n − 1)2 (n + 1)2

(7.6)

In most photorefractive materials, as in the case of sillenites, the index-of-refraction is rather high, as reported in the graphics of Fig. 1.9, so losses by reflection at the interfaces should be carefully accounted for. The angular coefficients in the graphics of Fig. 2.31 are 0.00142 for the low irradiance limit: Pt ∕P0 = (1 − R)2 e−𝛼0 d = 0.00142 ⇒ 𝛼0 = 754.4 m−1 and for the high irradiance limit: a −(𝛼0 + )d b = 5.7 × 10−4 ⇒ 𝛼0 + a = 867.7 m−1 Pt ∕P0 = (1 − R)2 e b From these data the parameters 𝛼0 and a∕b in Eqs. (2.93) and (2.94) can be computed. Thus we get 𝛼0 = 754.4 m−1 and a∕b = 112.8 m−1 . The mathematical data fitting to Eq. (2.92) along the entire range in Fig. 7.4 allows us to get the parameter c∕b as well. Figure 7.2 shows the time evolution of light-induced absorption (photochromic darkening) on a typical undoped Bi12 TiO20 crystal at a fixed incident irradiance. Steady-state results are

7.2 Light-Induced Absorption

760

Figure 7.2 Evolution of the absorption coefficient in an undoped B12 TiO20 crystal (labeled BTO-010) under uniform illumination of I0 ≈ 2 mW/cm2 at 𝜆 = 532 nm.

α (m–1)

720 680 640 600

0

Figure 7.3 Light-induced absorption: transmitted It versus incident I0 irradiances measured using an uniform beam of 532 nm wavelength on the same sample BTO-010 as Fig. 7.2. The dashed lines are the best fit at the limit I → 0 (with an angular coefficient of 0.00299) and for saturation with an angular coefficient of 8.78 × 10−4 . Reproduced from [39].

25

50 75 Time (min)

Absorption coefficient α (m–1)

1100

1000

1000

900

900

800

800

700

0.1

1 10 I0 (W/m2)

100

700

0

125

90

120

0.08 0.04 0

1100

100

0.12

I t (W/m2)

560

25

0

50 I0 (W/m2)

30

60 I0 (W/m2)

75

100

Figure 7.4 Light-induced absorption of undoped Bi12 TiO20 (sample labeled BTO-013) at 𝜆 = 514.5 nm as a function of the incident irradiance measured in the air. The left-hand side graphics is in semi-log scale for detailed view at low irradiances. The continuous curve on the right-hand side graphics is the best fitting to Eq. (2.90) with the following parameters: 𝛼0 = 789 m−1 , a = 1.4 × 10−6 m/(s2 W), b = 4.91 × 10−9 m2 /(W s2 ) and c = 7.48 × 10−9 s−2 .

159

160

7 General Electrical and Optical Techniques

shown in Fig. 7.3 on the same sample, where the nonlinear relation between the incident and transmitted irradiances is evident. Data fit to Eq. (2.92), with R = 0.2 and 𝜃 ≈ 0 give a c (7.7) = 198 m−1 = 0.75 W∕m2 𝛼0 = 662 m−1 b b These results can be formulated in terms of the two-center model parameters in Eqs. (2.83)– (2.87) with Eq. (2.27) as: + )s1 Φ𝛼0 ≈ (ND1 − ND1

(7.8)

+ (ND1 − ND1 )𝜏1 r2 s1 a = ND2 s2 + b (ND1 − ND1 )𝜏1 r2 s1 + s2 𝛽2 ℏ𝜔 c∕b ≈ + 𝜏1 r2 (ND1 − ND1 )s1

(7.9) (7.10)

where Φ𝛼0 represents the fraction of 𝛼0 associated with excitation of electrons to the Conduction Band (CB). Other samples were measured and the results are displayed in Tables 7.3 and 7.4. Data reported in this table show that all doped and undoped BTO crystals exhibit a large light-induced absorption effect that is characterized (at saturation) by the a∕b ratio. This effect is a rather slow one compared to the recording of a photorefractive grating in these materials. For an irradiance much lower than c∕b there is a negligible light-induced absorption effect, whereas for a much larger irradiance the absorption becomes almost saturated. This limit irradiance is comparatively weak (a few hundreds of μW/cm2 ) for all doped and undoped BTO a fact that allows one to assume a nearly saturated light-induced absorption condition, even for the moderately large irradiances usually employed, provided that the experiment is allowed to last for a sufficiently long time to reach equilibrium. The measurement of absorption may be complicated by the presence of luminescence effects. Luminescence occurs when electrons are excited to the CB and undergo a radiative decay to intermediate states in the Band Gap emitting correspondingly associated photons. The latter Table 7.3 Absorption parameters for pure and doped BTO for 𝜆 = 532 nm. sample

BTO-010

BTO-011

BTO-013

BTO:Ce

BTO:Pb

𝛼0 (m−1 )

662

658

583

430

473

a∕b (m−1 )

198

242

226

138

250

0.75

0.7

0.34

0.7

0.29

2

c∕b (W∕m )

Table 7.4 Saturated absorption for sillenites. 𝜶0 + a∕b (m−1 ) at 𝝀 (nm) Sample

633

532

514.5

BTO

40−90

850

1160

BSO

–

374 [56]

–

BTO:Ce

–

570

–

BTO:Pb

–

720

–

BTO:V

135

–

–

7.3 Dark Conductivity

5 BTO-008 BTO Q BTO-8

4

αd

3 2 1 0

400

λ (nm)

600

800

Figure 7.5 Absorption coefficient-thickness 𝛼d measured for three different BTO samples (BTO-8, BTO-Q and BTO-008) as a function of wavelength. BTO-8 and BTO-Q were measured in a standard spectrophotometer whereas BTO-008 was measured with a photodetector placed about 1 cm behind the crystal.

are less absorbed and they are detected at the sample output as illustrated in Section 2.2. This luminescence-arising radiation may be very misleading for absorption coefficient measurement close to the CB edge where light is strongly absorbed, as reported in Fig. 7.5 where the experimental absorption coefficient for undoped sillenite crystals apparently decreases for wavelengths below 𝜆 ≈ 450 nm. This fact is illustrated in Figs. 2.7 and 2.8, showing the incident light centered at 𝜆 = 408 that is not detected at all at the output because of being completely absorbed and the 570 nm-centered luminescence-arising light effectively emerging at the sample output instead. Figure 7.5 reports the actual measurement of the 𝛼d parameter (d being the sample thickness) on three undoped Bi12 TiO20 crystals labeled BTO-008, BTO-8 and BTO-Q using a spectrophotometer with a nonwavelength selective photodetector. For sample BTO-008, the photodetector was placed at 10 mm behind the sample, whereas for samples BTO-8 and BTO-Q the photodetector was placed comparatively farther away. For the latter two, the 𝛼d curve shows saturation roughly for wavelengths lower than 450 nm. For sample BTO-008, which is very close to the output photodetector instead, 𝛼d is apparently decreasing for 𝜆 ≤ 450 nm. Such a different behavior is easily understood because luminescence is a scattering process so that its irradiance decays as the inverse square distance so that its effect is much easier to detect the closer the detector is to the sample.

7.3 Dark Conductivity Photorefractive materials are semiconductors of large Band Gap (for sillenites, Eg ≈ 3.2 eV) and dark conductivity 𝜎d is an energy-barrier controlled parameter following the relation: Ea k 𝜎d = 𝜎0 e B T −

(7.11)

where 𝜎0 is a constant, kB the Boltzmann constant, Ea the activation energy and T the absolute temperature. In thermal relaxation conditions, Ea is, in general, the energy gap between the Fermi level in the Band Gap and the bottom of the CB (if electrons are involved in dark conductivity) or the top of the Valence Band (if holes are involved instead). By plotting ln(𝜎d )

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7 General Electrical and Optical Techniques

–15

In (σd) (au)

162

Ea = 0.89 eV –20

–25 2.5

2.7

2.9 3.1 1000/T (K–1)

3.3

3.5

Figure 7.6 Arrhenius curve dark conductivity for BTO:V. Data fitting to Eq. (7.11) leads to Ea = 0.89 eV. Reproduced from [30].

as a function of 1∕T, a linear relation is obtained and from its angular coefficient it is possible to compute Ea as illustrated in Fig. 7.6. Dark ac conductivity 𝜎dac , however, may depend on frequency f following the complex relation [30] in: 𝜎dac (f ) = [𝜎d + Af s ] + 𝚤Bf

(7.12)

where 𝜎d is obviously the dc component with B and s being parameters to be adjusted while data fitting. The presence of a nonnegligible frequency-dependent term, if ever verified, in the real part of 𝜎dac (f ) may indicate a hopping mechanism [131, 132] instead of a band transport mechanism involved in the process. To illustrate the procedure, the absolute value |𝜎dac (f )| was measured for a V-doped BTO crystal as a function of frequency f , for different temperatures and reported in Fig. 7.7, where it is clearly shown that its dependence on frequency is less pronounced (thus indicating a reduced influence of hopping mechanism) as the temperature increases. The dc conductivity 𝜎dc obtained by extrapolating |𝜎dac (f )| to f = 0, for different temperatures, gives a typical Arrhenius curve such as the one reproduced in Fig. 7.6. It is already known that in sillenite crystals the Band Gap energy is Eg ≈ 3.2 eV and the dark conductivity is based on holes, thus indicating that the Fermi level is closer to the top of the Valence Band. The lower values for Ea in the 30–120∘ C temperature range compared to the ≈ 1 eV in the higher temperature ranges, at least for undoped BTO, may indicate that in the lower range dark conductivity is mainly controlled by hoping or by tunneling, whereas it is mainly by excitation to Extended (probably Valence Band) States at higher temperatures. The rather low Ea value for Ga-doped BTO in Table 7.5 is also remarkable.

7.4 Photoconductivity Photoconductivity is an important property and is especially relevant as far as photorefractive materials are concerned.

7.4 Photoconductivity

10–5 120°C 110°C 100°C 90°C 80°C

│σadc│(Ω–1 m–1)

10–6

10–7

70°C 60°C 50°C

10–8

40°C 30°C

10–9

10–10

0

10

20

30

40

50

Frequency (Hz)

Figure 7.7 Frequency-dependence of the absolute value |𝜎dac (f )| in Eq. (7.12) for different temperatures. Table 7.5 Dark conductivity 𝜎d measurement. 30–120∘ C

Sample

150–250∘

Higher than 300∘

activation energy Ea (eV) from Eq. (7.11)

BTO

0.83

1.06 [27]

0.99 [24]

BTO:V

–

–

0.89

BTO:Pb

0.80

–

–

BTO:Ga

0.66

–

0.48 [25]

Reproduced from [30]

7.4.1

Photoconductivity in Bulk Material

In bulk samples, in the transverse configuration as represented in Fig. 7.9 the irradiance along the sample thickness z-axis) varies considerably and therefore the photoconductivity also varies. The measured overall photocurrent is therefore a kind of weighted average along the sample thickness that is related to the photoconductivity that we want to calculate. The z-dependence photoconductivity 𝜎ph can be written from Eq. (2.49) as: 𝜎ph (z) = eph (z)𝜇

ph (z) = 𝜏Φ𝛼

I(z) h𝜈

(7.13)

where ph is the density of electrons due to the action of light and is derived from Eq. (2.30). For materials exhibiting light-induced absorption, the Φ𝛼 in Eq. (7.13) should be substituted for the expression in Eq. (2.52). It is interesting to define a so-called average photoconductivity: 𝜎 ≡ lim h𝜈 z→0

𝜎ph (z) I(z)

= e𝜇𝜏

∑ i

(Φ𝛼0 + 𝛼li (0))i

(7.14)

163

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7 General Electrical and Optical Techniques

+HV L SF Laser

BTO CH C R OA LA

Figure 7.8 Schematic setup for the electric measurement of photoconductivity. A laser beam is chopped CH at frequency Ω and the beam is filtered and expanded using a spatial filter SF and collimated using a lens L. The chopped expanded and uniform beam shines the sample that produces a photocurrent under the action of a voltage HV. An operational amplifier OA with a feedback resistance R and capacitor C transforms the current into a voltage that is read using a Ω-tuned lock-in LA (for the case of photoconductivity) or a simple dc voltmeter (for the case of dark conductivity). Reproduced from [39]. d

Figure 7.9 Typical crystal schema, in the so-called “Transverse Configuration”, with the electrodes (H × d area) on the lateral surfaces separated by a distance 𝓁, the thickness (along the light propagation) is d, the height is H and the illuminated surface is H × 𝓁. H

l

where of 𝜎. 7.4.2

∑

indicates that there is a number of i photoactive centers involved in the formulation

Alternating Current Technique

We shall here focus on an ac method that facilitates the detection of small photocurrent signals in a much larger nonphotoconductive current. The method is based on the use of a time-modulated spatially uniform illumination and the detection of the associated current using a phase-sensitive frequency-tuned lock-in amplifier. If we illuminate a photoconductive (not necessarily a photorefractive) material with a spatially uniform sinusoidally (amplitude) modulated light of angular frequency Ω and contrast |m|, I = I0 (1 + |m| cos(Ωt))

(7.15)

the Eq. (2.18) turns into the differential equation 𝜕 (x, t) Φ𝛼 +  ∕𝜏 = I (1 + |m| cos Ωt) 𝜕t h𝜈 0

(7.16)

7.4 Photoconductivity

where the dark conductivity is neglected. Its solution is  = 1 cos(Ωt + 𝜙𝜎 ) + 0 0 ≡

Φ𝛼 I h𝜈 0

(7.17) (7.18)

1 ≡ 0 |m| √

1

(7.19)

1 + Ω2 𝜏 2

tan 𝜙𝜎 ≡ −𝜏Ω

(7.20)

Therefore, a time-modulated photoconductivity of the form 𝜎 = 𝜎ph (1 + √

|m| 1 + Ω2 𝜏 2

cos(Ωt + 𝜙𝜎 ))

(7.21)

results [133], with 𝜏 being the photoelectron effective life-time as defined in Eq. (2.25), provided that we can neglect the response time of the measuring circuit itself. In fact, the response time of the measurement circuit can be neglected because the sample’s resistance (usually very high in most photorefractive materials even under illumination) is not related with the photocurrent generation as deduced from Eq. (7.16). Also, the input resistance of the operational amplifier OP-AMP used to convert the photocurrent into a voltage in Fig. 7.8 is always very low and the associated response time is accordingly very low too. The output OP-AMP resistance instead is usually very large but there are instrumental features able to strongly reduce this output impedance and thus reduce the associated response time. From the development before, it seems that ac-photocurrent measurement is likely to be limited by charge carriers’ lifetime in the extended states rather than by the response time of the instrument itself. It is easy to show that, for a rectangular time-modulated (with fundamental angular frequency Ω) spatially uniform illumination, the dc plus fundamental term of the irradiance has the form I = I0 + I0

2 cos(Ωt) 𝜋

(7.22)

If this chopped light is shining on a photorefractive (or simply photoconductive) crystal, a time-modulated photocurrent does also result where its fundamental harmonic term has the form 2∕𝜋 iph (t) = iph √ cos(Ωt + 𝜙𝜎 ) 1 + Ω2 𝜏 2

(7.23)

which can √ be measured using a lock-in amplifier tuned to Ω as depicted in Fig. 7.8. The term 1∕ 1 + Ω2 𝜏 2 is experimentally determined for the frequency Ω used in the experiment. Figure 7.10 shows a typically measured photocurrent iph versus I(0) for the BTO sample labeled BTO-010. The (○) represent the photocurrent measured for the nonexposed sample whereas the (•) show the data for the sample having just been previously exposed to saturation. The overall measured photocurrent iph corresponds to I0 (or more precisely to I(0), which is its value inside the sample) and is related to the material’s parameters by Eq. (7.28). From these experimental data, the photoconductivity average in Eq. (7.14) can be computed. Results are displayed in Table 7.7. For comparison, the photoconductivity was also measured at 514.5 nm (BTO-011) and at 633 nm (BTO-010) showing a large dependence on the wavelength, probably due to the wavelength dependence of the characteristic effective cross-sections s1 and s2 . Some results for doped crystals are also shown in Table 7.7.

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7 General Electrical and Optical Techniques

3000 40

pA

30

2000

20 10

iph (pA)

166

0

0

0.2

W/m2

0.4

0.6

1000

0

0

5

10 I(0) (W/m2)

15

20

Figure 7.10 Photocurrent (in pA) as a function of the incident irradiance on the input plane inside the crystal I(0), measured using a time-modulated uniform 532 nm wavelength laser beam onto crystal sample BTO-010 with 2000 V applied. The corresponding time modulated photocurrent was measured using a lock-in amplifier where (○) are data for a sample that has been kept in the dark for a long time and (•) are data for the previously light-saturated crystal. The dashed line for the (•) is the best fitting for the final linear range that gives an angular coefficient of 138 pA.m2 /W. The dashed line for the ( ○) in the inset represents the best fitting for the nonexposed sample at the limit I(0) → 0 condition giving an angular coefficient of 69.2 pA.m2 /W. From these data, the values in Table 7.7 were computed for BTO-010. Reproduced from [39].

7.4.3

Wavelength-Resolved Photoconductivity

Figure 7.11 shows a typical setup for wavelength-resolved photoconductivity (WRP) measurement using almost monochromatic LEDs [134]. This setup provides discrete wavelength only, which may introduce some extra uncertainties but provides convenient, rather strong intensities for each one of the available wavelengths. It also allows the sample to be pre-exposed to the strong illumination of a selected wavelength and for measurements to be carried out at another one, by rapidly rotating the LEDs on the disc. Measurements are carried either in the Transverse or in the Longitudinal configurations and both give different and useful information about the photoactive centers in the material Band Gap. In both configurations, it is always necessary to measure the actual photoconductivity as computed from the photocurrent measured for different applied electric fields on the sample and verifying Ohm’s law. 7.4.3.1 Transverse Configuration

In this case, the photocurrent is measured in the transverse direction to the light through the crystal as depicted in Fig. 7.9, and the actual photocurrent at z can be written in differential form as diph (z) = 𝜎ph (z)EHdz

(7.24)

where E is the transversally applied electric field, H is the height as shown in Fig. 7.9 and z is the coordinate along crystal thickness. The expression of 𝜎ph (z) is derived from Eq. (2.49) and Eqs. (2.52)–(2.54) to be 𝜎ph (z) = e𝜇𝜏(Φ𝛼0 + 𝛼li (z))

I(0) −(𝛼 + 𝛼 (z))z 0 li e ℏ𝜔

(7.25)

7.4 Photoconductivity

V L

BS

D

C

LED

D

LA

Figure 7.11 (Left) Photograph of the wavelength-resolved photoconductivity experimental setup using almost monochromatic LEDs ranging from near infrared to near ultraviolet wavelength and placed on the perimeter of a rotating disc driven by a computer-controlled stepping-motor. The light of the LED is collected by a system of lenses producing a uniform almost monochromatic illumination on the sample placed on a small home-made housing with shielded electrodes connected to an electric voltage source and adequately placed photodetectors to enable the measurement of incident and transmitted intensity on and through the sample. (Right) Schema of the setup with L: lens system, D: photodetectors, BS: beamsplitter, C: crystal sample, V: voltage source and LA: lock-in amplifier.

where Φ𝛼0 + 𝛼li (z) and 𝛼0 + 𝛼li (z) stand here to take into account the possibility of light-induced absorption. I(0) is the incident irradiance at the input surface inside the crystal. The overall photocurrent is computed by substituting Eq. (7.25) into Eq. (7.24) and integrating throughout the crystal thickness d: iph = EHe𝜇𝜏

I(0) z=d [Φ𝛼0 + 𝛼li (z)] e−(𝛼0 + 𝛼li (z))z dz ℏ𝜔 ∫z=0

(7.26)

Substituting the actual expression for 𝛼li in Eq. (2.90) into Eq. (7.26) the integration can be carried out either analytically or numerically. Otherwise, assuming the light-induced absorption (if any) to be constant throughout the crystal’s thickness or dealing with a sample thin enough so as to be able to assume: 𝛼li ≡ 𝛼li (0) ≈ 𝛼li (z) the integral in Eq. (7.26) is easily computed: ) Φ𝛼 + 𝛼li I(0) ( 1 − e−(𝛼0 + 𝛼li )d iph = EHe𝜇𝜏 0 𝛼0 + 𝛼li ℏ𝜔

(7.27)

(7.28)

From Eq. (7.28) we should define a phenomenological photoconductivity coefficient: ( ) iph 𝓁 ℏ𝜔 (𝛼0 + 𝛼li )d 𝜎≡ (7.29) Hd V I(0) 1 − e−(𝛼0 + 𝛼li )d where V is the applied voltage and 𝓁 the interelectrode distance. The quantity ) ( iph 𝓁 (7.30) Hd V in Eq. (7.29) represents the average photoconductivity and I(0) 1 − e−(𝛼0 + 𝛼li )d ℏ𝜔 (𝛼0 + 𝛼li )d

(7.31)

167

168

7 General Electrical and Optical Techniques

represents the number of photons absorbed per unit time in the unit crystal volume, so that 𝜎 actually represents an average normalized photoconductivity. Comparing Eqs. (7.28) and (7.29) we go to the useful relation: ∑ (Φ𝛼0 + 𝛼li )i (7.32) 𝜎 = e𝜇𝜏 i

∑ which is the same (with Eq. (7.27)) as in Eq. (7.14), with explicitly indicating that there are i photoactive centers involved, each one of them with their own characteristic parameter Φ𝛼0 + 𝛼li representing the absorbed photons effectively exciting electrons from the corresponding photoactive center to the CB, with Φ representing the quantum efficiency for photoelectron generation. Thus, while measuring iph to compute 𝜎 and plot the latter as a function of h𝜈 of the light shining on the sample, a rather sudden variation ∑ Δ𝜎 = e𝜇𝜏 [(Φ𝛼0 + 𝛼li )i+1 − (Φ𝛼0 + 𝛼li )i ] = Δ(Φ𝛼0 + 𝛼li ) (7.33) i

is expected in 𝜎 as a new photoactive center is included for h𝜈 going through the energetic position of a (even partially) filled photoactive center in the Band Gap, thus allowing one to detect and find out the position of such i-filled photoactive centers in the crystal Band Gap. Undoped Bi12 TiO20 The dependence of the photoconductivity coefficient 𝜎 on the light wavelength may provide important information about the photoactive Localized States (LS) in the Band Gap (BG) as shown in Figs. 7.12 and 7.13 for an undoped Bi12 TiO20 crystal under relaxed conditions in the dark for some hours, pre-exposed to h𝜈 = 2.3 eV light and still in normal intermediate conditions. The pre-exposure is carried out immediately before each one of the individual measurement for each one of the different wavelengths. The measurement was always carried out from the highest (lowest h𝜈) to lowest wavelength (highest h𝜈). Electrons can be excited, by the action of a sufficiently energetic photonic light, from the VB the Conduction Band (CB) or to a localized acceptor state in the BG, thus producing an equivalent number of holes free to move in the VB. Electrons can also be excited from a filled localized donor state in the BG to the CB where they are free to move. In both cases, the (photo)conductivity increases because of the increase of free carriers in the Extended (VB and/or CB) States. Each time the photonic energy is large enough to go through a charged photoactive center, a sudden variation of Φ𝛼0 + 𝛼li (0) in Eq. (7.32) occurs and a corresponding increase in 𝜎 should be detected. The wavelength-resolved photocurrent is therefore expected to show a rather step-like shape, if it is adequately wavelength-resolved. This technique is therefore a powerful tool for the study of localized states in the BG, and data in Figs. 7.12 and 7.13 are clear examples of the possibilities of this technique. We know that photoconductivity in sillenite crystals like BTO is electron-based so that the photonic energy h𝜈 in Figs. 7.12 and 7.13 is always referred to the bottom of the CB where electrons are excited. Steps in the experimental WRP spectrum of an undoped Bi12 TiO20 crystal in Fig. 7.12 indicate the presence of a photoactive center at about 1.9–2.0 eV (just the one needed for the 𝜆 = 780 nm light is able to record an electron-based hologram as reported in Fig. 8.8) that has certainly be electron-populated during normal operation and not necessarily by proposital pre-exposure process. Also, the pre-exposed sample curve reported in Fig. 7.12 exhibits clear steps in the range 1.2–1.9 eV, a large one by 1.9–2.0 eV and a much larger one at about 2.2 eV. The large step at 2.2 eV is also exhibited by the relaxed sample but it shows no steps below the one that is supposed to correspond to the Fermi level. The lack of steps at the lower h𝜈 in the relaxed sample indicates that there are empty (relaxed) photoactive donor centers

7.4.3.1.1

7.4 Photoconductivity

σ_ (sm/Ω)

10–27

10–29

Pre-exposed Normal

10–31 Relaxed 10–33 1.0

1.5

2.0

2.5

3.0

3.5

hν (eV) 5×10–30

_σ (sm/Ω)

4×10–30

3×10–30

Pre-exposed Relaxed

2×10–30

1×10–30

0 1.0

Normal

1.5

2.0

2.5

hν (eV)

Figure 7.12 Transverse configuration: coefficient 𝜎 on a logarithmic scale (upper graphics) and on a normal scale (lower graphics) for pre-exposed with h𝜈 = 2.4 eV light (•), normal (◽) and for relaxed (∘) undoped Bi12 TiO20 plotted as a function of h𝜈. Reproduced from [29].

in-between the Fermi level and the bottom of the CB that are detected only after been (at least partially) filled by pre-exposure. The expanded view in Fig. 7.13 shows a sharp step at about h𝜈 ≈ 2.5 eV for all three pre-exposed, partially and totally relaxed samples, and from there the photocurrent keeps steadily increasing without showing resolved steps. We may therefore conclude that this material has a strongly populated electron donor level at 2.2 eV (likely to be the Fermi level) and from there, plenty of populated levels as we approach the top of the VB. For a h𝜈 higher than the BG energy at 3.2 eV, we see a rapid decrease in 𝜎 because of the strong

169

7 General Electrical and Optical Techniques

1.0×10–28

0.8×10–28

σ_ (sm/Ω)

170

0.6×10–28 Normal 0.4×10–28 Pre-exposed

0.2×10–28

Relaxed 0 1.7

1.9

2.1

2.3

2.5

hν (eV)

Figure 7.13 Detailed view of Fig. 7.12 showing a strong increase in 𝜎 for all three curves at about h𝜈 ≈ 2.5 eV.

increase in optical absorption that prevents this higher photonic energy light from getting the sample’s volume to excite electrons out from the VB. V-Doped BTO The V-doped Bi12 TiO20 (BTO:V) crystal shows a Wavelength-Resolved Photoconductivity (WRP) spectrum in Fig. 7.14 that is different from that for the undoped crystal in Fig. 7.12:

7.4.3.1.2

• 𝜎 is roughly two orders of magnitude lower than for undoped BTO; • 𝜎 increases monotonically as a function of h𝜈, except for a degree at about 2.2 eV that presumedly corresponds to its Fermi level; • the degree at h𝜈 = 2.2 eV is sensibly less evident than for BTO; • pre-exposure to 2.4 eV illumination does not sensibly affect 𝜎. Most of the features of BTO:V may be understood following the discussion in Section 2.1.2.1. 7.4.3.2 Longitudinal Configuration

In the longitudinal configuration, the crystal plate is sandwiched between transparent conductive ITO electrodes as depicted in Fig. 7.15. Here, both the measured photocurrent iph and the light irradiance flow parallel to each other and perpendicularly to the front crystal plane (surface H𝓁) and iph can be formulated from Eqs. (2.22), (2.27) and (2.30) as: iph = 𝜎(z)E(z)H𝓁 I(z) h𝜈 −𝛼z I(z) = I(0) e

𝜎(z) = e𝜇𝜏Φ𝛼

(7.34) (7.35) (7.36)

where H, 𝓁 and d are described in Fig. 7.15. The photocurrent being continuous along the crystal plate thickness d, we may write from Eqs. (7.34) and (7.35): E(z)𝜎(z) ∝ E(z)I(z) = E(0)I(0)

(7.37)

7.4 Photoconductivity

1.0×10–29

σ_ (sm/Ω)

0.8×10–29

0.6×10–29

0.4×10–29

0.2×10–29

0 1.0

1.5

2.0

2.5

hν (eV)

Figure 7.14 𝜎 (s m/Ω) for thermally relaxed BTO:V (∘) and pre-exposed to h𝜈 = 2.4 eV (▴). Reproduced from [30]. Figure 7.15 Longitudinal configuration schema showing an externally polarized Bi12 TiO20 crystal plate sandwiched between ITO electrodes.

[010]

d

H

(001)

+ [100] –

ITO

and substituting I(0) from Eq. (7.36) into Eq. (7.37) we get: E(z) = E(0) e𝛼z

l

(7.38)

In the absence of a potential barriers and for a constant 𝛼, the voltage difference between both ITO electrodes would be: d e𝛼d − 1 V = E(z)dz = E(0) (7.39) ∫0 𝛼 𝛼d E(0) = (V ∕d) (7.40) 𝛼d e −1 and the photocurrent in Eq. (7.34) becomes: I(0) 𝛼d (V ∕d) H𝓁 (7.41) h𝜈 e𝛼d − 1 We know, however (see Section 2.7), that light may induce potential barriers both at the front and the rear crystal-electrode junctions. Because of the usually strong light absorption in the crystal volume, the potential barrier at the rear electrode is much weaker than at the front one, iph = e𝜇Φ𝛼

171

172

7 General Electrical and Optical Techniques

as represented in the schema of Fig. 7.16, so that an overall light-induced potential difference VSh between both electrodes results that should be considered in order to modify Eq. (7.41) accordingly: I(0) V + VSh 𝛼d H𝓁 (7.42) h𝜈 d e𝛼d − 1 From Eq. (7.42), we should write a phenomenological efficiency for the longitudinal configuration as: iph ∕e h𝜈 𝜂𝓁 = (7.43) V ∕d I0 H𝓁 iph = e𝜇𝜏Φ𝛼

which represents the number of electrons iph ∕e drifted per unit externally applied electric field V ∕d and per incident (not necessarily absorbed) photon I0 ∕(h𝜈) as measured in air with I0 ≥ I(0): the latter meaning the value at the input plane but inside the crystal. 7.4.3.2.1

Undoped BTO The parameter 𝜂 𝓁 as defined in Eq. (7.43) is measured and plotted

as a function of the photonic energy h𝜈 in Fig. 7.17 for a d = 0.81 mm thick undoped ITO-sandwiched Bi12 TiO20 crystal plate. Note that the curve arising from the positively polarized rear electrode is larger than for the reverse polarization as expected from the schema in Fig. 7.16. ITO electrode front face

ITO electrode rear face

Light

Voltage

Depth

Figure 7.16 Lateral view of the sandwiched BTO crystal plate showing the light-induced electric potential barriers at both electrodes with a schema of the electric potential distribution at the bottom.

7.5 Photo-Electric Conversion

50

2.8

η l (10–8 m/V)

3.0 1.4 20 0.7

0.0

𝛼 (cm–1)

40

2.1

10

1.0

1.5

2.0 2.5 h𝜈 (eV)

3.0

0 3.5

Figure 7.17 Plotting of 𝜂 𝓁 with positive polarization (ranging from 0 to 500 V) both at the front (◽) and at the rear (∘) electrode, as measured on the undoped Bi12 TiO20 crystal plate (labeled BTOJ18L and represented in Fig. (7.15) with d = 0.81 mm and ITO electrodes on the front and rear H𝓁 ≈ 50 mm2 surfaces. The dashed curves are the fitting of both efficiencies near their maximum using a second-order polynomial. The overall optical absorption coefficient 𝛼 is also shown (▴). Reproduced from [135].

Different from the transverse configuration, there is a trade-off here between the increasing photoconductivity as more photoactive centers are involved with increasing h𝜈, and the progressively reduced illumination (and associated overall photocurrent) throughout the crystal thickness due to the corresponding increase in the absorption coefficient 𝛼. The result is not a cumulative step-by-step increase in the overall photocurrent but a large and wide peak at some optimal value for h𝜈 arising from these two counterbalanced effects that, for the present conditions, appears to be at h𝜈 ≈ 2.5 eV in Fig. 7.17. Note also that the absorption coefficient 𝛼, which is also represented in Fig. 7.17 and steadily increases with h𝜈, apparently seems to decrease from about h𝜈 ≈ 3 eV on, because of the increasing luminescence (see Section 2.2) light as we approach the BG edge and is detected by the nonselective photodiodes at the crystal output, misleadingly indicating a nonreal decrease in 𝛼.

7.5 Photo-Electric Conversion Photoelectric conversion, which is to say, photocurrent flowing without any externally applied electric field, is expected to occur in an ITO-sandwiched photoconductive (in this case, a photorefractive) crystal plate in the longitudinal configuration because of the light-induced unbalanced front-to-rear potential barrier difference Vsh , as discussed in Section 7.4.3.2. 7.5.1

Wavelength-Resolved Photo-Electric Conversion (WRPC)

The light-induced photocurrent was measured in the longitudinal configuration without any externally applied electric field. In this case, the driving voltage may just be due to the photovoltaic effect (if any) and/or the already mentioned light-induced unbalanced front-to-rear potential difference Vsh in Eq. (7.42). In any case, an effective average electric field < E >: < E >≡ VSh ∕d

(7.44)

173

7 General Electrical and Optical Techniques

is to be considered here instead of V ∕d in Eq. (7.43), which is accordingly substituted for a new phenomenological efficiency for this light-induced photocurrent generation to be formulated as: i0ph h𝜈 𝜂0 ≡ (7.45) e I(0)H𝓁 h𝜈 ∑ (Φ𝛼)i < E(h𝜈) > (7.46) 𝜂0 = 𝜇𝜏 i=0

where Eq. (7.46) is its formulation in terms of material parameters. The term < E(h𝜈) > is written in this way to emphasize the fact that < E > depends on the photonic energy of the incident light. 7.5.1.1 Undoped BTO

Figure 7.18 shows the plot of 𝜂0 measured on an ITO-sandwiched undoped Bi12 TiO20 crystal plate as a function of h𝜈 that exhibits peaks at the positions where steps in the WRP transverse configuration should be expected. Also, the absorption coefficient 𝛼 is plotted here always showing the misleading decrease in 𝛼d as the BG edge is approached. The relation between WRPC in the longitudinal configuration (𝜂0 ) and WRP in the transverse configuration (𝜎) is clearly reported in Fig. 7.19 where we see that, as h𝜈 increases going through filled photoactive centers in the BTO BG, sharp steps appear in the WRP data at the same position where sharp peaks are detected for WRPC curves. Once more such different behavior is due to the different nature of both longitudinal and transverse configurations. The influence of the thickness of the BTO crystal plate in the WRPC spectrum is graphically reported in Fig. 7.20, where we see that the d = 0.81 mm BTO sample produces larger peaks than the thicker (d = 3 mm) sample everywhere, mainly for higher h𝜈; that is, for photoactive centers probably well below the bottom of the CB. It is worth pointing out that both the WRP and WRPC techniques are very useful to look for photoactive centers in photorefractive crystals and photoconductive materials in general whenever the adequate experimental conditions are determined, as discussed in this and previous sections. 5

24

4

18

3

𝛼d

η0 (10–3)

174

12 2 6

0 1.0

1

1.5

2.0 2.5 h𝜈 (eV)

3.0

0 3.5

Figure 7.18 Light-induced photoelectric conversion efficiency 𝜂0 measured (•) on an undoped sandwiched Bi12 TiO20 crystal (labeled BTOJ18L) in the longitudinal configuration together with the light absorption coefficient-thickness 𝛼d (∘). Reproduced from [135].

7.6 Modulated Photoconductivity

1.2

1.0–9

0.8

1.0–1.0

0.4

𝜎t

η0 (10–3)

1.0–8

1.0–11 1.0

1.5

2.0

0 3.5

3.0

2.5 h𝜈 (eV)

Figure 7.19 Comparative longitudinal 𝜂0 (without external applied field) (∘) and transverse 𝜎 (•) WRP, respectively, measured on an undoped Bi12 TiO20 crystal. Reproduced from [135]. 3×10–6 𝛼d (d = 3 mm) η0 (d = 3 mm) η0 (d = 0.81 mm) 𝛼d (d = 0.81 mm)

4 3

η0

𝛼d

2×10–6

5

2

1×10–6

1 0 1.0

1.5

2.0

2.5

3.0

0 3.5

h𝜈 (eV)

Figure 7.20 𝜂0 and 𝛼d measured on an ITO-sandwiched BTO with d = 3 mm and d = 0.81 mm under 𝜆 = 532 nm illumination chopped at 200 Hz.

7.6 Modulated Photoconductivity This method [136–138] consists of illuminating the sample with a dc and an ac time-modulated (frequency 𝜔) spatially uniform strong flux (F = I∕(h𝜈)) light: F = Fdc + Fac cos(𝜔t)

(7.47)

with a fixed photonic energy slightly higher than that of the Band Gap to produce band-to-band charge-carrier excitation. The dc flux fixes the recombination process, whereas the ac current generated by the ac flux reflects the trapping and release processes experienced by the

175

7 General Electrical and Optical Techniques

1011 NC/μ (cm–2 V eV–1)

176

0.29 eV

1010

109

130 K ≤ T ≤ 260 K CNbe = 2.5 × 1011 S–1

108 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Ebe – Eω (eV)

Figure 7.21 Modulated photocurrent data of an undoped Bi12 TiO20 crystal, with monochromatic light flux (I∕(h𝜈)) Fdc = 5 × 1014 cm−2 s−1 and Fac = 1013 cm−2 s−1 , where different shades correspond to different temperatures from 130 to 260 K varying in 5 K steps, and different symbols indicating frequencies varying from 12 Hz to 39.9 kHz. After multiple trials, the value Nbe C = 2.5 × 1011 s−1 was chosen, which leads to a rather good superposition of curves for different temperatures and frequencies at the same abscissa indicating a peak at ∣ Ebe − E ∣= 0.29 eV. Reproduced from [29].

ac generated carriers. The alternative current iac allows computing the quantity: N(E)C∕𝜇 =

sin 𝜙 2 AeE0 G 𝜋kB T iac

(7.48)

with N(E) being the density of states at energy E, C the capture coefficient of the probed states having a density N 1 at energy E, 𝜇 the mobility of majority carriers in the Extended State, A the cross-sectional area for photocurrent flowing, e the electron charge, E0 the applied electric field, G the charge carriers generation rate, 𝜙 is the phase shift between the excitation light and resulting photocurrent and the 2∕𝜋-term arises from the rectangular chopping of illumination (see Eq. (7.22)). A relation between E and the excitation frequency 𝜔 exists: ∣ Ebe − E ∣= kB T ln(Nbe C∕𝜔)

(7.49)

where Nbe C is called the “attempt-to-escape frequency”, Nbe being the equivalent density of states at the band edge Ebe (bottom of the CB or top of the VB). From Eq. (7.49), it is clear that a spectroscopy relation between DOS (Density of States) and energy E can be put into evidence by varying either 𝜔 or T or both. Note that the value of ∣ Ebe − E ∣ may indicate the position of a State below the CB or above the VB as well. After several trials the better value Nbe C = 2.5 × 1011 s−1 was selected in Fig. 7.21 leading to ∣ Ebe − E ∣ = 0.29 eV and to NC∕𝜇 ≈ 1.1 × 1011 V/cm2 . Following the same procedure, other energy ranges were explored for this undoped BTO crystal, the results of which are displayed in Table 7.6. Additionally, the relaxation time 𝜏rlx of charge carriers in the State of energy E in the Band Gap, as displayed in the last column of Table 7.6 were estimated from: 𝜏rlx ≈ or

eE∕(kB T) 2𝜋 Nbe C e∣ Ebe − E ∣ ∕(kB T) ≈ 2𝜋 Nbe C

(7.50) (7.51)

using the results in the same Table 7.6. 7.6.1

Quantum Efficiency and Mobility-Lifetime Product

Photoconductivity in the presence of light-induced (photochromic) absorption allows one to easily compute the mobility-lifetime product 𝜇𝜏. Lifetime 𝜏 in photochromic materials, 1 For example, it is NC = 2.5 × 1019 cm−3 for Si at 300 K

7.6 Modulated Photoconductivity

Table 7.6 DOS for Bi12 TiO20 . From [29]. ∣ Ebe − E ∣

NC∕𝝁

Nbe C

(NC∕𝝁)∕(Nbe C)

𝝉rlx a)

(eV)

(109 × V/cm2 )

(109 × s−1 )

(Vs/cm2 )

(10−8 × s)

0.10

1.5

1.0

1.5

4.8

0.14

7

1.0

7

22.5

0.29

110

250

0.4

30

0.44

0.11

12.5

0.01

20 × 104

a) From data here and Eq. (7.51)

+ however, where light-dependent shallow traps concentration (ND2 ) plays an important role, is not constant any more, as described by Eq. (2.51) + 1∕𝜏 ≡ 1∕𝜏1 + r2 ND2 + because ND2 (and consequently 𝜏 too) is strongly affected, even by moderate light intensity as described by Eqs. (2.47) and (2.48). If we are nevertheless able to neglect light-induced effects on + 𝜏, by assuming 𝜏1 𝛾2 ND2 ≪ 1 in Eq. (2.51), then 𝜏 becomes approximately constant. Therefore, from Eq. (7.32) we may compute ( ) 1 lim 𝜎 − lim 𝜎 ∕ lim 𝛼li ≈ e𝜇𝜏 (7.52) I→0 I→∞ e I→∞ The quantum efficiency parameter Φ may be also computed from Eq. (7.32) and from the knowledge of 𝜇𝜏 obtained in Eq. (7.52):

lim 𝜎∕(e𝜇𝜏𝛼0 ) = Φ

(7.53)

I→0

The term 𝜇𝜏 computed from Eq. (7.52) and the Φ computed from Eq. (7.53) are displayed in −12 2 Table 7.7 for undoped and some doped BTO. The √ 𝜇𝜏 = 0.72 × 10 m ∕V value for BTO-011 leads to a charge carrier diffusion length LD = 𝜇𝜏kB T∕e = 0.14 μm (kB is the Boltzman constant, T is the absolute temperature and e is the charge of the electron), which is in excellent agreement with the value LD = 0.14 ± 0.01 μm reported elsewhere [139] using a quite different technique for undoped BTO. The Φ reported in Table 7.7 for undoped BTO is much lower than Table 7.7 Photoconductivity and derived parameters for BTO at 532 nm. Sample

lim I→0

𝜎ph (0) I(0) −12

(10

lim

BTO-010

BTO-011

BTO-013

BTO:Ce

BTO:Pb

47.5 (1.8 b))

52.2

65.5

0.56

53.5

122.6 (7.2 b))

127.4 (103 a))

317.7

8.72

230.7

b))

0.72

2.58

0.14

1.66

0.19 (0.16 b))

0.25

0.10

0.02

0.16

m∕(Ω W))

𝜎ph (0)

I→∞ I(0)

𝜇𝜏

0.88 (0.47

(m2 ∕ V × 10−12 ) Φ a) 𝜆 = 514.5 nm [17] b) 𝜆 = 633 nm

177

178

7 General Electrical and Optical Techniques

the Φ′ ≈ 0.37 ± 0.03 reported in [139] using a different technique and also differently defined [139] as: Φ′ (𝛼0 + 𝛼li ) = Φ𝛼0 + 𝛼li

(7.54)

(Φ′ − Φ) = (1 − Φ′ )𝛼li ∕𝛼0 ≥ 0

(7.55)

with Eq. (7.55) showing that it is always Φ ≥ Φ in rough agreement with our results. ′

7.7 Photo-Electromotive-Force Techniques (PEMF) The PEMF arises when light-induced charge carriers move in a stationary space-charge electric field [140] and is produced in photoconductors where, under the action of light, a distribution of free charge carriers in the extended states (conduction and/or valence band) is produced and a fixed spatial distribution of electric charges in traps and associated space-charge electric field are built up as already described in Chapter 3. If the pattern of light is moving faster than the response of the space-charge field but slower than the lifetime of free charges in the extended states, the free charges will follow the movement but the space-charge field will not. In this way, the free charges will not be in equilibrium any more and a current will appear. Such an effect can be also produced in a photorefractive material that is in fact a photoconductor also exhibiting electro-optic properties. Electro-opticity is not necessary here, but the large density of photoactive centers in the bandgap, which is a characteristic of photorefractive materials, leads to large space-charge fields that are necessary to produce large PEMF currents. In addition, recording in photorefractives is carried out in volume using light with photonic energies below that of the material Band Gap, thus allowing a deeper penetration and consequently access to a higher number of photoactive centers in the volume. That is why photorefractive materials are particularly interesting for applications involving PEMF effects. The first reports on this subject can be traced back to the late 1970s [141], 1980s [142, 143] and 1990s [144, 145] in the former USSR, and were related to holographic recording in bulk photorefractives. PEMF can be produced using either a speckle (see Fig. 7.24) or a pattern of interference fringes (see Fig. 8.25) projected onto the photoconductor or photorefractive crystal. The latter is traditionally referred to as “holographic photo-electric electromotive force” (holographic PEMF), although holography (in terms of recording and reconstruction of a wave) is not at all involved here. Only the recording of a pattern-of-fringes into a corresponding space-charge electric field distribution occurs. We shall nevertheless keep the traditional, somewhat misleading, term “holographic PEMF” and include this technique in Chapter 8 because most of the mathematical development of this technique actually arises from the corresponding ones for holography as far the recording of a space-charge electric field is concerned. Both speckle and holographic photoelectromotive force techniques are useful for mechanical (in-plane) vibration and deformation measurement [146–150] as well as for materials characterization [151–154]. 7.7.1

Speckle-Photo-Electromotive-Force (SPEMF) Techniques

The speckle pattern of light is formed by light scattered from a rough reflecting or transmitting surface and is formed by randomly distributed grains or speckles, each one showing an intensity

7.7 Photo-Electromotive-Force Techniques (PEMF) 1

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0.2

0.4

0.6

0.8

1

1.2

1.4

1

0.2

0.4

0.6

0.8

1

1.2

1.4

0.2

0.4

0.6

0.8

1

1.2

1.4

x

Figure 7.22 Plot of the Airy function (left), the equivalent Gaussian function (center) and the superposition of both (right), with x = ∕0 .

distribution closely represented by the Airy function [155, 156] ( ) J (𝜋1.22R∕R0 ) 2 z I(R) = 2 1 R0 ≡ 1.22𝜆 𝜋1.22R∕R0 Dl

(7.56)

where Dl is the diameter of the illuminated scattering surface, 𝜆 is the illumination light √ wavelength, z is the distance from the scattering surface to the observation plane and R = X 2 + Y 2 the radial coordinate in the observation plane. The Airy function in Eq. (7.56) goes to zero at R = R0 . The successive Airy function maxima are much weaker than the principal one at R = 0 so we may assume that almost all of the light is concentrated in a disk of radius R0 as seen in the observation plane, thus R0 represents the radius of a speckle. It is easy to show that the Airy function Eq. (7.56) can be approximately substituted by a Gaussian function of the form 2 2 e−R ∕W

W ≡ R0 ∕2

(7.57)

in the (0, 2W) range for R, as clearly illustrated in Fig. 7.22, where curves corresponding to Eq. (7.56) and (7.57) are simultaneously represented. It is much easier to operate with a Gaussian rather than with the Airy function, so we shall always use the former one to describe the light distribution in a single speckle, provided we restrict ourselves to R ≤ 2W = R0 . We shall therefore write the light distribution for a single speckle as 2 2 I = I0 e−R ∕W

7.7.1.1

for

R2 ∕W 2 ≤ 4

(7.58)

Speckle Pattern onto a Photorefractive Material: Stationary State

The speckle pattern of light described previously is projected in the volume of a photorefractive crystal. In this way, the light excites charge carriers, that diffuse and/or drift and are retrapped, as described by the well-known (see Section 2.3.2.1) equations: 𝜕ND+ 𝜕t

=G−

sI (N − ND+ ) + 𝛽(ND − ND+ ) h𝜈 D  ≡ 𝛾R ND+  1∕𝜏 = 𝛾ND+ G≡

where  is the density of free electrons in the conduction band (CB), ND+ and ND are, respectively, the ionized and total electron donor centers, s is the effective cross-section for photon-electron interaction, h𝜈 is the photonic energy of the light, 𝛽 is the dark electron

179

180

7 General Electrical and Optical Techniques

generation and 𝜏 is the photoelectron lifetime in the conduction band. For stationary conditions, we should state that: 𝜕ND+ =G−=0 (7.59) 𝜕t in which case, we deduce that: G=

(7.60) 2

2

 = 0 e−R ∕W + d

(7.61)

with 0 ≡

ND − ND+ sI0

(7.62)

𝛾ND+ h𝜈 ND − ND+ 𝛽 d ≡ 𝛾ND+

(7.63)

The stationary density current is therefore: ⃗j = e𝜇 E⃗ + eD∇ ⃗

(7.64)

2X x̂ + 2Y ŷ 2 2 0 e− ∕W (7.65) 2 W where e is the absolute value of the electric charge of the electron, 𝜇 and D are, respectively, the mobility and diffusion coefficients of the charge carriers (electrons in this case), and x̂ and ŷ are the unit vectors along coordinates X and Y . In the following, we shall use reduced coordinates x = X∕W and y = Y ∕W , so that Eqs. (7.64) and (7.65) should be rewritten as: ⃗ =− ∇

⃗j = e𝜇 E⃗ + e𝜇W ED ∇ ⃗

(7.66)

⃗ )y ⃗ = x̂ (∇ ⃗ )x + ŷ (∇ ∇

(7.67)

⃗ )x = −2x0 e−r (∇

(7.68)

where

⃗ )y = −2y0 e−r (∇

2

2

(7.69)

and r 2 ≡ x2 + y 2

(7.70)

D (7.71) 𝜇W with ED being the so-called diffusion field. The current density along x- and y-coordinates is then: 2 (7.72) jx = e𝜇 Ex − 2x𝜇eED 0 e−r = j0 ED ≡

2 jy = e𝜇 Ey − 2y𝜇eED 0 e−r = 0

(7.73)

The stationary condition leads to continuity in the current along the x-coordinate so that we state that jx = j0 where j0 is a constant. The y-direction in the crystal, instead, is open-looped,

7.7 Photo-Electromotive-Force Techniques (PEMF)

so that there is no current, and that is why we write jy = 0. From Eqs. (7.72) and (7.73) we should compute the corresponding stationary fields: Ex = Ey =

E0 2

e−r + Rd 2yED

+

2xED

(7.74)

2 1 + Rd er

(7.75)

2 1 + R er √ d Er = Ex2 + Ey2

(7.76)

with E0 ≡ j0 ∕𝜎0

𝜎0 ≡ e𝜇0

Rd ≡ d ∕0

(7.77)

Figure 7.23 shows the representation of Er ∕ED from Eq. (7.76) in the xy plane, for two different values of Rd , where the influence of Rd in space-charge field shaping is evident. Vibrating Speckle Pattern If the speckle pattern is vibrating along x-coordinate, with angular frequency Ω and reduced amplitude 𝛿, we should substitute everywhere:

7.7.1.1.1

x → x + 𝛿 sin Ωt

(7.78)

therefore, getting this new expression 2 2 I = I0 e−y + (x + 𝛿 sin Ωt)

(7.79)

2 2 I = I0 e−r + 𝛿 ∕2 I1 (x, 𝛿, Ω, t)

(7.80)

2 2  = 0 e−(r + 𝛿 ∕2) I1 (x, 𝛿, Ω, t) + d

(7.81)

2 2 ⃗ )x = −20 e−(r + 𝛿 ∕2) (xI1 (x, 𝛿, Ω, t) + 𝛿I2 (x, 𝛿, Ω, t)) (∇

(7.82)

2 2 ⃗ )y = −20 e−(r + 𝛿 ∕2) yI1 (x, 𝛿, Ω, t) (∇

(7.83)

and

where “” mean the time average and 2 I1 (x, 𝛿, Ω, t) ≡ e−2x𝛿 sin Ωt + (𝛿 ∕2) cos 2Ωt

4 Er /ED 3 2 1 0

(7.84)

1.5 Er /ED 1 2 0.5 0 y

–2 x

0 2

–2

–2

0

0 y

–2 x

0 2

–2

Figure 7.23 Plotting of Er ∕ED in the xy plane, for d = 0.001 (left) and 0.1 (right). Reproduced from [152].

181

182

7 General Electrical and Optical Techniques

iΩ

X = Δ sin Ω t A

Figure 7.24 Schematic representation of an ac photocurrent produced by a sinusoidally vibrating (with angular frequency Ω) speckle pattern of light on the surface of a photorefractive crystal with parallel in-plane electrodes (coplanar configuration). Reproduced from [148].

X = Δ sin Ω t

X

Free electron cloud in the conduction band

Stationary space-charge field

Figure 7.25 Stationary space-charge field arising from a speckle pattern of light vibrating faster than the response time of the space-charge field and slower than the lifetime of the free photoelectrons.

I2 (x, 𝛿, Ω, t) ≡ I1 (x, 𝛿, Ω, t) sin Ωt

(7.85)

If we substitute Eqs. (7.81)–(7.83) into Eqs. (7.66)–(7.73), calculate the time averages of jx and jy and assume that Ω𝜏SC ≫ 1, where 𝜏SC represents the response time for space-charge field build-up in the photorefractive material, then we get: ⃗ ) x ⟩ = j0 ⟨jx ⟩ = q𝜇⟨ ⟩Ex − 2xq𝜇W ED ⟨(∇

(7.86)

⃗ )y ⟩ = 0 ⟨jy ⟩ = q𝜇⟨ ⟩Ey − 2yq𝜇W ED ⟨(∇

(7.87)

7.7 Photo-Electromotive-Force Techniques (PEMF)

3 Er /ED 2

4

1

2

0 –4

0 –2 x

Er /ED

1.0 2

0.0 –4

y

0 –2

–2

0

4

0.5

x

2

–4

4

y

–2

0 2 4

–4

Figure 7.26 Plotting of Er ∕ED in the xy plane for a speckle pattern of light vibrating along coordinate x with reduced amplitude 𝛿 = 1 for d = 0.001 (left) and 0.1 (right). Reproduced from [152].

The stationary space-charge fields can be computed from Eqs. (7.86) and (7.87): Ex (x, y, 𝛿, Ω) =

2 E0 eb 2

+ 2ED

⟨I1 ⟩ + d eb y⟨I1 ⟩ Ey (x, y, 𝛿, Ω) = 2ED 2 ⟨I1 ⟩ + d eb

x⟨I1 ⟩ + 𝛿⟨I2 ⟩ 2 ⟨I1 ⟩ + d eb

b2 ≡ x2 + y2 + 𝛿 2 ∕2

(7.88) (7.89) (7.90)

The two-dimensional space-charge field is computed as: √ Er (x, y, 𝛿, Ω) = Ex (x, y, 𝛿, Ω)2 + Ey (x, y, 𝛿, Ω)2

(7.91)

and represented in Fig. 7.26, normalized by ED , for two different values of d , where we see the stretching of the stationary space-charge field pattern along the direction of vibration and the way it is affected by d . 7.7.1.1.2

Photocurrent Components The x-component of the photocurrent generated by the

vibrating speckle pattern is: ⃗ )x jx = e𝜇 Ex + e𝜇W ED (∇

(7.92)

2 2 E0 ∕ED + 2 e−b (x⟨I1 ⟩ + 𝛿⟨I2 ⟩) 2 b − 2jD e−b (xI1 + 𝛿I2 ) jx = jD (I1 + d e ) 2 ⟨I1 ⟩ + d eb

(7.93)

jD ≡ 𝜎0 ED

(7.94)

where Ex is computed from Eq. (7.88) and the dc component of the photocurrent density is jxDC

= jD d

2 E0 ∕ED eb + 2x⟨I1 ⟩ + 𝛿⟨I2 ⟩ 2 ⟨I1 ⟩ + d eb

and the time-dependent terms are: ( ) 2 2 E0 ∕ED + 2 e−b (x⟨I1 ⟩ + 𝛿⟨I2 ⟩) jx1 = jD I1 − 2x e−b 2 ⟨I ⟩ +  eb 1

d

(7.95)

(7.96)

183

184

7 General Electrical and Optical Techniques 2 jx2 = −2jD 𝛿 e−b I2

(7.97)

Integrating these density currents along the x-coordinate we get the linear-integrals: 𝓁∕2

jDC =

∫−𝓁∕2

jxdc d𝓁

(7.98)

𝓁∕2

j1 =

∫−𝓁∕2

jx1 d𝓁

(7.99)

jx2 d𝓁

(7.100)

𝓁∕2

j2 =

∫−𝓁∕2

If we assume 𝓁 ≫ 1, that is to say, if we integrate over actual distances much larger than the speckle radius, the integrals here are independent from 𝓁 and are proportional to the currents that can actually be measured, the proportionality constant being the sample cross-section for current flux and the number of speckles involved in the process. Harmonic Terms The normalized time-dependent periodic function j1 ∕jD + j2 ∕jD in Eqs. (7.99) and (7.100) can be written in terms of a Fourier series:

7.7.1.1.3

n ∑ j1 j + 2 = ao ∕2 + (an cos(nΩt) + bn sin(nΩt)) jD jD n=1

(7.101)

𝜋∕Ω

an =

Ω (j ∕j + j ∕j ) cos(nΩt)dt 𝜋 ∫−𝜋∕Ω 1 D 2 D

bn =

Ω (j ∕j + j ∕j ) sin(nΩt)dt 𝜋 ∫−𝜋∕Ω 1 D 2 D

(7.102)

𝜋∕Ω

(7.103)

For no external electric field (E0 = 0) we calculated the first b1 (n = 1) and second b2 (n = 2) harmonic terms previously and plotted them in Figs. 7.27 and 7.28, respectively, for different values of d . The harmonic coefficients a1 and a2 are zero. Figure 7.27 shows that b1 has a maximum (normalized) value for 𝛿 varying from about 0.9 (for d = 0) to 1.1 (for d = 1). For the same conditions, the coefficient b2 in Fig. 7.28 is roughly zero. The presence of this maximum in b1 (that is, in I Ω ), for sufficiently fast vibrations, at a fixed value of 𝛿 varying between 0.85 for Rd = 0.1 and 0.9 for Rd = 0, does apparently depend on the dark-to-photoconductivity ratio Rd only. The present model was already experimentally verified [148, 157] and results are reproduced in Figs. 7.32 and 7.33. The position of this maximum could be used as a reference point for calibrating the setup for lateral vibration measurements and such a practical possibility has already been experimentally demonstrated elsewhere [148]. Experimental Setup The experimental setup is shown in Fig. 7.29. A laser beam is directed onto a loudspeaker membrane (vibrating target under analysis) with a small retroreflective strip on its surface. The amplitude and frequency vibration of the speaker are controlled by a function generator, which also provides the reference signal to the lock-in amplifier (Model 5210 ECG Princeton Applied Research). A Doppler velocimeter (DV) is used for independent measurement of the vibration amplitude of the loudspeaker. The reflected speckle pattern beam is focused on the photorefractive crystal (without applied electric field) by means of a 25 mm diameter, 50 mm focal length photographic objective lens. The speckle pattern vibrates with the same frequency as the target surface, and produces a photocurrent that is pre-amplified using an electrometer class operational amplifier operating in transimpedance mode, and converted into a voltage, the first harmonic term of which 𝑣Ω = iΩ Rfb (Rfb = 100 MΩ being the feedback resistance in the pre-amplifier) is measured using

7.7.1.1.4

7.7 Photo-Electromotive-Force Techniques (PEMF)

0.20

b1 (au)

0.15

0.10

0.05

0

0.5

0

1.0 𝛿

1.5

2.0

Figure 7.27 Simulation of the first harmonic photocurrent coefficient b1 (in arbitrary units) as a function of 𝛿, for y = 0, ED = 1000 V/m, jD = 1 for d = 0 (•), d = 0.01 (◽) and d = 0.1 (∘). Reproduced from [157]. Rd = 0 Rd = 0.1

1×10–8

Rd = 1

|b2| (au)

8×10–9 6×10–9 4×10–9 2×10–9 0 0.0

0.2

0.4

0.6

0.8

1.0 𝛿

1.2

1.4

1.6

1.8

2.0

Figure 7.28 Simulation of the first harmonic photocurrent coefficient b2 as a function of 𝛿, for d = 0, 0.1 and 1.

185

186

7 General Electrical and Optical Techniques

VΩ

FG

LA

RG

iΩ

– +

crystal 633 nm

metallic housing

DV

Laser beam 532 or 1064 nm

Figure 7.29 Schematic representation of the experimental setup. A laser beam is directed to a vibrating target (commercial loudspeaker with a retroreflecting strip); the backscattered light in the form of an oscillating speckle pattern it is focused onto the photorefractive crystal (BTO with 𝜆 = 532 nm or CdTe with 𝜆 = 1064 nm) fixed on a plate in a metallic housing creating the PEMF effect. The loudspeaker is driven by a function generator FG that also provides the reference signal for the frequency-tuned phase-selective lock-in amplifier LA used for detecting the signal from the photorefractive crystal; the current (iΩ ) from the crystal is converted into a voltage signal by means of a pre-amplifier (an electrometer-class operational amplifier operating in transimpedance mode) fixed by the side of the crystal. A home-made laser Doppler vibrometer DV using a 633 nm laser beam is used for independent measurement of the loudspeaker vibration.

the lock-in amplifier referred to previously. The crystal and the pre-amplifier are fixed, close to each other, on the same fiberglass plate that is placed inside the metallic housing shown in Fig. 7.30. The objective lens is screwed at one end of the housing so that the speckle can be focused on the crystal in order to get a maximum photocurrent output. The pre-amplified signal in the housing is fed into the lock-in amplifier using a cooper shielded BNC cable that strongly improves signal-to-noise ratio (SNR). Two different (among many other possibilities) photorefractive sensors may be used: undoped titanosillenite Bi12 TiO20 (BTO) with a 𝜆 = 532 nm laser and a CdTe:V crystal when a 𝜆 = 1064 nm light from a Nd-Yag laser is used. Electrodes on the Crystal Sensor The PEMF current can be collected either on the crystal surface (just using two parallel electrode strips on the input crystal surface separated about 1 or 2 mm from each other, the so-called coplanar electrode configuration as shown in Fig. (7.24)) or on the bulk of the crystal (using electrodes on the lateral sides, the so-called lateral electrode configuration, which is currently used for holographic and related experiments), or even both electrode configurations at the same time. The coplanar configuration is generally used for very absorbing crystals whereas lateral electrodes are used for rather transparent materials. For Bi12 TiO20 crystals under 𝜆 = 532 nm illumination, the lateral electrodes are better performing. Note also that, whatever the electrode configuration used, the photocurrent is collected in a (total or partial) volume of the crystal, so that the actual experimental value is iΩ and not jΩ .

7.7.1.1.5

7.7 Photo-Electromotive-Force Techniques (PEMF)

Figure 7.30 Optical sensor in metallic housing (from Fig. 7.29) showing the separated components, from left to right: adjustable lens, lens adapting ring, main supporting housing with BNC connectors, photorefractive sensor housing.

Figure 7.31 Expanded front view of the photorefractive sensor housing (from Fig. 7.30) showing the photorefractive crystal sensor on a fiberglass plate with circuitry.

50 Hz 100 Hz

120

200 Hz

100

800 Hz

400 Hz 1200 Hz

iΩ (pA)

80

1600 Hz 2500 Hz

60

3200 Hz 3500 Hz

40 20 0 0.0

0.2

0.4

0.6

0.8 𝛿

1.0

1.2

1.4

Figure 7.32 First harmonic photocurrent as function of reduced vibration amplitude 𝛿 measured using a BTO crystal under 𝜆 = 532 nm illumination, for frequencies ranging from 50 to 3500 Hz.

187

7 General Electrical and Optical Techniques

400 ×

× × ×

×

300

×

×

Figure 7.33 Experimental first harmonic photocurrent IΩ measured on a CdTe:V photorefractive crystal as a function of 𝛿 for Ω = 200 Hz (∘), 400 Hz (◽), 615 Hz (▿), 1300 Hz (•) and 1700 Hz (×), with a I(0) = 3.48 mW/cm2 𝜆 = 1064 nm from a Nd-YAG laser. Reproduced from [157].

×

× ×

i Ω (pA)

188

× ×

200

× × ×

100

0

×

0

0.5

1.0 𝛿

1.5

2.0

However, as both quantities are proportional to each other with the proportionality depending on the crystal volume concerned, volume density of speckles, absorption coefficient at the working wavelength light and geometry and size of electrodes, we may report indistinctly either to the current or the current density. First Harmonic Term as a Function of 𝜹 Experimental data for the first harmonic photocurrent iΩ measured as a function of the reduced vibration amplitude 𝛿, for different target frequencies, using 𝜆 = 532 nm light on a BTO crystal (always E0 = 0) with a 380 mm distance between the target and the objective lens input plane are reported in Fig. 7.32, where curves clearly exhibit a maximum, in agreement with theoretical predictions (see Fig. 7.27), although the position of the experimental maximum at 𝛿 ≈ 0.7 is different from the predicted one at 𝛿 ≈ 0.9. First harmonic photocurrent iΩ data measured in a similar experiment on a CdTe:V photorefractive crystal under 𝜆 = 1064 nm illumination is shown in Fig. 7.33, where we clearly see that the characteristic maximum becomes increasingly evident as target frequency Ω increases approaching the theoretical condition 𝜏sc Ω ≥ 1 and reaching the theoretical value 𝛿 ≈ 0.9 for Rd ≈ 0.

7.7.1.1.6

189

8 Holographic Techniques 8.1 Holographic Recording and Erasing The building up of a spatially modulated space-charge electric field, arising from the excitation, retrapping, diffusion and drifting of charge carriers (electrons and/or holes), is described in Chapter 3. Hole-electron competition is discussed in Section 3.4.1 for the general case of running holograms and mathematically formulated in Section 3.4.1.1. When holographic recording (and erasure) is carried out by exciting and retrapping charge carriers on two (or more) LS, two (or more) holograms are recorded that are not independent but electrically coupled, no matter whether charge carriers on both LSs are of the same or different signs, as mathematically described in Section 3.4.1.1 for the case of electrons on one LS and holes on the other. If a single LS is involved, then a single hologram is recorded but if electrons and holes participate in the process, electrons and holes are electrically coupled too and the dynamics of recording and erasure of this hologram depends not only on electrons and holes but also on their mutual coupling. Most applications of photorefractive crystals involve holographic recording so that holographic techniques themselves are particularly suited for materials characterization as far as holographic recording is concerned. Some of these techniques are well known and have already been extensively used in the past: holographic recording and holographic erasure time constants, diffraction efficiency, amplitude gain in two-wave mixing and so on. These and similar ones might be considered direct holographic techniques and will be analyzed in the next section. Other methods requiring more sophisticated detection techniques will be considered further on in this chapter in Section 8.6. We shall throughout assume we are dealing with “thick” holograms in order to verify the Bragg selectivity condition and therefore have to deal with only one single diffraction order.

8.2 Direct Holographic Techniques The measurement of experimental quantities like diffraction efficiency, amplitude gain, holographic sensitivity and the time constant for recording and for erasure, among many other possibilities, are important because they will determine the applications of a certain material. These quantities will, at the same time, allow one to compute several fundamental material parameters as the diffusion length, Debye length (and density of donors), photocarriers’ mobility, quantum efficiency, dark conductivity, photoconductivity coefficient and so on. We shall mention next just a few examples of these direct methods. These measurements are carried out using a simple holographic (or interferometric or two-wave mixing) setup such as the one schematically represented in Fig. 8.1. Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

190

8 Holographic Techniques

Sh1

BS

Sh3

M1

laser

Sh2

M2

C

D1

D2

Figure 8.1 Holographic setup: a laser beam is divided by the beamsplitter BS, reflected by mirrors M1 and M2 and interfering with an angle 2𝜃. A sinusoidal pattern of light is produced in the volume where these two beams interfere. A photorefractive crystal C is placed in the volume where this pattern of light is produced. The irradiance of the two interfering beams are measured behind the crystal using photodetectors D1 and D2. Shutters Sh1, Sh2 and Sh3 are used to cut off the main and each one of the interfering beams if necessary.

8.2.1

Energy Coupling

Energy coupling appears as light diffracts through a phase grating because, in general, some light is transferred from one beam to the other direction at the output behind the grating according to the equation √ √ (8.1) IS = IS0 (1 − 𝜂) + IR0 𝜂 + 2 IS0 IR0 𝜂(1 − 𝜂) cos 𝜑 √ √ IR = IR0 (1 − 𝜂) + IS0 𝜂 − 2 IR0 IR0 𝜂(1 − 𝜂) cos 𝜑 (8.2) For such an energy transfer to occur, it is necessary that 𝜑 ≠ 𝜋∕2. In the case of photorefractive materials this is usually the case. Also, energy coupling in photorefractives is rather efficient because, if energy is transferred in the adequate direction (from the strongest to the weakest beam), the pattern-of-fringes visibility steadily increases from the input to the output, thus producing a nonlinear enhancement that is not evident in the simple formulation of Eqs. (8.1) and (8.2) but is explicitly included in the expression Eq. (4.84) IS (d) = IS0

1 + 𝛽2 1 + 𝛽 2 e−Γd

For 𝛽 2 ≫ 1, this expression simplifies to IS (d) = IS0 eΓd where Γ here assumes the role of an exponential gain. Computing Γ from the previous equation is particularly simple and precise because only one single photodetector is used: the one along direction S. Then, measuring the IS (d) without and with the other beam shining, one is able to compute IS0 and IS (d), always measure behind the sample, and from these data compute Γ. Figure 8.2 shows experimental data of energy transfer in a TWM experiment on BTO. Both IS (d) and IR (d) are measured simultaneously with one and the other beam being switched off as indicated in the figure.

8.2 Direct Holographic Techniques

Intensity (a.u.)

0.25

0.15

0.05

0

20

40

60

Time (s)

Figure 8.2 Energy transfer between interfering 𝜆 = 633 nm beams in the two-wave mixing experiment, represented in Fig. 8.1, on a BTO crystal (2.8 mm thick): The figure shows the overall irradiance at the crystal output with both beams onto the sample (shutters Sh1, Sh2 and Sh3 open) and when one (all open and Sh3 switched off ) and the other (all open and Sh2 switched off ) beam are alternatively switched off. From these data, and knowing the input recording beams irradiance ratio 𝛽 2 = 1.5, it is possible to compute the exponential gain coefficient Γ and also 𝜂.

Energy coupling has been already analyzed for a pure phase grating in Section 4.2.2 where the exponential energy (or amplitude) gain coefficient Γ was defined in Eq. (4.85) in terms of the imaginary part of Eeff 2𝜋n3 reff ℑ{Eeff } (8.3) 𝜆 cos 𝜃 ′ Substituting Eq. (3.64) into Eq. (8.3), in the absence of external field (E0 = 0), the energy gain can be written as 2𝜋n3 reff ED Γ= (8.4) 𝜆 cos 𝜃 ′ 1 + K 2 ls2 Γ=

From this expression, we deduce that the maximum for Γ is ΓM =

n3 reff 𝜋kB T for Kls = 1 ls 𝜆e cos 𝜃 ′

(8.5)

which is achieved for K 2 ls2 = 1. By measuring Γ in a TWM experiment, as a function of K, one may find its maximum value ΓM and from the relations in Eq. (8.5) one can also compute ls from the simple relation K 2 ls2 = 1. From the value of ls (and some other material parameters) we can compute the effective trap concentration (ND )eff from Eq. (3.49). In addition, the reff can be computed from Eq. (8.5) if the index-of-refraction n is known. A practical consequence stems from Eq. (8.5): maximum energy transfer can be achieved by adequately choosing the hologram wave vector to be K = 1∕ls . Figure 8.3 illustrates procedure referred to previously as applied to the Ti-doped KNSBN crystal described in Table 8.1 [158] with extraordinary polarized 514.5 nm light and the grating vector K⃗ parallel to the c-axis. In this case ΓM = 5.5 cm−1 and the corresponding ls = 0.20 μm are obtained. Note that, because of the polarization direction of the recording beams lying in the incidence plane and making

191

8 Holographic Techniques

Figure 8.3 Exponential gain coefficient Γ as a function of the external incidence angle 𝜃 measured for a KNSBN:Ti ⃗ crystal with its optical c-axis parallel to the grating vector K. Holographic recording is carried out with extraordinarily polarized (polarization direction along the c-axis) 514.5 nm wavelength laser line. Reproduced from [158].

6 Γ (cm–1)

192

4 ΓM = 5.1 cm–1

2

θM = 12.4° 0

10

0

20

30

40

θ (degrees)

Table 8.1 Properties of a KNSBN:Ti sample. KNSBN formula

(Kx Na1−x )2m (Sry Ba1−y )1−m Nb2 O6

Ti-dopant

0.36% wt

dimensions

5.5 × 5.1 × 4.9 mm

c-axis

along 5.5 mm side

𝛼 (514.5 nm)

0.15 cm−1

no (514.5 nm)

≈ 2.31*

ne (514.5 nm)

≈ 2.28*

*from Red Optics/USA datasheet

an angle 2𝜃 ′ between them as estimated inside the sample, all expressions for Γ before should be factored by cos 2𝜃 ′ . Exercise: From data in Fig. 8.2, show that Γ ≈ 0.68 cm−1 . 8.2.2

Diffraction Efficiency

Diffraction efficiency 𝜂 is conveniently computed, from the diffracted I d and the transmitted I t irradiances measured in the output beams behind the sample, as 𝜂=

Id Id + It

(8.6)

where bulk absorption and interfaces loses are conveniently not included so that the diffraction phenomenon itself is better analyzed. This measurement procedure, however, requires one of the recording beams to be switched off during the time the transmission and diffraction of the other beam is measured. This time should be short compared to the response time of the material under analysis for the hologram not to be sensibly erased in the meantime. It is also interesting to use energy coupling experiments to measure 𝜂. To do this, it is necessary to measure the two-wave mixed irradiance along one of the directions behind the sample, let us say IS , as formulated by Eq. (8.1) √ √ IS = IS0 (1 − 𝜂) + IR0 𝜂 + 2 IS0 IR0 𝜂(1 − 𝜂) cos 𝜑 The irradiance is also measured at the moment the other beam is switched off (IS )IR0 =0 = IS0 (1 − 𝜂)

(8.7)

8.2 Direct Holographic Techniques

From these two equations, we write √ √ IS − (IS )IR0 =0 IR0 𝜂 IR0 𝜂 = 0 +2 cos 𝜑 (IS )IR0 =0 IS 1 − 𝜂 IS0 1 − 𝜂

(8.8)

From this equation, 𝜂 can be computed if one knows the input irradiance ratio IR0 ∕IS0 and the 𝜑, which is usually 0 or 𝜋 for nonphotovoltaic materials in the absence of applied field. We assumed here parallel-polarized diffracted and transmitted output beams; if this is not the case, a correction should be made. This method is very interesting for relatively thick samples because, in these cases, the lack of perfect planicity of the sample’s surfaces may lead to a lenslike effect and the transmitted and diffracted beams may be focused/defocused while going through the sample. In this case, it is difficult to compare the irradiances of the diffracted and transmitted beams, along the two different directions behind the sample, to carry out the classic measurement of 𝜂 from Eq. (8.6). This energy-transfer method instead, only requires measurement along one single beam, always using one single photodetector measuring at one single position behind the crystal. The measurement of 𝜂 for thick samples and reversible materials is discussed in detail in the Appendix. Exercise: From Eq. (8.8) and data in Fig. 8.2, compute 𝜂. Verify the compatibility between this result and the value of Γ computed in the exercise in Section 8.2.1. 8.2.2.1

Debye Length Dependence on Light Intensity

As discussed in Section D.1.1, Debye screening length ls , in general, varies with the intensity of the light. Such a variation was reported [159] on a Bi12 TiO20 crystal by measuring the gain factor Γ and the diffraction efficiency 𝜂 at different light intensities in a two-wave mixing energy exchange experiment (see example in Section 8.2.1) with K = 12.8 μm−1 , 𝛽 2 = 25 and I(0) varying from 0 to 2.5 W/m2 with 𝜆 = 532 nm, obtaining the results reported in Table 8.2. Debye length dependence on light irradiance was also reported elsewhere [38] for BaTiO3 and for undoped Sn2 P2 S6 [160] too. 8.2.3

Holographic Sensitivity

The speed at which holograms are recorded can be characterized by the so called “sensitivity” , which is defined as the refractive index time variation (∣ 𝜕Δn∕𝜕t ∣) per unit absorbed light power in the unit crystal volume (Iabs ∕d) per unit light pattern modulation (∣ m ∣) at the initial recording stage (t = 0) and in the thin crystal limit d → 0̂ [161]: [ ] 𝜕 ∣ Δn ∣ d (8.9)  = lim d→0 |m|Iabs 𝜕t t=0 From the expression of n1 = Δn in Eq. (3.54), to perform the time derivative in Eq. (8.9) and the expression for Esc (t) in Eq. (3.42) with Esc (0) = 0, and Eeff substituted from Eq. (3.170), we get Table 8.2 Debye length on illumination for Bi12 TiO20 . Experiment

𝚪

𝜼

ls0

0.026

0.044

(μm)

Isat NA ∕ND

3.6

5.8

(W/m2 )

Units

193

194

8 Holographic Techniques

the final expression for : | | E0 + Ephv + iED | | (8.10) | | 2 | 𝜏M (1 + K 2 L − iKLE − iKLphv ) | | | D In the absence of externally applied field (E0 = 0) and for a nonphotovoltaic material (Ephv = 0),  simplifies down to: | n3 reff d sI0 ∕(h𝜈) || ED |  = lim (8.11) | | 2 d→0 2Iabs sI0 ∕(h𝜈) + 𝛽 | 𝜏 (1 + K 2 L ) | M | | D ( ) 𝜎d n3 reff kB T e𝜇𝜏Φ || K + = (8.12) | | 2𝜖𝜀0 e 1 + K 2 L2D I0 𝛼 h𝜈 | with Iabs ≈ I0 𝛼 assumed to be constant throughout the thin crystal sample, 𝜏M substituted from Eq. (3.48) and ED as defined in Eq. (3.21). The plot of  as a function of K (or of the incidence angle 𝜃) is similar in shape (see Fig. 8.12) to that of Γ in Fig. 8.3, except for the fact that  is maximum for KLD = 1: ( ) n3 reff kB T 𝜎d e𝜇𝜏Φ M = (8.13) + for KLD = 1 𝜖LD 4e𝜀o I0 𝛼 h𝜈 n3 reff d sI0 ∕(h𝜈) d→0 2Iabs sI0 ∕(h𝜈) + 𝛽

 = lim

The linear plot of  in Eq. (8.12) as a function of 1∕I0 𝛼 allows computing some material parameters. In fact, the corresponding angular coefficient and independent term are, respectively n3 reff kB T K 𝜎 2𝜖𝜀o e 1 + K 2 L2D d n3 reff kB T e𝜇𝜏Φ K 2𝜖𝜀o e 1 + K 2 L2D h𝜈

(8.14) (8.15)

and from these parameters, 𝜎d and 𝜇𝜏Φ can be easily computed. Table 8.3 shows some values for the M (and also for ΓM ) for different materials, as measured by ourselves or extracted from the literature. Table 8.3 Holographic sensitivity and gain for some materials. M (10−7 cm3 ∕mJ)

𝚪M (cm−1 )

reff (pm/V)

KNSBN [162]

1.7

2.1

23.4

KNSBN:Ti* [158]

4.1

5.1

55.1

KNSBN:Cu [162]

0.6

7.2

27.4

KNSBN:Cr [162]

2.3

11

92.8

SBN:Rh [163]

0.02

70

-

BSO

20 [164]

3 [164]

5.0 [161]

BTO

7.7

8

5.6 [161]

BaTiO3 [161]

5.4

50

97+

LiNbO3 :Fe [165]

0.1

GaAs [164]

500 [161]

0.4 [166]

30.8 [1] 1.7

CdTe:V

4000 [167]

1 [168]

5.5

Unless stated otherwise, data refer to extraordinary polarization at 𝜆 = 514.5 nm except for GaAs and CdTe, which is 𝜆 = 1064 nm * described in Table 8.1; + Red Optronics Co. datasheet

8.4 Hologram Erasure

8.2.3.1

Computing 

The definition of holographic sensitivity  and the material parameters associated to it are discussed in Section 8.2.3 but a question of practical interest arises: how to actually measure it? We may do it from the evolution of the diffraction efficiency 𝜂. In fact, for the initial stage of recording when 𝜂 ≪ 1, the latter can, from Eqs. (3.53) and (3.54), be approximated to | 𝜋Δn(t)d |2 | 𝜂(t) ≈ || | | 2𝜆 cos 𝜃 ′ | and we can compute [ √ ] ] [ [ √ ] 𝜕 𝜂(t) 𝜕 𝜂(t) 𝜕 ∣ Δn(t) ∣ = 𝜕t 𝜕 ∣ Δn(t) ∣ 𝜕t t=0 t=0 [ t=0 ] 𝜕 ∣ Δn(t) ∣ 𝜋d = 2𝜆 cos 𝜃 ′ 𝜕t t=0 Therefore, a phenomenological equation for  can be written as: [ √ ] 2𝜆 cos 𝜃 ′ 𝜕 𝜂(t)  = lim d→0 |m|𝜋I0 𝛼d 𝜕t

(8.16)

(8.17)

t=0

8.3 Hologram Recording Electric coupling of charge carriers, as developed in Section 3.4.1.1, is always to be considered for holograms recorded on different LS, either involving same charge carriers (only electrons or only holes) or different ones (electrons on one LS and holes on the other). Also, when a hologram is recorded on a single LS but involves both electrons and holes, they are electrically coupled too. In any case, such coupling is to be considered for dynamics of holographic recording and/or erasing. The building up of a spatially modulated space-charge electric field is described in Chapter 3, based on a set of rate equations for excitation, retrapping, diffusion and drifting of charge carriers (electrons and/or holes). Hole-electron competition is discussed in Section 3.4.1 for the general case of running holograms and mathematically formulated in Section 3.4.1.1.

8.4 Hologram Erasure Different from for recording (see Section 8.3), erasure of holograms recorded on two different LS, both based on charge carriers of the same or of different sign (holes on one and electrons on the other) can be mathematically described by two uncoupled differential equations, one for electrons and another for holes see Eqs. (3.132) and (3.133) as reported in Section 3.4.1.1.4. This feature, together with the fact that erasure is free from environmental phase perturbations (while recording is certainly not), makes holographic erasure a very interesting technique for material characterization. In particular, for the measurement of conductivity, because the hologram characteristic response time 𝜏sc , as defined in Eq. (3.43), is proportional to 𝜏M , which according to Eq. (3.48) is inversely proportional to the conductivity 𝜎 and directly proportional to the dielectric permittivity. Therefore, aside from parameters K 2 L2D , K 2 ls2 and similar ones, the relaxation of a hologram, under the action of light or in the dark, allows computing the respective conductivities in the sample’s volume without needing electrodes at all, as far as the dielectric constant of the material is known. It is nevertheless important to emphasize that the

195

8 Holographic Techniques

two independent differential equations referred to previously do not in general allow parameters to be computed for one charge carrier separately from the other, except for very special conditions leading to Eqs. (3.145) and (3.146).

8.4.1

One Single Photoactive Center Involved

If a hologram is recorded on a single photoactive center (localized state, LS), whether one single charge carrier, electrons or holes or both are involved, a single space-charge electric field modulation is produced and diffraction efficiency 𝜂 erasure will show a single time-decaying exponential (see Section 3.4.1): 𝜂 = 𝜂0 e−2t∕𝜏sc 𝜏sc = 𝜏M

(8.18)

1 + K 2 L2D

(8.19)

1 + K 2 ls2 𝜖𝜀0 𝜏M = 𝜎d + 𝜎ph I 𝜎ph = e𝜇𝜏Φ𝛼 h𝜈

(8.20) (8.21)

as in the example of Fig. 8.4. Note that the light intensity I in Eq. (8.20) is measured outside the crystal sample but corrected for, at least, the first air–crystal interface reflection (1 − R) as well as for the angle 𝜃 (cos 𝜃) of the erasing beam after refraction at that interface. The corresponding time constant 𝜏sc is associated to the charge carrier involved and, if holes and electrons are involved, 𝜏sc will depend on the mutually charge carrier electric coupling in the process. The experimental single exponential decay shown in Fig. 8.4, for example, does not allow one to decide whether we are dealing with one single type or both types of charge carriers but it is certain that just one single LS is involved. 8.4.1.1 Bulk Absorption

In the presence of bulk light absorption, however, the intensity varies along the sample thickness so that 𝜏M and 𝜏sc in Eqs. (8.19) and (8.20) are no longer constant. Therefore, Eq. (8.18) should be now written as √ √ ∫ e−t∕𝜏sc (z) dz 𝜂 = 𝜂0 0 d d

(8.22)

80

Figure 8.4 White light hologram erasure in LiNbO3 :Fe: The erasure data (•), measured using one of the 514.5 nm recording beams, adequately fit a single exponential (dashed curve) law as described by Eq. (8.26) with a = 1.06 rad and b = 180 min.

60 ɳ(%)

196

40 20 0

0

100

200

Time (min)

300

8.5 Materials

If we assume 𝜂d ≪ 𝜂ph in Eq. (8.21), then Eq. (8.22) becomes ( )] √ [ ( ) 𝜂 √ t t Ei − 𝜂= − Ei − 𝛼d 𝜏sc (0) 𝜏 (0) e𝛼d

(8.23)

sc

with Ei (x) being the so-called “exponential integral function” [169]: ∞ e−u E(x) = du ∫x u 8.4.2

(8.24)

Two (or More) Photoactive Centers (Localized States) Involved

Different from for recording, holographic erasure involving two (or more) LSs with charge carriers of the same or different signs, are described by two independent (uncoupled) differential equations, one for each LS, as reported in Section 3.4.1.1.4. Erasure will, therefore, show an overall amplitude evolution of the form (see Section 3.4.1.1.4) A e−ra t + B e−rb t

(8.25)

where the coefficients ra and rb depend on the electrical coupling of charge carriers in both LS as well as on the relaxation time constants of charge carriers in each on of both LS. For particular conditions, however (see Section 3.4.1.1.4), it will be possible to separately compute the individual time constants. But in general, we should be aware of the complex problem represented by the presence of more than one photoactive center and more than one type of charge carrier involved. 8.4.2.1

Same Charge Carriers

If the same charge carriers (either electrons or holes) are involved in both (assuming two) LS, then in-phase holograms will result, with A and B in Eq. (8.25) being of the same sign, and erasure will look like the experimental curve reported in Fig. 8.6 for Pb-doped Bi12 TiO20 under 𝜆 = 633 nm illumination. 8.4.2.2

Holes and Electrons on Different Photoactive Centers

If charge carriers in both (always assuming two) LS are of opposite signs (electrons on one LS and holes on the other), the two corresponding holograms will be π-shifted (see Section 3.4.1.1.4) with A and B in Eq. (8.25) of the opposite sign. In this case, erasure will look like the examples of the curves in Figs. 8.5, 8.7, 8.8 and 8.10 showing local maxima and/or minima.

8.5 Materials This section reports some examples of recording and/or erasing of holograms in some materials to illustrate the way these processes may help to find out the photoactive centers and charge carriers involved as well as to give some information about the materials themselves. 8.5.1

Fe-doped LiNbO3 : Hologram Erasure under White Light Illumination

Figure 8.4 shows the white-light erasure of a previously recorded (using 514.5 nm light) hologram in Fe-doped LiNbO3 crystal. Diffraction efficiency measurement during erasure was carried out using short pulses of one of the 𝜆 = 514.5 nm wavelength recording beams, short

197

8 Holographic Techniques

0.3 476 nm 634 nm 670 nm 593 nm 524 nm

η (au)

0.2

0.1

0 0

50

100

150

200

250

Time (s)

Figure 8.5 The graph shows the erasure of holograms in undoped BTO under 10–15 min ≈1 mW/cm2 pre-illumination with light of different wavelengths as indicated in the graph. The recording and erasure were always carried out with 𝜆 = 780 nm. Measurement along the other direction behind the crystal showed similar shapes. Erasure curves are artificially shifted in time for better observation.

Figure 8.6 Hologram diffraction efficiency (arbitrary units) decay during 𝜆 = 633 nm light erasing of a hologram previously recorded with the same light on a Pb-doped Bi12 TiO20 (BTO:Pb) crystal. Erasure monotonically decreases and adequately fits the double exponential in Eq. 8.27 leading to A1 = 0.37, A2 = 0.28, 𝜏sc1 = 34.0 s, 𝜏sc2 = 5.47 s and background light C = 0.0078.

04 03 I d (au)

198

02 01

–50

0 Time (s)

50

enough not to sensibly interfere in the erasure. In this case, 𝜂 is rather large and its expression is the one reported in Eq. (3.53), with the time-evolution being described by 𝜂 ∝ sin2 [a(1 − e−t∕b )]

(8.26)

The rather good data fit to a monoexponential expression in Fig. 8.4 indicates that a single photoactive level is actually involved here.

8.5 Materials

0.45 1

2

η (au)

0.30

0.15

0

0

50

100

150

200

Time (s)

Figure 8.7 Diffraction efficiency (𝜂 in arbitrary units) during erasure of a hologram in a Pb-doped BTO (same sample as in Fig. 8.6) measured along both directions (along the reference beam and along the signal beam) at the crystal output. Both erasure curves (squares and circles) are artificially shifted in time for better observation. The crystal was pre-exposed for a few minutes to a uniform light at 𝜆 = 532 nm. Pre-exposure was switched off immediately before holographic recording started using an He-Ne laser line of 𝜆 = 633 nm. The hologram was erased with one of the in-Bragg recording beams. No external electric field was applied. Experimental data were fitted (continuous curves) with Eq. (8.28) and the resulting parameters reported in Table 8.4.

8.5.2

Bi12 TiO20 (BTO)

Hole-electron competition has been detected on undoped sillenites, mainly Bi12 TiO20 . Such competition seems to be enhanced by some dopants such as Pb (see Section 8.5.2.2) and V (see Section 8.5.2.3). 8.5.2.1

Undoped BTO under 𝝀 = 780 nm Illumination

Figure 8.5 shows the erasing of holograms recorded with low photonic energy 𝜆 = 780 nm (h𝜈 = 1.6 eV) light and pre-exposed (for 10–15 min) with spatially uniform light of different wavelengths. Holographic recording and erasing without pre-exposure leads to a mono-exponential slow decaying curve, similar to the monoexponentials in Fig. 8.8 for nonpre-exposed Pb-doped BTO, which is likely to be a hole-based (because being slow) hologram recorded on the [Bi3+ +h+ ] center at the Fermi level at ≈ 1 eV above the VB. Pre-exposure recording with low energy photons h𝜈 ≤ 1.85 eV (𝜆 ≥ 670 nm) apparently has no effect since diffraction efficiency erasure is always a mono-exponential monotonically decreasing curve. For pre-exposure in the range h𝜈 = 2.6 down to at least 2 eV, erasure curves show the typical pattern of hole-electron competition with a hole-based and an electron-based hologram on two different LS, as also shown in Fig. 8.8. It is then evident that h𝜈 ≈ 2 eV is the minimum photonic energy required to fill in electrons into LS at the 1.6 eV (or lower) energy gap from the bottom of the CB to allow for an electron-based hologram to be recorded by the low photonic energy 𝜆 = 780 nm beams besides the always-present hole-based one. Recording with 𝜆 = 1064 nm (h𝜈 = 1.17 eV) was unsuccessful whatever the pre-exposure illumination used either for undoped or for Pb-doped BTO, maybe because it was too close to the limit of the 1 eV energy gap from Fermi level to the top of the VB. Note that direct recording and erasure with the more energetic light (e.g. 𝜆 = 514.5 nm or 63 nm) always shows a monotonically decaying erasure curve, for undoped BTO, no matter if there is pre-exposure or not, and independently of the photonic energy of any pre-exposure. We should therefore conclude that more energetic radiation does record a holeand electron-based hologram, both on the same LS. Similar results were already reported [170] as well as for Bi12 TiO20 , but for 𝜆 = 1064 nm recording and white light pre-exposure.

199

8 Holographic Techniques

Diffracted light (au)

200

0.15

0.15

0.10

0.10

0.05

0.05

0

0

50

100 Time (s)

150

200

0 0

50

100 Time (s)

150

200

Figure 8.8 Erasure of holograms in Pb-doped BTO (same sample as in Fig. 8.6) recorded over 2 min with a diode laser of 780 nm wavelength, observed along the reference beam direction (left-hand graph) and along the signal beam (right-hand graph) using one of the recording beams. Curves showing a local maximum result from 3 min pre-exposure at 𝜆 = 524 nm (h𝜈 ≈ 1.37 eV) light from a LED and were fitted with Eq. (8.28) leading f s f to a fast grating characteristic time of 𝜏sc ≈ 13 − 16 s and a corresponding value 𝜏sc ≈ 35𝜏sc for the slow grating. The monotonically decreasing curves were not pre-exposed and actually verify a monoexponential law with a 𝜏sc ≈ 100 s. Reproduced from [29].

8.5.2.2 Bi12 TiO20 :Pb (BTO:Pb)

Doping with M2+ does not affect photoactive centers much as shown for Pb-doped BTO in Table 7.7 where its photoconductivity is not appreciably different from that of the undoped material. Also, this Pb2+ dopant seems not greatly to reduce Bi3+ in the formula of Eq. (2.6) because holes and electrons are clearly being excited with h𝜈 = 1.6 eV (𝜆 = 780 nm) light from different centers (probably Bi3+ for electrons and [Bi3+ + h+ ] for holes) as shown by the holographic erasure curve in Fig. 8.5. It seems that Bi3+ centers are normally empty (and must be filled by optical pumping) because the experiment referred to in Fig. 8.7 requires the sample to be pre-exposed (at h𝜈 ≈ 2.3 eV) before hologram recording (and erased) with h𝜈 ≈ 2 eV light. Without pre-exposure, a single charge carrier excited from different centers is involved as shown in Fig. 8.6. Recording with h𝜈 ≈ 1.6 eV and pre-exposing at h𝜈 ≈ 2.4 light shows (see Fig. 8.8) electrons and holes excited from different centers, but without pre-exposure only one center is involved, probably [Bi3+ + h+ ], because only slow (hole-based) holograms appear. The erasure of holograms recorded on a Pb-doped BTO is studied here. As mentioned in Section 3.4.1, a combined hologram, made of electrons and holes excited from the same localized state in the Band Gap, produces a single hologram because just one photoactive center will be modulated by the action of light (one single hologram built up) and the erasure will consequently show a single decaying exponential. If electrons and holes are excited from different LSs instead, two different holograms will be recorded, one based on electrons and the other on holes that are partially or totally shifted, and the erasure will look like Fig. 8.8, producing two decaying (an electron-based one usually faster and a hole-based one usually slower) exponential terms.

8.5 Materials

Under certain conditions (see Section 3.4.1.1.4), erasure has the characteristic shape shown in Fig. 8.6 because of the presence of two distinct holograms with wide different characteristic exponential decay times: the faster time approximately corresponds to that of electron-based space charge build-up time 𝜏SC1 (see Eq. (3.145)) whereas the slower one 𝜏sc2 ∕(1 − 𝜅12 𝜅21 ) (see Eq. (3.146)) depends on the hole-based space charge build-up time (𝜏SC2 ) and the coupling constants 𝜅12 and 𝜅21 . BTO:Pb under 𝝀 = 633 nm Light Holographic recording and erasing with 𝜆 = 633 nm (h𝜈 = 1.96 eV) on Pb-doped BTO shows a decaying curve in Fig. 8.6 that better fits a double exponential of the form 𝜂 = |A e−t∕𝜏sc1 + A e−t∕𝜏sc2 |2 + C (8.27)

8.5.2.2.1

1

2

than a single one. This indicates that two photoactive centers are involved with same charge carriers. BTO:Pb Recorded with 𝝀 = 633 nm and 𝝀 = 532 nm Pre-exposure Pre-exposure with 𝜆 = 532 nm (h𝜈 ≈ 2.3 eV) before recording and erasing with 𝜆 = 633 nm produces electron- and hole-based holograms on different LS as shown in Fig. 8.7. Pumping with 2.3 eV involves energy large enough to fill in intermediate LS from the Fermi level to allow the 𝜆 = 633 nm light to record an electron-based hologram in these filled LS and record an another hole-based one on the [Bi3+ +h+ ] centers. Note that a combined photorefractive (refractive index modulation) plus photochromic (absorption modulation) holograms may produce a pattern of erasure similar to the one shown in Fig. 8.7 in one of the directions behind the crystal but not in both (see Eqs. (4.28)–(4.30)), which is the case for hole-electron competition. The curves in Fig. 8.7 show the erasure of previously recorded holograms using 633 nm wavelength beams, previously exposed to uniform 532 nm light. These curves were fitted with

8.5.2.2.2

f

s 𝜂 = |Af e−t∕𝜏sc ei𝜑 − As e−t∕𝜏sc |2

(8.28)

that is derived from Eq. (3.137), where Af and As are the amplitudes of the fast and the slow f s holograms, 𝜏sc and 𝜏sc are their respective holographic response time. Note the negative sign indicating the 𝜋-phase shift between both gratings due to the opposite electric charge involved in each one, plus the arbitrary phase shift 𝜑 probably arising from environmental perturbations on the setup during recording. The results from fitting on curves 1 and 2 are shown in Table 8.4. The same experiment (recording and erasure with 𝜆 = 633 nm and pre-exposure with 𝜆 = 532 nm) on undoped BTO always led to only a monotonically decreasing 𝜂. 8.5.2.2.3

BTO:Pb Recorded with 𝝀 = 780 nm Illumination With and Without 𝝀 = 524 nm Pre-exposure

Figure 8.8 shows the effect of short wavelength pre-exposure on holographic recording (and erasure) on BTO:Pb (same sample as in Fig. 8.6) with 𝜆 = 780 nm. Table 8.4 Hole-electron competition in BTO:Pb – data from Fig. 8.7. Curve

Af (au)

As (au)

As ∕Af

f 𝝉sc (s)

s 𝝉sc (s)

s f 𝝉sc ∕𝝉sc

𝝋 (rad)

1

0.72

0.35

0.49

3.2

172

54

0.89

2

0.71

0.36

0.51

3.6

184

51

0.87

201

8 Holographic Techniques

8.5.2.3 Bi12 TiO20 :V (BTO:V)

The recording (Fig. 8.9) and erasure (Fig. 8.10) of a hologram on V-doped BTO, using 𝜆 = 514.5 nm laser beams show a strong hole-electron competition with a fast hologram and a slower one, probably electron and hole-based ones, respectively, the latter being sensibly larger in size with holes and electrons on different photoactive centers and spatially separated too. Hologram Recording Holographic recording on vanadium-doped BTO (up to ≈ 0.1% in weight of vanadium) exhibits an interesting example of strong hole-electron competition with electrical coupling. Vanadium produces photosensitive centers in the Band Gap (BG) at less than 1.64 eV from the bottom of the CB. It also exhibits a strong hole-electron competition, with holes predominating over electrons and being generated from different centers in the Band Gap [30]. It was also reported [171] that V in BTO is in its 5+ valence state and occupies tetrahedral sites. Figure 8.9 shows fitting of experimental diffraction efficiency data for this material to Eq. (8.29) that arises from Eq. (4.124), taking into account that the resulting space-charge electric field here is the one arising from hole-electron competition with electrical coupling as reported in Eqs. (3.123) and (3.124), so we should write: ( )2 𝜋n3 reff 2 2 2 st st 2 |m|2 ||Eeff |2 = |Est |2 = |Esc1 + Esc2 | lim 𝜂 = |m| |Eeff | d 𝜂→0 2𝜆 cos 𝜃 st st 2 ∝ |Esc1 + Esc2 | (8.29)

8.5.2.3.1

This fitting clearly demonstrates the importance of adjusting the parameters 𝜁1,2 (see Section 3.4.1.1) of the coupling coefficients for adequately representing this material behavior. Hologram Erasing Figure 8.10 shows the typical diffraction efficiency decaying for hologram with hole-electron competition arising from holes and electrons on two different photoactive centers (LS), which follows the mathematical formulation in Eq. (8.28) but including a background term C:

8.5.2.3.2

f

s 𝜂(t) = |Af e−t∕𝜏sc ei𝜑 − As e−t∕𝜏sc |2 + C

(8.30)

4 3

η(10–6)

202

2 1 0

0

2

4

6

8

10

12

E0 (kV/cm)

Figure 8.9 Diffraction efficiency (recorded and measured using 𝜆 = 514.5 nm laser beams [87]) as a function of the applied electric field measured (•) on a V-doped BTO (0.30% V in weight) with hole-electron competition. The continuous curve is the theoretical fit (a single factoring parameter in ordinates was used for data fitting) assuming hole- and electron-charge carriers from different photoactive centers with ls1 = 0.164 μm, 𝜁1 = 0.99, ls2 = 0.163 μm and 𝜁2 = 0.88. The dashed curve is for 𝜁1 = 𝜁2 = 1 (see Eqs. (3.125) and (3.126)), which represents holes and electrons at the same position in space, all other parameters unchanged.

8.5 Materials

η (au) 0.06 0.05 0.04 0.03 0.02 0.1

0.2

0.5

1

2

5

10

20

Time (s)

Figure 8.10 Diffraction efficiency (au) as a function of time (seconds, in logarithmic scale) (•) during erasure with 𝜆 = 514.5 nm light of a hologram recorded on BTO:V using same wavelength coherent laser beams, without externally applied electric field. Curve fitting to Eq. (8.30) leads to: Af = 0.17, As = 0.25, 𝜏f = 0.28 s, 𝜏s = 20 s and background C = 0.011. Reproduced from [30].

with the simplifying assumptions in Eqs. (3.142)–(3.144): e 𝜏f ≈ 𝜏sc

(8.31) [

e e 𝜏sc ≡ 𝜏M

1 + K 2 L2D 1 + K 2 ls2

] ≈ e

𝜖𝜀0 𝛼d [1 + K 2 L2D ]e −𝛼d 𝜎e (0)(1 − e )

𝜏sch 𝜏s ≈ 1 − 𝜅12 𝜅21 [ ] 1 + K 2 L2D 𝜖𝜀0 𝛼d h h h ≈ 𝜏M 𝜏sc ≡ 𝜏M = 2 2 1 + K ls 𝜎h (0)(1 − e−𝛼d )

(8.32) (8.33) (8.34)

with 𝜎(0)(1 − e−𝛼d )∕d in Eqs. (8.32) and (8.34) being the average of 𝜎(z) throughout the sample’s thickness. From Fig. 8.10, the sensitivity  was computed (see Section 8.2.3.1) as well as the fast-to-slow sensitivity ratio f ∕s from Eq. (8.30) as f ∕s =

Af ∕𝜏f As ∕𝜏s

(8.35)

for BTO:V and similarly for other doped and undoped BTO, and is reported in Table 8.5. The relative photoconductivity 𝜎(0)h𝜈∕I(0) is also reported as computed from Fig. (8.10) and from wavelength-resolved photoconductivity (WRP) experiment. Note that the quantity 𝜎(0)h𝜈∕I(0) is very different as measured from holographic and from WRP experiments, and this is certainly due to the way 𝛼 is measured in both experiments, with a photodetector very close to the sample back surface in WRP experiments, thus allowing for some luminescence light to be included; which is not the case for holographic experiments, so we should conclude that the latter is a far more reliable technique here. 8.5.2.4

Holographic Relaxation in the Dark: Dark Conductivity

Hologram relaxation in the dark depends on the dark conductivity 𝜎d that is an energy-barrier controlled phenomenon that follows an Arrhenius-type exponential law. The 𝜏sc ∝ 𝜎d is easily

203

8 Holographic Techniques

Table 8.5 Sensitivity and relative photoconductivity: doped and undoped BTO at 𝜆 = 514.5 nm. BTO:V



Units

Fast

10−10 m3 /J

32

f ∕s

Slow

BTO

BTO:005*

BTO:Pb

0.65

53

52

60

—

60

95

60

180

—

80

49

𝜎(0)h𝜈∕I(0)

10−30 s m/Ω

HOLO.†

27

‡

WRP

9

From [30] † : Holographic techniques; ‡ : Wavelength-resolved photoconductivity * produced in the former USSR

100

𝜏S0 (min)

204

10

1 3.0

2.8

3.2 100/T (K–1)

Figure 8.11 Hologram relaxation in the dark: exponential time as a function of inverse temperature for hologram relaxation in the dark. The hologram was recorded using 𝜆 = 514.5 nm light onto an undoped BTO sample (BTO-8) approximately 1 mm thick. Diffraction efficiency was measured from time to time using one of the in-Bragg recording beams during a very short time and correcting data for the effect of exposure to light. From the Arrhenius-type curve, an activation energy of 1.04 eV was computed.

measured during dark holographic relaxation and obviously follows the Arrhenius law too: 𝜏sc =

0 𝜏sc

Ea k e BT −

(8.36)

that allows computing the energy of the barrier-controlling energy Ea as illustrated in Fig. 8.11 for a hologram recorded on undoped BTO (sample BTO-8) using 𝜆 = 514.5 nm (h𝜈 = 2.41 eV) laser light. The result of 1.04 eV here is close to the p-type dark conductivity already measured on this material using purely electric techniques, and is close to the energy gap between the Fermi level and the top of the VB, thus indicating that the decaying hologram was recorded on the Fermi level that is known to be 2.2 eV below the bottom of the CB.

8.6 Phase Modulation Techniques

The practical measurement of the evolution of 𝜂 in the dark that is required here is somewhat complicated by the fact that light is always required to measure 𝜂. It is always possible to use very weak and very short light pulses at sufficiently large intervals and always correcting the effect of these short pulses of light. It is not recommended to use an auxiliary beam of different 𝜆 (usually a much larger one) to measure 𝜂 because it is difficult to match Bragg conditions between the beam and the grating being measured on one side, and because no one knows what might be the effect of such wavelength radiation on the complex nature of some materials.

8.6 Phase Modulation Techniques In this section, we shall describe some examples illustrating the possibilities of phase modulation (described in Section 4.3) for materials characterization. The use of phase modulation has two main advantages: • It is a real-time nonperturbating technique that allows to carry out measurements without disturbing the recording itself. • It allows, in general, to operate in self-stabilized recording mode where either the first (I Ω ) or the second (I 2Ω ) harmonic term is used as the error signal for operating the stabilization, in which case the other harmonic is available for measurement. In this condition, the phase shift 𝜑 between the transmitted and the diffracted beams along the same direction at the sample output is known a priori because it is fixed by the stabilization loop. 8.6.1

Holographic Sensitivity

The relevance of holographic sensitivity () both for material research and applications has been already discussed in Section 8.2.3. We will just point out here the interest of phase-modulation techniques for the practical measurement of this quantity [158]. In Section 8.2.3, we have shown how it is possible to compute  from the evolution of 𝜂. Here, we shall show how to compute  from the time evolution of the I 2Ω -signal, when the I Ω is used as an error signal in a self-stabilization setup, in which case we should write ] [ √ ] [ [ 2Ω ] 𝜕 𝜂 𝜕I 𝜕I 2Ω = (8.37) √ 𝜕t t=0 𝜕 𝜂 t=0 𝜕t t=0 √ where 𝜕I 2Ω ∕𝜕 𝜂, is computed from Eq. (4.173) as ] [ √ 𝜕I 2Ω = 4J2 (𝜓d )J0 (𝜓d ) ISo IRo (8.38) √ 𝜕 𝜂 t=0 where 𝜂 in Eq. (4.173) was substituted by 𝜂J02 (𝜓d ) to account for the effect of pattern-of-fringes vibration onto 𝜂, so that 𝜂 represents here its nonperturbed value. Also, we substitute 𝜑 = 0 into the expression for I 2Ω because the setup is self-stabilized. Self-stabilization strongly contributes to reduce phase perturbations and improve measurement dispersion but one should keep in mind that the 𝜑 = 0 value fixed by self-stabilization should correspond to the unconstrained recording condition, because otherwise the recording would be modified by self-stabilization itself, as already discussed in Chapter 6. Substituting Eqs. (8.38) and (8.16) into Eq. (8.37) we get ] [ [ 2Ω ] √ 𝜕 ∣ Δn ∣ 𝜋d 𝜕I 0 0 = 2J2 (𝜓d )J0 (𝜓d ) IS IR (8.39) 𝜕t t=0 𝜆 cos 𝜃 ′ 𝜕t t=0

205

8 Holographic Techniques

Figure 8.12 Photorefractive sensitivity  data (∘) as a function of the external incidence angle 𝜃 for the KNSBN:Ti sample of Table 8.1 in the same optical and recording configuration as in Fig. 8.3. From these data we compute LD = 0.18 μm.

5 × 10–7

S (cm3/mJ)

4 × 10–7 3 × 10–7 2 × 10–7

SM = 4.1 × 10–7 cm3/mJ

θM = 13.0°

1 × 10–7 0

0

10

20 θ (degrees)

30

40

8 V2Ω (V)

206

Figure 8.13 Second harmonic evolution for KNSBN:Ti for the same sample and experimental conditions as for Fig. 8.12 with 𝜃 = 15o and IS0 + IR0 ≈ 3 mW/cm2 .

6 4 2 0

0

100

200

300

400

500

t (s)

From Eqs. (8.39) and (8.9) and the definition of |m|, we get [ 2Ω ] 𝜆 cos 𝜃 ′ 𝜕I 1 = √ 𝜋𝛼I0 d 𝜕t t=0 0 0 4J2 (𝜓d )J0 (𝜓d ) IS IR

(8.40)

The use of I 2Ω therefore allows measuement of the sensitivity during holographic recording without perturbing it. Figure 8.12 shows the experimental sensitivity measured at 𝜆 = 514.5 nm for the same KNSBN:Ti sample reported in Table 8.1, and same optical configuration and experimental setup as for Fig. 8.3. In this case, however,  was computed from Eq. (8.40) and the evolution of I 2Ω for the limit of t → 0. A typical experimental plot of V 2Ω ∝ I 2Ω for this material is shown in Fig. 8.13 where the low data dispersion is due to the fact that the setup was self-stabilized using the first harmonic term I Ω as the error signal, as will be discussed further in Chapter 9. From data in Fig. 8.12, the maximum sensitivity M = 4.1 × 10−7 cm3 /mJ and corresponding LD = 0.18 μm are computed. In this case, as for the case of Γ in Section 8.2.1, all previously expressions for  should be also factored by cos 2𝜃 ′ . 8.6.2

Holographic Phase-Shift Measurement

The explicit formulation of the phase shift 𝜙P of the stationary space-charge electric field grating in Eq. (3.56) allows one to compute ls from the experimental plot of tan 𝜙P versus E0 . The parameter ls is related to the effective density of traps as indicated by Eq. (3.49). The measurement of 𝜙P itself is also of relevance because it allows one to know whether we are dealing with a purely diffusion-arising recording mechanism (𝜙P = ±𝜋∕2), a photovoltaic material (𝜙P ≈ 0 or 𝜋) or some mixed effects. In fact, this technique has been already used to show that Fe-doped

8.6 Phase Modulation Techniques

BaTiO3 crystals exhibit photovoltaic effects [172]. The problem is that 𝜙P is hardly directly available [111, 173] from the experiments. One possibility is to compute 𝜙 from the output beam phase shift 𝜑. In the absence of wave mixing, the simple relation in Eq. (4.178) 1 = − tan 𝜑 tan 𝜙 does hold, where tan 𝜑 can be computed from Eq. (4.174), so we should write ISΩ J2 (𝜓d ) 1 = 2Ω tan 𝜙 IS J1 (𝜓d )

(8.41)

where the quantities on the right-hand side in Eq. (8.41) are measured in a phase modulated two-wave mixing experiment. Note that self-stabilization is not recommended here because it will, in general, affect the recording process and consequently the phase shift would be also affected. It is possible, however, to stabilize the recording pattern of fringes using the interference of the transmitted and reflected beams on a small glassplate tightly fixed by the side of the sample under analysis, as discussed in Chapter 9. 8.6.2.1

Wave-Mixing Effects

In the presence of self-diffraction effects, however, the simple relation in Eq. (4.178) does not hold. The much more complicated relation in Eq. (4.128) should be used instead, where 𝜙P is implicitly indicated by the parameters Γ and 𝛾 and their relation tan 𝜙P = Γ∕𝛾. Figure. 8.14 clearly illustrates this point: the directly experimentally obtained tan 𝜑 versus E0 data (squares) are plotted; the −1∕ tan 𝜙 computed from tan 𝜑 using Eqs. (4.128), (4.85), (4.90), (4.86), (3.44) and (4.91) is also plotted in the same figure. We see that, except for low E0 , tan 𝜑 ≠ −1∕ tan 𝜙, as discussed previously. The continuous curve is the fitting of Eq. (3.56), which is clearly in good agreement with the −1∕ tan 𝜙 data. From this fit we get ls . 8.6.3

Photorefractive Response Time

It is possible to use phase modulation techniques in two-wave mixing (TWM) experiments for hologram response time measurement, as an alternative to conventional hologram erasure techniques [174]. The basic idea in this method is simple: a small phase modulation in one of the interfering beams makes the interference pattern on the crystal to vibrate with the modulation frequency as described in Section 4.3. For a frequency much smaller than the frequency response of the crystal, the recorded hologram vibrates synchronously with the pattern of light and no modulation signal is detected along any of the two-wave mixed beams behind the crystal. For a much higher modulation frequency instead, the hologram is comparatively too slow to Figure 8.14 Evolution of the −1∕ tan 𝜙 accounting on self-diffraction effects as described in the text (∘), as a function of the applied field E for a 2 mm thick nominally undoped Bi12 TiO20 sample with K = 7.08 μm−1 , 𝛽 2 = 9 and I0 ≈ 4 mW∕cm2 using the 514.5 nm wavelength laser line. The crystal is in a transverse electro-optical configuration with the (110)-plane perpendicular to the incidence plane and the [001]-axis perpendicular to the grating ⃗ Data fitting leads to l = 0.027 μm. In the same figure, (◽), vector K. s the directly measured tan 𝜑 is plotted.

10 8

tan φ

6 4

–1/ tan ϕ

2 0

0

2

4 6 E (kV/cm)

8

10

207

208

8 Holographic Techniques

move while the pattern oscillates, and therefore the resultant modulation signal is independent from the crystal response and dithering frequency. For intermediate frequency values, the modulation signal behind the crystal depends on the crystal response: from these data, its response time may be obtained. In a TWM setup such as the one depicted in Fig. 4.19, one of the interfering beams (S in this case) is phase modulated with angular frequency Ω and phase amplitude 𝜓d . The interference pattern of light onto the crystal therefore vibrates with frequency Ω as described before in Eq. (4.151) I(x, t) = I0 + I0 ∣ m ∣ cos(Kx + 𝜙 + 𝜓d sin Ωt) The light pattern modulation ∣ m ∣ is assumed to be constant throughout the sample thickness. The time evolution of the space-charge electric field modulation amplitude Esc , for a purely diffusion-arising (that is, without external field E0 = 0) photorefractive hologram produced by this pattern of light is ruled by Eqs. (3.42)–(3.44) 𝜕Esc Esc im(t)ED =− + 𝜕t 𝜏sc 𝜏sc (1 + K 2 ls2 )

(8.42)

with E0 = 0. Assuming a pattern of light vibrating with frequency Ω and amplitude 𝜓d : m(t) = ∣ m ∣ exp(𝚤𝜓d sin Ωt) = ∣ m ∣

∞ ∑

JL (𝜓d ) exp 𝚤(LΩt + 𝜙)

(8.43)

L=−∞

and substituting m(t) in Eq. (8.42) for its expression in Eq. (8.43), we find the solution for Esc (t): Esc (t) = −𝚤 ∣ m ∣

∞ ∑

ED 1+

K 2 ls2 L=−∞

JL (𝜓d ) exp 𝚤(LΩt) 1 + 𝚤LΩ𝜏sc

(8.44)

For a GaAs crystal with its [001]-axis perpendicular to the incidence plane, the [110]-axis parallel to K⃗ and the interfering beams being linearly polarized along the direction of the [001]-axis, as depicted in Fig. 8.15, the diffracted beams behind the crystal are linearly polarized along the ⃗ K-direction, which is orthogonally polarized to the transmitted beams [58, 175]. The overall irradiance along the direction S behind the crystal may be therefore written as √ IS ≈ ∣ ŝS0 exp 𝚤(𝜓d sin Ωt) + 𝚤̂rR0 𝜂 e𝚤(𝜑 − 𝜋∕2) cos 𝛾∣2 (8.45) ŝ and r̂ are unit vectors parallel to the polarization of the transmitted and diffracted beams, respectively, with ŝ.̂r = 0 in this particular experiment. For low diffraction efficiency ∣ 𝜂 ∣ ≪ 1 and from Eq. (4.124) it is √ 𝚤(𝜑 − 𝜋∕2) 𝜋n1 d 𝜂e ≈m and mn1 = −n30 reff Esc ∕2 𝜆 cos 𝜃 [001]

(8.46)

K

[110]

Figure 8.15 Two-wave mixing experiment in a photorefractive GaAs intrinsic crystal with mutually orthogonally polarized diffracted and transmitted beams. The polarization direction is represented by the black arrows: the input and transmitted beam polarization is along the [001]-axis, whereas the diffracted is perpendicular to the [001]-axis.

8.6 Phase Modulation Techniques

From Eq. (8.44), n1 can be written as n1 =

∞ ∑ JL (𝜓d ) ED i 3 exp 𝚤(LΩt) n0 reff ∣ m ∣ 2 2 2 1 + K ls L=−∞ 1 + iLΩ𝜏sc

(8.47)

where n0 is the bulk index-of-refraction and reff is the electro-optic coefficient. Substituting the modulated beam in Eq. (8.45) with its Bessel development ∞ ∑

S0 exp(𝚤𝜓d sin Ωt) = S0

JN (𝜓d ) exp(iNΩt)

(8.48)

N=−∞

and inserting Eqs. (8.47) and (8.48) into Eq. (8.45), the resultant expression is developed into IS = C +

𝜋dn30 reff 𝜆 cos 𝜃

∞ ∑

∣m∣

ED 1 + K 2 ls2

JN (𝜓d )JL (𝜓d )

N,L=−∞

S0 R0 ŝ.̂r ×

LΩ𝜏sc sin[(N − L)Ωt] − cos[(N − L)Ωt] 1 + (LΩ𝜏sc )2

(8.49)

where C is a constant. For 𝜓d ≪ 1, Eq. (8.49) may be limited to the second-order Bessel function, in which case we should also substitute the Bessel functions for their approximate expressions: J1 (𝜓d ) ≃ 𝜓d ∕2

J0 (𝜓d ) ≃ 1

J2 (𝜓d ) ≃ 𝜓d2 ∕8

(8.50)

In this case, the final expression for the second harmonic in Ωt is obtained [IS2Ω ]n1 = |[IS2Ω ]n1 | cos(2Ωt + 𝜙n1 ) |[IS2Ω ]n1 | = |m|S0 R0 ŝ.̂r𝜓d2 tan 𝜙n1 =

𝜋dn3o reff ED 2𝜆 cos 𝜃(1 + K 2 ls2 )

2(Ω𝜏sc )2 − 1 3Ω𝜏sc

(8.51) √

2 Ω2 𝜏sc 2 2 (1 + Ω2 𝜏sc )(1 + 4Ω2 𝜏sc )

(8.52) (8.53)

From Eq. (8.52), the response time 𝜏sc may be computed. Experimental results for a GaAs semi-insulating crystal are shown in Fig. 8.16. A similar approach may be used for computing the amplitude of the first harmonic for an index-of-refraction (of amplitude n′1 ) grating resulting from the photoactive centers modulation during photorefractive recording (that is different from the photorefractive index-of-refraction grating) and is in-phase with the pattern of light onto the sample. In this case, we should assume that n′1 ∝ ND+ A(t)

(8.54)

where ND+ A(t) is related to Esc (t) in Eq. (3.39) 𝚤K𝜖𝜀o Esc (t) ≈ eND+ A(t) It is therefore enough to substitute Esc (t) in Eq. (8.44) into this expression to get n′1

∝

ND+ A(t)

∞ ∑ JL (𝜓d ) 𝜖𝜀0 kB T K 2 =−∣m∣ exp 𝚤(LΩt) e e 1 + K 2 ls2 L=−∞ 1 + 𝚤LΩ𝜏sc

(8.55)

with √

𝜂n′1 e

𝚤𝜑n′1

∝−∣m∣

∞ ∑ JL (𝜓d ) 𝜋d 𝜖𝜀0 kB T K 2 exp 𝚤(LΩt) 2 2 𝜆 cos 𝜃 e e 1 + K ls L=−∞ 1 + 𝚤LΩ𝜏sc

(8.56)

209

8 Holographic Techniques

40

Figure 8.16 Second harmonic response curves for an undoped semi-insulating GaAs crystal illuminated with a 1.06 μm laser wavelength line, with |m| = 1 and an angle 𝛾 = 10∘ between the transmitted beam ⃗ polarization direction and the grating vector K. Theoretical fit to data (◽) for K = 3.5 μm−1 and I0 ≈ 118 mW/cm2 lead to 𝜏sc = 0.22 ms; fit to data (∘) for K = 2.1 μm−1 and I0 ≈ 168 mW/cm2 lead to 𝜏sc = 0.1 ms.

30 I 2Ω (a.u.)

210

20 10 0

0

1000

2000

3000

4000

5000

Modulation frequency Ω/2π (Hz)

Processing in a similar to that before, but for the first harmonic in Ωt, we get an expression for an in-phase index-of-refraction grating ∣ [ISΩ ]n′1 ∣∝ 2𝜓d ∣ m ∣ S0 R0

Ω𝜏sc 𝜖𝜀o kB T K 2 𝜋d √ q q 1 + K 2 ls2 𝜆 cos 𝜃 1 + Ω2 𝜏 2 sc

(8.57)

The same procedure as before is employed for computing ∣ IS2Ω ∣ for a pure amplitude (photochromic) grating in-phase, associated with the photorefractive effect (that is, arising from photoactive trap absorption modulation). In this case, √ 𝜂A e𝚤𝜑A =

𝛼1′ d 2 cos 𝜃

for

∣ 𝜂A ∣ ≪ 1

with 𝛼1′ ∝ ND+ A(t) and the final result is 2 𝜓2 𝜖𝜀 k T K 2 Ω2 𝜏sc d ∣ [IS2Ω ]𝛼1′ ∣ ∝ d ∣ m ∣ S0 R0 o B √ 2 q q 1 + K 2 ls2 cos 𝜃 (1 + 4Ω2 𝜏 2 )(1 + Ω2 𝜏 2 ) sc sc

(8.58) (8.59)

(8.60)

Note that for the case of a nonphotorefractive index-of-refraction and an absorption gratings, there is no anisotropic diffraction so the transmitted and diffracted beams are always parallel polarized and ŝ.̂r = 1, which is not the case for the photorefractive index-of-refraction grating. 8.6.4

Selective Two-Wave Mixing

Selective two-wave-mixing (S2WM) [176] is a simple two-wave mixing (TWM) experiment where the irradiance levels along both directions behind the crystal are simultaneously measured using the same phase-modulation techniques described previously. This method takes advantage of differences in symmetry properties of amplitude and phase gratings, concerning energy exchange as described in Section 4.2.1. In fact, any phase shift of the pattern of light referred to a phase grating produces an asymmetric change in the energy of the beams along both directions behind the sample: if the intensity increases in one direction, it necessarily decreases in the other, due to energy conservation. For an amplitude grating instead, energy conservation does not verify, and the intensity change in both directions is the same: a phase shift produces an increase or a decrease in the intensity along both directions at the same time. For low diffraction efficiency gratings, we may neglect phase coupling and assume that the effect of simultaneous amplitude and index-of-refraction gratings (both in- or out of phase from the

8.6 Phase Modulation Techniques

recording pattern of fringes) are not coupled [176], so the expression for the output irradiance in Eq. (8.45) can be approximately written as √ √ √ IS ≈ ∣ ŝS0 e𝚤𝜓d sin Ωt + 𝚤̂rR0 𝜂P e𝚤𝜙P + 𝚤̂sR0 𝜂n e𝚤𝜙n + ŝR0 𝜂a e𝚤𝜙a ∣2 (8.61) where the sub-indices “P”, “n” and “a” indicate the diffraction efficiency and the corresponding holographic phase shift of a photorefractive index-of-refraction grating, a nonphotorefractive one and an absorption grating, respectively. Similarly, we find for the other direction √ √ √ IR = ∣ r̂ S0 e−𝚤𝜓d sin Ωt + 𝚤̂sS0 𝜂P e−𝚤𝜙P + 𝚤̂rS0 𝜂n e−𝚤𝜙n + r̂ S0 𝜂a e−𝚤𝜙a ∣2 (8.62) Developing Eqs. (8.61) and (8.62) in terms of Ωt and limiting ourselves to the first and second harmonic terms, we get the final expressions for these terms measured along both directions behind the sample: √ √ √ ISΩ = 2𝜓d ∣ S0 ∣∣ R0 ∣ ( 𝜂a sin 𝜙a + ŝ.̂r 𝜂P cos 𝜙P + 𝜂n cos 𝜙n ) (8.63) √ √ √ IRΩ = 2𝜓d ∣ S0 ∣∣ R0 ∣ ( 𝜂a sin 𝜙a − ŝ.̂r 𝜂P cos 𝜙P − 𝜂n cos 𝜙n ) IS2Ω = IR2Ω =

𝜓d2 2 𝜓d2

√ √ √ ∣ S0 ∣∣ R0 ∣ ( 𝜂a cos 𝜙a − ŝ.̂r 𝜂P sin 𝜙P − 𝜂n sin 𝜙n )

(8.64) (8.65)

√ √ √ ∣ S0 ∣∣ R0 ∣ ( 𝜂a cos 𝜙a + ŝ.̂r 𝜂P sin 𝜙P + 𝜂n sin 𝜙n )

(8.66) 2 All cross terms (corresponding to twice diffracted beams) before were neglected because diffraction efficiencies involved are assumed to be very small. The difference Δ and the sum Σ between corresponding terms previously leads to √ √ IΔΩ ≡ ISΩ − IRΩ = 4𝜓d ∣ S0 ∣∣ R0 ∣ (̂s.̂r 𝜂P cos 𝜙P + 𝜂n cos 𝜙n ) (8.67) √ √ IΔ2Ω ≡ IS2Ω − IR2Ω = −𝜓d2 ∣ S0 ∣∣ R0 ∣ (̂s.̂r 𝜂P sin 𝜙P + 𝜂n sin 𝜙n ) IΣΩ ≡ ISΩ + IRΩ = 4𝜓d ∣ S0 ∣∣ R0 ∣

√ 𝜂a sin 𝜙a

IΣ2Ω ≡ IS2Ω + IR2Ω = 𝜓d2 ∣ S0 ∣∣ R0 ∣

√

𝜂a cos 𝜙a

(8.68) (8.69) (8.70)

From this set of equations, we see that the difference terms are a function of index-of-refraction gratings only (photorefractive and nonphotorefractive), whereas addition terms depend only on amplitude gratings. This is a consequence of the difference in symmetry properties of these kinds of gratings, as already stated in Section 4.2.1. Such properties are very important for separately measuring amplitude and phase effects in a continuous nondestructive way. S2WM also allows one to operate a self-stabilized holographic setup just using amplitude or just phase effects, or stabilizing the setup on the amplitude grating while following the evolution of the phase grating recording and so on. It is still possible to further specialize Eqs. (8.67)–(8.70) for the case of low diffraction efficiency coefficients when phase-coupling effects can be neglected: in this case, photorefractive gratings (in the absence of externally applied field) are nonlocalized with 𝜙P ≈ ±𝜋∕2. Instead, the other two gratings, either arising from the photoactive trap modulation or direct modulation of the material by the action of light, are usually localized ones with 𝜙n = 𝜙a ≈ 0 (or 𝜋). Substituting these values into Eqs. (8.67)–(8.70) we get √ IΔΩ ≈ 4𝜓d ∣ S0 ∣∣ R0 ∣ 𝜂n (8.71)

211

8 Holographic Techniques

√ IΔ2Ω ≈ ∓𝜓d2 ∣ S0 ∣∣ R0 ∣ ŝ.̂r 𝜂P IΣΩ = 4𝜓d ∣ S0 ∣∣ R0 ∣ IΣ2Ω ≈ 𝜓d2 ∣ S0 ∣∣ R0 ∣

√

(8.72)

𝜂a sin 𝜙a ≈ 0

√ 𝜂a

(8.73) (8.74)

A similar S2WM technique was first reported by Boothroyd and coworkers [177], which is based on the symmetric-antisymmetric effects of phase and absorption gratings for a running hologram moving along one direction and along the opposite one. 8.6.4.1 Amplitude and Phase Effects in GaAs

GaAs crystals (see Section 1.5) have an electrooptic tensor identical to that of sillenites, but they do not have optical activity [178]. This fact largely simplifies experiments with these materials. In sillenites, GaAs and similar materials, photorefractive index-of-refraction effects can be distinguished from amplitude and nonphotorefractive effects of any other nature by the fact that the former exhibits anisotropic diffraction whereas the latter ones do not. For the (110)-crystal ⃗ and cut with the [001]-axis orthogonal to the incident plane containing the grating-vector K, the incident beam polarized along the [001]-axis, as shown in Fig. 8.15, the photorefractive diffracted output beam is orthogonally polarized to the transmitted one [58]. The amplitude and nonphotorefractive index-of refraction gratings, if any, have parallel-polarized diffracted and transmitted beams. The measurement of polarization properties at the output, therefore, may allow one to distinguish between both kinds of grating. However, this technique will not enable amplitude effects to be distinguished from nonphotorefractive index-of-refraction modulation, such as that due to photorefractive trap centers modulation, or any other effect leading to a modulation in electrical polarizability not related to electrooptic properties. S2WM instead is specifically concerned with symmetry in energy exchange and may therefore allow one to separate amplitude from phase effects, no matter the origin of the latter. The two-wave mixing experiment indicated in Fig. 8.17 to be carried out on a GaAs sample could allow using S2WM and polarization properties to detect the simultaneous presence of amplitude and photorefractive phase effects as already reported elsewhere [179]. Here, we shall only report the data measured along one single direction behind a polarizer placed at the sample output, as depicted in the schema of Fig. 8.18, where the first harmonic in Eq. (8.63) becomes √ √ √ ISΩ = 2𝜓d ∣ S0 ∣∣ R0 ∣ (sin2 𝛾 𝜂a sin 0 + sin 𝛾 cos 𝛾 𝜂P cos 𝜋∕2 + sin2 𝛾 𝜂n cos 0) (8.75) ISΩ = 2𝜓d ∣ S0 ∣∣ R0 ∣ sin2 𝛾

√ 𝜂n

(8.76)

and the second harmonic in Eq. (8.65) becomes IS2Ω =

𝜓d2 2

∣ S0 ∣∣ R0 ∣ (sin2 𝛾

√ √ √ 𝜂a cos 0 − sin 𝛾 cos 𝛾 𝜂P sin 𝜋∕2 − sin2 𝛾 𝜂n sin 0) (8.77)

D2

GaAs (001)

212

|110|

P

D1

Figure 8.17 Two-wave mixing experiment in a photorefractive GaAs intrinsic crystal with incident and transmitted beams polarized along the [001]-axis of the GaAs crystal. The polarization of the diffracted beams (the shorter arrows) at the crystal output depends on the nature of the diffraction grating in the GaAs. A polarizer (P) and two photodetectors with a summation/subtraction device produce the adequate electric signal for TWM processing.

8.6 Phase Modulation Techniques

[001]

[001]

γ

K

[110] [110]

Figure 8.18 Two-wave mixing experiment in a photorefractive GaAs intrinsic crystal as for Fig. 8.15, but with a polarizer at the crystal output where its transmitted polarization direction makes an angle 𝛾 with the crystal axis [110]. 0.20

0.05

0.15

I 2Ω

0.03 0.10

IΩ

0.02

0.05

0.01 0

IΩ (au)

I2Ω (au)

0.04

0

30

60

0 90

γ (degrees)

Figure 8.19 Plot of the first IΩ (Eq. (8.76)) and second I2Ω (Eq. (8.78)) harmonic terms after fitting the corresponding actual data in GaAs as a function of the polarization angle 𝛾 behind the crystal (see Fig. 8.18) during steady-state multiple nature holograms recorded with 𝜆 = 1064 nm and K = 2.1 μm−1 .

IS2Ω =

𝜓d2 2

∣ S0 ∣∣ R0 ∣ (sin2 𝛾

√

𝜂a ∓ sin 𝛾 cos 𝛾

√ 𝜂P )

(8.78)

where 𝛾 is the angle between the polarization direction transmitted by the polarizer behind the sample, and the [110]-axis of the GaAs. The experimental first and second harmonics were measured on an intrinsic photorefractive GaAs crystal, in the setup schematically depicted in Fig. 8.18, as a function of the polarization angle 𝛾. These data did accurately fit Eqs. (8.76) and (8.78), respectively, and the latter fitting is reproduced in Fig. 8.19. From these fittings were obtained the values of all three different kind of gratings simultaneously present in this √ material after holographic recording using 𝜆 = 1064 nm light: a photorefractive grating with 𝜂P = 1%, √ a nonphotorefractive index-of-refraction grating with 𝜂n = 0.20% and an absorption grating √ with 𝜂a = 0.05%. More details can be found in the literature [179]. In this case, it was not necessary to use the S2WM technique to separately measure the three different gratings because the conditions were particularly simple, but in more complex cases S2WM should be required. The different effects of index-of-refraction and absorption gratings on the first and second harmonic terms in phase-modulated TWM have been already reported for separately detecting these two effects in photorefractive quantum wells [180, 181] The symmetric/asymmetric diffraction effects in TWM underlying the S2WM technique have already been used to assess a purely (or largely predominantly) photorefractive nature

213

214

8 Holographic Techniques

to the dark build-up of a relatively large grating in BaTiO3 after holographic recording with 𝜆 = 488 nm and switching off the recording beams [182]. This technique was also used [119] to separately detect the relatively weak absorption grating (compared to the simultaneously recorded photorefractive grating) in Fe-doped LiNbO3 and use this grating as a reference for self-stabilized recording a photorefractive hologram with an index-of-refraction modulation exceeding the value for 𝜂 = 1, which would have been impossible to achieve for self-stabilized recording on the photorefractive hologram itself, due to the upper 𝜂 = 1 limitation of this recording technique, as already discussed in Section 6.3.2. The use of S2WM has also enabled the simultaneous presence of amplitude effects and photorefractive hole-electron competition to be detected in an undoped Bi12 TiO20 crystal at a rather low irradiance level (200 to 300 μW∕cm2 ) [183]. 8.6.5

Running Holograms

Running holograms were theoretically discussed in Section 3.4. These kinds of hologram have been extensively applied to image processing and other applications but hardly used for material characterization, probably because of the inherent complexity of their nature and the number of independent parameters characterizing the process. Here, we shall focus on the possibilities of running holograms as a tool for materials characterization. To experimentally study these moving holograms, it is very convenient to use phase modulation techniques that allow measurement pf several properties of the hologram without perturbing the process. The adequate setup is schematically depicted in Fig. 8.20 where the first and second temporal harmonic signals are used to simultaneously compute the diffraction efficiency 𝜂 and the phase-shift 𝜑 between the beams behind the sample. The oscillator produces the dither signal of frequency Ω that is used for detection, and the high voltage source is used to feed a saw-tooth electric signal to the piezo-mirror so as to produce a moving pattern of fringes onto the sample. It has already been reported [78, 184] that bulk optical absorption and optical activity strongly influence the amplitude gain in running holograms. It has also been shown [185] that the typical asymmetric shape of the diffraction efficiency versus velocity curve is mainly due to the bulk absorption and the higher spatial harmonic components whenever present. For the purpose of materials characterization, however, it is interesting to use a low value for the pattern-of-fringes modulation coefficient to fulfill the first spatial harmonic approximation [186] that leads to a comparatively simple set of equations, as reported in Section 3.3, describing the wave coupling in the crystal volume that facilitates the moving grating analysis [187]. It is also convenient to M

D2

Figure 8.20 Experimental setup for the generation and measurement of running holograms.

O IR

Laser 532 nm

D1

G BS BTO O

IS

LA–2Ω

VA

LA–Ω

PZT OSC

+ HV

VΩ

V2Ω

8.6 Phase Modulation Techniques

use a relatively thin sample to be able to neglect the combined effect of optical activity (for the case of sillenites) and birefringence [185]. We shall show some results from experiments that were carried out on a thin photorefractive Bi12 TiO20 (BTO) crystal and measure the diffraction efficiency and the output beams phase-shift as a function of the pattern-of-fringes movement. We show that the analysis of the experimental results might lead to erroneous conclusions because of the presence of even law concentration of minority hole-photoactive centers in this sample. Following the considerations in the paper by Shamonina et al. [185] we shall neglect birefringence and optical activity (as we are concerned with a thin BTO sample), take into consideration bulk absorption (that is particularly large in BTO for the 514.5 nm wavelength) and consider self-diffraction effects. Charge carriers’ excitation and transport in photorefractives are complex phenomena sometimes involving more than one type of photoactive center [188], centers with more than one valence state [18] and hole-electron competition [189, 190]. Here, we shall assume the simplest “one center and one charge-carrier” model where the possible presence of holes will be treated as a perturbation. Experimental conditions will be looked for to minimize such a perturbation. In these conditions the steady state space-charge electric field is described in Section 3.4. In the presence of self-diffraction, the diffraction efficiency is described by Eq. (4.123) which for a sufficiently small crystal thickness d simplifies to Eq. (4.124) (( )2 ( )2 ) 2𝛽 𝛾d Γd 2 𝜂≈m ∣m∣= + 4 4 1 + 𝛽2 The phase-shift 𝜑 between the transmitted and diffracted beams behind the sample is reported in Eq. (4.128). In the presence of bulk absorption, however, the expressions for 𝜂 and tan 𝜑 should be accordingly modified as described in Section 4.2.2.3. Here, we shall focus on sillenites that are known to exhibit hole-conductivity in the dark and electron-conductivity under the action of light [161]. It is not possible to exclude some degree of hole-electron competition in the deep trap level too, at least for some samples [183, 190]. As shown next, even a comparatively small density of hole-photoactive centers may considerably affect the shape of the diffraction efficiency curve. In order to mathematically simulate this effect, we assume the simple model in Section 3.4.1.1 where independent photoactive deep centers for electrons and for holes are present without any interaction among them, except for the fact that at each point in the crystal volume the charge density is built up by the superposition of both holes and electrons. For the sake of simplicity, here we shall assume that the effective field st Esc can be separately computed for electrons and for holes, as formulated in Eq. (3.127), using their corresponding assumed material parameters (LD , ls etc.). The integrals in Eqs. (4.143) and (4.144) are computed for electrons (𝛾 e and Γe ) and for holes (𝛾 h and Γh ), and the expressions for the diffraction efficiency in Eq. (4.145) and phase-shift in Eq. (4.146) can be thus computed separately for the electrons, for the holes and for the superimposed hole-electron condition (using the quantities 𝛾 e + 𝛾 h and Γe + Γh ). Such a simulation for 𝜂 and for tan 𝜑 was computed assuming some typical parameters for the electron-photoactive centers in BTO and assuming a much lower quantum efficiency and concentration for holes: 100-fold lower quantum efficiency and approximately 25-fold lower concentration representing a roughly five-fold higher Debye length, with the condition LD > ls . The result is depicted in Fig. 8.21 for K = 2.55 μm−1 and in Fig. 8.22 for K = 11.3 μm−1 were the values for K in our experiment. In the latter figure, we see that even such a small hole-photoactive center concentration (with an electron-to-hole hologram diffraction efficiency ratio of 𝜂e ∕𝜂h ≈ 17 at K𝑣 = 0) can produce a pronounced effect on the shape of the overall 𝜂. Something similar is seen in Fig. 8.21 for K = 2.55 μm−1 except that here the effect

215

8 Holographic Techniques

η

0.015

0.01 0.005

0

–5

0

5

15

Kv (rad/s)

4

tan φ

216

2

–5

0

5

15

Kv (rad/s)

Figure 8.21 Diffraction efficiency (left) and tan 𝜑 (right) as a function of Kv computed with the experimental parameters K = 2.55 μm−1 , 𝛼 = 11.65 cm−1 , 𝜉E0 = 4.55 KV∕cm and I0 = 17.5 mW/cm2 . The material parameters are LD = 0.22 μm, ls = 0.03 μm, 𝛽 2 = 40 and Φ = 0.4 for electrons (continuous curve), whereas for holes they are LDh = 0.16 μm, lsh = 0.15 μm and Φh = 0.004 (dashed curve). The resulting electron-to-hole diffraction efficiency ratio at K𝑣 = 0 is 𝜂e ∕𝜂h ≈ 2.4. The thick continuous curve is the overall result. Reproduced from [191].

of holes is not so weak: 𝜂e ∕𝜂h ≈ 2.4. In both cases, the effect of holes becomes weaker as we go further away from K𝑣 = 0 along an increasing K𝑣. The different effect of hole-electron competition for different values of K is due to the fact that for small K values it is K 2 ls2 ≪ 1 both for holes and for electrons. This means that the hologram build-up is not limited by its respective photoactive centers density (see Eq. (3.49)), so similarly strong holograms result both for holes and for electrons as seen in Fig. 8.21. Instead, for a larger value of K it should still be K 2 ls2 ≪ 1 for electrons but not for holes, which may approach saturation and develop a weaker hologram as seen in Fig. 8.22. The effect of holes on the tan 𝜑 as seen in Figs. 8.21 and 8.22 is much less relevant (except near K𝑣 = 0) than for 𝜂. As for the latter, the effect of hole-electron competition becomes negligible for K𝑣 > 0 sufficiently far from the origin. We should also note that it is not reliable to compute LD for the case where K 2 L2D ≫ 1 because in these conditions (the movement of charge-carriers being much larger than the fringe period) there is a kind of randomization of the charge-carriers in the volume of the sample and the movement of the hologram is no longer dependent on the value of this parameter. This conclusion is supported by numerical simulations too: some simulations have shown (although the physical reason is not yet evident) that tan 𝜑 is also not very dependent on LD whatever the value of K. It is clear that, with the restrictions discussed previously, both 𝜂 and tan 𝜑 data

8.6 Phase Modulation Techniques

0.006

η

0.004

0.002

0

–2

–1

0

1

2

1

2

Kv (rad/s)

tan φ

2

1

0

–2

–1

0 Kv (rad/s)

Figure 8.22 Diffraction efficiency (left) and tan 𝜑 (right) as a function of Kv computed with K = 11.3 μm−1 . All other experimental and material parameters and the meaning of thick, thin and dashed curves are the same as for Fig. 8.21 with 𝜂e ∕𝜂h ≈ 17 for K𝑣 = 0. Reproduced from [191].

should be used to compute the material parameters, mainly for K𝑣 > 0 to minimize eventual hole-perturbations. A two-wave mixing running hologram experiment was carried out using the 514.5 nm wavelength laser line and a 2.05 mm thick nominally undoped photorefractive Bi12 TiO20 crystal growth by the Czochralski technique [192] (𝛼 = 11.65 cm−1 ) with the [001] crystal axis perpendicular to the incidence plane and the hologram wavevector K⃗ parallel to the [110]-axis in a configuration similar to the one depicted in Fig. 8.15 for GaAs. An external electric field is applied to the crystal along K⃗ by means of silver-painted electrodes. The interfering incident beams are expanded and collimated so that uniform irradiances over the sample (less than 10% variation) result. The beams are linearly polarized with their polarizations selected to be at 45∘ to the [001]-axis at the mid-crystal plane, in which case the transmitted and diffracted beams behind the crystal are parallel polarized [193] as illustrated in Fig. 5.5. The setup is adjusted so that energy is transferred from the signal (IS ) to the pump (IR ) beam in order to characterize a negative gain process. Negative gain produces a lower 𝜂 but was shown [94] to lead to more stable experiments that facilitate the measurement. A piezoelectric-supported mirror (PZT) fed with an electric ramp signal of adjustable slope produces a detuning K𝑣 on one of the interfering beams. To produce negative K𝑣 values, the sense of the applied field E0 is just reversed. Data measured at K𝑣 = 0 were verified not to depend on the direction (direct or reversed) of the applied field E0 . The diffraction efficiency 𝜂 and output phase-shift 𝜑 can be written in terms

217

8 Holographic Techniques

Figure 8.23 Diffraction efficiency 𝜂 experimental data (spots) as a function of detuning K𝑣 and best theoretical fit (continuous curve) to Eq. (4.145) for 𝜉 = 0.96, K = 2.55 μm−1 , E0 = 7.3 KV/cm, 𝛽 2 = 41.2 and I0 = 22.5 mW/cm2 . The resulting best fitting parameters are LD = 0.14 μm, and Φ = 0.45. Data for K𝑣 < 0 (small spots) were not used for the fit. Reproduced from [191].

η

0.01

0 0.005

0

–5

5 Kv (rad/s)

10

15

Figure 8.24 Tan 𝜑 experimental data (spots) as a function of K𝑣 for the same conditions as in Fig. 8.23, with data (large spots) fitted to Eq. (4.146) (continuous curve) and the resulting parameter being Φ = 0.41. Data for K𝑣 < 0 (small spots) are also not considered for the fit here. Reproduced from [191].

4 3 tan φ

218

2 1

0

5 Kv (rad/s)

10

of the first and second harmonic terms as )2 ( )2 ( ⎡ ⎤ 1 VΩ 2V 2Ω ⎢ ⎥ 𝜂= + Ω Ω IS IR (KdΩ )2 ⎢ AΩ 2𝑣d KPZT A2Ω (𝑣d KPZT )2 ⎥ ⎣ ⎦ ] [ Ω 2Ω K Ω 𝑣 V A PZT d tan 𝜑 = 2Ω Ω 4 V A

(8.79)

(8.80)

Figures 8.23 and 8.24 show typical experimental results (spots) for 𝜂 and tan 𝜑, respectively, as functions of K𝑣 for K = 2.55 μm−1 , 𝛽 2 = 41.2 and I0 = 22.5 mW∕cm2 . From Fig. 8.23, we got LD = 0.14 μm and Φ = 0.45, whereas from Fig. 8.24 we got Φ = 0.41. Note that the large spots for K𝑣 ≥ 0 were actually used for fitting, whereas the small spots (K𝑣 < 0 side) are just included to appreciate their differences with the theoretical fit. It was not possible to fit the whole experimental data (small and large spots) set to theory in any of our experiments. It is well known that the effective field inside the sample may be different from its nominal value E0 [72, 194], so the latter should be substituted by 𝜉E0 everywhere, with 𝜉 being an experimentally evaluated effective field coefficient as discussed in chapter C. In this work, 𝜉 was computed from an auxiliary experiment, where 𝜂 was measured as a function of E0 at 𝑣 = 0 and these data fit theory to get the corresponding 𝜉 values in Table 8.6. Other experiments were carried out to measure 𝜂 and tan 𝜑 for the same K = 2.55 μm−1 and also for K = 11.3 μm−1 . The theoretical fitting of the data from each one of these experiments allows to obtain some material parameters that are displayed in Table 8.6 together with their corresponding 𝜉 values. The average results for this sample were: LD ≈ 0.14 μm, ls = 0.03 ± 0.01 μm and Φ = 0.4 ± 0.1.

8.7 Holographic Photo-Electromotive-Force (HPEMF) Techniques The photo-electromotive-force (PEMF) is produced in photoconductors or photorefractive materials where, under the action of light, a distribution of free charge carriers in the extended

8.7 Holographic Photo-Electromotive-Force (HPEMF) Techniques

Table 8.6 Running hologram: undoped BTO at 𝜆 = 514.5 nm. Diffraction efficiency −1

K 𝛍m

I0 mW/cm

𝜷

2.55

22.5

2.55

22.5

11.3 11.3

Phase-shift

LD 𝛍m

ls

𝚽

LD 𝛍m

ls

𝚽

𝜼(𝒗 = 0) 𝝃

41.2

0.14

–

0.45

–

–

0.41

0.96

39

0.14

–

0.63

–

–

0.48

0.78

24.3

30.8

–

0.015

–

–

0.032

0.38

0.90

19.4

26.7

–

0.028

–

–

0.036

0.30

0.90

2

2

states (conduction and/or valence band) is produced and a fixed spatial distribution of electric charges in traps and associated space-charge electric field are built up as already discussed in Section 7.7.1. Here, there is a pattern of interference fringes (instead of a speckle) of light that is projected onto the photorefractive crystal as illustrated in Fig. 8.25. As for the speckle, if the pattern of light is moving faster than the response of the space-charge field, but slower than the lifetime of free charges in the extended states, the free charges will follow the movement but the space-charge field will not and a photocurrent will appear. The present PEMF is not at all concerned with holography in the sense that it does not require the recording of and index-of-refraction modulation hologram at all, but it does rely on the mathematical development describing the holographic space-charge electric field build-up and also requires a typical holographic setup to produce the required holographic pattern-of-fringes to be projected onto the photorefractive crystal, so it seems useful to include this technique in this chapter. In nonphotovoltaic photorefractive crystals, in the absence of any externally applied electric field, the recorded space-charge field modulation and the free charge-carrier (for simplicity assumed to be just electrons) distribution in the conduction band are mutually 𝜋∕2-phase shifted so that the electric current averaged along the interelectrode distance is zero. However, if the pattern of fringes is moved along the grating wavevector, the previously referred to phase shift is modified and a pulse, ac or a dc current may appear, according to the way the pattern of fringes is moved. Let us assume a sinusoidal pattern of fringes with K⃗ along the interelectrode coordinate x, as described by Eq. (3.6) I = I0 (1+ ∣ m ∣ cos(Kx + 𝜙)) = I0 + (I0 ∕2)[m eiKx + m∗ e−iKx ] PZT

O

IS

R-feedback (001)

coherent laser beams

|110|

O

IR

BTO

+ – OA

output

Figure 8.25 Holographic photoelectromotive force current setup schema: a laser beam of 514.5 nm wavelength is divided in two, filtered, expanded, collimated and made to interfere over the BTO sample. A piezoelectric-supported mirror PZT in one of the beams is vibrating with angular frequency Ω. A lock-in amplifier measuring current, and schematically represented by the operational amplifier with feedback, is ⃗ tuned to Ω in order to measure the first harmonic component iΩ of the photocurrent along the K-direction in the sample’s volume. Reproduced from [153]

219

220

8 Holographic Techniques

⃗ with amplitude Δ, so that m should be substiwith the fringes sinusoidally vibrating along K, tuted as follows m ⇒ m(t) = |m| ei𝜙 eiKΔ sin Ωt (8.81) Similar to the development in Section 8.6.3, the following relation holds eiKΔ sin Ωt =

+∞ ∑

Jl (KΔ) eilΩt

(8.82)

l=−∞

where Jl () is the ordinary Bessel function of order l. Therefore, the modulation in Eq. (8.81) is written as +∞ ∑ m(t) = |m| ei𝜙 Jl (KΔ) eilΩt (8.83) l=−∞

Let us assume the first spatial harmonic approximation (see Section 3.3) that allows one to consider the linearized expressions in Eqs. (3.23)–(3.25)  (x, t) =  +  ∕2[a(t) eiKx + a∗ (t) e−iKx ] 0

0

ND+ (x, t) = ND+ + ND+ ∕2[A(t) eiKx + A∗ (t) e−iKx ] E(x, t) = E + (1∕2)[E (t) eiKx + E∗ (t) e−iKx ] 0

sc

sc

The equations here substituted into Eqs. (2.18)–(2.23) were shown to lead to the following relations for Esc (t) in Eq. (3.41), for a(t) in Eq. (3.37) and for A(t) in Eq. (3.39), as follows 𝜏sc

𝜕Esc (t) + Esc (t) = −m(t) Eeff 𝜕t

a(t) =

Esc (t)𝚤K𝜇𝜏 + m(t) sI0 ∕(sI0 + 𝛽) − A(t)ND ∕(ND − ND+ ) 1 + 𝚤e∕q K𝜏𝜇E0 + K 2 𝜏

𝚤K𝜖𝜀o Esc (t) ≈ qND+ A(t) From the last two equations and Eq. (8.83) we get a(t) = 𝚤

K 2 L2D − K 2 ls2 Esc (t) 1 + K 2 L2D

ED

+

+∞ |m| e𝚤𝜙 ∑ Jl (KΔ) e𝚤lΩt 1 + K 2 L2D l=−∞

(8.84)

where we have assumed that sI0 ≫ 𝛽 and E0 = 0. On the other hand, the solution of the differential equation before for the Esc (t) is +∞ ∑ Jl (KΔ) e𝚤lΩt Esc (t) = −|m|Eeff 1 + 𝚤lΩ𝜏sc l=−∞

(8.85)

where we have assumed that the electric grating has been recorded for the pattern of fringes fixed at 𝜙 = 0. Substituting Eq. (8.85) into (8.84) we get the expression a(t) =

+∞ +∞ 2 2 2 2 ∑ Jl (KΔ) e𝚤lΩt |m| ei𝜙 ∑ 𝚤lΩt − 𝚤 K LD − K ls |m| Eeff J (KΔ) e l 2 2 ED l=−∞ 1 + 𝚤lΩ𝜏sc 1 + K 2 LD l=−∞ 1 + K 2 LD

(8.86)

It is possible to show [140] that the total current density flowing through the electrodes at the ⃗ can be written as ends of the sample, along K, L

j(t) =

1 e𝜇 (x, t)Esc (x, t)dx L ∫0

8.7 Holographic Photo-Electromotive-Force (HPEMF) Techniques

where L is the interelectrode distance. In our special case, the previously formula simplifies to ( )  E (t)∗ E (t) j(t) = e𝜇 0 a(t) sc + a(t)∗ sc (8.87) 2 2 2 We should also write a(t) and Esc (t) in Eqs. (8.86) and (8.85), respectively, in terms of their harmonic components in Ω as follows +∞ ∑

a(t) =

alΩ eilΩt

(8.88)

l=−∞ +∞

Esc (t) =

∑

lΩ ilΩt Esc e

(8.89)

l=−∞

with the following first few parameters for a(t) K 2 L2D − K 2 ls2 E |m| ei𝜙 a0 = J (KΔ) − i |m| eff J0 (KΔ) 0 ED 1 + K 2 L2D 1 + K 2 L2D aΩ =

K 2 L2D − K 2 ls2 E J (KΔ) |m| ei𝜙 J (KΔ) − 𝚤 |m| eff 1 1 2 2 ED 1 + 𝚤Ω𝜏sc 1 + K 2 LD 1 + K 2 LD

(8.90)

(8.91)

a2Ω =

K 2 L2D − K 2 ls2 E J (KΔ) |m| e𝚤𝜙 J (KΔ) − 𝚤 |m| eff 2 2 2 2 2 2 ED 1 + 𝚤2Ω𝜏sc 1 + K LD 1 + K LD

(8.92)

alΩ =

K 2 L2D − K 2 ls2 E J (KΔ) |m| e𝚤𝜙 J (KΔ) − 𝚤 |m| eff l 2 l 2 2 2 ED 1 + 𝚤lΩ𝜏sc 1 + K LD 1 + K LD

(8.93)

and for Esc (t) 0 Esc = −|m|Eeff J0 (KΔ)

Ω Esc = −|m|Eeff

(8.94)

J1 (KΔ) 1 + 𝚤Ω𝜏sc

(8.95)

J2 (KΔ) 1 + 𝚤2Ω𝜏sc

(8.96)

2Ω = −|m|Eeff Esc

Jl (KΔ) (8.97) 1 + 𝚤lΩ𝜏sc The expression in Eq. (8.87) can be also written in terms of its harmonics, in the same way as already done for E,  and other parameters, as lΩ Esc = −|m|Eeff

j0 jΩ 𝚤Ωt j2Ω i2Ωt + e + e + ... + cc 2 2 2 where the few first coefficients are 𝜎 0 ∗ 0 Ω ∗ Ω 2Ω ∗ 2Ω ) + (a0 )∗ Esc + aΩ (Esc ) + (aΩ )∗ Esc + a2Ω (Esc ) + (a2Ω )∗ Esc + ...) j0 = 0 (a0 (Esc 2 𝜎 −Ω ∗ Ω 0 ∗ 0 Ω ∗ −2Ω ∗ jΩ = 0 (a0 (Esc ) + (a0 )∗ Esc + aΩ (Esc ) + (a−Ω )∗ Esc + a2Ω (Esc ) + a−Ω (Esc ) 2 2Ω −Ω + (aΩ )∗ Esc + (a−2Ω )∗ Esc ...) 𝜎0 2Ω −2Ω ∗ 0 ∗ 2Ω Ω −Ω ∗ Ω 0 0 ∗ j = (a0 (Esc ) + (a ) Esc + a (Esc ) + (a−Ω )∗ Esc + (a−2Ω )∗ (Esc ) + a2Ω (Esc ) ...) 2 j(t) =

(8.98)

(8.99)

(8.100) (8.101)

221

222

8 Holographic Techniques

where 𝜎0 = e𝜇0 . After substituting and rearranging terms we got the following expression for the dc component ∗ 𝜎0 |m| J0 (KΔ)2 J1 (KΔ)2 Eeff m∗ + Eeff m ∗ ∗ (E m + E m) − 𝜎 |m| 0 eff 2 2 1 + K 2 L2D eff 1 + K 2 L2D 1 + Ω2 𝜏sc ∗ J (KΔ)2 Eeff m∗ + Eeff m − 𝜎0 |m| 2 2 1 + K 2 L2D 1 + 4Ω2 𝜏sc

j0 = −

(8.102)

as well as for the first harmonic jΩ =

𝜎0 |m| J0 (KΔ)J1 (KΔ) 𝚤Ω𝜏sc ∗ (Eeff m∗ − Eeff m) 2 1 + K 2 L2D 1 + 𝚤Ω𝜏sc 𝜎 |m| J1 (KΔ)J2 (KΔ) i3Ω𝜏sc ∗ + 0 m) (Eeff m∗ − Eeff 2 2 1 + K 2 LD (1 − 𝚤Ω𝜏sc )(1 + 𝚤2Ω𝜏sc )

(8.103)

and for the second harmonic |m| J0 (KΔ)J2 (KΔ) ∗ (Eeff m∗ + Eeff m) 2 1 + K 2 L2D ∗ m 𝜎 |m| J0 (KΔ)J2 (KΔ) Eeff m∗ + Eeff − 0 2 2 1 + i2Ω𝜏sc 1 + K 2 LD 2 E m∗ + E ∗ m 𝜎 |m| J1 (KΔ) eff eff + 0 2 2 2 1 + K LD 1 + iΩ𝜏sc

j2Ω = − 𝜎0

+ i𝜎0 |m|J0 (KΔ)J2 (KΔ) −i

K 2 L2D − K 2 ls2 |Eeff |2 m − m∗ ED 1 + i2Ω𝜏sc 1 + K 2 L2D

K 2 L2D − K 2 ls2 |Eeff |2 m − m∗ 𝜎0 |m| J1 (KΔ)2 2 ED (1 + iΩ𝜏sc )2 1 + K 2 L2D

(8.104)

In the absence of external field (E0 = 0), it is Eeff =

𝚤ED 1 + K 2 ls2

that substituted in these equations leads to the following simplified expression for the dc term j0 = −𝜎0 |m|2

J0 (KΔ)2

ED sin 𝜙 (1 + K 2 L2D )(1 + K 2 ls2 ) ED J1 (KΔ)2 − 2𝜎0 |m|2 sin 𝜙 2 2 2 2 2 (1 + K LD )(1 + K ls ) 1 + Ω2 𝜏sc ED J2 (KΔ)2 − 2𝜎0 |m|2 sin 𝜙 2 (1 + K 2 L2D )(1 + K 2 ls2 ) 1 + 4Ω2 𝜏sc

(8.105)

as well as for the first harmonic J0 (KΔ)J1 (KΔ)

Ω𝜏sc ED cos 𝜙 2 2 1 + iΩ𝜏 1+ sc 1 + K ls 3Ω𝜏sc ED J (KΔ)J2 (KΔ) − 𝜎0 |m|2 1 cos 𝜙 2 2 1 + K LD (1 − iΩ𝜏sc )(1 + i2Ω𝜏sc ) 1 + K 2 ls2

jΩ = −𝜎0 |m|2

K 2 L2D

(8.106)

8.7 Holographic Photo-Electromotive-Force (HPEMF) Techniques

and for the second harmonic term J (KΔ)J2 (KΔ) ED sin 𝜙 j2Ω = − 𝜎0 |m|2 0 1 + K 2 L2D 1 + K 2 ls2 J (KΔ)J2 (KΔ) ED 1 − 𝜎0 |m|2 0 sin 𝜙 1 + K 2 L2D 1 + K 2 ls2 1 + i2Ω𝜏sc ED J (KΔ)2 1 − 𝜎0 |m|2 1 sin 𝜙 2 2 1 + iΩ𝜏 2 2 1 + K LD 1 + K ls sc − 2𝜎0 |m|2 J0 (KΔ)J2 (KΔ) + 𝜎0 |m|2 J1 (KΔ)2

K 2 L2D − K 2 ls2

ED 1 sin 𝜙 1 + K 2 L2D 1 + K 2 ls2 1 + i2Ω𝜏sc

K 2 L2D − K 2 ls2 1+

K 2 L2D

ED

1 sin 𝜙 1 + K 2 ls2 (1 + iΩ𝜏sc )2

(8.107)

In order to understand the meaning of Eqs. (8.105)–(8.107) it is necessary to keep in mind that in the absence of perturbations it should be 𝜙 = 0, in which case the dc and the second harmonics are null. Note also that 𝜎0 does depend on the irradiance and may therefore vary along the sample thickness in absorbing materials. The first harmonic can be also written in its binomial form as follows jΩ = jRΩ + ijIΩ

(8.108)

jRΩ = Ω𝜏sc J1 (KΔ) ×

2 [2J0 (KΔ) + 3J2 (KΔ)](1 + 2Ω2 𝜏sc ) − J0 (KΔ) 2 2 (1 + Ω2 𝜏sc )(1 + 4Ω2 𝜏sc )

𝜎0 |m|2 ED cos 𝜙

(8.109)

(1 + K 2 L2D )(1 + K 2 ls2 )

2 jIΩ = −Ω2 𝜏sc J1 (KΔ)

2 3J2 (KΔ) + J0 (KΔ)(1 + 4Ω2 𝜏sc )

𝜎0 |m|2 ED cos 𝜙

2 2 (1 + Ω2 𝜏sc )(1 + 4Ω2 𝜏sc )

(1 + K 2 L2D )(1 + K 2 ls2 )

(8.110)

The real part of the first harmonic is jΩ iΩt (jΩ )∗ −iΩt jΩ + (jΩ )∗ jΩ − (jΩ )∗ e + e = cos Ωt + i sin Ωt 2 2 2 2 √ = jRΩ cos Ωt − jIΩ sin Ωt = (jRΩ )2 + (jIΩ )2 cos(Ωt + 𝜑Ω )

ℜ{jΩ } =

(8.111)

The actual measured value (using a lock-in amplifier tuned to Ω-frequency, for example) is the amplitude of this real signal, that is √ (8.112) |jΩ | = (jRΩ )2 + (jIΩ )2 The latter theoretical equation is plotted in Fig. 8.26, as a function of KΔ, for some particular conditions. The expression of |jΩ | is particularly interesting for the case of low Ω𝜏sc (Ω𝜏sc ≪ 1) where it assumes the form |j0Ω | ≈ Ω|m|2 ED cos 𝜙 J1 (KΔ)(J0 (KΔ) + 3J2 (KΔ))

(8.113)

with ≡

𝜎0 𝜏sc (1 +

K 2 L2D )(1

+

K 2 ls2 )

≈

𝜖𝜀0 (1 + K 2 ls2 )2

(8.114)

223

224

8 Holographic Techniques

Figure 8.26 |jΩ | (in arbitrary units) as a function of the vibration amplitude KΔ (in radians) for Ω𝜏sc = 1000, 5, 1 and 0.1 rad, from the finest to the coarsest dashed curves, respectively, always without an externally applied field.

|j Ω| (au) 0.3 0.23 0.2 0.15 0.1 0.05 1

𝜏sc 𝜎0 = 𝜖𝜀o

2

3

1 + K 2 L2D − iKLE 1 + K 2 ls2 − iKlE

4

≈ 𝜖𝜀o

K∆ (rad)

1 + K 2 L2D 1 + K 2 ls2

(8.115)

where the approximate relations in the right side in Eq. (8.115) are for no externally applied field E0 = 0. Note that  in Eq. (8.114) is independent of the irradiance and the light absorption and consequently, the expression in Eq. (8.113) is also constant along the sample’s thickness in these conditions. For the opposite limit condition Ω𝜏sc ≫ 1 we also get a simplified formulation of |jΩ | Ω |j∞ |≈

 |m|2 ED cos 𝜙 J0 (KΔ)J1 (KΔ) 𝜏sc

(8.116)

that does not depend any more on Ω. Holographic PEMF, in a two-wave interferometric setup [142–144, 153], allows transducing the phase modulation in one of the interfering beams into an oscillating pattern-of-fringes projected onto a suitable photorefractive or just a plain photoconductive material. This oscillation produces an alternating current in the photoconductive sample that depends on the amplitude of the pattern-of-fringes oscillation, among other parameters, and may be therefore used, with adequate scaling, to measure the longitudinal oscillation amplitude of the device producing the phase modulation. This technique is a self-calibrating one because the size of the signal is easily related to the spatial period of the pattern of fringes which is straightforwardly computed from the geometry of the setup. Self-calibration has facilitated the application of this technique to wide different fields [195, 196], including the measurement of mechanical vibrations amplitude [197] as well as materials characterization [198–200]. Here, we shall focus on this latter point because it is the subject of this part of the book. The experimental setup is schematically shown in Fig. 8.25 where a sinusoidal pattern of fringes is projected onto the (110) crystallographic plane of an undoped BTO crystal, with its [001]-axis perpendicular to the plane of incidence ⃗ The angle between the interfering beams (in air) is 51∘ , in this case, and the wavelength and to K. is 𝜆 = 0.5145 μm. The visibility coefficient |m| of the pattern of fringes is computed from the ratio of amplitudes of the interfering waves, taking also into account their polarization direction that are in the plane of incidence. A piezoelectric supported mirror (PZT), placed in one of the interfering beams, is driven by a sinusoidal voltage 𝑣(t) = 𝑣d sin Ωt

(8.117)

that produces a corresponding phase modulation of amplitude (in radians) KΔ KΔ = KPZT 𝑣d where KPZT is the response of the piezoelectric.

(8.118)

8.7 Holographic Photo-Electromotive-Force (HPEMF) Techniques

Light absorption effects cannot be neglected in most photorefractive materials. In fact, we also need to consider the effect of absorption on the holographic response time, besides its obvious effect on the photoconductivity. Because of bulk absorption, the irradiance decreases exponentially along the sample’s thickness coordinate z so that irradiance-dependent quantities such as photoconductivity and holographic response time also vary along z as follows (8.119) 𝜎 (z) = 𝜎 (0) e−𝛼z 0

0

𝜏sc (z) = 𝜏sc (0) e𝛼z

(8.120)

where 𝜎0 (0) and 𝜏sc (0) are the values at the input plane inside the sample and 𝛼 is the effective absorption coefficient. The expression for jΩ in Eqs. (8.108)–(8.110) is therefore also dependent on z, because of its dependence on 𝜎0 (z) and 𝜏sc (z), and should rather be written as jΩ (z) in order to explicitly indicate this dependence. The experimentally measured photocurrent value, accounting on an irradiance decrease along the sample’s thickness, is d

|iΩ | = H

∫0

|jΩ (z)|dz

(8.121)

where H is the height and d is the thickness of the sample. The first harmonic amplitude value |iΩ | was measured, using the direct current measurement facilities of an EG& G model 5210 lock-in amplifier, as a function of KΔ for different fixed values of Ω. Typical results for the same sample are shown in Figs. 8.28 and 8.29. |j Ω| (au) 0.34 0.32 2

4

6

8

10

Ω τsc (rad)

0.28 0.26 0.24 0.22

Figure 8.27 Computed |jΩ | (in arbitrary units) as a function of Ω𝜏sc in rad for a fixed amplitude KΔ = 1.1 rad. 4

i Ω (pA)

Figure 8.28 First harmonic component of the holographic current |iΩ | data (spots) as a function of the KΔ for I0 = IRo + ISo = 455 W/m2 . The continuous curves are the best fit to theory, from Ω∕2𝜋 = 980 Hz (thickest continuous) to 3.5 Hz (thinnest dashed). Data for 980, 546 and 349 Hz are omitted because are close to data for 152 Hz. Reproduced from [153].

2

0

0

1.25 KΔ (rad)

2.5

225

8 Holographic Techniques

2.0

j Ω (nA)

1.5

Ω = 129Hz

1.0

Ω = 268Hz Ω = 313Hz Ω = 496Hz

0.5

0

Ω = 695Hz

0

1

2

3

K∆ (rad)

Figure 8.29 First harmonic component of the holographic current |jΩ | data (spots) as a function of KΔ for I0 = IRo + ISo = 177 W/m2 . All data fit the same (not shown) curve. Reproduced from [153].

The curves in Fig. 8.28 represent the best fit of the theoretical equations to data (spots). Same procedure was followed for data in Fig. 8.29, although the fitting curve here is not shown. In both cases, there is an excellent agreement between theory and experimental data. Note that the maxima of all high frequency curves in Fig. 8.29 occur at KΔ = 1.1 in agreement with the position of the maximum for the product J0 (KΔ)J1 (KΔ) in Eq. (8.116). The maxima for the curves in Fig. 8.28 instead occur at KΔ = 1.1 for the higher frequencies and progressively shift to higher values of KΔ for the decreasing frequencies. This is probably due to the fact that the maximum (for the term J1 (KΔ)(J0 (KΔ) + 3J2 (KΔ))) for the low-frequency limit expression represented by the Eq. (8.113) occurs at KΔ ≈ 2, so the position of the maximum will depend on the frequency Ω.

+

+

+

+

+

+

+

4 +

j Ω (pA)

226

+

2

0 0

500

1000

Ω/2π (Hz)

Figure 8.30 |iΩ | data (spots) plotted as a function of Ω∕2𝜋, for KΔ = 1.1 rad: Ce-doped BTO (thickest curve), for Pb-doped BTO (thinnest curve) and undoped BTO (mid thickness curve).

8.7 Holographic Photo-Electromotive-Force (HPEMF) Techniques

Table 8.7 Best fitting parameters from HPEMF experiments [153]. BTO

BTO:Ce

BTO:Pb

I0

(W/m2 )

460

443

443

I(0)

(W/m2 )

360

346

346

0.6

0.55

0.55

d

(mm)

2.05

6.05

6.0

L

(mm)

6.2

6.05

6.05

H

(mm)

7.0

7.55

7.5

𝛼514.5nm

(m−1 )

1290

932

1013



(10−10 F/m)

2.51

0.15

1.86

ls

(μm)

0.05

0.2

0.07

m

𝜏sc (0)

(ms)

8.7

1.11

7.6

∕𝜏sc (0)

(10−10 F/sm)

260

130

250

𝜎0 (0)(1 + K 2 L2D )∕I(0)

(10−10 m/(Ω W)

1

2

1

Experimental |iΩ | data obtained for different frequencies at a fixed KΔ = 1.1 and fixed irradiance from Fig. 8.28 are plotted in Fig. 8.30. Curves for Ce-doped and Pb-doped BTO obtained in the same way are also shown in this figure. It is interesting to point out that the shape of these curves are quite similar to the theoretical one in Fig. 8.27. Note that, for the high-frequency range, the effect of bulk light absorption in Eq. (8.121) is just a term d

∫0

e−𝛼z dz =

1 − e−𝛼d 𝛼

(8.122)

From the results in this section it is clear that our theoretical model adequately describes the experimental phenomena involved. In order to illustrate the use of this technique for materials characterization, the theoretical Eq. (8.121) is fitted (continuous curve) to the experimental |iΩ | data (spots) in Fig. 8.30, and from this fitting we are able to find out the parameters  and 𝜏sc , or just ∕𝜏sc in the high-frequency limit. The experimental and the material parameters from fitting are reported in Table 8.7. Note that ls is mainly determined from , which is also reported in Table 8.7. The 𝜎0 (0) or 𝜎0 (0)∕I(0) instead (although associated with K 2 L2D , which means that an auxiliary experiment is still necessary to compute LD ) are derived either from 𝜏sc (0) or from ∕𝜏sc . Note that the latter can be directly obtained from the high-frequency range without caring about the frequency-dependent part of the curve.

227

229

9 Self-Stabilized Holographic Techniques This chapter will show some of the many interesting possibilities of self-stabilized holographic recording for the measurement of photorefractive materials parameters, although this technique is not limited to photorefractives. We shall give only a few examples, focusing on the few materials for which we are able to give direct first-hand actual experimental data.

9.1 Holographic Phase Shift The phase shift (𝜙I ) between the pattern of light onto a photorefractive crystal and the resulting hologram at the very beginning of the recording process in Eq. (3.73) is the same as the one for running holograms at resonance in Eq. (3.86), as is easily verified. Different from the steady-state case, the value 𝜙I can be easily obtained from 𝜑 in Eq. (4.178) because at the very beginning of the recording process the hologram is very weak so self-diffraction effects can be neglected, as described in Eq. (4.124), and in this case the simple relation 𝜑 = 𝜙I + 𝜋∕2 holds [201] for ∣ Γ ∣≪ 1 and ∣ 𝛾 ∣≪ 1. Instead, the measurement of the stationary holographic phase shift under self-stabilized conditions is, unfortunately, not possible because self-stabilization directly acts on the output phase 𝜑 and therefore also on the hologram phase-shift itself. It is, however, possible to stabilize the pattern of fringes using a reference placed close to the sample to be measured, which will minimize perturbations and get more reliable results. Here, we shall illustrate the use of such an external reference-based stabilization, but for the measurement of the initial phase-shift (𝜙I ) described in Section 3.3.2 the same procedure may be used for the stationary phase-shift. The external reference is produced by a small and thin glass plate, firmly fixed by the side of the crystal as already illustrated in Fig. 6.27. The interference of the transmitted and reflected beams generated by the glass plate exhibit harmonic terms in Ω, as for the beams through the crystal. The first term is selected using a lock-in amplifier tuned to Ω, electronically integrated (although integration is not fundamental in this case), amplified and used to operate a feedback stabilization loop. The gain–bandwidth product of the feedback loop is set to fix the light fringe pattern in a time interval much shorter than the one required to measure with the lock-in amplifier. In order to illustrate the method and the advantages of stabilized recording techniques, even for the measurement of the phase shift, we shall describe the measurement of the initial phase shift for a 2.05 mm thick nominally undoped Bi12 TiO20 crystal [72]. In this sample, the charge carriers are electrons without any noticeable hole competition. The experiment was carried out with a [001]-crystal axis perpendicular to the grating vector K⃗ that is parallel to the applied electric field direction, with the pattern of fringes projected onto the (110)-crystal face. The recording was carried out using a 532 nm wavelength laser line with the input beam polarization Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

9 Self-Stabilized Holographic Techniques

chosen to have the transmitted and diffracted beams approximately parallel-polarized behind the crystal [193]. The room temperature was kept fixed to 22 ± 1∘ C. The first I Ω and second I 2Ω harmonic terms of the irradiance behind the sample were separately detected using two lock-in amplifiers (tuned to Ω and 2Ω, respectively) for computing 𝜑 from Eq. (4.174) in Section 4.3.1.2. In the present experiment, we used an electronic processing circuit and a two-phase lock-in amplifier that essentially produces two signals in quadrature [112] VX = V0 sin 𝜑 and VY = V0 cos 𝜑 √ V0 = 𝜅 𝜂(1 − 𝜂) Ω

2Ω

Ω

(9.1) (9.2)

2Ω

But the usual V ∝ I and V ∝ I signals, reported in Section 6.2, could be used as well. Here, 𝜅 is a known setup constant and 𝜂 the diffraction efficiency of the hologram. In the present experimental conditions, self-diffraction can be neglected and the (initial) holographic phase-shift is straightforwardly computed from VX and VY as 1∕ tan 𝜙I = − tan 𝜑 = −VX ∕VY

(9.3)

Figure 9.1 illustrates typical VX and VY data recorded at a time interval of approximately one-tenth of the hologram response time 𝜏sc [85]. Figure 9.2 shows the tan 𝜑 data computed from VX and VY signals as for the ones shown in Fig. 9.1 and plotted against the applied electric field for three independent experiments. Unlike curves B and C, data plotted in curve A were measured with the stabilization loop switched off. All experimental data fit to theoretical Eq. (3.73) where the electric field E0 is substituted with its effective value 𝜉E0 , as discussed in Chapter C, for the region in the crystal where the measurement is carried out. The best theoretical fittings in Fig. 9.2 are represented by continuous curves that lead to a diffusion length value of LD and a parameter 𝜉 for each experiment, which are listed in Table 9.1. Computing 𝜑 in the nonstabilized experiment (curve A, measurement time ≈ 60 ms) produce a much larger data dispersion than for the stabilized experiments (curves B and C). Although all three experiments show a good agreement for the computed LD parameter, the scattered data in curve A do not lead to a good estimation for the effective field coefficient 𝜉, thus showing the relevance of stabilization techniques. Dashed lines in curves B and C were plotted to give 500 Look-in outputs (μV)

230

500

400

VY

300

300

200

200

100

VY

400

VX

100

VX

0

0 0

0.3

0.6 0.9 Time (s) (a)

1.2

0

0.3

0.6 0.9 Time (s)

1.2

(b)

Figure 9.1 Typical time evolution of the VX and VY signals (dots) at the initial stage of the recording process in Bi12 TiO20 for E = 0 (a) and E = 3.15 kV/cm (b). The ratio between the angular coefficients of the linear fittings (continuous curves) are used to compute 𝜑. The diffraction efficiencies at t = 1.2 s are 𝜂 ≈ 3 × 10−5 (a) and 𝜂 ≈ 5 × 10−5 (b), whereas the minimum detectable signal was estimated to correspond to 𝜂 ≈ 10−7 . Reproduced from [72].

tan φ

9.1 Holographic Phase Shift

0.50

0.50

0.40

0.40

0.30

0.30 A

0.20

0.20

0.10

0.10

0

0

0

1

2 3 4 5 E (kV/cm)

6

7

B

C

0

1

2 3 4 5 E (kV/cm)

6

7

Figure 9.2 Computed initial tan 𝜑 versus applied electric field data (spots) in Bi12 TiO20 . The best fits to theory are represented by the continuous curves. Curve A represents nonstabilized experiments, whereas curves B and C represent stabilized experiments. Experimental parameters and the values for LD and 𝜉 computed from data fitting are reported in Table 9.1. Dashed lines in curve B were plotted for LD = 0.13 μm (upper) and for LD = 0.14 μm (lower) and similarly in curve C for LD = 0.13 μm (upper) and for LD = 0.15 μm (lower), to approximately indicate the precision of the measurement. Reproduced from [72].

Table 9.1 Initial phase shift: for Bi12 TiO20 from data fitting in Fig. 9.2. IRo

ISo 2

𝝉sc

K −1

(s)

LD 𝝃

Curve

(mW/cm )

(𝛍m )

(𝛍m)

A

2.8

0.12

7.07

0.6

0.73

0.15

B

0.42

0.025

7.07

3.5

0.86

0.135

C

0.32

0.020

11.27

12

0.87

0.14

Curves A, B and C refer to Fig. 9.2. From [72]. 𝜏sc is the hologram response time measured for E = 0. 𝜉 is the electric field correction coefficient.

an idea of the data dispersion effect on LD measurement precision. These lines were plotted just introducing small variations in the best fitted LD value in order to wrap up most of the corresponding experimental data. We compared our results, for the same sample, wavelength and room temperature, with those obtained from two well-known techniques: Measurement of the hologram time constant versus spatial frequency [202, 203] leading to LD = 0.12 ± 0.04 μm and the measurement of the holographic sensitivity versus spatial frequency, as in Section 8.2.3 and published elsewhere [158, 204], lead to LD = 0.15 ± 0.04 μm. Results from both these techniques are clearly consistent with those reported in Table 9.1. The same stabilization technique reported previously also used for the measurement of the stationary phase-shift, for which the results are reported in Fig. 9.3. This figure shows experimental phase-shift data for the same Bi12 TiO20 sample and same configuration as for data in Fig. 8.14 but for 𝜆 = 532 nm. For stationary phase-shift, however, the calculations are more complex (as already discussed in Section 8.6.2) than for the initial phase because of self-diffraction effects that are almost absent for the latter.

231

9 Self-Stabilized Holographic Techniques

1.5

0.5 tan φF

232

–0.5

–1.5 –1.0 × 106

–0.5 × 106

0

0.5 × 106

10 × 106

E (V/m)

Figure 9.3 Output phase-shift 𝜑 versus applied electric field (E0 ) data (circles) for a 2.05 mm thick Bi12 TiO20 crystal and grating-vector K = 5.5 μm−1 for 𝛽 2 = 30, and 532 nm wavelength, with 𝛼 = 8.5 cm−1 . The continuous curve is the best fit to the theoretical equation in Eq. (4.128) that leads to ls = 0.03 with a field factor 𝜉 ≈ 0.74.

9.2 Fringe-Locked Running Holograms Self-stabilized or fringe-locked running holograms, as described in Section 6.2.2, were recorded on undoped Bi12 TiO20 and Bi12 SiO20 crystals. The expression for the hologram speed in Eq. (6.31) can be also written as [75, 164, 205] 𝑣=

2𝑣M EM E0

(9.4)

2 E02 + EM

2 = ED2 [1 + (KLD )−2 ] with EM

and 2EM 𝑣M =

e Φ Iabs h𝜈𝜀o 𝜀 K 2 d

(9.5) (9.6)

to better evidence the maximum speed 𝑣M that is achieved for an applied field E0 = EM . The Equation (9.6) is valid in the absence of bulk light absorption in the crystal (see Section 4.2.2.3) only. Note that the adaptive running hologram speed in Eq. (9.4) is different from the “free” resonance running hologram speed in Eq. (3.82). It is easy to realize that 𝑣 in Eq. (9.4) becomes independent of LD for both the cases KLD ≪ 1 and KLD ≫ 1, for the same reasons discussed in Section 8.6.5. Figure 9.4 shows some experimental results for a Bi12 SiO20 crystal, showing the theoretical fitting of Eq. (9.4) to experimental data, which allows computation of important transport parameters such as the diffusion length, LD , and quantum efficiency Φ. Experiments carried out on this Bi12 SiO20 sample in different conditions give very reproducible LD = 0.19 μm results with a precision better than 5% and a value of Φ ranging from 0.46 to 0.6 [164, 205]. It is unnecessary to point out that the use of adaptive techniques gives reliable and reproducible results. 9.2.1

Absorbing Materials

Moving holograms in strongly absorbing materials have been studied in Section 4.2.2.3 where we showed that in this case the parameters Γd and 𝛾d should be replaced everywhere with their

9.2 Fringe-Locked Running Holograms

50 40 V (10–3 μm/s)

Figure 9.4 Fringe-locked running hologram speed versus applied electric field for a 1.71 mm thick Bi12 SiO20 crystal with 𝛼 = 3 cm−1 for the 514 nm wavelength with m ≈ 0.3, IS = 12 μW∕cm2 , IR = 440 μW∕cm2 and K = 4.24 μm−1 . Theoretical fit (continuous curve) to experimental data (∘ ) leads to LD = 0.19 μm and 0.46 ≤ Φ ≤ 0.6 ranging from 0.6 to 0.46 with an estimated field factor of 0.87. Reproduced from [205].

30 20 10 0

0

2

4 6 E0 (KV/cm)

8

10

corresponding values integrated along the sample thickness, as described by Eqs. (4.143) and (4.144) [ ]z=d ( ) z=d bci 2aK𝑣 e𝛼z + b 2 arctan √ Γ(K𝑣, z)dz = Γd = 4𝑤 ai − √ ∫z=0 2c 𝛼 4ac − b2 4ac − b2 z=0 [ ] z=d c e2𝛼z + 4𝑤 i ln 2𝛼c a(K𝑣)2 e2𝛼z + bK𝑣 e𝛼z + c z=0 [ ]z=d ( ) z=d bcr 2aK𝑣 e𝛼z + b 2 arctan √ 𝛾(K𝑣, z)dz = 𝛾d = 4𝑤 ar − √ ∫z=0 2c 𝛼 4ac − b2 4ac − b2 z=0 [ ] z=d c e2𝛼z + 4𝑤 r ln 2𝛼c a(K𝑣)2 e2𝛼z + bK𝑣 e𝛼z + c z=0 In this case, the adaptive running hologram feedback condition is no longer 𝛾 ∝ ℜ{Eeff } = 0 as stated in Eq. (6.30), but is instead z=d

𝛾d ≡

∫0

z=d

𝛾dz ∝

∫0

st ℜ{Esc }dz = 0

(9.7)

Substituting the integral in Eq. (9.7) with its expression in Eq. (4.144), with 4ac ≥ b2 , we get [ ]z=d ( ) z=d bcr 2aK𝑣 e𝛼z + b 2 arctan √ 𝛾(K𝑣, z)dz = 𝛾d = 4𝑤 ar − √ ∫z=0 2c 𝛼 4ac − b2 4ac − b2 z=0 [ ] z=d c e2𝛼z + 4𝑤 r ln =0 (9.8) 2𝛼c a(K𝑣)2 e2𝛼z + bK𝑣 e𝛼z + c z=0 After rearranging terms, we get the final formulation √ 4a1 c1 − b21 𝜏M (0) K𝑣 ( e𝛼d − 1) 2c1 + 2a1 𝜏M (0)2 K 2 𝑣2 e𝛼d + b1 𝜏M (0) K𝑣( e𝛼d + 1) √ ( )⎤ ⎡ f 4a1 c1 − b21 2 2 2 2𝛼d + b1 𝜏M (0) K𝑣 e𝛼d + c1 ⎥ ⎢ 1 a1 𝜏M (0) K 𝑣 e = tan ⎢ 𝛼d − ln ⎥ 2 a1 𝜏M (0)2 K 2 𝑣2 + b1 𝜏M (0) K𝑣 + c1 ⎢ 2gc1 + f b1 ⎥ ⎣ ⎦ (9.9)

233

234

9 Self-Stabilized Holographic Techniques

where the parameters here are conveniently slightly differently defined to those in Section 4.2.2.3 as a1 = (K 2 L2D f )2 + (1 + K 2 L2D )2

(9.10)

b1 = 2f (K 2 lS2 − K 2 L2D )

(9.11)

c1 = (1 + K 2 lS2 )2 + ( f K 2 lS2 )2

(9.12)

g = K 2 L2D f 2 + K 2 L2D + 1

(9.13)

f =𝜉

E0 ED

(9.14)

It is worth pointing out that Eq. (9.9) brings about an implicit relation G(K𝑣, E0 ) = 0

(9.15)

between the speed (𝑣 or K𝑣) and the applied electric field E0 (included in parameters f , a1 , b1 , c1 and g previously) in substitution of the simplified explicit relation in Eq. (6.31) for absorptionless materials. 9.2.1.1 Low Absorption Approximation

For the case 𝛼d ≪ 1, we can write e𝛼d ≈ 1 + 𝛼d and also substitute tan x ≈ x for |x| ≪ 1 in Eq. (9.9) just to verify that the expression obtained K𝑣𝜏M (0) =

1+

K 2 L2D

f for 𝛼d ≪ 1 + K 2 L2D f 2

(9.16)

is the same already reported in Eq. (6.31) except for the additional factor f that now includes the coefficient 𝜉, taking into account the effectively applied electric field value. 9.2.2

Characterization of Materials

The self-stabilized (fringe-locked) running hologram is a powerful tool for material characterization and, under adequate conditions, one single experiment may provide most of the relevant material parameters and also the value of the effectively applied electric field coefficient, 𝜉. From a fringe-locked experiment it is possible to directly measure the detuning K𝑣 as a function of the applied field E0 . It is also possible to compute the diffraction efficiency 𝜂 from the same experimental run. From these two datasets (K𝑣 versus E0 and 𝜂 versus E0 ) we are able to determine the whole set of parameters LD , ls and Φ, plus the experimental coefficient 𝜉 [206]. Although the theoretical analysis of the effect of bulk light absorption on the material response time is not vital to understand running holograms or to understand the way self-stabilized running holograms are produced, such an analysis is essential to accurately fit the theoretical equation to the experimental data in absorbing materials, so as to enable their characterization. In fact, in view of the large number of parameters involved here, it is necessary to have an accurate theoretical function as well as accurate experimental data to enable fitting with high possibilities of convergence without multiple solutions and minimum uncertainties. In other words, there are too many parameters to be fitted and they are better fitted as the theoretical model is better adjusted to the experiment and the experimental data are as undispersive as possible. Let us recall that self-stabilized recording, either involving stationary or nonstationary holograms, inherently produces less dispersive data than nonstabilized recording. This technique

9.2 Fringe-Locked Running Holograms

is even less dispersive than stabilized nonself-stabilized holograms (stabilized on external references different from the hologram itself being recorded). 9.2.2.1

Measurements

We have already seen the importance of dealing with data with reduced dispersion, so it is worth spending some time to briefly explain how to measure to get adequate data for processing. Hologram Speed, K𝒗 The detuning K𝑣 can be computed, in a continuous and nonperturbative way, from the movement of the PZT-supported mirror; that is to say, from the voltage applied to this device after calibration. This is the easiest and more direct way, although it is not the best one, to measure the hologram speed because the movement of the PZT also accounts for the feedback correction of environmental perturbations and steady state drifts (produced by temperature, for example) on the setup, so data from PZT are usually rather noisy as seen on the typical results in Fig. 9.5. A better way to carry out such measurements is using the pattern of interference between the transmitted and reflected beams in a small thin glass plate placed by the side of the sample as already discussed and shown in Fig. 6.24. In this way, it is possible to follow the evolution of such fringes as reported in Fig. 6.25 and far less dispersive K𝑣 data are obtained, as becomes obvious from the results in Fig. 9.6.

9.2.2.1.1

Diffraction Efficiency The first and the second harmonic terms in Ω, respectively, reported in Eqs. (4.172) and (4.173) are √ √ and ISΩ = 4J1 (𝜑d ) (IR0 )t (IS0 )t 𝜂(1 − 𝜂) sin 𝜑 √ √ IS2Ω = 4J2 (𝜑d ) (IR0 )t (IS0 )t 𝜂(1 − 𝜂) cos 𝜑

9.2.2.1.2

which are detected along the IS -direction, behind the sample under analysis, using a photodetector and lock-in amplifiers tuned to Ω and 2Ω, respectively, so that the corresponding output 0.8

Kv (rad/s)

0.6 0.4 0.2 0 –0.2

0

2

4

6

E0 / ED

Figure 9.5 Fringe-locked running hologram experiment: frequency detuning K𝑣 (measured from the movement of the PZT-supported mirror) versus normalized applied field E0 ∕ED data from a typical fringe-locked running hologram experiment carried out on an undoped Bi12 TiO20 crystal using the 514.5 nm wavelength with K = 7.55 μm−1 , IRo = 21.5 μW/cm2 and ISo = 0.45 μW/cm2 [207]).

235

9 Self-Stabilized Holographic Techniques

1.05

Kv (rad/s)

236

0.70

0.35

0

0

1

2

3

4

5

E0 / ED

Figure 9.6 Fringe-locked running hologram experiment on undoped Bi12 TiO20 crystal using the 514.5 nm wavelength with K = 8.5 μm−1 , IRo + ISo = 52 μW∕cm2 and 𝛽 2 = 183: frequency detuning K𝑣 (measured from the interference pattern from an auxiliary glassplate) versus normalized applied field E0 ∕ED data.

signals

√ VSΩ = AJ1 (𝜓d ) 𝜂(1 − 𝜂) sin 𝜑

and

√ VS2Ω = AJ2 (𝜓d ) 𝜂(1 − 𝜂) cos 𝜑

(9.17)

are obtained, where A is the overall amplification that depends on the photodetectors, beams irradiances, amplifiers and on other experimental settings. The VSΩ signal is used as an error signal in the feedback loop so it is automatically set to 0, by imposing sin 𝜑 = 0 as a consequence of the feedback condition in Eq. (9.7) and the expression of 𝜑 in Eq. (4.146). For nonphotovoltaic crystals, in the absence of an externally applied electric field, the equilibrium value is 𝜑 = 0. However, in the presence of an external field, in general this is 𝜑 ≠ 0. By imposing the 𝜑 = 0 constraint, the pattern of fringes is put in movement with a speed 𝑣 that depends on the mismatch between the actual equilibrium 𝜑-value and the imposed 𝜑 = 0 as already discussed in Section 6.2.2. Under steady-state conditions, the photorefractive hologram moves synchronously with the pattern of fringes. For 𝜑 = 0, we then have √ and VS2Ω = AJ2 (𝜓d ) 𝜂(1 − 𝜂) (9.18) VSΩ = 0 Therefore, it is possible to measure 𝜂 from VS2Ω , in a continuous nonperturbative way during recording. Typical experimental results obtained for 𝜂 are plotted on the right-hand side in Fig. 9.7. The left-hand side shows K𝑣 data, computed as described before, for the same sample and experiment. 9.2.2.2 Theoretical Fitting

The theoretical expression of 𝜂 in Eq. (4.145), for the imposed feedback condition in Eq. (9.7), becomes 𝜂=

cosh(Γd∕2) − 1 2𝛽 2 , 1 + 𝛽 2 𝛽 2 exp(−Γd∕2) + exp(Γd∕2)

(9.19)

But Γd (see Eq. (4.143)) is dependent on E0 and on K𝑣, so 𝜂 in Eq. (9.19) is also implicitly dependent on E0 and K𝑣. The other consequence of the feedback condition, besides leading to Eq. (9.19), is to bring about an implicit relation between E0 and K𝑣 as formulated in Eq. (9.15) G(K𝑣, E0 ) = 0

9.2 Fringe-Locked Running Holograms

0.05 0.04

0.3 𝜂

Kv (rad/s)

0.4

0.2

0.03 0.02

0.1

0.01 0

1

2

3 4 E0 / ED

5

6

0

1

2

3 4 E0 / ED

5

6

Figure 9.7 K𝑣 and 𝜂 experimentally measured as function of E0 ∕ED on an undoped Bi12 TiO20 crystal 2.35 mm thick (labeled BTO-013) with IR0 + IS0 = 14 W/m2 , 𝛽 2 ≈ 48, K = 7.55 μm−1 and 𝛼 = 1041 m−1 at 514.5 nm wavelength.

which means that K𝑣 is no longer an independent variable but one determined by E0 . Such an implicit relation turns the 3D surface represented by Eq. (9.19) into a 3D curve actually representing the theoretical formulation of fringe-locked experiments. Because of the 3D nature of the theoretical formulation, and in order to facilitate data fitting, it is interesting to operate with 3D experimental data too. Therefore, instead of displaying data in 2D as in Fig. 9.7, we display the same data in 3D as shown in Fig. 9.8, where the continuous curve is the result of previous data fitted in Fig. 9.7, but direct experimental data without previous fitting also could be plotted instead. We may choose not to fit the 3D experimental data with the 3D theoretical curve arising from Eqs. (9.19) and (9.15), but instead just use the 3D surface represented by Eqs. (9.19). The consequence of this choice is that experimental data are fitted with a larger class of functions (a surface instead of a curve) but the handling of this 3D surface is easier than the 3D curve containing the implicit relation G(K𝑣, E0 ) = 0. The result of such a fitting is shown in Fig. 9.9 and the parameters obtained from this fit are reported in Table 9.2 0

Figure 9.8 3D plotting of experimentally measured eta and K𝑣 as function of E0 ∕ED from Fig. 9.7.

E0 / ED 2 4

0.02

η 0.01

0 0.4

0.3

0.2

Kv (rad/s)

0.1

0

237

238

9 Self-Stabilized Holographic Techniques

0 E0 / E D 2 4

0.04

Figure 9.9 3D surface plotting of 𝜂 and K𝑣 as function of E0 ∕ED from Eq. (9.19) with same experimental data as for Fig. 9.8 showing the best fit theoretical 3D-curve (continuous thick curve) from Fig. 9.8. The resulting best fitting parameters are reported in Table 9.2.

0.03 η 0.02

0.01

0 0.6 0.4 Kv (rad/s)

0.2 0

Table 9.2 Parameters from experimental 𝜂 and K𝑣 data fitting as function of E0 ∕ED for undoped Bi12 TiO20 from Fig. 9.9. LD (𝛍m)

Data

𝚽

𝝃

Variance

input

0.16

0.04

0.35

0.75

–

output

0.13

0.042

0.41

0.73

1.38 × 10−8

input

0.1

0.01

0.4

0.6

–

output

0.094

0.044

0.74

0.75

1.16 × 10−8

input

1

0.1

0.1

0.1

–

output(* )

1.26 × 106

0.27

1019

637

1.6 × 10−4

input

0.01

0.001

0.1

1

–

0.048

9.33

1.08

1.24 × 10−7

*

output( ) *

ls (𝛍m)

3

9 × 10

unacceptable output

Fitting requires some initial hint (“input”) for the parameters we are looking for (LD , ls , Φ and 𝜉) in order to obtain associated results (“output”). A few such inputs and resulting outputs are reported in Table 9.2. The last two inputs lead to unacceptable outputs, either because they lead to impossible (e.g. Φ > 1) results or because they lead to unrealistic values for one or more of the parameters we are looking for. The first two inputs (first two rows) instead are acceptable and actually lead to similar results for all parameters, except for Φ, which is found to be either 0.41 or 0.74. Note also that the acceptable outputs are also characterized by a much lower variance coefficient for the fit, thus indicating a statistically better fit compared to the two last ones. Among other means to decide what are good and what are unacceptable results, one should bear in mind that 0 ≤ Φ ≤ 1 and 0 ≤ 𝜉 ≤ 1, for example.

9.3 Characterization of LiNbO3 :Fe

In the present case, we believe that the output in the first row is the right one because both LD and Φ are closer to the available data in the literature for similar samples (LD = 0.14 μm [72, 191] and for the same wavelength (Φ = 0.45 [191]), although the differences between both acceptable outputs are not very significant for this kind of experiment. Because of the comparatively large number (four) of parameters involved, the fitting is particularly sensitive to the dispersion of the experimental data. Reducing dispersion, that is to say obtaining data with higher accuracy, may considerably reduce the number of possible acceptable results. It is also possible that using the actual 3D theoretical curve instead of the larger class of 3D theoretical surfaces may also reduce the number of multiple solutions in data fitting. In order to better understand what the actual possibilities are to obtain reliable values for such a relatively large number of parameters to be fitted, it is worth recalling here some facts [191] about the influence of LD and ls in running hologram phenomena: if K 2 L2D ≫ 1, the value of LD does not affect the dynamics of the recording process because in this case the large diffusing length compared to the hologram spatial period somewhat randomizes the position of the excited electron in the conduction band. On the other hand, K 2 ls2 ≪ 1 means that the material is very far from saturation and therefore the process does not depend on the density of photoactive centers that is related to 1∕ls2 . If any of these conditions are fulfilled, the fitting will not lead to the parameter involved, simply because it is not relevant for the recording process itself.

9.3 Characterization of LiNbO3 :Fe The first part of this chapter was devoted to illustrate the use of stabilized and self-stabilized techniques for the characterization of fast and low diffractive materials such as sillenites. We shall show now the use of self-stabilized holographic recording for the characterization of a wide different type of material: a slow and highly diffractive photovoltaic like LiNbO3 :Fe. As is usual with photovoltaic crystals, we shall operate in short-circuit mode (as illustrated in Fig. 3.28) and record a hologram, without applied field, with the hologram vector K⃗ parallel to the c-axis. Let us recall that self-stabilized recording on LiNbO3 :Fe, with any degree of oxidation, is carried out using I 2Ω as error signal as described in Section 6.3.2. In this case, if we operate with equal irradiance recording beams (𝛽 ≈ 1), 𝜂 is described by Eq. (6.58) 𝜂 = sin2 𝛾d∕4

for

𝛽2 = 1

with 𝛾∕4 =

𝜋n3eff reff |Esc | 2𝜆

and Γ = 0

(9.20)

Under self-stabilized recording, a steady-state nonstationary space-charge field arises in the form st −iK𝑣t Esc (t) = −mEsc e E (𝜔 + i𝜔I ) st Esc ≡ eff R 𝜔R + i(𝜔I − K𝑣)

as already described in Section 6.3.2.1, with Eeff ≡

E0 + Ephv + iED ND+

1 + K 2 ls2 − iKlE − iKlphv N

D

≈

Ephv N+

1 − iKlE − iKlphv ND

D

239

9 Self-Stabilized Holographic Techniques

as reported in Eq. (3.170), where the photovoltaic field Ephv is reported in Eq. (3.150) as: Ephv =

𝜅ph Iabs

𝜎 =∝ I

𝜅ph h𝜈r

[Fe3+ ] 𝜎d μe st 𝜏sc ≈ 𝜏M = 𝜖33 𝜀0 ∕𝜎 ≈

(9.21)

[Fe2+ ] [Fe3+ ]

(9.22)

st where 𝜖33 is the static dielectric constant along the c-axis and 𝜅ph is a photovoltaic transport coefficient [42]. The feedback stabilization (I 2Ω = 0) with 𝛽 2 =1 imposes the additional condition (see Section 6.3.2.2) st ℑ{Esc }=0

(9.23)

Accordingly, the time evolution of 𝜂 during self-stabilized recording with 𝛽 2 ≈1 can be formulated as [ 3 ] √ 𝜋neff reff d −t∕𝜏 st sc ) | 𝜂(t) ≈ | sin (9.24) m ℜ{Esc }(1 − e 2𝜆 The I 2Ω is used as an error signal in the feedback stabilization loop and [ ] √ I Ω ∝ 𝜂(1 − 𝜂) = sin B(1 − e−t∕𝜏M ) B≡

𝜋n3eff reff d 𝜆

st } m ℜ{Esc

(9.25) (9.26)

is used to follow the recording evolution. Figures 9.10–9.12 show some experimental results for different samples. It is important to emphasize the interest of self-stabilized recording here: this ensures holographic recording, with minimum environmental perturbations, for the very long recording time required for these very slow materials and forces the recording to occur in such a way as to verify the simple relation in Eq. (9.24). From the evolution of I Ω in Figs. 9.10–9.12 and the theoretical relations in Eqs. (9.24)–(9.26), it is possible to characterize some important material parameters. To do this, it is necessary to keep in mind that the index-of-refraction is dependent on the light wavelength and that it is, as well as the electro-optic coefficient, quite different for ordinary and extraordinary light polarization. 2.5

Figure 9.10 Characterization of reduced LiNbO3 :Fe (labeled LNB3): self-stabilized holographic recording on a d = 1.39 mm thick crystal (labeled LNB3) using ordinarily and extraordinarily polarized 𝜆 = 514.5 nm light (𝛽 2 ≈ 1 and IR0 +IS0 ≈ 16 mW/cm2 ) with an irradiance absorption 𝛼 = 7.5 cm−1 at this wavelength. The fitting of Eq. (9.25) to experimental IΩ data gives B and 𝜏M as reported in Table 9.3.

Extraordinary

2.0 I Ω (AU)

240

1.5

Ordinary

1.0 0.5 0

0

500

1000 Time (s)

1500

9.3 Characterization of LiNbO3 :Fe

Figure 9.11 Characterization of reduced LiNbO3 :Fe (labeled LNB5): self-stabilized holographic recording on a d = 0.85 mm thick crystal using extraordinarily polarized 𝜆 = 514.5 nm light with IR0 = 141.1 W/m2 and IS0 =116 W/m2 . Eq. (9.25) was fitted to data and the resulting parameters reported in Table 9.3. At the end of the cycle when ISΩ = 0, it was measured 𝜂 = 1. From [123] and [124].

8 6

Ω

IS

4 2

2Ω

IS

0 0

Figure 9.12 Characterization of oxidized LiNbO3 :Fe (labeled LNB1): self-stabilized holographic recording on a d = 1.5 mm thick crystal using extraordinarily polarized 𝜆 = 514.5 nm light (IR0 = 113.5 W/m2 and IS0 = 108.1 W/m2 ) and fitted with Eq. (9.25). The resulting parameters are reported in Table 9.3. Reproduced from [123].

50

200

100 150 Time (s)

250

8 I

6

Ω

4 2 I

0 0

2000

2Ω

4000

6000

Time (s)

Table 9.3 Parameters for LiNbO3 :Fe samples. S 𝜶d at 514.5

[Fe3+ ] ×1019 /cm3

[Fe2+ ]/ [Fe3+ ]

𝝉M (s)

10−12 m3 /J

Sample

Pol.

d (mm)

LNB5

ext

0.85

2

0.03

10.7

540

LNB1

ext

1.5

2

0.002

4.65

3708

10.2

827

6.4

10.6

6.46

1872

1.8

3.3

ext LNB3

1.39

1.04

2

B

Exp.

Theor.

10.6 10.6

0.013

ord

It is straightforward to deduce from Eq. (9.25) that • the maximum of I Ω is achieved for B(1 − e−t∕𝜏M ) = 𝜋∕2 and • I Ω goes down to zero for B(1 − e−t∕𝜏M ) = 𝜋. These simple relations allow direct computation of B and 𝜏M from the experimental data, although it is always possible to compute these parameters from the theoretical equation fit to the experimental data as well. From the B and 𝜏M parameters, it is possible to compute the so-called sensitivity, as defined in Section 8.2.3 that, for the particular case of a photovoltaic crystal, should be formulated as: st 𝜀o )  = n3eff reff 𝜅ph ∕(2𝜖33

(9.27)

241

242

9 Self-Stabilized Holographic Techniques

Table 9.4 LiNbO3 :Fe material parameters. no

=

2.33

ne

=

2.25

[209] [209] −12

r13

=

8.6 × 10

r33

=

30.8 × 10−12 m/V

m/V

[1]

[1]

st 𝜖33

=

32

[1]

𝜅ph

≈

1.7 × 10−11 m/V

[210]

For 𝜆 = 514.5 nm

Table 9.5 Sensitivity and relative photoconductivity for doped and undoped BTO. BTO

BTO:005∗

BTO:Pb

0.65

53

52

60

–

60

95

60

180

–

80

BTO:V Units



Fast

10−10 m3 /J

Slow

32 HOLOG. 27

𝜎e (0)h𝜈∕IR (0) 10−30 s m/Ω WRP

9

*using 𝜆 = 514.5 nm laser light

and can be computed from =

 𝜆 𝜏M 2𝜋mIabs

(9.28)

The parameters B, 𝜏M and  computed for the different samples in these experiments are reported in Table 9.3 together with some information about these samples. General data about LiNbO3 crystals from the literature are reported in Table 9.4. Table 9.5 shows sensitivity and relative photoconductivity values for doped and undoped BTO. The  values computed from our experimental data in Table 9.3 are shown to be in good agreement with the theoretical values computed from the available data in the literature listed in Table 9.4. Note that the theoretical development here is concerned with the so-called first harmonic approximation that is verified for m ≪ 1 but not necessarily valid for the m ≈ 1 value in most of the experiments described here, a fact that may explain the lack of a better agreement between experimental data and theory in Table 9.3. The presence of a strong light scattering effect must also be pointed out, which may have a sensible effect on the measured response time 𝜏sc [208] and may also interfere with a better agreement with theory.

243

Part IV Applications

244

Introduction

A large number of interesting applications for photorefractive materials have been already reported in the scientific literature and plenty of others were being described at the time this book was written. We do not intend to even mention them here because the highly dynamical nature of the research in this area is likely to render them outdated in the short term. Instead, we will focus on three specific applications that we consider particularly illustrative: • measurement of mechanical vibrations and deformations, concerning fast and low diffractive photorefractive materials, • fixing a hologram for fabrication of diffractive holographic optical components involving slow materials exhibiting high diffraction efficiencies and • light-induced photoelectric conversion via the Dember effect besides the already well-known possibilities due to photovoltaic effects of some materials.

Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

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10 Vibrations and Deformations Holographic interferometry enables the real-time measurement of vibrational modes and static deformations in surfaces using low-power laser illumination and a photorefractive crystal as the recording medium. Since Huignard and coworkers first demonstrated [211] the possibilities of holographic interferometry using photorefractives to measure mechanical vibrations, plenty of publications have appeared [212] in this field, although it took a long time until an efficient device was made available for this purpose [213, 214]. In this chapter, we describe a conventional setup using a nominally undoped photorefractive Bi12 TiO20 (BTO) crystal where most of the critical elements have been optimized: target illumination and backscattered light collection, distribution of light between the object and the reference beams. The novelty here is the use of self-stabilized holographic recording to improve the setup performance. The use of photorefractive materials as real-time, reversible holographic recording media has been shown to eliminate most of the handicaps of holography, thus providing with a practical tool for vibration and deformation measurement. Low-frequency perturbations and changes in the setup are adaptively coped with because of the relatively fast response of these materials. Higher-frequency perturbations can instead be compensated by the use of an active stabilization feedback opto-electronic loop as we reported in Chapter 6 and described in detail in what follows. The efficient illumination of the target surface and the collection of the backscattered light from the surface is very important for maximizing the intensity of the holographically reconstructed object wave containing the required information about vibration and deformation. Such optimization needs the retro-reflectivity of the target surface to be taken into account. The negative feedback opto-electronic loop used for stabilizing the setup has been rearranged in order to decrease the level of parasitic signals in it.

10.1 Measurement of Vibration and Deformation Several techniques allow the measurement of vibrations and deformations using holography. We report here the time-average holographic interferometry for vibrations [211] and double holographic exposure for deformations and tilting [215]. These are general methods that have been adapted to the special features of photorefractive recording media. The good setup performance is due to the particular features of the setup, including efficient target illumination and light distribution, as well as the use of a negative opto-electronic feedback loop for stabilizing the setup. The use of a retroreflective painting on the target surface largely contributes to the performance of the setup. Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

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10.2 Experimental Setup The actual experimental setup used in these experiments is described in Fig. 10.1. The input laser beam (with irradiance I0 ) is divided into a reference and an object one, using a polarization beam-splitter cube (PBS). The amplitude ratio R between both output beams is controlled using a half-wave retardation plate HWP at the PBS input. The polarization direction of the PBS-exiting beams are made parallel by the use of another HWP at one of the outputs. A low-power microscope objective lens L1 is used to expand the object beam in order to illuminate the whole target surface that is a loudspeaker in this case. A device formed by a PBS, two HWP and one quarter wave retardation plate QWP are used to direct all the light onto the target surface and then allow the whole backscattered collected light to get through the PBS directly onto the recording photorefractive crystal (BTO) with minimum losses. Two photographic good quality objectives are used to produce a reduced target image onto the BTO, and then an adequately sized image onto the ccd-camera for observation and/or image acquisition and processing. The reference beam is also directed onto the BTO to interfere with the object beam to produce the required hologram for recording. The hologram is produced in real time in the BTO crystal and at the same time is reconstructed by the same reference beam used for Loudspeaker Laser PZT

PBS

HWP PBS HWP HWP

EOM

QWP

PLC

PBS L1

M

L2 PBS M BTO

D

P2 L4

SF

HWP PBS HWP

L3 LA

L5

P1 CCD

INT

HV

Figure 10.1 Schematic diagram of the experimental holographic setup: PBS: polarizing beamsplitter cube; HWP and QWP: halfwave and quarterwave retardation plates, respectively; M: first surface mirrors; PZT: piezo-electric supported mirror; PLC: path length compensator; EOM: electro-optical modulator; SF: spatial filter; BTO: photorefractive Bi12 TiO20 crystal; D: photodetector; P1 e P2: polarizers; CCD: image detector; LA: lock-in amplifier; INT: integrator; HV: high voltage source for the PZT.

10.2 Experimental Setup

recording it: the diffracted reference beam is actually the reconstructed object beam carrying all the information needed about the target vibration or deformation. 10.2.1

Reading of Dynamic Holograms

The reading of holograms written in real-time reversible recording media as photorefractive crystals requires special techniques because the uniform reference beam erases the hologram during reading. Several possibilities exist for reading these so-called dynamic holograms. We have chosen an efficient technique based on the anisotropic diffraction properties of some crystals, among which are the sillenites and in particular the Bi12 TiO20 (BTO) used in these experiments. In fact, under certain experimental conditions, the transmitted and diffracted (holographically reconstructed) beams are orthogonally polarized following the procedures discussed in Chapter 5 and in the literature [216]. In this case, the diffracted beam carrying the necessary information about vibration and deformation can be separated from the transmitted beam that carries no information, using just a simple polarizer (P1 in Fig. 10.1). 10.2.2

Optimization of Illumination

The amount of light available for illuminating the target, recording the hologram in the crystal and reading it is limited by the power of the laser source being used. A powerful source is interesting because: • it speeds up the holographic recording because the recording time is roughly inversely proportional to the average light onto the crystal • the speed-up of recording allows one to adaptively cope with perturbations of higher frequency • it allows the illumination of a larger target surface. In order to optimize the available amount of light, we must efficiently illuminate and collect the light from the target, and adequately divide the input beam between the object and reference beams in the setup. 10.2.2.1

Target Illumination

The illumination and light collection from the target surface is described in Fig. 10.1: a polarization beamsplitter cube PBS, a halfwave retardation plate HWP and a quarterwave plate QWP are used. The incident light (TE-polarized) is completely reflected toward the target by the PBS and on its way forth and back from it crosses the QWP twice, thus rotating its polarization direction by 90∘ and therefore being transmitted through the PBS to the crystal. In this way, the limited available light from the laser source is efficiently used. To further improve light collection from the target, the latter is painted with a special thin retro-reflective ink film. 10.2.2.2

Distribution of Light among Reference and Object Beams

Figure 10.3 shows a simplified schema of the light distribution between the reference and the object beams in the setup that allows the calculation of the diffracted reference beam intensity IRD that is to be maximized. The latter is computed from the relations that follow IRD = 𝜂IR0

𝜂 = 𝜂0 m2

m=

2𝛽 (1 + 𝛽 2 )

𝛽 2 ≡ IR0 ∕IS0

R ≡ IS1 ∕IR1

(10.1)

where 𝜂0 is the maximum diffraction efficiency that can be obtained for a hologram in the crystal. I0 = IS1 + IR1

IR0 = fIR1

IS0 = 𝜁 IS1

(10.2)

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Figure 10.2 (a) Lateral view of the holographic setup: CCD camera (1), output polarizer (2), photographic objective lens for imaging the hologram onto the CCD (3), photorefractive crystal in its nylon holder (4), photographic objective lens for imaging the target onto the crystal (5), target painted with retroreflective ink (6) and 633 nm He-Ne laser (7). (b) Detailed view of the photorefractive crystal in its nylon holder, between the two photographic objective lenses and the output polarizer.

10.2 Experimental Setup

Figure 10.3 Simplified schema showing the distribution of incident light (I0 ) between reference and object beams: BS, beamsplitter; M mirror; IR1 and IS1 reference and object beams at the BS output; IR0 and IS0 , reference and object beams effectively incident on the crystal.

I0

I1R

M

BS

I1S

I 0R ID R

IS0

Target

BTO

40 ID R (AU)(mV)

Figure 10.4 Optimization of the target illumination: IRD , diffracted reference beam measured (in arbitrary units) as a function of R = IS1 ∕IR1 (∘), and the best fitting to theory (continuous line). From fitting, we get f ∕𝜁= 1.15 for our retro-reflective painted loudspeaker membrane.

30 20 10 0

IRD = 4𝜂o I0 f

R f ∕𝜁 (R + f ∕𝜁 )2 (1 + R)

2

R

6

10

(10.3)

In this expression (10.3) for IRD , 𝜂0 depends on the BTO, I0 is the available laser irradiance, f and 𝜁 depend on mirror M and on the target. The only parameter that is possible to adjust over a large range in the expression is the distribution of light R at the beamsplitter BS. From the relations before, we see that IRD is maximum for √ f ∕𝜁 = R2 (1 + 1 + 2∕R + 1∕R2 ) (10.4) Figure 10.4 shows the values measured for IRD (∘) as a function of R for a loudspeaker membrane painted with retro-reflected ink. The continuous curve is the best theoretical fitting with Eq. (10.3) and from this fitting the value f /𝜁 = 1.15 is obtained. From the independently measured f = 0.14 value, the effectively collected retro-reflected light can be estimated to be 𝜁 = 0.12 for that target in our setup. As seen from data in Fig. 10.4, the maximum value for IRD is obtained for R = 0.61, in good agreement with the value that can be deduced from Eq. (10.4). 10.2.3

Self-Stabilization Feedback Loop

The light intensity propagating along the object beam direction, behind the crystal, can be written as: √ √ IS = IS0 𝜂 + IR0 (1 − 𝜂) ± 2 g 𝜂(1 − 𝜂) IS0 IR0 cos 𝜑 (10.5) where g is a parameter depending on mutual polarization and coherence relations between reference and object beams, and all other parameters have the usual meaning in this book. All measurements are carried out behind the crystal, so bulk absorption need not be considered throughout. If no external electric field is applied to the crystal (the present case), we can show that 𝜑 = 0. For actively stabilizing the setup, it is necessary to modulate the phase of one of the

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interfering beams (the reference one in our case) in order to produce the harmonic term that is selectively detected and amplified for use as error signal in the feedback loop. In fact, a phase modulation of amplitude 𝜓d and frequency Ω (2π10 kHz here) in the phase 𝜑 of Eq. 10.5 will result in harmonic components in Ω where the amplitude of the first two are √ √ ISΩ = 4 g J1 (𝜓d ) 𝜂(1 − 𝜂) IS0 IR0 sin 𝜑 (10.6) IS2Ω = 4 g J2 (𝜓d )

√

𝜂(1 − 𝜂)

√ IS0 IR0 cos 𝜑

(10.7)

with J1,2 being the Bessel function of order 1 or 2. The phase modulation in 𝜑 is produced with the help of an electro-optical modulator (EOM in Fig. 10.1) placed in one of the interfering beams. In nonperturbed conditions and in the absence of an externally applied field on the crystal, 𝜑 = 0 and consequently ISΩ = 0. As soon as a perturbation on the setup appears that is faster than the crystal’s response time, the phase shift becomes 𝜑 ≠ 0 and also ISΩ ≠ 0. Therefore the latter signal can be used as error signal in the negative feedback stabilization loop. The operation of the self-stabilization feedback loop is discussed in Section 6.2 and is represented by the same block-diagram in Fig. 6.7. In this case, the integrated error signal is a fundamental feature that largely improves the stabilization performance because it allows to keep the phase-shift condition 𝜑 = 0 and still cope with steadily growing perturbations. An element contributing to the good performance of stabilization is the choice of an error signal that is obtained from the object beam behind the crystal with the help of the polarizing components HMP, PBS and HWP at the crystal output in Fig. 10.1). Although this sampling of the output object beam will somewhat decrease the final IRD , it will avoid detecting the direct transmitted reference beam that is phase-modulated and residually amplitude-modulated, to some extent, due to unavoidable misalignment of the EOM. Such an amplitude-modulated (at the same frequency Ω) signal in the feedback loop would seriously interfere in the stabilization process. The experimental setup used in this experiment is relatively complicated but its operation is very simple and can be carried out by nonspecialized technicians, once the optical components are adjusted and fixed. The only adjustment left for the operator is to place the target in the correct position to have it adequately focused on the TV screen and sometimes to correct the illumination of the target by gently acting on the screw of a mirror in the setup. The analysis of the pattern of fringes can be carried out on its photographic image or alternatively, the pattern can be transferred to a personal computer for analysis with an adequate standard commercial software. The use of a photorefractive crystal acting as a (nearly) real-time holographic recording reversible medium is essential in this experiment and allows overcoming most of the handicaps of classical holography. Thus, the operator can forget that a hologram is being recorded somewhere, and all changes in the target can be observed almost in real-time. The setup allows to measure alternatively vibrations or deformations, with a very simple modification in the operation procedure, without any change in the setup. The photorefractive crystal (undoped Bi12 TiO20 ) used in this instrument has been chosen because of its advantageous properties compared to other possible materials: suitable spectral sensitivity, recording speed, diffraction efficiency, optical quality, availability on the international market and so on, plus other specific properties (anisotropic diffraction) that make it particularly interesting for our purposes. The actively stabilized opto-electronic circuit described in this book is also essential to enable the operation of this instrument in moderately perturbated environment. The use of self-stabilization (and not just external reference-based stabilization) is essential to allow observing well defined interference patterns that would be otherwise hard to observe except for occasional moments during the experiment.

10.2 Experimental Setup

10.2.4

Vibrations

The measurement of vibrations by time-average holographic interferometry is based on the fact that the diffraction efficiency of the hologram recorded (during a time much larger than the period of the vibration under analysis) by the light backscattered from a surface, vibrating with amplitude d and frequency Ω, can be written as: 𝜂(d) = 𝜂0 |m|2 J02 (4𝜋d∕𝜆)

m=

2𝛽 (1 + 𝛽 2 )

𝛽 2 ≡ IR0 ∕IS0

(10.8)

where 𝜂o is the diffraction efficiency (using equal intensity recording beams) of the surface at rest, |m| is the value of the visibility of the interference fringes, IR0 and IS0 are the intensities of the reference and object beams incident on the crystal, and J0 is the Bessel function of the order zero. The holographically reconstructed target surface image is therefore superimposed to a pattern of dark and bright fringes corresponding to the different maxima and minima of the Bessel function as shown in Fig. 10.5. The position of these fringes allows computing the map of the local values of the amplitude vibration d over the target surface with the help of a table of Bessel functions as shown in the table of Fig. 10.5. Note that each point of local maximum amplitude of vibration in the membrane is at the center of a pattern of approximately concentric fringes. To evaluate the performance of this technique, the response of some points of local maximum amplitude of vibration in a loudspeaker membrane as a function of the applied voltage is shown in Figs. 10.6 and 10.7 for two different frequencies. The vibration of a thin (0.2 mm thick) phosphorous-bronze metallic plate was also visualized using the real-time holographic interferometry technique referred to before. The external plate border was tightly fixed to the external metallic ring of a commercial loudspeaker, using a plastic (PVC) double ring with a clear 79.5 mm internal diameter. The density of the plate was 6.24 g/cm3 . The plate was painted with a thin retro-reflective ink film to increase the amount of backscattered light collected by the optical setup and focused onto the photorefractive crystal. Vibration amplitudes d (nm) 0 120.9 191.4 277.05 352.6 435.7 511.3 594.4 670.0 750.6 831.2

ZERO MAX radians x x J02(x) — 0 1 2.4 — — — 3.8 0.16 5.5 — — — 7.0 0.09 8.65 — — — 10.15 0.062 11.8 — — — 13.3 0.048 14.9 — — — 16.5 0.038

Figure 10.5 Loudspeaker membrane (left) driven at 3.0 kHz and analyzed by the time-average holographic interferometry technique. The brighter areas are those at rest, the first dark fringe indicates a vibration amplitude of 0.12 μm, the second one 0.28 μm, the third one 0.44 μm and so on according to data in the table (right) showing the amplitude d of the vibration associated with the minima (for J0 (x) = 0) and maxima in the pattern of fringes.

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10 Vibrations and Deformations

Amplitude (µm)

0.8

Figure 10.6 Amplitude of vibration at a point of local maximum in the membrane of a loudspeaker as a function of the applied voltage for a signal of 4.2 kHz.

0.6 0.4 0.2 0

100

200

300

Applied voltage (mV)

0.8

Amplitude (µm)

252

Figure 10.7 Amplitude of vibration at two different points of local maximum in the membrane of a loudspeaker as a function of the applied voltage for a signal of 1.4 kHz.

C

0.6

A

0.4 0.2 0

0

0.2

0.4

0.6

0.8

1.0

Voltage (V)

The loudspeaker was used to excite the plate. Figures 10.8–10.10 show the interference patterns obtained for the frequencies of the electric signals feeding the loudspeaker that lead to the first, third and fourth normal vibration modes, respectively. The amplitudes of the local maxima can be approximately estimated from the number of fringes and the table of Fig. 10.5. 10.2.5

Deformation and Tilting

Photorefractive crystals can be used as double exposure recording media because the recording takes a finite time (inversely proportional to the total amount of light onto the crystal), so it is possible to record the first image of the target under study and then the second image of the deformed target as is usually done in classical interferometry. In the case of photorefractive crystals, however, the latter one has been recorded while the former one begins to fade: at a certain moment in this process, both images reach similar intensities and a maximum contrast of the fringes arising from the interference of both wavefronts (first and second object images) is obtained as shown in Figs. 10.11–10.13. This pattern-of-fringes image can be recorded using a ccd TV camera (as is the case here), or any other adequate device, for further processing. The BTO crystal used in these experiments is particularly well suited because it exhibits a rather low dark conductivity that grants no sensible changes in the already recorded image once the light is switched off between both exposures.

10.2 Experimental Setup

Figure 10.8 Time-average holographic interferometry pattern of a thin phosphorous-bronze metallic plate tightly fixed by its external border to a loudspeaker vibrating at 255 Hz.

Figure 10.9 Time-average holographic interferometry pattern of a thin phosphorous-bronze metallic plate tightly fixed by its external border to a loudspeaker vibrating at 600 Hz.

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Figure 10.10 Time-average holographic interferometry pattern of a thin phosphorous-bronze metallic plate tightly fixed by its external border to a loudspeaker vibrating at 800 Hz.

Figure 10.11 Double exposure holographic interferometry of a tilted rigid plate. The smallest diagonal dimension of the small cells printed in the plate is approximately 4.8 mm.

10.2 Experimental Setup

Figure 10.12 Double exposure holographic interferometry of a rigid plate that was less tilted than in Fig. 10.11. The smallest diagonal dimension of the small cells printed in the plate is approximately 4.8 mm.

Figure 10.13 Double exposure holographic interferometry of a rigid plate that was more tilted than in Fig. 10.11. The smallest diagonal dimension of the small cells printed in the plate is approximately 4.8 mm.

Exercise: The fringes in the pattern of Fig. 10.11 are not parallel, thus indicating there is also a deformation besides the tilting of the plate. But those in Figs. 10.12 and 10.13 are rather regular, thus indicating that there was only tilting in these cases. Knowing that the light in the setup was the 633 nm He-Ne laser line and that the distance along the diagonal of the squares printed on the plate is 4.8 mm, calculate the angle of tilting for the three figures referred to here and verify that they are, respectively, ≈ 5.4, 15 and 66 μ rad.

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10.2.5.1 Applications of PEMF to Mechanical Vibration Measurements

One of the most interesting applications of this effect is related to mechanical vibration measurement [197]. Further publications [217] showed that PEMF could also be produced by a vibrating speckle pattern of light. The description of PEMF effects in a strongly absorbing photorefractive material by an interference pattern of light vibrating with a rather large amplitude was formulated by Mosquera et al. [153] and has also shown to be useful for material characterization. A mathematical model was further developed by T.O. Santos [157] for large amplitude vibrating speckle pattern of light and also applied for material characterization [147, 151, 152].

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11 Fixed Holograms 11.1 Introduction Photorefractive materials are essentially reversible real-time recording materials and consequently are not at all suitable for storing information unless the crystal can be kept in the dark. It is nevertheless possible to fix holograms in some materials using special techniques, thus allowing the production of volume diffractive optical components to be used in practical applications under illumination. One such technique for hologram fixing is to use double-doped LiNbO3 [218–220], where recording occurs at the shallower traps and the hologram is transferred, during recording, to the deeper traps where the hologram cannot be erased during readout with the recording (or larger) wavelength. Fixing by inducing ferroelectric domain inversion by the combined action of light and an applied electric field was also demonstrated in Srx Ba1−x Nb2 O6 crystals [221, 222] and in doped KSBN. Another fixing technique uses high-temperature compensation in order to substitute the initially recorded photosensitive hologram with an opposite sign complementary ionic (assumed to be H+ ) nonphotosensitive one in LiNbO3 :Fe [120, 223–226]. A similar procedure was successfully applied on undoped Bi12 SiO20 [227] and Bi12 TiO20 [228]. The development of a complementary fixed grating has been reported to occur even at room temperature in Bi12 SiO20 [229, 230]. The recording and compensation processes may be carried out simultaneously at high temperature and, in this way, very good results (up to fixed 𝜂 = 0.16) were obtained in KNbO3 :Fe [231]. We shall focus on a modification of the latter technique [232] where holographic recording and compensation are simultaneously carried out at moderately high temperature but recording is carried out in self-stabilized mode.

11.2 Fixed Holograms in LiNbO3 Fixed holograms in iron-doped LiNbO3 are made possible by first recording a large grating arising from electron photoexcitation at room temperature and then heating the sample to 120–135∘ to increase the mobility of positive H+ -ions in the crystal. In this way, the H+ ions completely neutralize the holographic recording space-charge grating. The sample is then allowed to cool down to room temperature and a strong spatially uniform white light is projected onto the sample to erase and phase-shift (because of the photovoltaic field on the electrons during white light excitation) the electronic grating to some extent. The result is an overall positive ion grating that is stable to illumination because H+ is not photosensitive. Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

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The result is better if oxidized samples are used [120, 124, 233], although the holographic recording is more time-consuming [123]. The use of self-stabilized recording is important here for reducing environmental perturbations during the long time recording (as is the case with LiNbO3 and similar crystals) but it does not allow the recording of an electronic grating with an index-of-refraction modulation larger than that corresponding to 𝜂 = 1 [123]. In fact, the first and the second harmonic terms in Eqs. (3.32) and (6.33), which should be used as error signals in self-stabilization, become both null at 𝜂 = 1 and then become useless for self-stabilization, thus fixing an upper index-of-refraction modulation limit for self-stabilization recording that could otherwise go far beyond. This means that after compensation and development of the self-stabilized recorded grating, a lower (sometimes a much lower indeed) efficiency fixed grating results [124]. In order to overcome such a limitation, Breer and coworkers [232] proposed to carry out the simultaneous recording and compensation at moderately high temperature during self-stabilized holographic recording. In this way, if the operating temperature is adequately chosen, 𝜂 = 1 is never reached because of the simultaneous compensation process. In this case, the compensated recording can, in principle, continue up to the maximum possible index-of-refraction modulation the material may allow, until the exhaustion of either the H+ or the iron dopant. 11.2.1

Simultaneous Recording and Compensation

We describe here use of the previously mentioned simultaneous recording and fixing process in crystals [234] that would otherwise result in a rather low fixed hologram using the conventional three-step process (recording at room temperature – compensation at high temperature – development again at room temperature). It is essential here to select the adequate operating temperature: neither too high to compensate for almost all the electronic grating so there would be no diffracted light left to operate the feedback stabilization loop, nor too low to slow down the process and make it impractical. 11.2.1.1 Theory

The simultaneous recording and compensation process should be carried out at a temperature high enough to increase the mobility of H+ (with volume concentration  + ) and allow compensating for the electron traps (donors ND − ND+ =[Fe2+ ] and acceptors ND+ =[Fe3+ ]) spatial modulation arising from photoelectron excitation and retrapping. The evolution of the first + spatial harmonic components for traps ND1 and ions 1+ , respectively, are ruled by the following coupled differential equations + (t) 𝜕ND1

𝜕t 𝜕1+ (t)

+ (t) + 𝜉H 1+ (t) = k m e−iK𝑣t + 𝛾e (1 + 𝜉e )ND1

(11.1)

+ (t) = 0 (11.2) + 𝛾H (1 + 𝜉H )1+ (t) + 𝜉e ND1 𝜕t as already reported by Sturman and coworkers [235] and slightly modified here for the case of a recording pattern of light, with visibility m and moving with speed 𝑣 along the grating ⃗ In fact, the use of a self-stabilized holographic recording setup produces, in genvector K. eral, a running pattern of fringes and corresponding running hologram because of the phase mismatch between the unconstrained nonstabilized recorded hologram and the stabilized hologram as already reported in Section 6.2.1 and particularly in Section 6.3.2. The parameters 𝛾e (1 + 𝜉e ) and 𝛾H (1 + 𝜉H ) in Eqs. (11.1) and (11.2) represent the response constants associated to grating build-up by electrons and H+ -ions, respectively, whereas 𝜉e and 𝜉H represent

11.2 Fixed Holograms in LiNbO3

the corresponding electric coupling, which values depend on crystal parameters, and k is a constant. These parameters are defined elsewhere [235] as: e𝜇 n 𝛾e ≈ e 0 (11.3) 𝜖𝜀0 e𝜇  𝛾H = H 0 (11.4) 𝜖𝜀0 𝜉e ≈ −i

Eph [Fe3+ ]

Eq [Fe] ED (ND )eff 𝜉H = Eq 0 where [Fe]≡ [Fe2+ ]+[Fe3+ ] and e(ND )eff Eq = (ND )eff = [Fe2+ ][Fe3+ ]∕[Fe] K𝜖𝜀0

(11.5) (11.6)

(11.7)

where 𝜇H and 0 are the mobility and average concentration of ions respectively, q the value of the electronic charge, 𝜖 the dielectric constant and 𝜀0 the permittivity of vacuum. The solution of these equations leads to transient and stationary terms (11.8) N + = N e−iK𝑣t + transients D1

st

1+ = st e−iK𝑣t + transients

(11.9)

We are just interested in the stationary terms, for which the amplitudes are k m (𝛾H (1 + 𝜉H ) − i K𝑣) 𝛾e 𝛾H (𝜉e + 𝜉H ) − K 2 𝑣2 − i K𝑣(𝛾e + 𝛾H ) k m 𝛾H st = − 𝛾e 𝛾H (𝜉e + 𝜉H ) − K 2 𝑣2 − i K𝑣(𝛾e + 𝛾H )

Nst =

From Eqs. (11.10) and (11.11), we compute st e𝜇H 0 ∕(𝜖𝜀0 ) 𝛾H =− ≈− Nst 𝛾H (1 + 𝜉H ) − iK𝑣 e𝜇H 0 ∕(𝜖𝜀0 ) − iK𝑣

(11.10) (11.11)

(11.12)

where the approximate sign on the right-hand side is for the 𝜉H ≪ 1 condition. Note that the gratings from electrons and from ions are phase-shifted because of the term iK𝑣. Otherwise, for standing holograms (𝑣 = 0), the phase shift would be exactly 𝜋; that is to say, counterphase. In any case, and for a sufficiently large ion concentration, e𝜇H 0 ≫∣ K𝑣 ∣ (11.13) 𝜖𝜀0 the relation in Eq. (11.12) simplifies to st ≈ −1 Nst

(11.14)

which means that, in these conditions, the electron-based grating can be completely compensated during self-stabilized recording, even for the case of moving holograms, provided the ions concentration and their mobility are high enough. This means that an electron donor trap spatial modulation and a nonphotosensitive ionic spatial modulation that move synchronously with the pattern of fringes are produced. This is the fundamental feature enabling simultaneous recording and compensation at high temperature.

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11 Fixed Holograms

11.2.1.2 Experiment: Simultaneous Recording and Compensating

Simultaneous recording and compensation were carried out on a 1.4 mm thick Fe-doped LiNbO3 crystal with [Fe2+ ]/[Fe3+ ] = 0.013, total iron concentration [Fe]≈ 2 × 1019 cm−3 and total hydrogen-ion concentration [H+ ]=3.2 × 1017 cm−3 . The sample was short-circuited (as is usual) with silver conductive glue and was placed in a copper holder surrounding the sample and in good thermal contact with a temperature-controlled heated massive copper cylinder as schematically represented in Fig. 11.1. A thin, approximately 8 cm diameter, hollow Pyrex glass cylinder around the sample (with a flat heat-isolating cover that is not shown in the figure) minimizes heat losses and convection but allows the recording laser beams to go through. The recording light was the 514.5 nm expanded an collimated beam from an Ar+ laser with fringes modulation |m| ≈ 1 and K = 10 μm−1 . The recording was carried out in the usual self-stabilized mode already described in Section 6.3.2, which allows one to carry out holographic recording even for hours without being affected by environmental perturbations. The error signal necessary to operate the feedback loop in the setup arises from the diffracted light (interfering with the other transmitted beam along the same direction behind the sample) from the remaining electronic grating that is not completely compensated by the nonphotosensitive ionic grating. It is therefore essential to keep the operating temperature high enough for the compensation to occur efficiently, but not too high to avoid the recorded grating being completely compensated. A preliminary nonself-stabilized recording experiment was carried out on a more oxidized sample ([Fe2+ ]/[Fe3+ ] ≈ 0.006) with similar total iron concentration in order to find out the adequate operating temperature. This experiment showed that 150∘ C is too high a temperature because the stabilization setup was not adequately operating, probably because the ionic compensation of the photosensitive electronic grating was too complete and there was not enough overall remaining grating to diffract light in order to operate the feedback loop in the stabilization setup. The 130–135∘ C temperature instead apparently allows a much higher remaining hologram, which was actually perfectly suitable for operating the feedback. During self-stabilized holographic recording at high temperature, the second harmonic (I 2Ω ) term in the irradiance behind the crystal is used as an error signal (see Section √ 6.3.2), so it is kept Ω at approximately zero by the feedback loop whereas the first harmonic (I ∝ 𝜂(1 − 𝜂)) shows √ the evolution of the overall (electronic plus ionic) hologram 𝜂 as reported in Fig. 11.2 for a typical experiment. Note that I Ω begins growing and after a while reaches a roughly constant value, indicating an equilibrium between the electronic grating recording and ionic compensation rates. After some time recording at 130–135∘ C on our 1.4 mm-thick crystal, the whole chamber was allowed to cool down to room temperature. The sample was then developed using a powerful white light spatially uniform source illuminating both sample sides and the diffraction efficiency W

C

L

H

L

S

Figure 11.1 Experimental setup: S: massive copper cylinder with temperature-controlled heating element in direct thermal contact with the copper holder H supporting and surrounding the sample C. A thin pyrex glass cylinder W to minimize heat losses and thermal convection, around the sample, allows laser beams L to go through. A flat heat-isolating plate (not seen) covers the upper cylinder side.

11.2 Fixed Holograms in LiNbO3

1.5

Figure 11.2 Evolution of I Ω and I 2Ω during high temperature self-stabilized holographic recording (and compensation) for a typical experiment.

IΩ

I Ω and I 2Ω (au)

1.0

0.5

0 I 2Ω –0.5

0

500

1000 Time (s)

1500

2000

70 60

η (%)

50 40 30 20 10 0

30

0

60

90

120

150

180

Time (min)

Figure 11.3 Diffraction efficiency of the overall grating during white-light development as a function of development time. Note that the time scale depends on the overall development light intensity on the sample.

was measured from time to time, during development, using one of the recording beams that were automatically in the Bragg condition. The variation between 130–135∘ C during recording and room temperature for development, however, usually produces some mechanical displacements requiring the angular adjustment of the sample in order to match Bragg conditions before diffraction efficiency measurement. The diffraction efficiency was measured as 𝜂=

Id

Id + It

(11.15)

where I d and I t are the diffracted and transmitted beams. In this way, interface loses and bulk absorption do not affect 𝜂. Figure 11.3 shows one such measurement leading to a stationary final value of 𝜂 ≈ 0.66 for the fixed grating. The coupled constant-thickness parameter is computed from the final 𝜂 √ (11.16) [𝜅d]fix = sin−1 ( 𝜂fix ) for the fixed grating. The experimental [𝜅d]fix for different recording/compensation times and same sample are reported in Table 11.1. It is interesting to note that the sample used in this

261

262

11 Fixed Holograms

Table 11.1 Fixed grating diffraction efficiency. Recording time (min)

Fixed 𝜿d

Fixed 𝜼

8.5

0.22

5

17

0.36

11.8

34

0.52

25

34

0.55

26.3

60

0.95

66

120

0.95

66

3000

0.81

64

experiment reached only 𝜂 = 3% after fixing in the usual three-step (recording at room temperature, compensation at high temperature, development at room temperature) procedure whereas it achieved 𝜂 ≈ 0.66 (or [𝜅d]fix = 0.95 rad) after 1 h of a simultaneous self-stabilized recording/compensation process. From the data in Table 11.1, we may deduce that the sample becomes exhausted after 60 min recording and that the saturation value for the fixed grating is actually 𝜂 ≈ 64–66%. A question arises about the possible interference of an absorption grating in the self-stabilization holographic recording process: is it easy to show that there is none.? In fact, the absorption grating arises from trap modulation, that is to say, from the modulation of Fe2+ and Fe3+ concentration in the crystal. Is it known, however, that, in the framework of first harmonic approximation [161] the space-charge field Esc (in phase with the photorefractive grating) and the trap density ND+ (in phase or counterphase with the absorption grating) are related by −iK𝜀0 𝜖Esc = eND+

(11.17)

which shows they are 𝜋∕2-shifted, that is to say that the photovoltaic holographic grating is 𝜙ph ≈ 𝜋, whereas the absorption grating holographic phase shift is 𝜙a ≈ ±𝜋∕2. The corresponding phase shifts between the transmitted and diffracted beams at the crystal output are, therefore 𝜑ph = 𝜙ph ± 𝜋∕2 ± 𝜋∕2

(11.18)

𝜑a = 𝜙a = ±𝜋∕2

(11.19)

Because the second harmonic term I 2Ω ∝ cos 𝜙 is the error signal in the feedback loop, it is straightforward to realize that the contribution of the absorption grating to I 2Ω is null. So, we should not expect any interference by an absorption grating in the self-stabilization holographic recording here. The interest of volume hologram fixing for nonvolatile optical memories [236] and optical component fabrication [237] is obvious and photorefractives (specially LiNbO3 ) look particularly suitable for these purposes. The high angular and wavelength Bragg selectivity forms the basis of the main interest in these components but they have also found rather unconventional and interesting applications, for example as sources of light masks for atomic nanolithography [238].

263

12 Photoelectric Conversion Photorefractive crystals may be useful for photoelectric conversion because some of them exhibit photovoltaic effects. We have also reported in Section 2.7 that it is possible to take advantage of Dember effect in these materials for photoelectric conversion, although neither of these two effects are, so far, comparable to the conversion efficiency of commercial semiconductor-based devices.

12.1 Photoelectric Conversion Efficiency: Dember and Photovoltaic Effects The simultaneous presence of photovoltaic and Dember effects in some photorefractive crystals was already discussed in Section 2.7, where the photocurrents arising from these two effects in an ITO-sandwiched thin BTO crystal slab were measured and separately plotted in Fig. 2.37, and from this figure their respective photoconversion efficiencies were computed and are displayed in Table 12.1 for different illumination chopped frequency. Table 12.1 Photoelectric conversion efficiency. Effect: Chop. freq. (Hz)

20

84.3

200

300

Photovolt. I0 (mW/cm2 )

Dember

Conv. Eff. (pA cm2 /mW)

1.02

1.30

0.81

12.8

0.93

0.61

1276

1.55

0.75

1.02

1.30

0.81

12.8

0.93

0.61

1276

1.46

0.80

1.02

1.30

0.81

12.8

0.93

0.61

1276

1.27

0.765

1.02

1.30

0.81

12.8

0.93

0.61

1276

1.06

0.66

Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

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12 Photoelectric Conversion

Several features can be concluded from the data in Table 12.1: • Efficiency does not sensibly vary with frequency for low light intensity for both Dember and photovoltaic effects. • For the highest intensity, the photovoltaic effect response monotonically decreases from around 20 to 300 Hz by a total amount of 46%, whereas the Dember effect response decreases by only 14% over the same range, showing a moderate relative maximum at 84.3 Hz. • In the whole frequency and intensity ranges, Dember and photovoltaic efficiencies are of a similar order-of-magnitude, although the former is systematically lower.

265

Part V Appendix

266

Introduction

This appendix is intended to provide some general and practical tools for those who are willing to start with experimental work and are still not familiar with the problems involved in handling these complex materials and measuring some of their basic properties, like diffraction efficiency. There are two final sections, one dealing with a rather theoretical subject (the physical meaning of some material parameters) and the other one providing general information about the operation of diode photodetectors, which are the most widespread and inexpensive tool for light measurement nowadays.

Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

267

Appendix A Reversible Real-Time Holograms While recording a photorefractive hologram, one should keep in mind that we are dealing with an almost real-time and reversible process. It is essentially different from recording on a photographic plate or any other nonreversible material. The speed of recording or erasing in photorefractives is roughly proportional to the total recording irradiance, so that, as we try to illuminate the hologram to observe diffraction, we may be erasing it. A low irradiance may slow down the process and by this means facilitate the observation of the phenomenon but, besides weakening the displayed hologram itself, it makes the recording proportionally more exposed to environmental perturbations, thus leading to a more unstable recording and consequently a poorer recording and a weaker hologram. Holographic recording is very sensitive to environmental perturbations and we can take advantage of this feature to detect the presence of a hologram. Detection of diffraction is particularly difficult in relatively fast and poorly diffractive materials such as GaAs and Bi12 TiO20 , or any other sillenite-type crystal where diffraction efficiency may be 𝜂 ≈ 0.01 or even lower. Detecting a hologram is performed differently, depending on whether one is doing it by direct naked-eye observation or by an instrumental-assisted technique. We shall, in the following, briefly describe both cases.

A.1 Naked-Eye Detection The direct qualitative detection of the actual presence of a hologram can be carried out by naked eye and is extremely useful because it is the first means we have for guiding our handling and adjusting of the setup for recording. Once a hologram, even a very weak one, is qualitatively detected, instrumental-assisted quantitative means can be used to optimize the setup. A.1.1

Diffraction

For the case of slow and highly diffracting materials such as LiNbO3 , the detection of a hologram being recorded is very simple because it is enough to switch off one of the recording beams and watch the diffraction of the other one. For faster materials, however, such a simple technique is not possible because the hologram is usually rapidly erased while exposed to one single beam, and also because diffraction efficiency in faster materials is usually rather weak so its visual detection may be jeopardized by the scattering of light from the sample itself or from other parts in the setup. It is also very difficult to detect the diffracted beam along with the transmitted one propagating along the same direction without switching the latter off: for the example of 𝜂 ≤ 0.01, the transmitted beam is more than 100-fold larger than the diffracted one! In order to perform detection during recording, it is therefore necessary to reduce somehow the transmitted beam without affecting the input recording beams. Sillenite-type crystals Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

268

Appendix A Reversible Real-Time Holograms

are particularly suited for such a task because of their anisotropic diffraction properties (see Chapter 5) that allow one to adjust the input polarization condition so as to produce diffracted and transmitted beams with different (even mutually orthogonal) polarization directions at the crystal output. In this case, it is enough to put a simple polarizer sheet behind the crystal and adjust it so as to minimize the transmitted direct beam through the sample while the other beam is switched off. After adjusting the polarizer, the other beam is switched on to allow the recording to proceed and, if a hologram is recorded, you should be able to see an unstable fluctuating light behind the polarizer that only appears when both beams are shining the crystal. If the response time is too fast and/or the environment is not noisy enough, you should swiftly knock down on the setup table to artificially perturb the recording and see some fluctuation in the diffracted light. In practice, it is not easy to have exactly orthogonally polarized beams and it is even not necessary: for instance, if we assume 70∘ between the transmitted and diffracted beams’ polarization directions, instead of 90∘ , the transmitted beam can be cut off while the diffracted beam will be reduced in 1 − cos 20∘ ≈ 6% only. A.1.2

Interference

It may sometimes be easier, however, to detect phase perturbations rather than diffraction efficiency variations. Let us recall the expression of the overall irradiance along the signal beam behind the sample as formulated in Eq. (6.1) √ √ IS = ISo (1 − 𝜂) + IRo 𝜂 + 2 𝜂(1 − 𝜂) ISo IRo cos(𝜑 + 𝜑N ) where we have substituted the phase modulation by a phase noise 𝜑N and assumed that the transmitted and diffracted beams are parallel-polarized and of similar irradiances. In this case, and always for our 𝜂 ≈ 0.01 example, the diffracted beam (the second term in the right-hand side) is 100-fold lower than the transmitted one (the first term), whereas the interference term (the third one), which is the only one where the phase parameter 𝜑 shows up, is roughly five-fold weaker than the transmitted beam. It is, however, still hard to see phase fluctuations in the interference term in such conditions without further reducing the transmitted beam. We should do it by just operating with orthogonally (or nearly orthogonally) polarized diffracted and transmitted beams condition. In this way, it is possible to use a polarizer almost aligned with the diffracted beam polarization direction (and therefore almost perpendicular to that of the transmitted beam) at the crystal output. The weak diffracted beam is almost not affected but the transmitted beam is strongly reduced, the relative size of the interference term is therefore increased and phase fluctuations in 𝜑N and/or in 𝜂 in the interference term are likely to be observed.

A.2 Instrumental Detection By this, we mean using a photodetector connected to an oscilloscope to detect fluctuations in the overall beam behind the crystal. Such √ fluctuations arise from variations in 𝜂 in the diffracted beam (usually rather small) and/or in 𝜂 in the interference term. Variations in the phase shift 𝜑 are also detected in the interference term and may be much faster than those in 𝜂. Instrumental detection does not require a large visibility (that is, a comparatively large interference term) as in the case of direct visual detection because the oscilloscope is able to operate in ac mode so as to reject the dc signal from the stronger transmitted beam, provided the photodetector feeding the oscilloscope does not become saturated by the overall irradiance shining on it.

A.2 Instrumental Detection

It is always possible to use the more sophisticated phase modulation techniques described in Section 4.3, which are particularly suitable for the detection of the interference term. This technique is very sensitive and allows for the detection of extremely weak signals out from very large background, almost dc signals. This technique is particularly convenient for crystals not exhibiting anisotropic diffraction effects, so the transmitted and diffracted beams are always parallel-polarized and there is therefore no possibility to play with the difference in polarization of the output transmitted and diffracted beams.

269

271

Appendix B Diffraction Efficiency Measurement Diffraction efficiency is an important quantity in holography and is therefore something to be measured to start characterizing the hologram under analysis. Unfortunately, its measurement is usually much harder to carry out than most people believe it to be. The difficulties arise from: • the very high angular Bragg selectivity of the hologram • the rather high average index-of-refraction of the material and • the reversible nature of the photorefractive recording process.

B.1 Angular Bragg Selectivity Diffraction efficiency of volume gratings has been an active subject of research [70, 239–241] in the last decades and, aside from its academic interest and practical applications, its measurement is of the highest importance for the characterization of photosensitive materials in general and photorefractives in particular. Diffraction efficiency (𝜂) measurement in volume holograms, however, is usually neither straightforward nor free of errors. A usual source of error arises from the high Bragg angular selectivity intrinsic to thick volume holograms, which leads to lower apparent efficiency values. It is usual to measure 𝜂 using a so-called “probe” beam, either with the same or a different wavelength from the one used for holographic recording. In any case, the incidence of this beam with wavelength 𝜆 should be adjusted to fulfill the Bragg condition as discussed in Section 4.1.2 K = 2k sin 𝜃

K = 2𝜋∕Δ

k = 2𝜋∕𝜆

(B.1)

where Δ is the spatial period of the hologram and 𝜃 the incidence angle. The advantage of using a probe beam is that the irradiance can be chosen to be weak enough not to sensibly affect the recording/erasure process and a wavelength can even be chosen having a minimum effect on the material so that measurement can be performed even in reversible recording materials without perturbing the already recorded hologram or the hologram being recorded. However, a simple calculation shows that for a 2 mm thick grating with K = 10 μm−1 (Δ ≈ 0.63 μm), for example, the angular Bragg selectivity (reducing 𝜂 from 1 to 0.5 in Eq. (4.21) is approximately 0.1 mrad, which is too restrictive for the usual angular divergence (1 to 2 mrad) of commercial He–Ne lasers. It is possible to show that, except for low efficiency gratings, the diffracted probe beam is not even proportional to 𝜂, in which case the probe-beam technique is not even useful for qualitative purposes either.

Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

272

Appendix B Diffraction Efficiency Measurement

B.1.1 In-Bragg Recording Beams It follows that for rather thick volume holograms, diffraction efficiency can be measured accurately, only using the recording beams themselves, because they are automatically “in-Bragg” whatever their wavefront. The handicap here arises from the fact that a sensible erasure may occur during measurement in reversible materials as photorefractives are. It is usually not even possible to decrease the irradiance using a filter, because by doing so we may slightly change the beam wavefront and such a change may be enough to produce a sensible mismatch with the recorded hologram and therefore show a reduced 𝜂. One possibility to reduce erasure during measurement without using a probe beam is to use a shutter to produce short pulses to minimize light exposure on the photosensitive material. B.1.2 Probe Beam The diffraction efficiency of purely index-of-refraction thick volume gratings was formulated by Kogelnik [70] and is ruled by √ sin2 𝜉 2 + 𝜈 2 (B.2) 𝜂= 1 + 𝜉 2 ∕𝜈 2 𝜉 ≈ 𝜃Kd∕2 𝜈=

(B.3)

𝜋n3eff reff Esc 2𝜆 cos 𝜃

(B.4)

d

as reported in Section 4.1.2 where 𝜃 is the angular departure from the Bragg condition, d is the hologram thickness and 𝜆 is the wavelength of the light used to measure diffraction. The value of the grating modulation is 𝜈, which is equivalent to ∣ 𝜅 ∣ as derived from Eq. (4.86), because it is formulated here for the particular case of photorefractive materials, with reff and neff being the effective electro-optic coefficient and index-of-refraction, respectively, and Esc being the space-charge electric field modulation amplitude. The diffraction efficiency actually measured must take into account the angular spectrum of plane waves of the laser beam used for measurement, which, because of the Gaussian shape of the beam intensity distribution, would be better formulated in a Gaussian form too [155] 2 2 A = A0 e−(𝜃 − 𝜃) ∕a

(B.5)

where a is the angular spectrum bandwidth. For a commercial 10 mW He-Ne laser of 0.7 mm beam diameter from Uniphase, for example, the technical datasheet informs a ≈ 1.2 mrad. The practical diffraction efficiency measurement of a grating is computed as a coherent summation along the full spatial angular spectrum. For the case of a low divergence beam with a rapidly varying phase difference term being averaged out using the stationary phase theorem [4], the diffraction efficiency expression is simplified to 𝜂 = I d ∕I t0

(B.6) 𝜃+𝜋∕2

with I d = A0

∫𝜃−𝜋∕2 𝜋∕2

and I t0 = A0

∫−𝜋∕2

2 2 𝜂 e−(𝜃 − 𝜃) ∕a d𝜃

2 2 e−𝜃 ∕a d𝜃

(B.7) (B.8)

where I d is the diffracted irradiance and I t0 represents the whole beam irradiance behind the sample.

B.1 Angular Bragg Selectivity

Of course, it is always possible to expand and carefully collimate the laser beam to closely fulfill Bragg condition but this procedure is rarely employed because it requires high-quality components, is time-consuming and is rather cumbersome. In order to illustrate these calculations, we carried out an experiment on a thick volume (1.5 mm) holographic grating of K ≈ 10∕μm recorded in a Fe-doped lithium niobate photorefractive crystal (labeled LNB1 in Table 6.1) using the ordinary polarization of a 541.5 nm wavelength line of an Ar+ laser. After being recorded, the grating was fixed using the process reported in Chapter 11, which results in the substitution of the original electronic grating by a nonphotosensitive positive ionic one. The fixed grating is carefully replaced at the same position in the recording setup with the help of a specially prepared support. The measurement of diffraction efficiency was then carried out on the fixed grating without any risk of partially erasing the grating during measurement, using the same ordinary polarized 514.5 nm wavelength beams previously employed for recording, which ensures full in-Bragg condition, following the method described in Section B.3. The result was o = 0.352 𝜂514

(B.9)

which, from Eq. (B.2), resulted in a grating modulation of o 𝜈514 = 0.635

(B.10)

The latter was converted for the 633 nm wavelength o o 𝜈633 = 𝜈514 514.5∕633 = 0.516

(B.11)

using the relation in Eq. (B.4) where the relatively small effect of the wavelength on the refractive index was neglected for the sake of simplicity. Substituting the value in Eq. (B.11) into Eq. (B.2), with 𝜉 = 0, we got the theoretically fully Bragg-matched value o 𝜂633 = 0.244

(B.12)

for the 𝜆 = 633 nm ordinarily polarized light. The sample was taken out from the holographic recording setup and placed in an auxiliary setup to measure the diffraction efficiency with an ordinarily polarized direct 633 nm He-Ne laser beam. The experimentally measured value was o ]exp = 0.16. Substituting the latter value into Eq. (B.6) together with the already computed [𝜂633 value in Eq. (B.11) and solving the corresponding equation using an appropriate algorithm for 𝜃 = 0, we found out the beam divergence a = 0.35 mrad for this laser beam. We are now able to check our results for the extraordinary polarization of the 633 nm laser beam. To do this we rotated the laser for the extraordinary polarization and carried out diffraction efficiency measurements for different values of 𝜃 as represented by the spots in Fig. B.1. 1

Figure B.1 Diffraction efficiency as a function of out-of-Bragg angle 𝜃 in mrad for the measured data (•), theoretically computed for a = 0.35 mrad (continuous curve) and for a → 0 (dashed curve). From [242].

0.8

η

0.6 0.4 0.2 0

0.5

1 θ‾ (mrad)

1.5

2

273

Appendix B Diffraction Efficiency Measurement

2 1.5 Y (rad)

274

1 0.5 0

0

0.5

1

1.5

2

Y (rad)

Figure B.2 𝜈, computed from Eq. (B.15), as a function of 𝜈 for in-Bragg condition and same parameters as in Fig. B.1. From [242].

Then, we converted the previously computed ordinary grating modulation in Eq. (B.11) into the extraordinary polarization as follows e o = 𝜈633 br 𝜈633

(B.13)

br = (ne ∕no )3 (r33 ∕r13 ) cos(2𝛽 ′ ) = 2.57

(B.14)

with [1]: ne = 2.2, no = 2.286, r33 = 30.9 pm/V and r13 = 9.6 pm/V. We replaced the resulting e value into Eq. (B.6) and plotted (continuous curve) the diffraction efficiency in Fig. B.1 as 𝜈633 a function of 𝜃 using the previously computed value a = 0.35 mrad, which characterizes the angular divergence of the laser beam in this setup. Figure B.1 shows a good agreement between experimental data and theory. Note that the theoretical curve is not mathematically fitted to but just plotted together with the experimental data in Fig. B.1. The dashed curve in the same figure represents what would have been the theoretically measured value using a hypothetically zero divergence (a → 0) probe laser beam. It may be somewhat surprising that such a low angular divergence as a = 0.35 mrad may lead to considerable errors if not adequately considered, as illustrated in Fig. B.1. This fact is also clearly illustrated in Fig. B.2, where the apparent 𝜈 modulation computed from the measured (average) 𝜂 √ 𝜈 = arcsin 𝜂 (B.15) without taking into account the finite angular divergence of the measurement probe beam, is plotted as a function of 𝜈. Figure B.2 clearly shows that 𝜈 is different from 𝜈 and, still worse, that they are not even proportional except for low values of 𝜈.

B.2 Reversible Holograms The reversibility of photorefractive materials is an interesting and useful property but represents a serious drawback for 𝜂 measurement because of erasure during measurement. The use of continuous non-perturbative measurement methods based on the in-Bragg recording beams is the best alternative. In this case, phase modulation, with the frequency Ω of the modulation being very fast Ω𝜏sc ≫ 1 compared to the material response time 𝜏sc , as described in

B.3 High Index-of-Refraction Material

Section 4.3, is one of the best-suited techniques. This technique allows measuring the first and second harmonic terms, along any one of the beams behind the crystal, which are formulated in Eqs. (4.172) and (4.173), respectively √ √ ISΩ = −4J1 (𝜓d ) ISo IRo 𝜂(1 − 𝜂) sin 𝜑 √ √ IS2Ω = 4J2 (𝜓d ) ISo IRo 𝜂(1 − 𝜂) cos 𝜑 tan 𝜑 = −

ISΩ J2 (𝜓d ) IS2Ω J1 (𝜓d )

If the phase modulation amplitude is sufficiently small, 𝜓d ≪ 1, the recording will not be sensibly affected. From these harmonic terms, it is possible to compute 𝜂 straightforwardly: [ ]2 [ ]2 ISΩ IS2Ω + = ISo IRo 𝜂(1 − 𝜂) (B.16) 4J1 (𝜓d ) 4J2 (𝜓d ) The advantage of this technique is its real-time online capabilities. Its main drawback arises from its dependence on some parameters such as the frequency response of the phase modulator, which is particularly delicate in the case of piezoelectric modulators because they are not very stable.

B.3 High Index-of-Refraction Material Diffraction efficiency measurement, even using the in-Bragg recording beams, may be jeopardized by the relatively large thickness of the sample under analysis, with an enhanced effect due to the relatively high index of refraction exhibited by most photorefractive materials. In fact, in these conditions and for faces even slightly deviating from the perfect parallel-plane condition, a lenslike effect is to be expected and the beam through the sample may be focused or defocused in different proportions for each one of both recording beams because they go through slightly different paths along the crystal. It is also possible that the recording beams may not be perfectly collimated ones, as illustrated in Fig. B.3. In this case, it is flawed to compute 𝜂 from the values of diffracted and transmitted irradiances because their values may be strongly dependent on their way through the sample and on the position along their propagation direction behind the sample where the detector is placed. It is, however, always possible to carry out these measurements using the total power of the beams, but for this purpose some focusing lenses should be used at the output of the sample or, alternatively, the recording beams should be reduced in size to be entirely collected Figure B.3 Measurement of diffraction efficiency: The recording beams are not collimated and the sample adds focusing/defocusing effects. The output irradiance along each one of the incident directions is the coherent addition of the transmitted and the diffracted beams. The two different detectors, with different responses, should be centered on the same spot of the crystal. From [242].

Is Ds

crystal

DR

O

IR

O

IS IR

275

276

Appendix B Diffraction Efficiency Measurement

into the detectors behind the sample. Neither of these possibilities is always practical, mainly if the recording process is taking place and the same setup is used for recording and measurement. In fact, a reduced illuminated area in a photovoltaic or a photorefractive crystals under applied field is highly undesirable because of the build-up of screening charges. On the other side, the use of a focusing lens may be interesting but could interfere with some online processing at the sample output. We shall show here that it is always possible to use the recording beams for directly computing 𝜂, even on lenslike samples. Let us assume the overall transmitted plus coherently added diffracted beams along any of the directions behind the sample can be written as √ √ (B.17) IS = ISt0 (1 − 𝜂) + IRt0 𝜂 − 2 𝜂(1 − 𝜂) ISt0 IRt0 cos 𝜑 √ √ IR = IRt0 (1 − 𝜂) + ISt0 𝜂 + 2 𝜂(1 − 𝜂) ISt0 IRt0 cos 𝜑

(B.18)

where 𝜑 is the phase shift between the transmitted (ISt0 (1 − 𝜂) or IRt0 (1 − 𝜂)) and diffracted (IRt0 𝜂 or ISt0 𝜂) beams, ISt0 and IRt0 are the respective transmitted intensities through the sample in the absence of diffraction and 𝜂 is the diffraction efficiency to be computed from 𝜂=

Iit0 𝜂 Iit0 𝜂 + Iit0 (1 − 𝜂)

i=S

or

R

(B.19)

We assume high Bragg angular selectivity gratings exhibiting only one diffracted order. The formulation in Eq. (B.19) is usually employed [94, 115, 120, 173] to get rid of bulk absorption, scattering in the sample volume and interfaces losses that are not related to diffraction itself. The responsivity of each one of the detectors DR and DS in Fig. B.3 are KR and KS , respectively, here including their (possible) different nature, electronics and aging. In this case, the voltages measured at each one of the detectors DS and DR are, respectively VS = KS IS

(B.20)

VR = KR IR

(B.21)

It is important to realize that whatever the shape of the beams, the transmitted and the diffracted ones (along the same direction) have the same shape because the latter is just the holographic reconstruction of the transmitted wave. That is to say, that each detector is always measuring a wavefront having a constant shape as shown in Fig. B.3. In order to measure 𝜂 for the case of uncalibrated photodetectors (only linearity of the response is assumed) and in the presence of lenslike effect, we should proceed as follows: 1. First, shut off the incident beam IR0 and let both detectors measure the respective transmitted and diffracted signals VSS = KS ISt0 (1 − 𝜂)

(B.22)

VRS = KR ISt0 𝜂

(B.23)

2. Then, shut off the other beam and repeat the measurement on the other detector VSR = KS IRt0 𝜂

(B.24)

VRR = KR IRt0 (1 − 𝜂)

(B.25)

B.3 High Index-of-Refraction Material

If the measurement is carried out during recording or on a reversible recording material, the operations here should be obviously carried out fast enough so as not to allow the hologram to be sensibly erased. Then we may substitute VSS and VRS instead of ISt0 (1 − 𝜂) and IRt0 𝜂, respectively (or VRR and VSR instead of IRt0 (1 − 𝜂) and IRt0 𝜂, respectively), into Eq. (B.19) to compute 𝜂 or, alternatively, from Eqs. (B.22)–(B.25) compute [ ]2 VSS VSR 1−𝜂 ∕ = (B.26) 𝜂 VRS VRR which is a second-order equation in 𝜂 that is straightforwardly solved out without any concern about the detectors’ responsivities or the beams’ shapes. The only concern here is about the linear dc-response of each detector (including its electronics), which is usually quite accurate for adequately designed photodetectors.

277

279

Appendix C Effectively Applied Electric Field The use of material characterization techniques requiring the application of an external electric field on the sample has a serious drawback: a considerable discontinuity in the electric field inside the sample may be produced by electrode-contact problems. Some nonuniformity in the electric field may also arise from nonuniform photoconductivity in the sample. In fact, under the action of a gaussian-shape spatial cross section light irradiance I(x), as illustrated in Fig. C.1, a similar Gaussian cross-section for the photoconductivity 𝜎(x) ∝ I(x) is produced. A lower value for the associated electric field E(x) results at the region where the sample is more illuminated, according to the relation j = 𝜎(x) E(x) = constant as illustrated in Fig. C.1. In this case, the field is not constant across the interelectrode distance and cannot be calculated from the applied voltage simply as voltage-to-interelectrode distance ratio. It is therefore necessary to define a parameter 𝜉 to account for the actual value of the effective field at any point in the sample. It is not possible to theoretically predict the value of 𝜉: In general, it is experimentally computed from the measurement of some well-known parameter or from the fitting of some quantity from which the theoretical equation is reasonably well known. One such possibility is to measure the electro-optic coefficient at different points along the interelectrode distance [243], and its variation allows one to directly deduce the variation of the effectively applied field, since the electro-optic coefficient should actually be constant. Let us analyze an example, such as the one in Eq. (3.56), describing the holographic phase shift 𝜙P as a function of the applied field E0 = V0 ∕𝓁 and other parameters, where V0 is the applied voltage and 𝓁 is the interelectrode distance. According to the development in this section, the effective electric field at the point we are measuring is probably different from the theoretical value E0 and a coefficient 𝜉 should be included to take account of the nonuniform photoconductivity in the material. Equation (3.56) should therefore be written as tan 𝜙P =

1 + K 2 ls2 + K 2 ls2 (𝜉E0 ∕ED )2 𝜉E0 ∕ED

(C.1)

where the coefficient 𝜉 should be evaluated from the fit of Eq. (C.1) to actual experimental data, as reported in Sections 9.1 and 9.2, or with the help of an auxiliary experiment as in Section 3.4.

Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

280

Appendix C Effectively Applied Electric Field

1.0 0.8

Figure C.1 Effective field coefficient: the figure shows a Gaussian cross-section irradiance I(x) illuminating a photoconductive material in steady-state regime with constant photocurrent j(x) = j, showing the resulting photoconductivity distribution 𝜎(x) and associated electric field E(x). The coordinate x (in arbitrary units) is along the two electrodes on the sample and all quantities represented (in ordinates) are also in arbitrary units.

I

0.6 0.4

σ

0.2 0

E 0

0.5

1.0

1.5

2.0

281

Appendix D Physical Meaning of Some Parameters It is interesting to get some insight into the meaning of some of the parameters that were defined during the development of the fundamental mathematical relations in the first part of this book. We shall focus on the Debye screening length, the diffusion coefficient and the diffusion length. We shall not provide careful mathematical demonstrations but simply just discuss their meaning and where they originate from.

D.1 Temperature A gas in thermal equilibrium has molecules of mass m moving randomly in all directions with velocities u having different values. These velocities are likely to follow a so-called Maxwellian distribution that, for one-single dimension model, takes the form 1 mu2 f (u) = A e 2 kB T −

+∞

n=

∫−∞

f (u)du

kB = 1.38 × 10−23 J∕o K √ m A=n 2𝜋kB T

(D.1) (D.2)

where n is the number of molecules per unit volume and f du is the number of molecules per unit volume with velocities between u and u + du. The width of the distribution is characterized by the constant T that we call “absolute temperature”. Let us compute the average kinetic energy in this distribution +∞

Eav =

∫−∞ (mu2 ∕2)f (u)du +∞ ∫∞

f (u)du

=

1 k T 2 B

(D.3)

Defining an average velocity uth as 1 2 mu 2 th we deduce that √ kB T uth = m Eav =

Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

(D.4)

(D.5)

282

Appendix D Physical Meaning of Some Parameters

We should extend this result to 3D, in which case f (u) and Eav become 1 − m(u2 + 𝑣2 + 𝑤2 )∕kB T 2 f (u, 𝑣, 𝑤) = A3 e Eav = D.1.1

( A3 = n

m 2𝜋kB T

3 k T 2 B

)3∕2 (D.6) (D.7)

Debye Screening Length

The matter untangles the question about how mobile charge carriers do distribute in an electric potential field if there are restrictions for the obvious solution that they all move to one or the other pole of the field [244]. Take charge carriers of one single sign, free to move inside a material with one of its plane faces placed close to, but not in contact with, a charged metallic plate with infinite dimensions along coordinates y and z. Under the action of the electric field from the charged plate, charges inside the material move and accumulate close to the boundary near the charged plate with concentration c(x), thus producing a concentration gradient of mobile carriers and a consequent diffusion current dc(x) (D.8) dx where D is the diffusion coefficient and the movement is assumed to occur along coordinate x. An opposite drift current under the effect of the local field E(x) jdiff = −qD

jdrift = q𝜇c(x)E(x)

(D.9)

arises that, once equilibrium is achieved, leads to jdrift + jdiff = 0 dc(x) = q𝜇c(x)E(x) dx Substituting E = −dV (x)∕dx and D∕𝜇 = kB T∕q here, we get qD

−

dV (x) kB T 1 dc(x) kB T dln(c(x)) = = dx q c(x) dx q dx

with the solution: kB Tln(c(x)) + qV (x) = constant = UF

(D.10) (D.11)

(D.12) (D.13)

where T is the electron gas temperature as computed from Section D.1 and UF is a constant along x and is called the “electrochemical potential energy” or “Fermi energy”. The expression in Eq. (D.13) can be written as c(x) = c0 e−(qV (x) − UF )∕(kB T)

(D.14)

Let us assume a small perturbation c1 (x) in c(x) c(x) = c0 + c1 (x) with c1 (x) ≪ c0

(D.15)

that substituted into Eq. (D.14) leads to c(x)∕c0 = 1 + c1 (x)∕c0 = e−(qV (x) − UF )∕(kB T) qV (x) − UF ≈1− kB T c1 (x)∕c0 ≈ −(qV (x) − UF )∕(kB T)

(D.16) (D.17)

D.1 Temperature

Deriving Eq. (D.17) twice in x we get: −

d2 V (x) kB T d2 c1 (x) = dx2 qc0 dx2

(D.18)

Taking into account Gauss’s theorem, we get −

d2 V (x) qc(x) = dx2 𝜖𝜀0

(D.19)

leading to q 2 c0 d2 c(x) = c(x) dx2 𝜖𝜀0 kB T

(D.20)

The solution of the differential equation in Eq. (D.20) is c(x) = c1 (0) e−x∕ls + c0

(D.21)

with

√ U − qV (0) c1 (0) ≡ c0 F kB T

ls ≡

𝜖𝜀0 kB T q 2 c0

(D.22)

In conclusion, we should say that for the case of a material having a surplus (and electrically isolated) charge at a definite position (the phase boundary of a precipitate, a charged grain boundary in a crystal or simply a point charge somehow held isolated at a fixed position somewhere in the material), it would be surrounded by a cloud of opposite (in this case electrons) charge carriers so as to shield it. The size of this cloud is determined by the kinetic energy distribution of the carriers at a given temperature T: the most energetic carriers standing farther away from the positive charge. This cloud of shielding charges decreases the electric potential produced by the positive charge following the relation in Eq. (D.21) so that you will not “feel” it any more if you are some Debye lengths ls away. The latter is therefore the typical distance needed to screen the surplus charge by the mobile carriers in the material. D.1.1.1

Debye Length in Photorefractives

For the case of photorefractives, ls as defined in Eq. (3.49), does not arise from charged plasma considerations that lead to Eq. (D.22) but from the mathematical development describing the recording of a space-charge modulation. It is considered a constant just because it is usually assumed to be computed for the low light irradiance limit as lim (ND )eff = lim I→0

I→0

ND+ (ND − ND+ ) ND

=

NA− + (ND − NA− ) ND

≡ (ND )0eff

(D.23)

but in general (ND )eff depends on the light intensity, so ls depends on this too. The expression in Eq. (D.22) can be formulated in terms of the (ND )eff → (ND )0eff if we substitute c0 by the density of free charge carriers (here electrons)  . In fact, and for the low irradiance limit, we should write, from Eq. (2.37):  ≈ (ND − NA− )

sI∕(h𝜈) 𝛾NA−

(D.24)

where 𝛽 in Eq. (2.37) was neglected here on the assumption that 𝛽 ≪ sI∕(h𝜈). Substituting  from Eq. (D.24), in place of c0 in the formulation of ls in Eq. (D.22), results in ls = ls0 C(I)

(D.25)

283

284

Appendix D Physical Meaning of Some Parameters

√ ls0 ≡ C(I) ≡

𝜖𝜀0 kB T

(D.26)

q2 (N )0 √ D eff Isat NA−

(D.27) I ND 𝛾NA− h𝜈 (D.28) Isat ≡ and I ≪ Isat s For the large irradiance limit instead, the corresponding expression for  , in Eq. (2.38), substituted into Eq. (D.22) leads to √ (D.29) ls = ls0 NA− ∕ND with I ≫ Isat Note that ls0 in Eqs. (D.25) and (D.29) is exactly the expression for the Debye length as defined in Eq. (3.49) for photorefractives with (ND )eff → (ND )0eff , whereas the effect of light relies on the term C(I). From Eqs. (D.25) and (D.29) we conclude that ls should decrease as I increases and becomes constant (saturated) for the high light intensity limit. Experimental results reported in Section 8.2.2.1 do quantitatively confirm the present conclusions.

D.2 Diffusion and Mobility Figure D.1 schematically represents the flux Γ of electrons going through a volume of stationary neutral atoms with volume density ni and a cross-section s for fully absorbing the electron momentum. Because of absorption, the flux decreases along coordinate x as 𝜕Γ = −sni Γ 𝜕x

(D.30)

Γ(x) = Γ(0) e−x∕𝜆m

𝜆m =

1 ni s

(D.31)

where 𝜆m is called the “mean free path”. For electrons with velocity 𝑣, the mean time between collisions is 𝜏 = 𝜆m ∕𝑣

(D.32)

Averaging over electrons with all possible velocities, we compute the collision frequency average (D.33)

f = ni s𝑣 V Γ(x+dx)

A Γ(x) dx

Figure D.1 Volume A × dx with fixed ions of volume density ni of characteristic collision cross-section s, receiving a flux Γ of electrons of mass me and velocity 𝑣.

D.2 Diffusion and Mobility

From the equation of motion including these collisions, we compute [244] the mobility and the associated diffusion coefficients for the electrons, which turn out to be |q| (D.34) 𝜇= me f k T = B e (D.35) me f In order to estimate the collision cross-section (for fully absorbed electron momentum) s, let us assume a coulombian force F = −q2 ∕(4𝜋𝜀0 r2 ) acting during an average time r0 ∕𝑣 and producing a 90∘ deviation on the electron, so that Δ(me 𝑣) = me 𝑣 =∣ Fr0 ∕𝑣 ∣≈ r0 ≈

q2 4𝜋𝜀0 r0 𝑣

q2 4𝜋𝜀0 me 𝑣2

and s = 𝜋r02 ≈

(D.36) (D.37)

q4 16𝜋𝜀20 m2e 𝑣4

(D.38)

and the average collision frequency is f = ni s𝑣 ≈

ni q4 16𝜋𝜀2 m2e 𝑣3

From the definition of conductivity [245] √ 𝜎 = 𝜇qne = 16𝜋𝜀0 ne kB Te ∕me ls2 From the equations here, we conclude √ 16𝜋𝜀o kB Te 2 𝜇= l q me s ( ) k T 162 𝜋 2 𝜀0 kB Te L2D = 𝜏 = 𝜇𝜏 B e = ni ls6 |q| q q

(D.39)

(D.40)

(D.41) (D.42)

285

287

Appendix E Photodiodes Photodiodes are essentially semiconductor interfaces of the n- and p-type (p/n or n/p junction diodes) where electrons diffuse from the n-type to the p-type semiconductor and holes diffuse the other way, so a depletion layer is formed on both sides of the interface. Because of the depletion layer on both sides of the interface is oppositely charged, a space-charge electric field and associated electric potential barrier ΔV appear, as indicated in Fig. E.1. Under the action of light, with photonic energy high enough to produce intrinsic (band-to-band) excitation in the semiconductor, an hole-electron pair is formed. If they are formed outside the depletion layer, they are likely to recombine in a rather short time. However, if they are formed inside the depletion layer, the electron and the hole drift along opposite directions because of the space-charge field and a current i0 appears that is proportional to the irradiance I i0 = KI

(E.1)

Sometimes, an intermediate intrinsic layer is used (p-i-n diodes) that is intended to enlargen the depletion layer, as seen in Fig. E.2, to allow most of the photo-generated electron-hole pairs to be actually produced inside this layer, so they directly contribute to i0 . In the absence of light, a drift current id is also produced (which depends on the temperature and the semiconductors’ nature and doping) by the thermal generation of hole-electron pairs in the depletion layer and is considered as a “dark noise”. The equilibrium is reached when the diffusion current idiff becomes high enough to counterbalance the drift current Id , idiff = id

(E.2)

Under the action of a direct electric potential V , as shown by the dashed line in Fig. E.3, the potential barrier decreases from ΔV to ΔV − V (from a continuous to dashed curve) and the diffusion current increases accordingly as shown in the figure. Instead, under the action of a reverse potential, the potential barrier increases, the diffusion current decreases following the Shockley equation idiff = id e−eV ∕kB T

(E.3)

and the overall current i under light and reverse polarization is, therefore i = id e−eV ∕kB T − id − KI

Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

(E.4)

288

Appendix E Photodiodes – –– –– – – –– – –– – – –– –– – – ––

P

V

Figure E.1 np-junction showing the depletion layer and a diagram of the Schottky potential barrier.

++++ ++ ++++ +++ ++ + +++ ++++ ++ ++++ +++

– – – – – – – – – –

N

E d

P

– –– –– – – –– – –– – – –– –– – – ––

– – – – – – – – – –

V

++++ ++ ++++ +++ ++ + +++ ++++ ++ ++++ +++

i

Figure E.2 np-junction showing the depletion layer including the intrinsic layer and a diagram of the Schottky potential barrier. N

E d

Figure E.3 pn-junction showing the depletion layer including the intrinsic layer and a diagram of the Schottky potential barrier. The dashed curve shows the potential barrier under a direct bias potential V indicated by the dashed arrow.

symbol V

+

– id

– p-type

intrinsic

+

–

+

–

+

–

+

–

+

–

+

n-type

+

– Voltage

i0

idiff V

E.1 Photovoltaic Regime The so-called photovoltaic operation regime is illustrated in Fig. E.4. For the so-called loaded setup in A, Eq. (E.4) turns into V ∕RL = id ( eeV ∕kB T − 1) − KI

(E.5)

showing the nonlinear relation between the voltage V measured in the load resistance RL and the irradiance I. Linearity is only approximately achieved for V ≪ kB T∕e. For the so-called

E.2 Photoconductive Regime

Figure E.4 Photovoltaic mode operation for photodiodes. A shows its operation with a load RL , B shows the open-circuit operation and C shows the short circuit operation.

i

i

V

i0

i

V

i0

i0

RL

A

B

C

open-circuit setup in Fig. E.4 B, there is a logarithmic relation between I and the output voltage V : 0 = id ( eeV ∕kB T − 1) − KI ) ( kB T KI V = +1 ln e id

(E.6)

The image C in Fig. E.4 shows the so-called “short-circuit” operation where the current is actually exactly proportional to the irradiance i = id ( e0 − 1) − KI

(E.7)

i = −KI

E.2 Photoconductive Regime The photoconductive operation mode is shown in Fig. E.5 where a reverse bias voltage VB is allowed for and the corresponding equation is ⎛ e(V − VB ) ⎞ ⎜ ⎟ V = id ⎜e kB T − big ⎟ − KI RL ⎜ ⎟ ⎝ ⎠

(E.8)

The relation becomes linear only for the approximation V ≈ −id − KI RL

for

VB ≫ V

(E.9)

The term −id on the right-hand side is the noise. There is not such a noise in the photovoltaic short-circuit setup, so photoconductive diodes are considered to be noisier than photovoltaic, but they are also faster because the reverse bias field also considerably reduces the capacitance of the depletion layer and the time constant RC is also proportionally reduced. Figure E.5 Photoconductive mode operation for photodiodes. A reverse bias voltage VB (usually VB ≫ V) is applied as shown, to increase speed and improve linearity of the response.

i

V

i0 RL VB + –

289

290

Appendix E Photodiodes

E.3 Operational Amplifier The use of an operational amplifier (OA) in the photovoltaic short-circuit regime allows transformation of the current output into a voltage with controllable gain, always keeping the short-circuit operating regime, as illustrated in Fig. E.6. In fact, the virtual ground at the OA input grants the short-circuit operation of the photodiode (V ≈ 0) with the output voltage (V0 ) being proportional to the light irradiance (I): V0 = i Rf ∝ I. Also, the very low OA output resistance is an interesting feature of this photodiode-amplifier device.

i i0

i

V +

Figure E.6 Operational amplifier operated photodiode in the short-circuit photovoltaic regime. Rf

AMP-OP

–

–

V0

291

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305

Index a Absorption bulk, nonstationary holograms 106–110 light-induced 38, 50, 157–161 transmittance with 51 activity, Optical sillenites, in 157 Amplifier, operational 290 Anisotropic media wave propagation in 5

b 𝛽 meaning of 35, 36 𝛽 2 definition of 96 BaTiO3 sensitivity, holographic 194 BG: Band Gap 19 schema in: BTO 25 CdTe 21 Doped semiconductor 30 idem, under illumination 33 Intrinsic semiconductor 30 Bi12 TiO20 bandgap schema in 26 conductivity, dark 203–205 Debye length 219 illumination, effect on 192 diffusion length 219 electro-optic coefficients of 11 holograms erasure 196 hole-electron competition 201, 202 phase shift, initial 229 phase shift, stationary 231 sensitivity 194

light-induced absorption 159 luminescence in 28 mobility-lifetime 176–178 parameters, table 157, 177 photoactive centers 25 photochromism 48 energy, activation 49 photoconductivity 166 wavelength-resolved 166–173 photoconductivity, table 177 quantum efficiency 28, 176–178 refractive index 14 running holograms 72, 214–218 absorbing materials, in 232–234 fringe-locked 232–239 hole-electron competition, with 76–83 self-stabilized 130 Bi2 TeO5 17, 42 Bragg selectivity 92, 271–274 BSO Bi12 SiO20 sensitivity, holographic 194 BTeO Bi2TeO5 17 BTO:Ce light-induced absorption 159 mobility-lifetime 176–178 photoconductivity 176–177 quantum efficiency 176–178 BTO:Pb Bi12 TiO20 :Pb erasure, hologram 199, 200, 202 light-induced absorption, table 159 mobility-lifetime 176–178 photoconductivity 176–177 quantum efficiency 176–178 BTO:V Bi12 TiO20 :V hole-electron coupling in 202 hologram recording and erasing in 200, 202 WRP for 168

Photorefractive Materials for Dynamic Optical Recording: Fundamentals, Characterization, and Technology, First Edition. Jaime Frejlich. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

306

Index

c CB: Conduction Band 19 CdTe 15, 21 bandgap schema 21 conductivity, dark 21 electro-optic coefficients, of 15 photoactive centers 20 refractive index, chrom. displacive 15 CdTe:V dark conductivity 21 electro-optic coefficient 15 sensitivity, holographic 194 characterization, materials 152 BTO 199–205 BTO:Pb 200–201 BTO:V 202–203 holographic techniques 189–195 dark conductivity 203–205 diffraction efficiency 192–193 energy coupling 190–192 hole-electron competition, with 199 holographic erasure 195–197 holographic sensitivity 193–195 photo-emf, holographic 218 recording and erasure 195 LNbO lithium niobate 197–199 nonholographic optical methods 155 photoconductivity 162–173 in bulk material 163–164 running holograms 72 self-stabilized recording of 130–133, 232 sensitivity, holographic 193–195 materials, table for 194 speckle photo-emf techniques 178–188 Concentration, effective trap 67 Conductivity definition 36 dark 36 photo, one-center model 35–37 photo, two-center model 37–40 Photo 36 two center model CdTe in 25 dark 40 dopant, and 40 photo 39 Coupled Wave Theory anisotropic diffraction, with 121

diffraction efficiency with wave mixing 139–142 dynamic 92 coupling coefficient 81 Coupling amplitude 94 coefficient 81 constant 89 energy 94 phase 93 phenomenological parameter 82 Crystals index ellipsoid 6–7 light propagation in 5–7 wave equation, general 6 Czochralski 19

d Debye screening length 68, 282–284 light intensity, dependence on 193, 283 measurement of 193 Deformation measurement, 2D image 252 Dember effect 53,263 Diffraction efficiency 69 fixed grating, pure absorption 91 fixed grating, pure phase 89, 96 out of Bragg 91–92 phase-shifted grating with self-diffraction, for 104 Kogelnik formula 69 unshifted pure absorption grating 91 unshifted pure index-of-refraction grating 91 unshifted 69 measurement 192 nonstationary with bulk absorption 106–110 optical activity, with 123 wave-mixing, with 139–142 Diffusion 36, 284–285 coefficient (D) 36 electric field (ED) 64 length (LD) 68 DOS: Density of States 29 Drift length (LE) 68

e Effective pattern of fringes modulation

65

Index

space-charge electric field 67 trap concentration 67 Efficiency Diffraction: see Diffraction efficiency Electric field diffusion, (ED) 65 effective, (Eeff ) 67 photovoltaic 86 Electro-optic effect 8–10 coefficient measurement 155 KDP, in 16–17 lithium niobate, in 16 sillenite-type crystal, in 11–17 table, sillenites 157 Electron charge, Absolute value of 35 density of free (N) 29 lifetime, free (ô) 68 ellipsoid, index 6–7 applied field, with 14 Energy exchange 94 measurement 189 BTO, in 190

f Φ: see Quantum efficiency Fermi level BTO, in 24, 25 CdTe, in 21 photovoltaic 86 pinned 30 quasi- 30 Fixing, holograms 257–262 Four-wave mixing 119–120 Fringes visibility effective (meff ) 66

g GaAs 15 electro-optic coefficients, of 15 selective two-wave mixing 213 sensitivity, holographic 193–194 Gain Exponential 99 measurement 189 KNSBN 192 Gamma (Γ) 101

gamma (𝛾) 97, 101 Glass constant 40 Grating dynamically recording phase modulation in 116–119

h Harmonic terms 116 Hole-electron competition BTO, in 199 BTO:Pb, in 200–201 BTO:V, in 201 Hologram bending 94 diffraction efficiency 69 fixed LiNbO3, in 258 simultaneous recording and compensation 258–262 nonstationary, with bulk absorption 106–110 phase shift 70, 101 running 72–83, 214–218 hole-electron competition, with 76–83 quality factor 75 resonance frequency 73 resonance speed 74 stabilized 130–133 Holographic erasure 83 hole-electron competition, with 105 first spatial harmonic approximation 66–72 measurement of 193 phase shift 70 initial 71 measurement 206–207 running, for 76–83 stationary 69 photo-EMF techniques 218–227 recording 57 hole-electron competition, with 76–83, 202–203 time evolution 96–100 undepleted pump approximation 96–98 relaxation: dark, in 203–205 sensitivity 193–195 table, for some materials 194

307

308

Index

Holographic (contd.) BSO 194 BTO 194 GaAs 194 KNSBN 194 KNSBN:Cr 194 KNSBN:Cu 194 KNSBN:Ti 194 LiNbO3:Fe 194 SBN 194 table, for some materials time constant 68

fixed hologram: simultaneous record and compensation, by 258–262 material parameters for samples 241 generic 242 LS: Localized States 28, 29–31 Luminescence 28–29

194

i Index ellipsoid: see Ellipsoid, index InP 15 electro-optic coefficients of 15

m Maxwell relaxation time (𝜏M ) 67 Mobility (𝜇) 36, 284–285 experimental 176 Modulation, pattern of fringes (m) 63 effective (meff ) 66

n N- A

36

k KDP 16–17 electro-optic coefficients of 17 refractive index of 17 KNSBN 192, 205 sensitivity, holographic 194 table 192 Kogelnik 69, 89

o One center model 35–36 uniform illumination 36 at high irradiance 37 Optical activity 122–124 diffraction efficiency with 123

l

p

Length Debye screening, (ls) 68 diffusion, (LD) 68 experimental data for undoped BTO 238 drift (LE) 68 Lifetime, free electron, (ô) 68 Light propagation crystals, in 5–7 anisotropic media, in 5–6 wave equation, general 6 Light-induced absorption (𝛼li) 48–51, 157–161 BTO, table 160 photochromic effect, or 48–51 photovoltaic, and nonlinear 46–47 LNbO: lithum niobate (LiNbO3) 15–16, 28 electro-optic coefficients 15 erasure, holographic 197–199

Pattern of fringes phase position: (𝜑) 70 Phase modulation in two-wave mixing 115 dynamic gratings, in 116 signal beam, in 116–118 techniques, phase 205–218 shift, holographic (𝜑P ) 70 measurement 206–207 Photo-electromotive-force holographic 218–227 setup 218 speckle 178–188 setup 186 techniques 178–188, 218–227 Photoactive centers cadmium telluride 21–22 deep and shallow 20–28

Index

effective cross-section 36 lithium niobate 28 photoconductivity, and 19 sillenite-type crystals 11–17 Photochromic activation energy for sillenites 22 effect 48–51 light-induced absorption, Transmittance with 51 Photoconductivity 29–40, 162–173 bulk material, in 163–164 coefficient 166 Localized States 29–31 measurement of 166 ac technique 164–166 mobility-lifetime product 176–178 modulated 175–178 models, theoretical (WRP) 32–40 one-center 35–37 two-center 37–40 wavelength-resolved166–173 BTO, for 168 BTO:V, for 170 longitudinal configuration 170–173 transverse configuration 166–170 Photodiode 287–290 operational amplifier 290 photoconductive 289 photovoltaic 288–289 Photoelectric conversion 173–175 wavelength-resolved (WRPC) 173–175 Photoelectron cross-section for, generation 36 generation coefficient, thermal 36 lifetime 36 quantum efficiency, generation (Φ) 36, 176–178 computing of 177 recombination constant 36 Photoluminescence BTO, in 29 Photorefractive response time 207–210 Photovoltaic effect 40–44 light polarization dependent 43–44 nonlinear 44–48 materials 84–87 space-charge field, effective 86–87

transport coefficient Polarization output 123–124

41

q Quantum efficiency photoelectron generation, for (Φ)

36

r Recording hologram glassplate-stabilized 147–150 running 72–83 wave mixing and bulk absorption, with 106–110 Refractive index BTO, of 14 KDP, of 16–17 LMbO, of 15–16 modulation 60–63 Relaxation time, dielectric (TM) 68 bulk absorption, with 106–100 Running hologram: see hologram, running

s Saturation space-charge electric field 66 SBN sensitivity, holographic 194 Schottky barrier, light-induced 51 electric field 56 first spatial harmonic approximation 66–72 general formulation 63–66 saturation 68–72 steady-state, solution for 64–66 time (T sc ) 68 space-charge electric field, effective 63–66 Schottky effect, light-induced 51–54 Selective two-wave mixing 210–214 GaAs 212–214 Self-stabilized recording actual materials, in 135–150 arbitrary phase shift 133–135 equilibrium condition, stable 130

309

310

Index

Self-stabilized recording (contd.) formulation, mathematical 127–135 running hologram 232–239 adaptive speed 232–234 sillenites, in 136 Sillenite-type crystal 22–27 doped 25–27 electro-optic coefficients of 11 p-type conductivity in 24 parameters table 157 photoactive centers in 22 Space-charge field effective 68 first spatial harmonic approximation 66 nonstationary 106–110 photovoltaic donor density, influence on 86–87 two-center model, for 39 Stabilized recording 125 running holograms of 130–133

t Tilting measurement, 2D image

252

Time Maxwell relaxation 68 space charge 68 Transmittance light-induced absorption, with 51 trap concentration, effective 68

v VB: Valence Band

19

w Wave general, equation 6 mixing selective, two- 210–214 propagation anisotropic media, in 5–6 crystals, light 5–7

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