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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) CampinasSP, 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 printondemand. 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 CataloginginPublication 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), CampinasSP 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
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Contents List of Figures xi List of Tables xxxiii Preface xxxv Acknowledgments xxxvii
Fundamentals 1 Introduction 3
Part I
1
ElectroOptic 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 ElectroOptic 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 OxideBi2 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 SilleniteType 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
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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 OneCenter Model 35 TwoCenter/OneCharge 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 PolarizationDependent Photovoltaic Eﬀect 43 Nonlinear Photovoltaic Eﬀect 44 LightInduced Absorption and Nonlinear Photovoltaic Eﬀects 46 Deep and Shallow Centers 47 LightInduced Absorption or Photochromic Eﬀect 48 Transmittance with LightInduced Absorption 51 Dember or LightInduced Schottky Eﬀect 51 Dember and Photovoltaic Eﬀects 54 Holographic Recording 55 Introduction 56
Part II
3
Recording a SpaceCharge 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
IndexofRefraction Modulation 60 General Formulation 63 Rate Equations 64 Solution for SteadyState 64 First Spatial Harmonic Approximation 66 SteadyState Stationary Process 68 Diﬀraction Eﬃciency 69 Hologram Phase Shift 70 TimeEvolution Process: Constant Modulation 70 SteadyState Nonstationary Process: Running Holograms 72 Running Holograms with HoleElectron 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 PhaseAmplitude Stationary Gratings 92 Fundamental Properties 94 Irradiance 95 Pure Phase Grating 96 Time Evolution 96 Stationary Hologram 100 SteadyState Nonstationary Hologram with WaveMixing 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 TwoWave Mixing 110 Phase Modulation 115 Phase Modulation in Dynamically Recorded Gratings 116 Phase Modulation in the Signal Beam 116 Output Phase Shift 118 FourWave Mixing 119 Conclusions 120
5
Anisotropic Diﬀraction
5.1 5.2 5.2.1 5.2.2
121 CoupledWave 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 FringeLocked Running Hologram 132 SelfStabilized Recording with Arbitrarily Selected Phase Shift 133 SelfStabilized Recording in Actual Materials 135 SelfStabilized Recording in Sillenites 136 SelfStabilized Recording in LiNbO3 136 Holographic Recording without Constraints 137 SelfStabilized 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
ElectroOptic Coeﬃcient 155 LightInduced Absorption 157 Dark Conductivity 161 Photoconductivity 162 Photoconductivity in Bulk Material 163 Alternating Current Technique 164 WavelengthResolved Photoconductivity 166 Transverse Conﬁguration 166 Longitudinal Conﬁguration 170 PhotoElectric Conversion 173 WavelengthResolved PhotoElectric Conversion (WRPC) 173 Undoped BTO 174 Modulated Photoconductivity 175 Quantum Eﬃciency and MobilityLifetime Product 176 PhotoElectromotiveForce Techniques (PEMF) 178 SpecklePhotoElectromotiveForce (SPEMF) Techniques 178 Speckle Pattern onto a Photorefractive Material: Stationary State 179
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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 Fedoped 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 PhaseShift Measurement 206 WaveMixing Eﬀects 207 Photorefractive Response Time 207 Selective TwoWave Mixing 210 Amplitude and Phase Eﬀects in GaAs 212 Running Holograms 214 Holographic PhotoElectromotiveForce (HPEMF) Techniques 218
9
SelfStabilized 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 FringeLocked 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 SelfStabilization 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 RealTime Holograms 267 NakedEye 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 InBragg Recording Beams 272 Probe Beam 272 Reversible Holograms 274 High IndexofRefraction 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
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List of Figures Figure 1
Naturally birefringent uniaxial lithium niobate crystal view under converging white light between crossed polarizers with its caxis (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
Threedimensional sillenite structure: darker spheres represent Bi3+ ions and paler gray ones are O2− . Acknowledgments to Prof. Jesiel F. Carvalho, IF/UFGGoiâniaGO, 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 readytouse 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
Indexofrefraction 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 caxis. 16
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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 caxis. 16 Figure 2.1
Energy diagram for a typical CdTe crystal doped with vanadium, with the Te in the Cd antisites 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, ICMCBBordeaux, 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 loneelectron 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 doubleoctahedra units. 23
Figure 2.6
Localized states in the Band Gap of nominally undoped Bi12 TiO20 crystal, from [29]. Filled electrondonors are in gray and empty ones in white; the DOS (density of states) is qualitatively represented by the width of the fullline limited levels whereas the dashedline 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 BTO008. 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. occupationofstates diagram” (right side). 30
Figure 2.10 Doped semiconductor: Fermi level pinned at the position of the dopant in the BG. On the righthand side is the “energy vs. occupationofstates” diagram. 30 Figure 2.11 Doped semiconductor: Fermi Ef and quasistationary Fermi levels upon illumination. The “energy vs. occupationofstates” graphics is shown on the righthand 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 asgrown 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 ntype (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 reexcited 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 ptype. 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 bandtransport 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 BTOJ40), leaddoped Bi12 TiO20 (labeled BTOPb), undoped Bi12 SiO20 (labeled BSO) and photovoltaic irondoped 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 caxis 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 Polarizationdependent 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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List of Figures
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 (BTOJ40) 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 leaddoped Bi12 TiO20 (BTOPb) sample. The dashed line is only a guide for the eyes. 47 Figure 2.28 Average photovoltaic current density data, measured along the caxis, versus light (𝜆 = 532 nm) intensity (light polarization direction along crystal caxis) for an irondoped LiNbO3 (LNb) sample show a strict linear behavior with the continuous line being the best ﬁtting with Eq. (2.67). 48 Figure 2.29 Lightinduced 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 BTO010 using a 𝜆 = 532 nm Gaussian crosssection 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 Lightinduced Schottky barrier at the illuminated transparent conductive ITO electrodephotorefractive 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 Crosssection schema of the ITOsandwiched 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 lockin 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 leftside ordinate axis. 53 Figure 2.37 Photovoltaicbased current data (•, ◾ and ▴) computed from curves in Fig. 2.36 are plotted on the leftside ordinate axis, whereas computed Demberbased currents (⚬, ◽ and Δ) are plotted on the rightside 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+ (electrondonors), empty traps ND+ (electronacceptors) 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 spacecharge electric ﬁeld modulation. 59 The electric ﬁeld modulation may produce deformations in the crystal lattice. 59 If the photoconductive material is also electrooptic, that is to say it is photorefractive, the spacecharge ﬁeld may produce an indexofrefraction modulation in the crystal volume that is inphase (or counterphase) with the spacecharge ﬁeld modulation and is π∕2shifted 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 spacecharge electric ﬁeld modulation. 61 The electric ﬁeld modulation produces a indexofrefraction modulation (volume grating) via electrooptic 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 Spacecharge electric ﬁeld grating being recorded by the 𝜙shifted sinusoidal pattern of fringes. 63 Spacecharge 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 spacecharge ﬁeld with E0 = 0 and 𝜏sc = 10 au. 68 Indexofrefraction 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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List of Figures
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 spatialcharge ﬁeld and indexofrefraction 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 patternoffringes 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 (LDaxis) and ls (LSaxis) 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 Onespecies/twovalence/twocharge carrier model contributing to charge transport: one single spatial trap modulation structure is produced. 77 Twospecies/twovalence/twocharge carrier model contributing to charge transport: two distinct spatial trap modulation structures are produced. 78 Holeelectron 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 holedonor level or holes from the electrondonor level, because of energy considerations. In this case, an electronbased 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 reexcited 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 indexofrefraction hologram that is phaseshifted by 𝜙 = 𝜙P referred to the recording pattern of fringes with 2𝜃 being the angle inside the material. 90
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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 twowave mixing: in this example, the weaker beam receives energy from the stronger, but could also be the other way round. 94 Phase coupling in twowave 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 S2 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 S2 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 S2 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 S2 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 PZTsupported mirror in the holographic setup, on the diﬀraction eﬃciency (thin curve) of a running hologram recorded in a photorefractive BTOcrystal using the 514.4nm wavelength. The diﬀraction eﬃciency evolution to equilibrium is faster for the negativegain (lower graphics, with K = 2.55 μm−1 ) than for the positivegain (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 2.05mm 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 2.05mm 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 piezoelectricsupported mirror, D: photodetector, LAΩ and LA2Ω: lockin 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 Wavemixing 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 fourwave mixing showing the signal S and reference R beams interfering to produce a realtime 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 midcrystal 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 parallelpolarized 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 2Darray 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 sawtoothshape 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
Blockdiagram of a selfstabilized setup: D photodetector, LAΩ phase sensitive lockin 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
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Figure 6.5
Schematic description of the actual selfstabilized holographic recording setup: C photorefractive crystal, D photodetector, LAΩ and LA2Ω phase sensitive lockin ampliﬁers tuned to Ω and 2Ω, respectively, HV high voltage source for the piezoelectric 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 twowave mixing in the holographic setup. 128
Figure 6.7
Blockdiagram of fringelocked running hologram setup: same as for Fig. 6.4 with the addition of an integrator INT at the output of the lockin ampliﬁer. 130
Figure 6.8
Schematic actual setup for selfstabilized running hologram recording: same as for Fig. 6.4 with the addition of an integrator INT at the output of the lockin ampliﬁer. 131
Figure 6.9
Fringelocked running hologram speed: Kv (rad/s) versus feedback ampliﬁcation 𝜅f (arbitrary units) in a fringelocked 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 selfstabilized 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 dualphase lockin 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 patternoffringes 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 Selfstabilized 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 10mmthick crystal with the patternoffringes 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 selfstabilization 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 pztdriven mirror, OSC signal generator, HV high voltage source, INT integrator, D1,2 detectors, LAΩ and LA2Ω lockin 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 selfstabilized 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 selfstabilized conditions (IS2Ω = 0) and 𝛽 2 = 10. Note that 𝜙 rapidly shifts away from 𝜋 during recording. Reproduced from [123]. 144 Figure 6.21 Selfstabilized recording in the lessoxidized crystal (sample LNB5) with 𝛽 2 ≈ 1 (IR0 = 141.1 W∕m2 and IS0 = 116 W∕m2 ). The evolution of ISΩ during the selfstabilized 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 Selfstabilized 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 Selfstabilized 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 patternoffringe speed, corrected from scattering, and the dashed curve is only a guide for the eyes. 146 Figure 6.26 Selfstabilized 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
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Figure 6.28 Glassplatestabilized 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 nonselfstabilized 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) selfstabilization 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 electrooptic 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 lockin 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 BTO010) under uniform illumination of I0 ≈ 2 mW/cm2 at 𝜆 = 532 nm. 159
Figure 7.3
Lightinduced absorption: transmitted I t versus incident I0 irradiances measured using an uniform beam of 532 nm wavelength on the same sample BTO010 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
Lightinduced absorption of undoped Bi12 TiO20 (sample labeled BTO013) at 𝜆 = 514.5 nm as a function of the incident irradiance measured in the air. The lefthand side graphics is in semilog scale for detailed view at low irradiances. The continuous curve on the righthand 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 coeﬃcientthickness 𝛼d measured for three diﬀerent BTO samples (BTO8, BTOQ and BTO008) as a function of wavelength. BTO8 and BTOQ were measured in a standard spectrophotometer whereas BTO008 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
Frequencydependence 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
Ωtuned lockin 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 socalled “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 timemodulated uniform 532 nm wavelength laser beam onto crystal sample BTO010 with 2000 V applied. The corresponding time modulated photocurrent was measured using a lockin ampliﬁer where (⚬) are data for a sample that has been kept in the dark for a long time and (•) are data for the previously lightsaturated 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 BTO010. Reproduced from [39]. 166 (Left) Photograph of the wavelengthresolved 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 computercontrolled steppingmotor. 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 homemade 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: lockin ampliﬁer. 167 Transverse conﬁguration: coeﬃcient 𝜎 on a logarithmic scale (upper graphics) and on a normal scale (lower graphics) for preexposed 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 preexposed 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 lightinduced 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 Lightinduced photoelectric conversion eﬃciency 𝜂0 measured (•) on an undoped sandwiched Bi12 TiO20 crystal (labeled BTOJ18L) in the longitudinal conﬁguration together with the light absorption coeﬃcientthickness 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 ITOsandwiched 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 inplane electrodes (coplanar conﬁguration). Reproduced from [148]. 182 Figure 7.25 Stationary spacecharge ﬁeld arising from a speckle pattern of light vibrating faster than the response time of the spacecharge 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 frequencytuned phaseselective lockin 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 preampliﬁer (an electrometerclass operational ampliﬁer operating in transimpedance mode)
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ﬁxed by the side of the crystal. A homemade 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 NdYAG 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 twowave 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 caxis parallel to the grating ⃗ Holographic recording is carried out with extraordinarily polarized vector K. (polarization direction along the caxis) 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 preillumination 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
List of Figures
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 Pbdoped 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 Pbdoped 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 preexposed for a few minutes to a uniform light at 𝜆 = 532 nm. Preexposure was switched oﬀ immediately before holographic recording started using an HeNe laser line of 𝜆 = 633 nm. The hologram was erased with one of the inBragg 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 Pbdoped 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 (lefthand graph) and along the signal beam (righthand graph) using one of the recording beams. Curves showing a local maximum result from 3 min preexposure 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 preexposed 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 Vdoped BTO (0.30% V in weight) with holeelectron competition. The continuous curve is the theoretical ﬁt (a single factoring parameter in ordinates was used for data ﬁtting) assuming hole and electroncharge 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 (BTO8) approximately 1 mm thick. Diﬀraction eﬃciency was measured from time to time using one of the inBragg recording beams during a very short time and correcting data for the eﬀect of exposure to light. From the Arrheniustype 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 selfdiﬀraction 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 electrooptical 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 Twowave 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 semiinsulating 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 Twowave 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 Twowave 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 steadystate 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 electrontohole diﬀraction eﬃciency ratio at
List of Figures
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 piezoelectricsupported mirror PZT in one of the beams is vibrating with angular frequency Ω. A lockin 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 Kdirection 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: Cedoped BTO (thickest curve), for Pbdoped 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 phaseshift 𝜑 versus applied electric ﬁeld (E0 ) data (circles) for a 2.05 mm thick Bi12 TiO20 crystal and gratingvector 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 Fringelocked 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 Fringelocked running hologram experiment: frequency detuning K𝑣 (measured from the movement of the PZTsupported mirror) versus normalized applied ﬁeld E0 ∕ED data from a typical fringelocked 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 Fringelocked 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 BTO013) 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 3Dcurve (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): selfstabilized 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): selfstabilized 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
List of Figures
Figure 9.12 Characterization of oxidized LiNbO3 :Fe (labeled LNB1): selfstabilized 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: piezoelectric supported mirror; PLC: path length compensator; EOM: electrooptical modulator; SF: spatial ﬁlter; BTO: photorefractive Bi12 TiO20 crystal; D: photodetector; P1 e P2: polarizers; CCD: image detector; LA: lockin 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 HeNe 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 retroreﬂective painted loudspeaker membrane. 249 Figure 10.5 Loudspeaker membrane (left) driven at 3.0 kHz and analyzed by the timeaverage 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 Timeaverage holographic interferometry pattern of a thin phosphorousbronze metallic plate tightly ﬁxed by its external border to a loudspeaker vibrating at 255 Hz. 253 Figure 10.9 Timeaverage holographic interferometry pattern of a thin phosphorousbronze metallic plate tightly ﬁxed by its external border to a loudspeaker vibrating at 600 Hz. 253 Figure 10.10 Timeaverage holographic interferometry pattern of a thin phosphorousbronze metallic plate tightly ﬁxed by its external border to a loudspeaker vibrating at 800 Hz. 254
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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 temperaturecontrolled 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 heatisolating plate (not seen) covers the upper cylinder side. 260 Figure 11.2 Evolution of I Ω and I 2Ω during high temperature selfstabilized holographic recording (and compensation) for a typical experiment. 261 Figure 11.3 Diﬀraction eﬃciency of the overall grating during whitelight 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 outofBragg 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 inBragg 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 crosssection irradiance I(x) illuminating a photoconductive material in steadystate 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 crosssection s, receiving a ﬂux Γ of electrons of mass me and velocity 𝑣. 284
Figure E.1
npjunction showing the depletion layer and a diagram of the Schottky potential barrier. 288
Figure E.2
npjunction showing the depletion layer including the intrinsic layer and a diagram of the Schottky potential barrier. 288
Figure E.3
pnjunction 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 opencircuit 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 shortcircuit photovoltaic regime. 290
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List of Tables Table 1.1 Table 2.1
Index of refraction of KDP. 17 Photovoltaic transport coeﬃcient 𝜅phv for Fe and Cudoped 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 electrooptic 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
Holeelectron 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
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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: electroopticity 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, lightinduced 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 indexofrefraction (and sometimes also light absorption coeﬃcient) modulation in the material volume as a consequence of their electrooptic properties. Most of the recording is carried out using a spatially modulated interference (holographic) pattern of light, an indexofrefraction and sometimes associated absorption coeﬃcient volume grating results. The realtime 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 “selfstabilized recording”. Running holograms and selfstabilized 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 realtime measurement of outofplane mechanical vibration modes in 2D and inplane 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 submicroscopic 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. CampinasSP, 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, CampinasSP, 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 caxis (optical axis) laying perpendicular to the plane (upper) and on the plane (lower).
3
Introduction
Photorefractive crystals are electrooptic and photoconductive materials. An electric ﬁeld applied to an electrooptic 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 (LSphotoactive 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 spacecharge electric ﬁeld. The combination of both eﬀects gives rise to the socalled photorefractive eﬀect: the light produces a photoconductivebased electric ﬁeld spatial modulation that in turn produces an indexofrefraction modulation via the electrooptic 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 indexofrefraction 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 socalled “indexofrefraction” 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 spacecharge electric ﬁeld and the production of indexofrefraction changes via an electrooptic (or Pockels) eﬀect. We should therefore rather call them “photoelectrorefractive” 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 electrooptic eﬀect including a little bit of tensorial analysis. The eﬀect of an applied electric ﬁeld over the index ellipsoid of some usual electrooptic 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 lightinduced absorption and their relation with the localized states (photoactive centers) in the forbidden band.
5
1 ElectroOptic Eﬀect The electrooptic 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 electrooptic eﬀect in photorefractive materials is of the highest importance because it is at the origin of the “imaging” of a spacecharge ﬁeld modulation into an indexofrefraction 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 indexofrefraction in the volume of the sample. In these materials, such a modulation arises from the buildup of a modulated spacecharge ﬁeld that on its turn modulates the indexofrefraction via an electrooptic 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 ElectroOptic 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 indexofrefraction 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 ElectroOptic 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 ElectroOptic 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 ElectroOptic Eﬀect
The slight variation in the refractive index produced by an electric ﬁeld can be described by the thirdorder electrooptic 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 Btensor 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 electrooptic 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 indexofrefraction build up produced by the slow accumulation of electric charges. Therefore, all the electrooptic coeﬃcients referred to in this chapter are only the lowfrequency 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 ElectroOptic 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 Threedimensional sillenite structure: darker spheres represent Bi3+ ions and paler gray ones are O2− . Acknowledgments to Prof. Jesiel F. Carvalho, IF/UFGGoiâniaGO, 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 sixfold coordination, surrounded by an octahedron of anions, and the “A” cation in 12fold cuboctahedral coordination. Perovskites are the bestknown 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 wellknown 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 piezoelectric, electro and elastooptic, 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 socalled “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 slantedsliced sample in Fig. 1.6, the striations are not visible through the polished (110)face (see also Figs 1.7 and 1.8). The electrooptic tensor of this crystal family in the principalaxes 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 readytouse crystal with renamed axes (bottom).
(1.33) [001]
[100]
(0 11 )
[001] (110)
11
12
1 ElectroOptic 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 ElectroOptic Eﬀect
2.70
2.65
n
14
2.60
2.55
2.50 450
500
550
λ (nm)
600
650
700
Figure 1.9 Indexofrefraction 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 indexofrefraction 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 electrooptic 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 indexofrefraction 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 sixfold coordination instead of a 7–12fold coordination in perovskites [10]. The electrooptic 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 ElectroOptic Eﬀect
Figure 1.13 Lithium niobate crystal with an applied electric ﬁeld along the photovoltaic caxis.
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 caxis.
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 indexellipsoid 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 electrooptic tensor somewhat similar to that of sillenites. It has the following electrooptic 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 indexofrefraction 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 OxideBi2 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 electrooptic eﬀect. We have brieﬂy shown how to calculate these eﬀects in a few kinds of crystal having diﬀerent electrooptic 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 electrooptic 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 (spacecharge) ﬁeld that is able to modify the indexofrefraction via electrooptic eﬀect as explained in Section 1.3. The movement of charges under the action of the diﬀusion gradient is opposed by the growing spacecharge 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 spacecharge 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 socalled “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. Spacecharge ﬁelds of a few kV per cm are easily produced in this way and consequently rather large overall indexofrefraction 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 indexofrefraction 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 lightinduced 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 (indexofrefraction itself ) and the imaginary (extinction coeﬃcient or absorption) part of the complex indexofrefraction may be also modulated via trap modulation in the material’s volume. Such socalled traparising indexofrefraction and absorption modulation are related to the electroopticalbased eﬀect but they are nevertheless additional eﬀects of a diﬀerent nature. We shall see further on that the indexofrefraction modulation arising from trap polarizability modulation and the one arising from spacecharge modulation are mutually 𝜋∕2phaseshifted and both are, in general, also shifted from the recording pattern of light. There is still the possibility to ﬁnd an additional local indexofrefraction 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 traparising and the electric ﬁeldarising indexof 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 spacecharge 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 buildup 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 selfdiﬀraction eﬀects. The conductivity may also vary along the interelectrode 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 socalled “onecenter/twovalence/one charge carrier” model is simple to analyze. However, some materials may require a “twocenter”, or a “onecenter/threevalence” model and so on [18]. Shallow and deep traps may coexist, and even holeelectron 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 wellknown 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 facecentered 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 wellstudied 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. Cdvacancies give the ptype 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 Cdvacancies by annealing under Cdvapor 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 Cdvacancy 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 Tedonors or Cdvacancy 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 Cdvacancies, 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 10fold 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 antisites 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, ICMCBBordeaux, France. From the Arrhenius plot, the energy of the Fermi level EA = 0.83 eV is computed.
2.1.2
SilleniteType 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 loneelectron 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 loneelectron 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 loneelectron 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 doubleoctahedra 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 outofequilibrium conditions are required to produce sillenite crystals with useful BiM antisite 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 “antisite defect BiM ”. The density of these centers in Bi12 GeO20 is lower (2–3fold) 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 ptype 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 electronacceptor or holedonor 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 ptype conductivity as reported previously. This means that holographic relaxation in the dark is probably due to ptype conductivity, from the Fermi level down to the VB, roughly about 1 eV below. However, the photoconductivity of these materials is largely ntype, 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 ntype nature of sillenites under the action of light may be due to a larger crosssection 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 holeexcitation. Even with 𝜆 = 670 (h𝜈 ≈ 1.85 eV) preexposure only ptype holograms are recorded as seen in Fig. 8.5. But at least from preexposure 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 preexposure. Direct 𝜆 = 780 nm ptype recording certainly proceeds by direct excitation of holes from [Bi3+ + h+ ] to the VB at an energy gap lower than 1.6 eV. Preexposure acting on ndonor 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 nbased recording with 𝜆 = 780 nm be possible. The WavelengthResolved 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+ ] donoracceptor 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 onecenter model. In fact, the onecenter model assumes that moderately low intensity light onto the sample will not signiﬁcantly change the totaltoacceptor 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 twocenter model instead may allow for a kind of lightinduced 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 ptype 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 electrondonor 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. Electrondonor 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 loneelectron 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 Pbdoped 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 electrondonors are in gray and empty ones in white; the DOS (density of states) is qualitatively represented by the width of the fullline limited levels whereas the dashedline 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 preexposure 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 Vdoping 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 asgrown and in hydrogendoped) 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 quasimonochromatic LED, centered at 𝜆 = 408 nm (leftside 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 rightside 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 BTO008. 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 twocenter 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 twocenter 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 occupationofstates (from zero to one) diagram shown on the righthand 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, steadystate Fermi (or quasiFermi) levels for electrons Efn and for holes Efp are deﬁned [14] as depicted in Fig. 2.11 with the occupationofstates accordingly modiﬁed as represented by the righthand
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. occupationofstates diagram” (right side). Figure 2.10 Doped semiconductor: Fermi level pinned at the position of the dopant in the BG. On the righthand side is the “energy vs. occupationofstates” diagram.
CB
EF acceptors Fermi
Eg
donors 0
VB
1
Figure 2.11 Doped semiconductor: Fermi Ef and quasistationary Fermi levels upon illumination. The “energy vs. occupationofstates” graphics is shown on the righthand side.
ILLUMINATION CB Efn EF Fermi Eg
Efp 0 VB
1
2.3 Photoconductivity
CB bandtoband 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 asgrown 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 ntype 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 asgrown 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 ntype (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 reexcited 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 ptype. 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 crosssection 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: onecenter and twocenter, 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
OneCenter Model
For the simplest onecenter/one charge (electrons) carrier model, one single type photoactive + ) is assumed, in which case the charges and spacecharge 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 (electronacceptors) 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 crosssection 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 steadystate 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 “onecenter/onecharge carrier” model only. 2.3.2.2
TwoCenter/OneCharge 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 bandtransport 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 crosssection 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 lightinduced 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 lightdependent 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 spatialderivatives, 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 quasiequilibrium 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 holeelectron 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 steadystate value is reached. Unlike Eq. (2.27) for the OneCenter 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 TwoCenter 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 ﬁllin an equal number of shallow traps (−Δ(ND1 − ND1 ) = Δ(ND2 − ND2 )) that have a diﬀerent crosssection (s2 ) from that (s1 ) one of the former deeper ones. + ∕𝜕t = 0) for ND2 in In order to compute 𝛼li we assume the steadystate 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 electrondonors and ﬁlling up the holedonors and, by this means, considerably reducing shallow trapsarising 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 threefold from undoped to [Ru]=1019 cm−3 Rudoped 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 PN junction observed in semiconductors or metalsemiconductor interfaces. The photovoltaic eﬀect has interesting practical applications because it results in a higher spacecharge 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 socalled “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 Cudoped 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 aircrystal 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 Cudoped 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 BTOJ40), leaddoped Bi12 TiO20 (labeled BTOPb), undoped Bi12 SiO20 (labeled BSO) and photovoltaic irondoped 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 BTOJ40 BTOPb 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 caxis 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 Fedoped 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 “lonepairs” (pairs of valence electrons that are not shared with another atom and are also called “nonbonding” pairs) of both cations Bi3+ and Te4+ . These lonepairs (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 PolarizationDependent Photovoltaic Eﬀect
Experimental results have shown that the photovoltaic eﬀect may depend on the polarization direction of the incident light, as demonstrated for Fedoped 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 thirdorder 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 Polarizationdependent 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 lightelectric 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 lightinduced 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 “asgrown” positive (ND1 and ND2 ) ions in the crystal. Once steadystate
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 Fedoped [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, Fedoped LiNbO3 in Fig. 2.28 exhibits a perfect linear behavior and Pbdoped Bi12 TiO20 in Fig. 2.27 can hardly be considered to be nonlinear [36].
2.5.1
LightInduced Absorption and Nonlinear Photovoltaic Eﬀects
Although nonlinear photovoltaic and lightinduced 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 crosssections, 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 BTOPb, all exhibit strong lightinduced 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 lightinduced 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 (BTOJ40) 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 leaddoped Bi12 TiO20 (BTOPb) 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 caxis, versus light (𝜆 = 532 nm) intensity (light polarization direction along crystal caxis) for an irondoped 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 BTOJ40
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 LightInduced Absorption or Photochromic Eﬀect Lightinduced 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 onecenter model. Let us
2.6 LightInduced Absorption or Photochromic Eﬀect
therefore refer to the twocenter model, including shallow traps. In this case, the lightinduced 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 timederivative) 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 lightinduced 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 lightinduced 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 BTO8), 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 Lightinduced 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 LightInduced 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 BTO010 using a 𝜆 = 532 nm Gaussian crosssection 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 LightInduced Absorption
The formulation of the transmitted light in the presence of lightinduced 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 LightInduced Schottky Eﬀect The Schottky eﬀect is the buildup of an electric potential barrier occurring at the interface between semiconductors with diﬀerent dopants (nSi and pSi, for example), or between a metallic ﬁlm on a ntype semiconductor: electrons from the ntype semiconductor ﬂow into the ptype one or into the metallic electrode, leaving behind a positively charged depletion
51
52
2 Photoactive Centers and Photoconductivity ITO – + – electrode – + – – –– – – +++ – – – – + – –– + – – – – + + –– + – – + – – – – –+ + + – +– – – – + + –
Figure 2.32 Lightinduced Schottky barrier at the illuminated transparent conductive ITO electrodephotorefractive 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 Crosssection schema of the ITOsandwiched BTO plate indicating the photocurrent ﬂow under illumination.
iph
Light
ITO electrode front face – – + + – – + + – – + + + – – + – + – + + – – + + – – + + – – + + – + – + + – – + – – + + + – –
ITO electrode rear face + – – + + – + – +
+ +
+
–
– + – +
– –
+ – – + –
Voltage
Depth
2.7 Dember or LightInduced Schottky Eﬀect
Figure 2.35 ITO sandwiched 0.81 mm thick BTO crystal plate with electrodes wired to a lockin ampliﬁer.
front ITO electrode
back ITO electrode
(00 1) illumination
to lockin 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 leftside 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 socalled “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 ITOcrystal 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 crystalelectrode 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 lightinduced Schottky barrier at the front (strongly illuminated) crystalelectrode 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 Photovoltaicbased current data (•, ◾ and ▴) computed from curves in Fig. 2.36 are plotted on the leftside ordinate axis, whereas computed Demberbased currents (○, ◽ and △) are plotted on the rightside 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 ITOsandwiched 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 caxis whereas that arising from Dember eﬀect is determined by the way light illuminates the transparent electrodesandwiched crystal plate: by reversing illumination from the front to the rear electrode, the Demberbased photocurrent ﬂowing sense is also reversed without changing the sense of the photovoltaicbased one. The overall ac photocurrent on the d = 0.81 mm thick ITOsandwiched 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 lockin 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 photovoltaicbased and the Demberbased 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 realtime optical recording, reversibility, indeﬁnite number of recordingerasure cycles and very high spatial resolution. Also, the ﬁnal recording state does not depend on the irradiance and on the total energy (timeintegrated irradiance) but on the patternoflight 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 spacecharge electric ﬁeld without caring about the associated indexofrefraction modulation, whereas Chapter 4 is devoted to the buildup of an indexofrefraction modulation in the material’s volume; that is to say, a phase volume hologram. Because of the realtime nature of the recording process, the hologram does diﬀract the recording beams during recording, thus modifying their relative amplitudes and their mutual phaseshift, 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 wavemixing or selfdiﬀraction, is characteristic of realtime 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 feedbackcontrolled 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 SpaceCharge Electric Field This chapter focuses on the mechanisms responsible for the buildup of a modulated spacecharge electric ﬁeld under the action of a modulated pattern of light projected onto the sample, without considering wavemixing eﬀects; that is to say, without caring about the diﬀraction of the recording beams by the indexofrefraction modulation associated with the spacecharge ﬁeld hologram being built up. The theoretical model is based on the Band Transport Theory and rate equations proposed by Kukhtarev and coworkers [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 reexcited 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 spacecharge electric ﬁeld modulation, as illustrated in Fig. 3.3, which is π∕2phase shifted to the spatial modulation of charge because of the wellknown Poisson equation relating charge and electric ﬁeld. If the material is electrooptic, besides being photoconductive, then the spacecharge ﬁeld modulation produces a corresponding modulation in the indexofrefraction, 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 SpaceCharge 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+ (electrondonors), empty traps ND+ (electronacceptors) 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 SpaceCharge 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 spacecharge 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 electrooptic, that is to say it is photorefractive, the spacecharge ﬁeld may produce an indexofrefraction modulation in the crystal volume that is inphase (or counterphase) with the spacecharge ﬁeld modulation and is π∕2shifted to the recording pattern of light.
59
60
3 Recording a SpaceCharge 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 patternoffringes 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 IndexofRefraction Modulation Let us think about the way the spacecharge ﬁeld modulation may act on the index ellipsoid in order to produce the indexofrefraction 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 IndexofRefraction 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 spacecharge electric ﬁeld modulation.
Figure 3.11 The electric ﬁeld modulation produces a indexofrefraction modulation (volume grating) via electrooptic eﬀect.
direction of the reading beam (the incident beam that is diﬀracted by the grating in the crystal’s volume) the indexofrefraction will be changing as the spacecharge 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 indexofrefraction modulation is invariant for any polarization direction of the reading beam, as far as the electrooptic conﬁguration represented in Figs. 1.3, 1.10 and 1.11 is concerned.
61
62
3 Recording a SpaceCharge 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 spacecharge ﬁeld will change from the maximum value along x (Fig. 1.11) to the maximum along the other direction. Therefore, the indexofrefraction 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 indexofrefraction 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 indexofrefraction 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 Spacecharge electric ﬁeld grating being recorded by the 𝜙shifted sinusoidal pattern of fringes.
(3.5) (3.6)
⃗ S(0) 2θ
⃗ R(0)
spacecharge field grating and associated hologram
X
R⃗
ϕ pattern of light
Z ∆ d
S⃗ K = 2Π/∆
63
64
3 Recording a SpaceCharge Electric Field
where 𝜙 is the phase shift between the pattern of fringes and the spacecharge electric ﬁeld grating, with the following deﬁnitions: IS0 ≡ S0 2
IS ≡ S2
IR0 ≡ R0 2
IR ≡ R2
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 socalled complex patternoflight fringe modulation. 3.2.1
Rate Equations
Unless otherwise stated it the simplest “one center, two valence, one chargecarrier” 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 SteadyState
We should now ﬁnd a solution to the rate equations in Section 3.2.1 for the steadystate. In this case all variables are timeindependent 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 spacecharge electric ﬁeld j0 = e (x)𝜇E(x) + e
E(x) =
j0 e𝜇0 (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 xcomponent 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 e𝜇0
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 diﬀusionarising spacecharge 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 spacecharge ﬁeld. For the particular case of small patternoffringes 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 spacecharge ﬁeld for diﬀerent patternoffringes 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 “ﬁrstspatial approximation” only for m ≤ 0.3.
65
3 Recording a SpaceCharge 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 Spacecharge 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 spacecharge ﬁeld for an arbitrarily large patternoffringes 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 spacecharge 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 socalled “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)) = −𝜇0 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 spacecharge 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 SpaceCharge Electric Field
1.0
Figure 3.17 Simulated recording (from 0 to 20 au) and erasure (from 20 to 50 au) of a spacecharge 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 ∕(eμ0 )
(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 spacecharge 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
SteadyState Stationary Process
For stationary steadystate conditions, it is 𝜕Esc (t)∕𝜕t = 0, which substituted into Eq. (3.42) gives the stationary spacecharge 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 “steadystate” means that it has reached an equilibrium with a timeinvariant 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 spacecharge ﬁeld of amplitude Esc , which produces, via a linear electrooptic eﬀect, an indexofrefraction modulation of amplitude n1 , as deﬁned, for example, in Eqs. (1.45) and (1.46). The indexofrefraction modulation is always inphase or counter phase with the spacecharge 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 wellknown 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 electrooptic coeﬃcient for the given crystal conﬁguration, Esc  is the amplitude of the spacecharge 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 indexofrefraction 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
SPACECHARGE FIELD ∆n
0.5 0 –0.5 –1.0
0
0.5
1.0
1.5
Figure 3.18 Indexofrefraction 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 spatialcharge ﬁeld and indexofrefraction modulation in the absence of any externally applied electric ﬁeld (E0 = 0). All vertical coordinates are in “arbitrary units”.
69
70
3 Recording a SpaceCharge 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 indexofrefraction modulation pattern, for a pure refractive index hologram) is given by the phase of the complex Esc as the latter two are necessarily always inphase as indicated in Eq. (3.54). The socalled 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 steadystate conditions, as in Eq. (3.52), where it is Esc = −meﬀ Eeﬀ , the phaseshift 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
TimeEvolution Process: Constant Modulation
For the general case, the hologram being recorded (because of the growing of a spacecharge 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 coupledwave 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) = mEeﬀ  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) = mEeﬀ 
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 SpaceCharge 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 SteadyState 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 SteadyState 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 twowave 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 phaseshift that optimizes amplitude coupling. The research extended RUNNING HOLOGRAM
Figure 3.19 Schematic description of running hologram generation in photorefractives. A moving patternoffringes 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 SteadyState Nonstationary Process: Running Holograms
to semiinsulating 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 socalled spacecharge 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 spacecharge 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 patternoffringes 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 xaxis. 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 steadystate 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 SpaceCharge 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 unitmodulation steadystate 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 SteadyState 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 phaseshift” 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 steadystate 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 SpaceCharge 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 twowave mixing experiments, as discussed in Section 4.2.1.1. 3.4.1
Running Holograms with HoleElectron Competition
It is possible to have electrons and holes excited simultaneously, by the action of light, in order to produce their correspondingly associated spacecharge 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 oneexponential 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 spacecharge modulation to be built up. Figure 3.22 Plotting of Q as a function of LD (LDaxis) and ls (LSaxis) 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 SteadyState 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 Onespecies/twovalence/twocharge 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 buildup of two physically distinct (each one on diﬀerent centers) and opposite sign gratings, and the overall recording dynamics may follow a twoexponential 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 SpaceCharge Electric Field
ILLUMINATION
CB
–
– +
+ – + –
–
–
– + – +
+
+ –
+
+
– +
+ –
acceptor – + ND2 ND2 acceptor
+ NA 2 nonphotoactive
–+
+
+ –
+
+ + ND
nonphotoactive
– – +
+ + –
+
VB
– N– A
+– + –
+
–
–
– – +
+
+ –
–
+
ND– ND –
donor
+ ND2 donor
Figure 3.26 Twospecies/twovalence/twocharge carrier model contributing to charge transport: two distinct spatial trap modulation structures are produced.
CB
– +
– – – + +
+++ – –
+
+ –
+
–
+
–
+
–
– – +
+ –
– +
+ –
– +
+ –
VB
+ –
Figure 3.27 Holeelectron 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 holedonor level or holes from the electrondonor level, because of energy considerations. In this case, an electronbased 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 reexcited 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 SteadyState Nonstationary Process: Running Holograms
there is a mechanism for providing reversibility. The simultaneous presence of electron and holephotoactive 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 subindex “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) + e1 ∇ (x, t) j⃗1 = e (x, t)𝜇1 E(x,
(3.96)
and a similar independent set of equations for holes with subindex “2”: 𝜕(x, t) 1 = G2 − R2 − ∇ . j⃗2 𝜕t e − (x, t) 𝜕ND2 = G2 − R2 𝜕t ⃗ t) − e2 ∇(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 subindex “2”, whereas “1” is for the electrons. The solution of the previous system of coupled equations starts with the usual assumption of quasistationary condition: 𝜕 ∕𝜕t ≈ 𝜕∕𝜕t ≈ 0
(3.103)
79
80
3 Recording a SpaceCharge 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) − iK0 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) + iKo 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 + iKo 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 − iKo 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 SteadyState 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 steadystate 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 spacecharge 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 SpaceCharge 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)
SteadyState 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 spacecharge 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 electronbased 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 SteadyState Nonstationary Process: Running Holograms
Hologram Erasure The erasure of a hologram with holeelectron 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 holebased space charge ﬁeld is much larger than that (𝜏sc1 ) for the electronbased 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 holeelectron competition in Pbdoped Bi12 TiO20 is described in Section 8.4.2.2.
83
84
3 Recording a SpaceCharge Electric Field
3.5 Photovoltaic Materials Space charge electric ﬁeld buildup 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 xcoordinate 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 shortcircuited as well as for an opencircuited 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 buildup in the presence of a modulated pattern of light I(x) = Io (1 + m cos(Kx)) and the resulting selfdiﬀraction eﬀects make it diﬃcult to ﬁnd an analytical solution. In order to get such a solution, however, we shall neglect selfdiﬀraction 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 shortcircuited 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 SpaceCharge 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 electrondonor density involved here. Considering: sI 𝜏 h𝜈 to be substituted into Eq. (3.148) it turns out to be 𝜎 ≈ e𝜇ph
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 spacecharge ﬁeld timederivative 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 spacecharge electric ﬁeld modulation that is produced by the action of a spatially modulated pattern of light as described in Chapter 3 also produces a phaseshifted associated realtime indexofrefraction spatial modulation due to the electrooptic eﬀect analyzed in Chapter 1. As a consequence, a phase grating, because of its realtime buildup, 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 “selfdiﬀraction” 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 indexofrefraction 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 indexofrefraction 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 indexofrefraction 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 indexofrefraction hologram that is phaseshifted 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 TEpolarization, 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 indexofrefraction 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 outofBragg 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 realtime and reversible recording properties) make these materials particularly suitable as, for example, eﬃcient ﬁlters in extendedcavity semiconductor lasers for improving singlemode 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 2mmthick purely indexofrefraction 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 “selfdiﬀraction” (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 indexofrefraction 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 steadystate 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 indexofrefraction modulation Δn on the fringes visibility m. Δn = mn1
4.2.1
n1 ≡ −
Combined PhaseAmplitude Stationary Gratings
Therefore, for the general case of a dynamic grating exhibiting at the same time a phase modulation (with indexofrefraction 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)
93
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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 amplitudephase 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 phaseshifting in the following cases: 1. inphase (or counterphase) amplitude and ±90∘ shifted phase grating, 2. inphase (or counterphase) amplitude grating and inphase phase grating (meaning no beam coupling) with equal input irradiance beams (ISo = IRo ). Any one of these conditions will make the righthand 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 phasecoupling and therefore no hologram “bending” due to selfdiﬀraction. Figure 4.4 Amplitude coupling in twowave 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 twowave 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).
95
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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 buildup of a spacecharge electric ﬁeld and the associated indexofrefraction modulation and corresponding phase grating. For a purely phasemodulated (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 indexofrefraction modulation amplitude for m = 1. Assuming no absorption at all, neither bulk nor modulation eﬀects, the Kogelnik coupledwave 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 Timederiving 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 indexofrefraction grating, and state the socalled “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 secondorder 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 zeroorder 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)
97
98
4 Volume Hologram with Wave Mixing
from Eqs. (4.55) and (3.52). Note that the holographic phaseshift 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 “twowave 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 twowave 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 S2 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 S2 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 S2 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 rampshaped voltage (thick curve) applied to a PZTsupported 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 negativegain experiment (lower graphics) shows a much faster and less oscillatory evolution to equilibrium than the positivegain 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 S2 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 PZTsupported mirror in the holographic setup, on the diﬀraction eﬃciency (thin curve) of a running hologram recorded in a photorefractive BTOcrystal using the 514.4nm wavelength. The diﬀraction eﬃciency evolution to equilibrium is faster for the negativegain (lower graphics, with K = 2.55 μm−1 ) than for the positivegain (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 phaseshift of the pure phase hologram. We shall only consider an indexofrefraction modulation that, in photorefractive crystals, arises from the electrooptic properties of these materials. In fact, a modulated pattern of spacecharge 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 electrooptic eﬀect a corresponding modulated pattern of indexofrefraction 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 phasemodulated (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 indexofrefraction 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
102
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 patternoflight phase planes and consequently the hologram phase planes too. Equation (4.96) represents a patternoffringes 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
104
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.
Phaseshifted holograms We shall now deal with an arbitrarily phaseshifted 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 wellknown 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 SteadyState Nonstationary Hologram with WaveMixing 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 buildup 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 selfdiﬀraction 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 farfromsaturation 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 K𝑣value 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
107
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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 PhaseShift 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 phaseshift 𝜑 between the transmitted and diﬀracted beams behind the sample is aﬀected too. Its formulation in the presence of selfdiﬀraction 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 514.5nm 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 twowave 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
109
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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 selfdiﬀraction 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 selfdiﬀraction and absorption, the movement of the hologram (and consequently the value of the hologram phaseshift) 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 TwoWave Mixing
As already mentioned before, photorefractive twowave 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 2.05mm 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 2.05mm thick and 𝛼 = 1165 m−1 .
113
114
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 twowave with energy ﬂowing from the stronger pump to the weaker signal beam does behave like a negative feedback ampliﬁer. Additional photorefractive twowave 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 piezoelectricsupported mirror, D: photodetector, LAΩ and LA2Ω: lockin 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 twowave mixing is produced by phasemodulating (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 timeaverage 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 timeaverage 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 timedependent terms in Bessel series and making the timeaverage < cos(𝜓d sin Ωt) > = J0 (𝜓d ) + 2J2 (𝜓d ) < cos 2Ωt > +2J4 (𝜓d ) < cos 4Ωt > +... = J0 (𝜓d )
(4.152)
115
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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 phasemodulated 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 phasesensitive frequencytuned lockin 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 spacecharge 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 phasemodulated dynamic grating. We shall here describe in detail the accurate formulation of temporal harmonic components in phase modulated photorefractive twowave 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 twowave 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 𝜙shifted 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 twowave 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 twicediﬀracted 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 coworkers [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 + IΩ sin Ωt + I2Ω cos 2Ωt + ...
(4.158)
IΩ = − 4J1 (𝜓d )ℑ{S∗ S(0)S }
(4.159)
I2Ω = 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
IΩ (z) = 4J1 (𝜓d )IS (0) 𝛽2
(4.164)
Γ Γ z − z e 2 + e2
and
I2Ω = −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 IΩ
(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 Sbeam 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 FourWave Mixing
Figure 4.20 Wavemixing 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 (socalled 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 selfdiﬀraction 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 FourWave 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 fourwave mixing showing the signal S and reference R beams interfering to produce a realtime 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 “twowave mixing” or TWM. We can similarly mix four waves instead of two and this is called “fourwave 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 realtime (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 realtime hologram as shown in the righthand 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 socalled “phaseconjugate” 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 realtime recording, • the low pass ﬁltering arising from the ﬁnite material response time, • the amplitude coupling or energy transfer derived from the phaseshifted 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 sillenitetype crystals and this feature derives from the structure of their electrooptic tensor. In fact, let us recall the expressions for the indexofrefraction 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 CoupledWave 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 sillenitetype 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 ycomponents 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 parallelpolarized 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 socalled “selfstabilized” 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 selfstabilized recording in microelectronics grade photoresists: The large heighttowidth ratio of the structures shown in these ﬁgures could never have been obtained without using selfstabilized 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 selfstabilized 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 2Darray 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 phasemodulated 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 phaseselective frequency tuned ampliﬁer known as a “lockin” 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 sawtoothshape 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 selfstabilized 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 selfstabilization 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 phaseshift 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 selfstabilization experiments. The setup is schematically represented in the blockdiagram 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 twowave 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 Blockdiagram of a selfstabilized setup: D photodetector, LAΩ phase sensitive lockin 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 selfstabilized holographic recording setup: C photorefractive crystal, D photodetector, LAΩ and LA2Ω phase sensitive lockin ampliﬁers tuned to Ω and 2Ω, respectively, HV high voltage source for the piezoelectric 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 twowave 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 Ωtuned lockin ampliﬁer LAΩ 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 voltagetophase 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 voltagetophase 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 patternoffringes position and their diﬀerence is the socalled hologram phaseshift 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 steadystate conditions, 𝜙P and 𝜓0 are constants and the corresponding steadystate 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 spacecharge electric ﬁeld amplitude, which are inphase and 𝜋∕2shifted, 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 steadystate 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 spacecharge 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 selfstabilized way is the central point here. Such holograms are also known as “fringelocked” 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 rewritten 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 Blockdiagram of fringelocked running hologram setup: same as for Fig. 6.4 with the addition of an integrator INT at the output of the lockin 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 selfstabilized running hologram recording: same as for Fig. 6.4 with the addition of an integrator INT at the output of the lockin ampliﬁer.
the actual setup schema is depicted in Fig. 6.8. An integrator was included here between the lockin 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 steadystate 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 steadystate (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 FringeLocked 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 selfstabilized (also known as fringelocked) 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 514.5nm wavelength and a nominally applied ﬁeld of 4.7 kV/cm as seen in Fig. 6.9, where it is shown that for a 50fold 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 toolarge ampliﬁcation may produce instabilities and even drive the setup to an oscillatory behavior. 6.2.3
SelfStabilized Recording with Arbitrarily Selected Phase Shift
The selfstabilized holographic recording setup described previously suﬀers from a serious limitation: the phaseshift 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 Fringelocked running hologram speed: Kv (rad/s) versus feedback ampliﬁcation 𝜅f (arbitrary units) in a fringelocked 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 selfstabilized 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 dualphase lockin 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 coworkers [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 irradiancetovoltage 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 SelfStabilized Recording in Actual Materials
The same signal 𝑣D (t) is also ﬁltered to extract the Ω component (bandpass ﬁlter BP Ω), which is phaseshifted 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 lockin ampliﬁer LA2Ω 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 lockin ampliﬁer LA2Ω, 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 lockin 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 lockin 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 phaseshift 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∕2shifted 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 selfstabilized 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 selfstabilized.
6.3 SelfStabilized Recording in Actual Materials We shall apply the selfstabilization 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 selfstabilized 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 nonselfstabilized regime. In order to understand this feature, one should ﬁgure out that the feedbackdriven 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. Selfstabilized 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 longterm duration recording materials as this one certainly is. 6.3.1
SelfStabilized Recording in Sillenites
The presently analyzed selfstabilization 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 “errorsignal” 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 selfstabilized mode. Because the freeand feedbackconstrained 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 piezomirror drives the system back to the I Ω = 0 stable position. A lockin 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 selfstabilized 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
SelfStabilized Recording in LiNbO3
Selfstabilized 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 selfdiﬀraction, 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 patternoffringes 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 Selfstabilized 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 10mmthick crystal with the patternoffringes on the (110) plane and the hologram vector K⃗ perpendicular to the [001]axis and parallel to [110].
Signal (Volts)
6.3 SelfStabilized 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 selfstabilization 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, selfstabilization 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 phasecoupling 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 pztdriven mirror, OSC signal generator, HV high voltage source, INT integrator, D1,2 detectors, LAΩ and LA2Ω lockin ampliﬁers tuned to Ω and 2Ω, respectively.
SpaceCharge Electric Field Buildup Because the selfstabilized setup produces a running hologram, in general, here we shall refer to a moving (with speed 𝑣 along the grating ⃗ spacecharge electric ﬁeld grating that is, neglecting selfdiﬀraction 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 movingpattern term and excluding the external ﬁeld E0 . ND+ and ND are the concentration of the empty (electronacceptors 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 SelfStabilized Recording in Actual Materials
where the ﬁrst righthand term represents the stationary spacecharge wave moving synchronously along with the patternoffringes, 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 quasistationary (a slowly st was substituted by the slowly varying time timedependent) steadystate 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 selfstabilization, 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 selfdiﬀraction (wavemixing) 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 quasisteady state (that is to say, slowly timedependent) recorded grating may be assumed to be formulated as for the steadystate 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 indexofrefraction 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 SelfStabilized 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 (outofBragg) uniform stationary grating where, for 𝛽 2 ≠ 1, it is always 𝜂 < 1. For the special case of 𝛽 2 = 1 an inBragg 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 SelfStabilized Recording
In a selfstabilized 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. Selfstabilized 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 (nonselfstabilized 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) IΩ = 4J1 (𝜓d )IS (0)
𝛽2
I2Ω = −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 phaseselective 2Ωtuned lockin 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) selfstabilized 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 SelfStabilized 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 selfstabilized 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 feedbackconstrained 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 selfstabilized 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 selfstabilized 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 selfstabilized regime), which is also the (openloop, 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 selfstabilized 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 selfstabilized 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 openloop 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 shortcircuited 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 selfstabilization 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 inBragg 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 selfstabilized 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 SelfStabilized Recording in Actual Materials
8 IsΩ (au)
6
4
2 Is2Ω
0 0
50
100
150
200
Time (s)
Figure 6.21 Selfstabilized recording in the lessoxidized crystal (sample LNB5) with 𝛽 2 ≈ 1 (IR0 = 141.1 W∕m2 and IS0 = 116 W∕m2 ). The evolution of ISΩ during the selfstabilized 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 Selfstabilized 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 Selfstabilized 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. Selfstabilized 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 patternoffringes), by the eﬀect of selfstabilization, 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 selfstabilization loop. The timeevolution of IG and the K𝑣 computed from these data for a typical selfstabilized experiment are shown in Fig. 6.25. The same experiment was carried out on a reduced sample (labelled 7581) 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 indexofrefractions and electrooptic 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 patternoffringe speed, corrected from scattering, and the dashed curve is only a guide for the eyes.
6.3 SelfStabilized Recording in Actual Materials 2.5
Figure 6.26 Selfstabilized 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 selfstabilized 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 selfstabilized recording with ordinarily polarized light allows the achievement of roughly a threefold higher spacecharge 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 fourfold faster than the former. More details about the use of light polarization to improve the indexofrefraction modulation recorded in LiNbO3 has been published elsewhere [124]. The larger time (fourfold) compared to the (threefold) spacecharge 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 10fold 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 Fedoped 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. GlassplateStabilized 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 Rbeam and the phasemodulated Sbeam 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 optoelectronic 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 selfstabilization. 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 selfstabilization, 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 glassplatestabilized experiment didn’t work: the recording was so noisy that stabilization (and recording) in these conditions was impossible. ISΩ
LAΩ
M 4
1 r5
nm
D1 IS
IR0
se
La
LA2 Ω C
BS
G
IS0 IS0 Ω OSC
IS2Ω
IG D2
LAΩ
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 Glassplatestabilized 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 SelfStabilized 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 selfstabilized 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].
Selfstabilized 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) selfstabilization 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 nonselfstabilized 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 selfstabilized experiment where 𝜂 steadily grows up to the limit (because of selfstabilization constraints) of 𝜂 = 1. We should conclude that, by keeping 𝜑 = 𝜋∕2, selfstabilization somehow keeps the hologram inBragg 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 selfstabilization 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 selfstabilized 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 ﬁrsthand experimental results. Selfstabilization 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 ElectroOptic Coeﬃcient The electrooptic or Pockels coeﬃcient is one of the most important parameters of photorefractive materials. Although an eﬀective value of the electrooptic 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 lightinduced spacecharge ﬁeld opposing the externally applied ﬁeld during electrooptic 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 electrooptic coeﬃcient should therefore take into account optical activity and should use the lowest possible illumination intensity and/or be fast enough to avoid the buildingup of a lightinduced electric ﬁeld that may jeopardize measurements. A practical setup the measurement of the unclumped electrooptic coeﬃcient reﬀ = r41 = r52 = r63 in sillenite crystals published by de Oliveira et al. [125] is schematically described in Fig. 7.1. Almost monochromatic lightemitting 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 lockin 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
lockin amplifier
oscilloscope
Figure 7.1 Schema of the experimental setup for electrooptic 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 lockin 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 shortcircuited under illumination for a while, for spacecharge 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 spacecharge 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 shortcircuiting) 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 wavelengthdependent electrooptic coeﬃcients. We may conclude then, together with other researchers [129, 130] that sillenite crystals do not
7.2 LightInduced Absorption
Table 7.1 Eﬀective electrooptic 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 wavelengthdependent eﬀects on their electrooptic coeﬃcients, and if ever such a dependence appears, attention should be paid to the jeopardizing eﬀect of lightinduced spacecharge ﬁeld due to photoconductivity that is eﬀectively strongly wavelengthdependent [125]. Dopants, instead, may have a somewhat sensible eﬀect on reﬀ , which is not surprising because they may aﬀect crystal structures.
7.2 LightInduced Absorption This subject was already theoretically developed in Chapter 2, where Eq. (2.57) describes the lightinduced 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 lightinduced 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 indexofrefraction samples. Figure 2.31 shows transmittance data (circles) for an undoped BTO crystal sample (labeled BTO010, 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 indexofrefraction 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 lightinduced absorption (photochromic darkening) on a typical undoped Bi12 TiO20 crystal at a ﬁxed incident irradiance. Steadystate results are
7.2 LightInduced Absorption
760
Figure 7.2 Evolution of the absorption coeﬃcient in an undoped B12 TiO20 crystal (labeled BTO010) under uniform illumination of I0 ≈ 2 mW/cm2 at 𝜆 = 532 nm.
α (m–1)
720 680 640 600
0
Figure 7.3 Lightinduced absorption: transmitted It versus incident I0 irradiances measured using an uniform beam of 532 nm wavelength on the same sample BTO010 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 Lightinduced absorption of undoped Bi12 TiO20 (sample labeled BTO013) at 𝜆 = 514.5 nm as a function of the incident irradiance measured in the air. The lefthand side graphics is in semilog scale for detailed view at low irradiances. The continuous curve on the righthand 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 twocenter 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 lightinduced 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 lightinduced 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 lightinduced 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
BTO010
BTO011
BTO013
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 BTO008 BTO Q BTO8
4
αd
3 2 1 0
400
λ (nm)
600
800
Figure 7.5 Absorption coeﬃcientthickness 𝛼d measured for three diﬀerent BTO samples (BTO8, BTOQ and BTO008) as a function of wavelength. BTO8 and BTOQ were measured in a standard spectrophotometer whereas BTO008 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 luminescencearising 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 nmcentered luminescencearising 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 BTO008, BTO8 and BTOQ using a spectrophotometer with a nonwavelength selective photodetector. For sample BTO008, the photodetector was placed at 10 mm behind the sample, whereas for samples BTO8 and BTOQ 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 BTO008, 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 energybarrier 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 frequencydependent 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 Vdoped 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 Gadoped 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 Frequencydependence 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 zaxis) 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 zdependence photoconductivity 𝜎ph can be written from Eq. (2.49) as: 𝜎ph (z) = eph (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 lightinduced absorption, the Φ𝛼 in Eq. (7.13) should be substituted for the expression in Eq. (2.52). It is interesting to deﬁne a socalled 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 Ωtuned lockin 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 socalled “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 timemodulated spatially uniform illumination and the detection of the associated current using a phasesensitive frequencytuned lockin 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 timemodulated 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 lifetime 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 OPAMP 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 OPAMP 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 acphotocurrent 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 timemodulated (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 timemodulated 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 lockin 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 BTO010. 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 (BTO011) and at 633 nm (BTO010) showing a large dependence on the wavelength, probably due to the wavelength dependence of the characteristic eﬀective crosssections 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 timemodulated uniform 532 nm wavelength laser beam onto crystal sample BTO010 with 2000 V applied. The corresponding time modulated photocurrent was measured using a lockin ampliﬁer where (○) are data for a sample that has been kept in the dark for a long time and (•) are data for the previously lightsaturated 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 BTO010. Reproduced from [39].
7.4.3
WavelengthResolved Photoconductivity
Figure 7.11 shows a typical setup for wavelengthresolved 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 preexposed 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 wavelengthresolved 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 computercontrolled steppingmotor. 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 homemade 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: lockin ampliﬁer.
where Φ𝛼0 + 𝛼li (z) and 𝛼0 + 𝛼li (z) stand here to take into account the possibility of lightinduced 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 lightinduced 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 iﬁlled 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, preexposed to h𝜈 = 2.3 eV light and still in normal intermediate conditions. The preexposure 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 wavelengthresolved photocurrent is therefore expected to show a rather steplike shape, if it is adequately wavelengthresolved. 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 electronbased 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 electronbased hologram as reported in Fig. 8.8) that has certainly be electronpopulated during normal operation and not necessarily by proposital preexposure process. Also, the preexposed 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
Preexposed 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
Preexposed 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 preexposed with h𝜈 = 2.4 eV light (•), normal (◽) and for relaxed (∘) undoped Bi12 TiO20 plotted as a function of h𝜈. Reproduced from [29].
inbetween the Fermi level and the bottom of the CB that are detected only after been (at least partially) ﬁlled by preexposure. The expanded view in Fig. 7.13 shows a sharp step at about h𝜈 ≈ 2.5 eV for all three preexposed, 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 Preexposed
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. VDoped BTO The Vdoped Bi12 TiO20 (BTO:V) crystal shows a WavelengthResolved 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; • preexposure 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 preexposed 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 crystalelectrode 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 lightinduced 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 ITOsandwiched 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 lightinduced electric potential barriers at both electrodes with a schema of the electric potential distribution at the bottom.
7.5 PhotoElectric 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 secondorder polynomial. The overall optical absorption coeﬃcient 𝛼 is also shown (▴). Reproduced from [135].
Diﬀerent from the transverse conﬁguration, there is a tradeoﬀ 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 stepbystep 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 PhotoElectric Conversion Photoelectric conversion, which is to say, photocurrent ﬂowing without any externally applied electric ﬁeld, is expected to occur in an ITOsandwiched photoconductive (in this case, a photorefractive) crystal plate in the longitudinal conﬁguration because of the lightinduced unbalanced fronttorear potential barrier diﬀerence Vsh , as discussed in Section 7.4.3.2. 7.5.1
WavelengthResolved PhotoElectric Conversion (WRPC)
The lightinduced 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 lightinduced unbalanced fronttorear 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 lightinduced 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 ITOsandwiched 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 Lightinduced photoelectric conversion eﬃciency 𝜂0 measured (•) on an undoped sandwiched Bi12 TiO20 crystal (labeled BTOJ18L) in the longitudinal conﬁguration together with the light absorption coeﬃcientthickness 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 ITOsandwiched 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 timemodulated (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 bandtoband chargecarrier 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 crosssectional 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∕𝜋term 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 “attempttoescape 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 MobilityLifetime Product
Photoconductivity in the presence of lightinduced (photochromic) absorption allows one to easily compute the mobilitylifetime 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 lightdependent 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 lightinduced 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 BTO011 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
BTO010
BTO011
BTO013
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 PhotoElectromotiveForce Techniques (PEMF) The PEMF arises when lightinduced charge carriers move in a stationary spacecharge 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 spacecharge electric ﬁeld are built up as already described in Chapter 3. If the pattern of light is moving faster than the response of the spacecharge ﬁeld but slower than the lifetime of free charges in the extended states, the free charges will follow the movement but the spacecharge 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 electrooptic properties. Electroopticity is not necessary here, but the large density of photoactive centers in the bandgap, which is a characteristic of photorefractive materials, leads to large spacecharge 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 photoelectric 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 patternoffringes into a corresponding spacecharge 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 spacecharge electric ﬁeld is concerned. Both speckle and holographic photoelectromotive force techniques are useful for mechanical (inplane) vibration and deformation measurement [146–150] as well as for materials characterization [151–154]. 7.7.1
SpecklePhotoElectromotiveForce (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 PhotoElectromotiveForce 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 wellknown (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 crosssection for photonelectron 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 = −2x0 e−r (∇
(7.68)
where
⃗ )y = −2y0 e−r (∇
2
2
(7.69)
and r 2 ≡ x2 + y 2
(7.70)
D (7.71) 𝜇W with ED being the socalled diﬀusion ﬁeld. The current density along x and ycoordinates 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 xcoordinate so that we state that jx = j0 where j0 is a constant. The ydirection in the crystal, instead, is openlooped,
7.7 PhotoElectromotiveForce 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 ≡ e𝜇0
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 spacecharge ﬁeld shaping is evident. Vibrating Speckle Pattern If the speckle pattern is vibrating along xcoordinate, 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 = −20 e−(r + 𝛿 ∕2) (xI1 (x, 𝛿, Ω, t) + 𝛿I2 (x, 𝛿, Ω, t)) (∇
(7.82)
2 2 ⃗ )y = −20 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 inplane electrodes (coplanar conﬁguration). Reproduced from [148].
X = Δ sin Ω t
X
Free electron cloud in the conduction band
Stationary spacecharge field
Figure 7.25 Stationary spacecharge ﬁeld arising from a speckle pattern of light vibrating faster than the response time of the spacecharge 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 spacecharge ﬁeld buildup 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 PhotoElectromotiveForce 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 spacecharge 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 twodimensional spacecharge 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 spacecharge ﬁeld pattern along the direction of vibration and the way it is aﬀected by d . 7.7.1.1.2
Photocurrent Components The xcomponent 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 timedependent 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 xcoordinate we get the linearintegrals: 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 crosssection for current ﬂux and the number of speckles involved in the process. Harmonic Terms The normalized timedependent 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 darktophotoconductivity 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 lockin 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 preampliﬁed 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 preampliﬁer) is measured using
7.7.1.1.4
7.7 PhotoElectromotiveForce 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 frequencytuned phaseselective lockin 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 preampliﬁer (an electrometerclass operational ampliﬁer operating in transimpedance mode) ﬁxed by the side of the crystal. A homemade laser Doppler vibrometer DV using a 633 nm laser beam is used for independent measurement of the loudspeaker vibration.
the lockin ampliﬁer referred to previously. The crystal and the preampliﬁer 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 preampliﬁed signal in the housing is fed into the lockin ampliﬁer using a cooper shielded BNC cable that strongly improves signaltonoise 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 NdYag 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 socalled coplanar electrode conﬁguration as shown in Fig. (7.24)) or on the bulk of the crystal (using electrodes on the lateral sides, the socalled 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 PhotoElectromotiveForce 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 NdYAG 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 spacecharge electric ﬁeld, arising from the excitation, retrapping, diﬀusion and drifting of charge carriers (electrons and/or holes), is described in Chapter 3. Holeelectron 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 twowave 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 twowave 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 patternoffringes 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 twowave 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 indexofrefraction 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 Tidoped KNSBN crystal described in Table 8.1 [158] with extraordinary polarized 514.5 nm light and the grating vector K⃗ parallel to the caxis. 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 caxis parallel to the grating vector K. Holographic recording is carried out with extraordinarily polarized (polarization direction along the caxis) 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
Tidopant
0.36% wt
dimensions
5.5 × 5.1 × 4.9 mm
caxis
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 twowave 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 parallelpolarized 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 energytransfer 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 twowave 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 mIabs 𝜕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 m𝜋I0 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 spacecharge 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). Holeelectron 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 spacecharge electric ﬁeld modulation is produced and diﬀraction eﬃciency 𝜂 erasure will show a single timedecaying 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 socalled “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 inphase 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 Pbdoped 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 πshifted (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
Fedoped LiNbO3 : Hologram Erasure under White Light Illumination
Figure 8.4 shows the whitelight erasure of a previously recorded (using 514.5 nm light) hologram in Fedoped 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 preillumination 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 Pbdoped 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 timeevolution 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 Pbdoped 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 preexposed for a few minutes to a uniform light at 𝜆 = 532 nm. Preexposure was switched oﬀ immediately before holographic recording started using an HeNe laser line of 𝜆 = 633 nm. The hologram was erased with one of the inBragg 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)
Holeelectron 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 preexposed (for 10–15 min) with spatially uniform light of diﬀerent wavelengths. Holographic recording and erasing without preexposure leads to a monoexponential slow decaying curve, similar to the monoexponentials in Fig. 8.8 for nonpreexposed Pbdoped BTO, which is likely to be a holebased (because being slow) hologram recorded on the [Bi3+ +h+ ] center at the Fermi level at ≈ 1 eV above the VB. Preexposure recording with low energy photons h𝜈 ≤ 1.85 eV (𝜆 ≥ 670 nm) apparently has no eﬀect since diﬀraction eﬃciency erasure is always a monoexponential monotonically decreasing curve. For preexposure in the range h𝜈 = 2.6 down to at least 2 eV, erasure curves show the typical pattern of holeelectron competition with a holebased and an electronbased 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 electronbased hologram to be recorded by the low photonic energy 𝜆 = 780 nm beams besides the alwayspresent holebased one. Recording with 𝜆 = 1064 nm (h𝜈 = 1.17 eV) was unsuccessful whatever the preexposure illumination used either for undoped or for Pbdoped 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 preexposure or not, and independently of the photonic energy of any preexposure. We should therefore conclude that more energetic radiation does record a holeand electronbased 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 preexposure.
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 Pbdoped 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 (lefthand graph) and along the signal beam (righthand graph) using one of the recording beams. Curves showing a local maximum result from 3 min preexposure 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 preexposed 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 Pbdoped 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 preexposed (at h𝜈 ≈ 2.3 eV) before hologram recording (and erased) with h𝜈 ≈ 2 eV light. Without preexposure, a single charge carrier excited from diﬀerent centers is involved as shown in Fig. 8.6. Recording with h𝜈 ≈ 1.6 eV and preexposing at h𝜈 ≈ 2.4 light shows (see Fig. 8.8) electrons and holes excited from diﬀerent centers, but without preexposure only one center is involved, probably [Bi3+ + h+ ], because only slow (holebased) holograms appear. The erasure of holograms recorded on a Pbdoped 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 electronbased one usually faster and a holebased 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 electronbased space charge buildup time 𝜏SC1 (see Eq. (3.145)) whereas the slower one 𝜏sc2 ∕(1 − 𝜅12 𝜅21 ) (see Eq. (3.146)) depends on the holebased space charge buildup 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 Pbdoped 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 Preexposure Preexposure with 𝜆 = 532 nm (h𝜈 ≈ 2.3 eV) before recording and erasing with 𝜆 = 633 nm produces electron and holebased 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 electronbased hologram in these ﬁlled LS and record an another holebased 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 holeelectron 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 𝜋phase 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 preexposure 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 Preexposure
Figure 8.8 shows the eﬀect of short wavelength preexposure on holographic recording (and erasure) on BTO:Pb (same sample as in Fig. 8.6) with 𝜆 = 780 nm. Table 8.4 Holeelectron 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 Vdoped BTO, using 𝜆 = 514.5 nm laser beams show a strong holeelectron competition with a fast hologram and a slower one, probably electron and holebased 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 vanadiumdoped BTO (up to ≈ 0.1% in weight of vanadium) exhibits an interesting example of strong holeelectron 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 holeelectron 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 spacecharge electric ﬁeld here is the one arising from holeelectron 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 m2 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 holeelectron 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 Vdoped BTO (0.30% V in weight) with holeelectron competition. The continuous curve is the theoretical ﬁt (a single factoring parameter in ordinates was used for data ﬁtting) assuming hole and electroncharge 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 fasttoslow 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 wavelengthresolved 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 energybarrier controlled phenomenon that follows an Arrheniustype 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; ‡ : Wavelengthresolved 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 (BTO8) approximately 1 mm thick. Diﬀraction eﬃciency was measured from time to time using one of the inBragg recording beams during a very short time and correcting data for the eﬀect of exposure to light. From the Arrheniustype 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 barriercontrolling energy Ea as illustrated in Fig. 8.11 for a hologram recorded on undoped BTO (sample BTO8) using 𝜆 = 514.5 nm (h𝜈 = 2.41 eV) laser light. The result of 1.04 eV here is close to the ptype 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 realtime nonperturbating technique that allows to carry out measurements without disturbing the recording itself. • It allows, in general, to operate in selfstabilized 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 phasemodulation 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 selfstabilization 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 patternoffringes 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 selfstabilized. Selfstabilization strongly contributes to reduce phase perturbations and improve measurement dispersion but one should keep in mind that the 𝜑 = 0 value ﬁxed by selfstabilization should correspond to the unconstrained recording condition, because otherwise the recording would be modiﬁed by selfstabilization 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 selfstabilized 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 PhaseShift Measurement
The explicit formulation of the phase shift 𝜙P of the stationary spacecharge 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 diﬀusionarising 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 Fedoped
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 righthand side in Eq. (8.41) are measured in a phase modulated twowave mixing experiment. Note that selfstabilization 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
WaveMixing Eﬀects
In the presence of selfdiﬀraction 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 twowave 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 twowave 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 selfdiﬀraction 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 electrooptical 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 spacecharge electric ﬁeld modulation amplitude Esc , for a purely diﬀusionarising (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 ⃗ Kdirection, 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 Twowave 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 indexofrefraction and reﬀ is the electrooptic 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 secondorder 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  = mS0 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 semiinsulating crystal are shown in Fig. 8.16. A similar approach may be used for computing the amplitude of the ﬁrst harmonic for an indexofrefraction (of amplitude n′1 ) grating resulting from the photoactive centers modulation during photorefractive recording (that is diﬀerent from the photorefractive indexofrefraction grating) and is inphase 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 semiinsulating 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 inphase indexofrefraction 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 inphase, 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 indexofrefraction 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 indexofrefraction grating. 8.6.4
Selective TwoWave Mixing
Selective twowavemixing (S2WM) [176] is a simple twowave mixing (TWM) experiment where the irradiance levels along both directions behind the crystal are simultaneously measured using the same phasemodulation 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 indexofrefraction 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 subindices “P”, “n” and “a” indicate the diﬀraction eﬃciency and the corresponding holographic phase shift of a photorefractive indexofrefraction 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 indexofrefraction 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 selfstabilized 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 phasecoupling 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 symmetricantisymmetric 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 indexofrefraction 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 gratingvector 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 indexof refraction gratings, if any, have parallelpolarized 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 indexofrefraction 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 twowave 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 Twowave 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 Twowave 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 steadystate 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 indexofrefraction 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 indexofrefraction and absorption gratings on the ﬁrst and second harmonic terms in phasemodulated 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 buildup 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 Fedoped LiNbO3 and use this grating as a reference for selfstabilized recording a photorefractive hologram with an indexofrefraction modulation exceeding the value for 𝜂 = 1, which would have been impossible to achieve for selfstabilized 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 holeelectron 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 phaseshift 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 sawtooth electric signal to the piezomirror 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 patternoffringes 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 phaseshift as a function of the patternoffringes 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 holephotoactive 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 selfdiﬀraction 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 holeelectron competition [189, 190]. Here, we shall assume the simplest “one center and one chargecarrier” 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 spacecharge electric ﬁeld is described in Section 3.4. In the presence of selfdiﬀraction, 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 phaseshift 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 holeconductivity in the dark and electronconductivity under the action of light [161]. It is not possible to exclude some degree of holeelectron competition in the deep trap level too, at least for some samples [183, 190]. As shown next, even a comparatively small density of holephotoactive 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 phaseshift in Eq. (4.146) can be thus computed separately for the electrons, for the holes and for the superimposed holeelectron 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 electronphotoactive centers in BTO and assuming a much lower quantum eﬃciency and concentration for holes: 100fold lower quantum eﬃciency and approximately 25fold lower concentration representing a roughly ﬁvefold 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 holephotoactive center concentration (with an electrontohole 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 electrontohole 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 holeelectron 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 buildup 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 holeelectron 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 chargecarriers being much larger than the fringe period) there is a kind of randomization of the chargecarriers 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 holeperturbations. A twowave 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 silverpainted 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 midcrystal 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 piezoelectricsupported 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 phaseshift 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 PhotoElectromotiveForce (HPEMF) Techniques The photoelectromotiveforce (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 PhotoElectromotiveForce (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
Phaseshift
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 spacecharge 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 spacecharge ﬁeld, but slower than the lifetime of free charges in the extended states, the free charges will follow the movement but the spacecharge 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 indexofrefraction modulation hologram at all, but it does rely on the mathematical development describing the holographic spacecharge electric ﬁeld buildup and also requires a typical holographic setup to produce the required holographic patternoffringes 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 spacecharge ﬁeld modulation and the free chargecarrier (for simplicity assumed to be just electrons) distribution in the conduction band are mutually 𝜋∕2phase 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
Rfeedback (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 piezoelectricsupported mirror PZT in one of the beams is vibrating with angular frequency Ω. A lockin 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 Kdirection 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) = −mEeﬀ 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 PhotoElectromotiveForce (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 = −mEeﬀ J0 (KΔ)
Ω Esc = −mEeﬀ
(8.94)
J1 (KΔ) 1 + 𝚤Ω𝜏sc
(8.95)
J2 (KΔ) 1 + 𝚤2Ω𝜏sc
(8.96)
2Ω = −mEeﬀ 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 = −mEeﬀ
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 = e𝜇0 . 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 mJ0 (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 m2
J0 (KΔ)2
ED sin 𝜙 (1 + K 2 L2D )(1 + K 2 ls2 ) ED J1 (KΔ)2 − 2𝜎0 m2 sin 𝜙 2 2 2 2 2 (1 + K LD )(1 + K ls ) 1 + Ω2 𝜏sc ED J2 (KΔ)2 − 2𝜎0 m2 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 m2 1 cos 𝜙 2 2 1 + K LD (1 − iΩ𝜏sc )(1 + i2Ω𝜏sc ) 1 + K 2 ls2
jΩ = −𝜎0 m2
K 2 L2D
(8.106)
8.7 Holographic PhotoElectromotiveForce (HPEMF) Techniques
and for the second harmonic term J (KΔ)J2 (KΔ) ED sin 𝜙 j2Ω = − 𝜎0 m2 0 1 + K 2 L2D 1 + K 2 ls2 J (KΔ)J2 (KΔ) ED 1 − 𝜎0 m2 0 sin 𝜙 1 + K 2 L2D 1 + K 2 ls2 1 + i2Ω𝜏sc ED J (KΔ)2 1 − 𝜎0 m2 1 sin 𝜙 2 2 1 + iΩ𝜏 2 2 1 + K LD 1 + K ls sc − 2𝜎0 m2 J0 (KΔ)J2 (KΔ) + 𝜎0 m2 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 m2 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 m2 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 lockin ampliﬁer tuned to Ωfrequency, 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  ≈ Ωm2 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∞ ≈
m2 ED cos 𝜙 J0 (KΔ)J1 (KΔ) 𝜏sc
(8.116)
that does not depend any more on Ω. Holographic PEMF, in a twowave interferometric setup [142–144, 153], allows transducing the phase modulation in one of the interfering beams into an oscillating patternoffringes 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 patternoffringes 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 selfcalibrating 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. Selfcalibration 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 PhotoElectromotiveForce (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 irradiancedependent 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 lockin 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 lowfrequency 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: Cedoped BTO (thickest curve), for Pbdoped BTO (thinnest curve) and undoped BTO (mid thickness curve).
8.7 Holographic PhotoElectromotiveForce (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 Cedoped and Pbdoped 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 highfrequency 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 highfrequency 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 highfrequency range without caring about the frequencydependent part of the curve.
227
229
9 SelfStabilized Holographic Techniques This chapter will show some of the many interesting possibilities of selfstabilized 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 ﬁrsthand 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 steadystate 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 selfdiﬀraction 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 selfstabilized conditions is, unfortunately, not possible because selfstabilization directly acts on the output phase 𝜑 and therefore also on the hologram phaseshift 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 referencebased stabilization, but for the measurement of the initial phaseshift (𝜙I ) described in Section 3.3.2 the same procedure may be used for the stationary phaseshift. 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 lockin 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 lockin 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 SelfStabilized Holographic Techniques
chosen to have the transmitted and diﬀracted beams approximately parallelpolarized 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 lockin 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 twophase lockin 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, selfdiﬀraction can be neglected and the (initial) holographic phaseshift 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 onetenth 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 Lookin 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 wellknown 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 phaseshift, for which the results are reported in Fig. 9.3. This ﬁgure shows experimental phaseshift data for the same Bi12 TiO20 sample and same conﬁguration as for data in Fig. 8.14 but for 𝜆 = 532 nm. For stationary phaseshift, however, the calculations are more complex (as already discussed in Section 8.6.2) than for the initial phase because of selfdiﬀraction eﬀects that are almost absent for the latter.
231
9 SelfStabilized 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 phaseshift 𝜑 versus applied electric ﬁeld (E0 ) data (circles) for a 2.05 mm thick Bi12 TiO20 crystal and gratingvector 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 FringeLocked Running Holograms Selfstabilized or fringelocked 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 FringeLocked Running Holograms
50 40 V (10–3 μm/s)
Figure 9.4 Fringelocked 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 SelfStabilized 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 selfstabilized (fringelocked) 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 fringelocked 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 selfstabilized 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 selfstabilized recording, either involving stationary or nonstationary holograms, inherently produces less dispersive data than nonstabilized recording. This technique
9.2 FringeLocked Running Holograms
is even less dispersive than stabilized nonselfstabilized 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 PZTsupported 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 lockin 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 Fringelocked running hologram experiment: frequency detuning K𝑣 (measured from the movement of the PZTsupported mirror) versus normalized applied ﬁeld E0 ∕ED data from a typical fringelocked 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 SelfStabilized Holographic Techniques
1.05
Kv (rad/s)
236
0.70
0.35
0
0
1
2
3
4
5
E0 / ED
Figure 9.6 Fringelocked 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 𝜑value and the imposed 𝜑 = 0 as already discussed in Section 6.2.2. Under steadystate 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 righthand side in Fig. 9.7. The lefthand 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 FringeLocked 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 BTO013) 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 fringelocked 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 SelfStabilized 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 3Dcurve (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 selfstabilized techniques for the characterization of fast and low diﬀractive materials such as sillenites. We shall show now the use of selfstabilized 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 shortcircuit mode (as illustrated in Fig. 3.28) and record a hologram, without applied ﬁeld, with the hologram vector K⃗ parallel to the caxis. Let us recall that selfstabilized 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 selfstabilized recording, a steadystate nonstationary spacecharge 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 SelfStabilized 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 caxis 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 selfstabilized 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 selfstabilized 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 indexofrefraction is dependent on the light wavelength and that it is, as well as the electrooptic coeﬃcient, quite diﬀerent for ordinary and extraordinary light polarization. 2.5
Figure 9.10 Characterization of reduced LiNbO3 :Fe (labeled LNB3): selfstabilized 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): selfstabilized 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): selfstabilized 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 socalled 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 SelfStabilized 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 socalled 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 • lightinduced photoelectric conversion via the Dember eﬀect besides the already wellknown 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 realtime measurement of vibrational modes and static deformations in surfaces using lowpower 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 selfstabilized holographic recording to improve the setup performance. The use of photorefractive materials as realtime, 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. Lowfrequency perturbations and changes in the setup are adaptively coped with because of the relatively fast response of these materials. Higherfrequency perturbations can instead be compensated by the use of an active stabilization feedback optoelectronic 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 retroreﬂectivity of the target surface to be taken into account. The negative feedback optoelectronic 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 timeaverage 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 optoelectronic 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 beamsplitter cube (PBS). The amplitude ratio R between both output beams is controlled using a halfwave retardation plate HWP at the PBS input. The polarization direction of the PBSexiting beams are made parallel by the use of another HWP at one of the outputs. A lowpower 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 ccdcamera 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: piezoelectric supported mirror; PLC: path length compensator; EOM: electrooptical modulator; SF: spatial ﬁlter; BTO: photorefractive Bi12 TiO20 crystal; D: photodetector; P1 e P2: polarizers; CCD: image detector; LA: lockin 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 realtime 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 socalled 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 speedup 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 (TEpolarized) 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 retroreﬂective 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 HeNe 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 retroreﬂective 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 retroreﬂected 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 retroreﬂected 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
SelfStabilization 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 electrooptical 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 selfstabilization feedback loop is discussed in Section 6.2 and is represented by the same blockdiagram 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 phaseshift 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 phasemodulated and residually amplitudemodulated, to some extent, due to unavoidable misalignment of the EOM. Such an amplitudemodulated (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) realtime 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 realtime. 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 optoelectronic circuit described in this book is also essential to enable the operation of this instrument in moderately perturbated environment. The use of selfstabilization (and not just external referencebased 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 timeaverage 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 m2 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) phosphorousbronze metallic plate was also visualized using the realtime 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 retroreﬂective 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 timeaverage 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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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 patternoffringes 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 Timeaverage holographic interferometry pattern of a thin phosphorousbronze metallic plate tightly ﬁxed by its external border to a loudspeaker vibrating at 255 Hz.
Figure 10.9 Timeaverage holographic interferometry pattern of a thin phosphorousbronze metallic plate tightly ﬁxed by its external border to a loudspeaker vibrating at 600 Hz.
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Figure 10.10 Timeaverage holographic interferometry pattern of a thin phosphorousbronze 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 HeNe 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 realtime 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 doubledoped 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 hightemperature 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 selfstabilized mode.
11.2 Fixed Holograms in LiNbO3 Fixed holograms in irondoped 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 spacecharge 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 phaseshift (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 timeconsuming [123]. The use of selfstabilized 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 indexofrefraction 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 selfstabilization, become both null at 𝜂 = 1 and then become useless for selfstabilization, thus ﬁxing an upper indexofrefraction modulation limit for selfstabilization recording that could otherwise go far beyond. This means that after compensation and development of the selfstabilized 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 selfstabilized 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 indexofrefraction 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 threestep 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 selfstabilized 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 buildup 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 righthand side is for the 𝜉H ≪ 1 condition. Note that the gratings from electrons and from ions are phaseshifted 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 electronbased grating can be completely compensated during selfstabilized 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.
259
260
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 Fedoped LiNbO3 crystal with [Fe2+ ]/[Fe3+ ] = 0.013, total iron concentration [Fe]≈ 2 × 1019 cm−3 and total hydrogenion concentration [H+ ]=3.2 × 1017 cm−3 . The sample was shortcircuited (as is usual) with silver conductive glue and was placed in a copper holder surrounding the sample and in good thermal contact with a temperaturecontrolled 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 heatisolating 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 selfstabilized 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 nonselfstabilized 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 selfstabilized 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 mmthick 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 temperaturecontrolled 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 heatisolating 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 selfstabilized 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 whitelight 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 constantthickness 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 threestep (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 selfstabilized 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 selfstabilization 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 spacecharge 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∕2shifted, 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 selfstabilization 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 semiconductorbased 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 ITOsandwiched 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 orderofmagnitude, 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 RealTime Holograms While recording a photorefractive hologram, one should keep in mind that we are dealing with an almost realtime 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 sillenitetype 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 nakedeye observation or by an instrumentalassisted technique. We shall, in the following, brieﬂy describe both cases.
A.1 NakedEye 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, instrumentalassisted 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 100fold 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. Sillenitetype 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 RealTime 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 parallelpolarized and of similar irradiances. In this case, and always for our 𝜂 ≈ 0.01 example, the diﬀracted beam (the second term in the righthand side) is 100fold 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 ﬁvefold 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 parallelpolarized 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 indexofrefraction 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 socalled “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 probebeam 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 InBragg 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 “inBragg” 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 indexofrefraction 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 electrooptic coeﬃcient and indexofrefraction, respectively, and Esc being the spacecharge 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 HeNe 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 highquality components, is timeconsuming 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 Fedoped 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 inBragg 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 Braggmatched 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 HeNe 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 outofBragg 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 inBragg 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 nonperturbative measurement methods based on the inBragg 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 IndexofRefraction Material
Section 4.3, is one of the bestsuited 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 realtime 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 IndexofRefraction Material Diﬀraction eﬃciency measurement, even using the inBragg 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 parallelplane 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 buildup 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 IndexofRefraction 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 secondorder 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 dcresponse 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 electrodecontact problems. Some nonuniformity in the electric ﬁeld may also arise from nonuniform photoconductivity in the sample. In fact, under the action of a gaussianshape spatial cross section light irradiance I(x), as illustrated in Fig. C.1, a similar Gaussian crosssection 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 voltagetointerelectrode 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 wellknown parameter or from the ﬁtting of some quantity from which the theoretical equation is reasonably well known. One such possibility is to measure the electrooptic 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 electrooptic 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 crosssection irradiance I(x) illuminating a photoconductive material in steadystate 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 socalled Maxwellian distribution that, for onesingle 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 spacecharge 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 crosssection 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 crosssection 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 crosssection (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 ptype (p/n or n/p junction diodes) where electrons diﬀuse from the ntype to the ptype 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 spacecharge 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 (bandtoband) excitation in the semiconductor, an holeelectron 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 spacecharge ﬁeld and a current i0 appears that is proportional to the irradiance I i0 = KI
(E.1)
Sometimes, an intermediate intrinsic layer is used (pin diodes) that is intended to enlargen the depletion layer, as seen in Fig. E.2, to allow most of the photogenerated electronhole 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 holeelectron 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 npjunction showing the depletion layer and a diagram of the Schottky potential barrier.
++++ ++ ++++ +++ ++ + +++ ++++ ++ ++++ +++
– – – – – – – – – –
N
E d
P
– –– –– – – –– – –– – – –– –– – – ––
– – – – – – – – – –
V
++++ ++ ++++ +++ ++ + +++ ++++ ++ ++++ +++
i
Figure E.2 npjunction showing the depletion layer including the intrinsic layer and a diagram of the Schottky potential barrier. N
E d
Figure E.3 pnjunction 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
– ptype
intrinsic
+
–
+
–
+
–
+
–
+
–
+
ntype
+
– Voltage
i0
idiff V
E.1 Photovoltaic Regime The socalled photovoltaic operation regime is illustrated in Fig. E.4. For the socalled 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 socalled
E.2 Photoconductive Regime
Figure E.4 Photovoltaic mode operation for photodiodes. A shows its operation with a load RL , B shows the opencircuit operation and C shows the short circuit operation.
i
i
V
i0
i
V
i0
i0
RL
A
B
C
opencircuit 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 socalled “shortcircuit” 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 righthand side is the noise. There is not such a noise in the photovoltaic shortcircuit 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 shortcircuit regime allows transformation of the current output into a voltage with controllable gain, always keeping the shortcircuit operating regime, as illustrated in Fig. E.6. In fact, the virtual ground at the OA input grants the shortcircuit 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 photodiodeampliﬁer device.
i i0
i
V +
Figure E.6 Operational ampliﬁer operated photodiode in the shortcircuit photovoltaic regime. Rf
AMPOP
–
–
V0
291
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303
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Index a Absorption bulk, nonstationary holograms 106–110 lightinduced 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 electrooptic coeﬃcients of 11 holograms erasure 196 holeelectron competition 201, 202 phase shift, initial 229 phase shift, stationary 231 sensitivity 194
lightinduced absorption 159 luminescence in 28 mobilitylifetime 176–178 parameters, table 157, 177 photoactive centers 25 photochromism 48 energy, activation 49 photoconductivity 166 wavelengthresolved 166–173 photoconductivity, table 177 quantum eﬃciency 28, 176–178 refractive index 14 running holograms 72, 214–218 absorbing materials, in 232–234 fringelocked 232–239 holeelectron competition, with 76–83 selfstabilized 130 Bi2 TeO5 17, 42 Bragg selectivity 92, 271–274 BSO Bi12 SiO20 sensitivity, holographic 194 BTeO Bi2TeO5 17 BTO:Ce lightinduced absorption 159 mobilitylifetime 176–178 photoconductivity 176–177 quantum eﬃciency 176–178 BTO:Pb Bi12 TiO20 :Pb erasure, hologram 199, 200, 202 lightinduced absorption, table 159 mobilitylifetime 176–178 photoconductivity 176–177 quantum eﬃciency 176–178 BTO:V Bi12 TiO20 :V holeelectron 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 electrooptic coeﬃcients, of 15 photoactive centers 20 refractive index, chrom. displacive 15 CdTe:V dark conductivity 21 electrooptic 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 holeelectron competition, with 199 holographic erasure 195–197 holographic sensitivity 193–195 photoemf, 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 selfstabilized recording of 130–133, 232 sensitivity, holographic 193–195 materials, table for 194 speckle photoemf techniques 178–188 Concentration, eﬀective trap 67 Conductivity deﬁnition 36 dark 36 photo, onecenter model 35–37 photo, twocenter 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 phaseshifted grating with selfdiﬀraction, for 104 Kogelnik formula 69 unshifted pure absorption grating 91 unshifted pure indexofrefraction grating 91 unshifted 69 measurement 192 nonstationary with bulk absorption 106–110 optical activity, with 123 wavemixing, 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
spacecharge electric ﬁeld 67 trap concentration 67 Eﬃciency Diﬀraction: see Diﬀraction eﬃciency Electric ﬁeld diﬀusion, (ED) 65 eﬀective, (Eeﬀ ) 67 photovoltaic 86 Electrooptic eﬀect 8–10 coeﬃcient measurement 155 KDP, in 16–17 lithium niobate, in 16 sillenitetype 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 Fourwave mixing 119–120 Fringes visibility eﬀective (meﬀ ) 66
g GaAs 15 electrooptic coeﬃcients, of 15 selective twowave 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 Holeelectron 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 holeelectron competition, with 76–83 quality factor 75 resonance frequency 73 resonance speed 74 stabilized 130–133 Holographic erasure 83 holeelectron 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 photoEMF techniques 218–227 recording 57 holeelectron 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 electrooptic 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 electrooptic 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 Lightinduced 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 electrooptic coeﬃcients 15 erasure, holographic 197–199
Pattern of fringes phase position: (𝜑) 70 Phase modulation in twowave mixing 115 dynamic gratings, in 116 signal beam, in 116–118 techniques, phase 205–218 shift, holographic (𝜑P ) 70 measurement 206–207 Photoelectromotiveforce 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 crosssection 36 lithium niobate 28 photoconductivity, and 19 sillenitetype crystals 11–17 Photochromic activation energy for sillenites 22 eﬀect 48–51 lightinduced 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 mobilitylifetime product 176–178 modulated 175–178 models, theoretical (WRP) 32–40 onecenter 35–37 twocenter 37–40 wavelengthresolved166–173 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 wavelengthresolved (WRPC) 173–175 Photoelectron crosssection 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 spacecharge ﬁeld, eﬀective 86–87
transport coeﬃcient Polarization output 123–124
41
q Quantum eﬃciency photoelectron generation, for (Φ)
36
r Recording hologram glassplatestabilized 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 spacecharge electric ﬁeld 66 SBN sensitivity, holographic 194 Schottky barrier, lightinduced 51 electric ﬁeld 56 ﬁrst spatial harmonic approximation 66–72 general formulation 63–66 saturation 68–72 steadystate, solution for 64–66 time (T sc ) 68 spacecharge electric ﬁeld, eﬀective 63–66 Schottky eﬀect, lightinduced 51–54 Selective twowave mixing 210–214 GaAs 212–214 Selfstabilized recording actual materials, in 135–150 arbitrary phase shift 133–135 equilibrium condition, stable 130
309
310
Index
Selfstabilized recording (contd.) formulation, mathematical 127–135 running hologram 232–239 adaptive speed 232–234 sillenites, in 136 Sillenitetype crystal 22–27 doped 25–27 electrooptic coeﬃcients of 11 ptype conductivity in 24 parameters table 157 photoactive centers in 22 Spacecharge ﬁeld eﬀective 68 ﬁrst spatial harmonic approximation 66 nonstationary 106–110 photovoltaic donor density, inﬂuence on 86–87 twocenter 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 lightinduced 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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