HOLOGRAM STORAGE BY THE PHOTOREFRACTIVE. M. Gamal Moharam. B.Sc. (Hon), Alexandria University (Egypt), 1971 A THESIS SUBMITTED IN PARTIAL FULFILLMENT

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1 HOLOGRAM STORAGE BY THE PHOTOREFRACTIVE EFFECT by M. Gamal Moharam B.Sc. (Hon), Alexandria University (Egypt), 1971 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Electrical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1978 M. Gamal Moharam

2 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Electrical Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date June % (?, 1Q7R

3 ABSTRACT Exposure of some insulating crystals such as lithium niobate to light of appropriate wavelength induces small changes in the refractive index. This effect has been named the photorefractive effect. It allows phase holograms to be stored in these crystals. The work to be described was undertaken in order to obtain a better understanding of the hologram storage process which is believed to involve the spatial redistribution of photoexcited electrons among traps. This causes a space charge field to develop which modulates the refractive index via the linear electro-optic effect. A new reliable criterion for deciding whether the Raman-Nath or the Bragg regime of diffraction w i l l be observed with a given hologram was proposed. It is shown that the distinction between "thin" and "thick" holograms is invalid as a criterion for which regime operates. The new bulk photovoltaic effect proposed by Glass et al. is an important mechanism in the photorefractive effect in ferroelectric crystals. It is shown that as formulated by Glass et al. i t is formally equivalent to a fictional "virtual field" acting on the photo-liberated electrons provided that their migration length is short compared to the grating spacing. Hologram writing by the photorefractive effect was modelled in progressive stages of complexity. A l l the models were based on the assumption that the transport length of the free electrons is short compared to the grating spacing. This appeared to be a generally accepted assumption. The f i r s t treatment allowed for the feedback effect of the space charge field and for the dark conductivity. It was for uniform illumination and constant 1 applied voltage. The effects of the modulation ratio and the applied field were investigated. This treatment was then modified to allow for the effect of the absorption constant in reducing the intensity of the light as i t i i

4 propagates through the crystal. It was shown that the hologram becomes nonuniform through the crystal thickness as a result of this effect. Hologram writing with one-dimensional Gaussian beams was modelled allowing for the feedback effect of the space charge field. A large scale space charge field associated with the envelope of the light pattern was shown to affect the writing process. It was found that an increase in the fractional illumination of the crystal improves the writing process. The dark conductivity is shown to have an important effect on the process. The final model was again for uniform illumination and allowed not only for the feedback effect of the photoinduced field and the effect of the dark conductivity and absorption but also for the interaction between the hologram being written and the light pattern which is writing i t. This causes energy transfer between the two writing beams, thus modifying the light pattern. Optical erasure of holograms with the light incident either on and off the Bragg angle was modelled. The treatment allows for the feedback effect of the space charge fields and for the effect of the absorption in reducing the light intensity. The model allowed for the interaction between the diffracted and the reading beams for the case of incidence at the Bragg angle. The resulting interference pattern writes a new hologram which may add to or subtract from the hologram to be erased. An experimental method is described in which a normally incident ancillary light beam of different wavelength than that used to write the hologram allows the diffraction efficiency to be determined without errors due to multiple internal reflections. A limited experimental investigation was made of hologram storage in LiNbO^. Photocurrent and optical measurement were carried out on the same crystal. Almost 100% relative diffraction efficiency was observed. i i i

5 The value of the virtual field obtained from holographic measurement was found to agree within 10% with the value obtained from photocurrent measurements. During hologram writing, energy transfer of up to 70% between the two writing beams was observed. However, since the "virtual" field in these experiments was much larger than the diffusion equivalent field, the model predicted only about 5% energy transfer. It i s, therefore, suggested that the transport length of the photoexcited electrons in the crystal used, was not short compared to the grating spacing. It is also shown that light induced scattering can cause serious error in measuring the diffraction efficiency expecially during optical erasure. iv

6 TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS v LIST OF ILLUSTRATIONS v i i i ACKNOWLEDGEMENTS x i 1. INTRODUCTION 1 2. THEORY OF LIGHT DIFFRACTION BY PERIODIC PHASE GRATINGS Introduction Theoretical Analysis Discussion Summary MECHANISMS OF THE PHOTOREFRACTIVE EFFECT Introduction The Electro-optic Nature of the Photorefractive Effect Chen's Internal Field Model Johnston's Polarization Model Defect Sites and Impurities The Transport Length Introduction Amodei's Model for Short Transport Length Young et al.'s Model with Arbitrary 29 Transport Length Discussion The Bulk Photovoltaic Effect Discussion The Bulk Photovoltaic Effect Diffusion THEORY OF HOLOGRAM WRITING BY THE PHOTOREFRACTIVE EFFECT Introduction The Feedback Effict of the Space Charge Field 42 v

7 Page Introduction Model ' Calculated Results and Discussion The Time Development of the Space Charge Field The Effect of the Modulation Ratio The Diffraction Efficiency Summary The Effect of Light Loss due to Absorption Model Discussion Summary The Effects of Beam Coupling During Hologram Writing Introduction Model Algorithm Calculated Results and Discussion Introduction The Nonuniformity of the Grating The Diffraction Efficiency Summary HOLOGRAM WRITING WITH GAUSSIAN BEAMS Introduction Model Calculated Results and Discussion Introduction The Effect of the Ratio of Crystal Length to Beam Width The Effect of the Dark Conductivity Summary READING AND OPTICAL ERASURE OF HOLOGRAMS STORED BY THE PHOTOREFRACTIVE EFFECT Introduction 1^ 6.2 Model Off Bragg Angle Incidence Bragg Angle Incidence Calculated Results and Discussion Summary 118 vi

8 7. EXPERIMENTAL CONSIDERATIONS. FOR HOLOGRAM FORMATION Introduction The Optical System 122 Page 7.3 Influence of Multiple Internal Reflections PHOTOCURRENT AND HOLOGRAPHIC MEASUREMENTS Introduction Photocurrent Measurements Experimental Procedure Results and Discussion Holographic Measurements Experimental Procedure Results and Discussion CONCLUSIONS Suggestions for Further Research 151 REFERENCES 152 APPENDIX A PROPERTIES OF LITHIUM NIOBATE 157 A.l Crystal Growth 157 A.2 Miscellaneous Physical Properties 157 A.3 Thermal Bleaching and Fixing of Holograms in LiNb APPENDIX B THE ELECTRO-OPTIC EFFECT IN LITHIUM NIOBATE 159 APPENDIX C READ-WRITE HOLOGRAPHIC OPTICAL MEMORY SYSTEM vii

9 LIST OF ILLUSTRATIONS Figure Page 2.1 System configuration for light diffraction by periodic phase gratings The intensity of the f i r s t four diffracted modes vs. the grating strength in the Raman-Nath regime The intensity of the zero and f i r s t order modes vs. the grating strength in the Bragg regime Optically induced birefringence change caused by a circular beam Configuration for hologram recording Relation of the crystal axes and the two writing beams for different configurations of forming holograms Calculated time development of the fundamental and f i r s t four harmonic components of the photoinduced space charge field Calculated time development of the fundamental component of the field for different values of the effective modulation ratio The dependence of the steady state space charge field on the effective modulation ratio The dependence of the steady state fundamental component of the field on the dark conductivity Calculated time development of the diffraction efficiency during hologram writing for different values of the applied field Calculated time development of the diffraction efficiency for different values of the absorption constant The effect of the absorption constant on the time development of the diffraction efficiency Configuration for hologram writing with constant voltage applied to the c-faces of the crystal The time development of the amplitude of the fundamental component of the change in the refractive index at different depths in the crystal 78 v i i i

10 Figure Page 4.10 The time development of the spatial phase shift of the change in the refractive index at different depths in the crystal The time development of the diffraction efficiency during hologram writing for different values of total applied field The dependence of the time development of the diffraction efficiency during writing on the absorption constant The time development of the diffraction.'. efficiency during writing on the dark conductivity Spatial distribution of the Fourier component of the photoinduced space charge field The dependence of the local dc component of the space charge field on the dark conductivity for different values of the crystal fractional illumination The Fundamental Fourier component of the field due to drift vs. the ratio of light to dark carrier concentration for different values of the crystal fraction illumination The dependence of the f i r s t harmonic component of the field due to drift on the dark conductivity for different values of the crystal fraction illumination Experimental observation of the dependence of the photoinduced space charge field on the crystal fractional illumination and the light intensity (Cornish et al.) The dependence of the fundamental and f i r s t harmonic component of the photoinduced field on the dark conductivity A schematic representation of a qualitative model for the optical erasure process I l l 6.2 The time development of the absolute diffraction efficiency during hologram write-erase cycle The time development of the absolute diffraction efficiency and the fundamental component of the photoinduced field at different depths in the crystal 116 ix

11 Figure Page 6.4 Optical erasure characteristics off and on the Bragg angle Interference pattern of two plane waves Diffraction of the reference beam by the hologram Experimental arrangement for measuring the diffraction efficiency Alternative arrangement for measuring the diffraction efficiency The intensity of the writing beams with and without the plexiglass table cover An intensity scan across the diameter of the spatially filtered collimated beam The dependence of the multiple internal reflection correction factor on the optical thickness and on the intrinsic diffraction efficiency Experimental arrangement used to determine the intrinsic diffraction efficiency using an ancillary third beam Experimental results on the effective and intrinsic diffraction efficiency The dependence of photocurrents in Fe-doped lithium niobate on the light intensity and the applied voltage Measured time development of the relative diffraction efficiency of holograms stored in lithium niobate crystal Observed relative intensities of the transmitted and diffracted beams during optical erasure on the Bragg angle The transmitted and diffracted beams' intensities during destructive reading 147 G.l A schematic of read-write-erase holographic optical memory 164 x

12 ACKNOWLEDGEMENTS I am most grateful to my supervisor, Dr. L. Young for his encouragement and guidance during the course of this research. I would like to thank Dr. W. D. Cornish for helpful suggestions and technical assistance. I wish to express my appreciation to Mrs. A. Semmens for typing the thesis, to Mr. A. Mackenzie for drawing many of the graphs and to Mr. J. Stuber for his assistance in the machine shop. The National Research Council of Canada (Grant No. A3392 and scholarship awarded ) and The University of British Columbia (graduate fellowship awarded ) are gratefully acknowledged for their financial support. xi

13 To Gehan x i i

14 1. CHAPTER I INTRODUCTION In Fe-doped lithium niobate and similar crystals, exposure to light of the appropriate wavelength induces small changes in the refractive index. This phenomenon is sometimes called the photorefractive effect. It is believed that the mechanism of the photorefractive effect in these crystals is broadly as follows. The incident light causes photoexcitation of electrons from traps. The electrons drift and diffuse and are subsequently retrapped. In ferroelectric crystals a new type of photoeffect is also involved, as will be discussed later. The spatial redistribution of elec trons among traps sets up a space charge field which modulates the refrac tive index of the crystal via the electro-optic -effect. The optically induced changes in refractive index may be removed either by uniform illumi nation or by heating. These treatments cause the electrons to be optically or thermally excited from traps and uniformly redistributed, so that the modulation in the refractive index is removed. The photorefractive effect was f i r s t encountered in electrooptic modulators and frequency doublers where the optically induced inhomogeneities caused scattering and decollimation of the light, thus degrading the performance of the devices (Ashkin et al. 1966). It was later recognized that this photorefractive effect could be useful as a new means of generating phase holograms. Volume phase holograms have been stored by the photorefractive effect in the ferroelectric crystals lithium niobate (Chen et al. 1968), strontium barium niobate (SBN) (Thaxter 1969), barium titanate (Townsend et al. 1970), barium sodium niobate (Amodei et al. 1971c) and

15 2. lead zirconate titanate (PLZT) (Micheron et al. 1974). Photorefractive materials are potential candidates for the storage media in holographic memory systems. They do not require development or bleaching processes and, therefore, can be used in real time, in contrast to photographic emulsions and thermo-plastics. They can be used for readwrite applications because the hologram can be optically or thermally erased and a new hologram written. Diffraction efficiencies approaching 100% are theoretically possible and have been observed experimentally in lithium niobate (Chapter 8). (For definitions of diffraction efficiency see Chapter 7). At present, photorefractive media have lower sensitivity than might be desired but improvement is possible and steps have been taken in that direction, especially with lithium niobate (Phillips et al. 1974). Volume holographic storage media have, in theory, the attractive characteristic of very high data packing density. It was shown by van Heerden (1963) that the theoretical ultimate storage capacity of a volume 3 hologram is V/X bits where. V is the volume and/a. is the wave length of light. This means that theoretically more than 10 bits can be stored in 3 a 1 cm crystal. However, the practical limit set by other optical parameters of the storage system is lower than this (van der Lugt 1973). The feasibility of read-in and read-out requiring no mechanically moving parts and the high packing density of optical systems promise to offer large capacity with relatively fast access compared to other large capacity systems. A number of review articles are available which discuss the advantages and limitations of optical memories (Rajchman 1970, King 1972, Anderson 1972, H i l l 1972, Kiemle 1974, Chen and Zook 1975). The topic is also discussed in Appendix C. The work described here was undertaken in order to obtain a better 12

16 3. understanding of hologram storage by the photorefractive effect in connection with possible engineering applications. Although extensive experimental investigation of the hologram storage process has been carried out, a l l theoretical analysis so far has been limited in applicability by reasons of certain simplifications. A more rigorous model of hologram writing and erasure processes was needed to investigate the mechanisms of the photorefractive effect by comparison with experimental results. Such a model should allow one to obtain the physical parameters of the crystal and to determine the usefulness of a crystal in a specific application. The a b i l i ty to predict the diffraction efficiency under a given optical exposure is also important for the design of the system and the prediction of the system performance. The work was carried out with special interest in Fe-doped lithium niobate because i t seemed the most promising material for applications and because high quality crystals are readily available. However, the theoretical analysis should apply to other ferrolectric photorefractive materials which exhibit the new bulk photovoltaic effect observed in lithium niobate (Glass et.al. 1974). The analysis should also apply to paraelectric materials, such as potassium tantalate niobate (KTN) which exhibit a comparable photorefractive effect i f an electric field is applied to the crystal (Chen 1967). For holograms stored by the photorefractive effect to play a major role in holographic memory systems, they should exhibit, among other things, high diffraction efficiency and angular and wavelength selectivities. That i s, they should operate in the Bragg regime of diffraction. In Chapter 2, the theory of light diffraction by phase gratings (holograms) is outlined and a new reliable criterion to determine whether the phase hologram is operating in the Bragg or the Raman-Nath regimes of diffraction is proposed.

17 4. It is shown that thickness of the grating does not enter into the criterion and, therefore, the distinction between "thick" and "thin" gratings or holograms is invalid as a description of whether the grating is in the Bragg or the Raman-Nath regime of diffraction. It is generally recognized that drift and diffusion are involved in the transport mechanism of the photorefractive effect (Amodei 1971a, 1971b). Glass, von\der Linde and Negran (1974, 1975) have proposed that the photorefractive effect involves also an entirely new transport mechanism which they have labelled the "bulk photovoltaic effect". This mechanism is thought to be responsible for the photocurrents which were previously attributed to internal fields of pyroelectric origin. These mechanisms are outlined in Chapter 3 and i t is shown that this new photovoltaic effect, as represented by a current density proportional to the light intensity, is formally equivalent to a fictional "virtual field" acting on the photoreleased electrons provided that their migration length is short compared to the grating spacing. In Chapter 8, some experimental results which bear upon the photovoltaic effect are discussed. The feedback effect of the photoinduced space charge field on the redistribution of electrons during hologram writing is considered in Chapter 4. A new model for hologram writing applicable over the entire range of exposure allowing for this feedback effect as well as the dark conductivity is developed for uniformly illuminated crystal under constant applied voltage. Short transport length of the free electrons is assumed. It is shown that both the drift and diffusion produced components of the photoinduced space charge field evolve with the same time constant and thus the spatial phase shift of the index modulation from the incident intensity pattern is constant in time and its value is determined by the relative contribution of

18 5. drift and diffusion. The dependence of the space charge field on the modulation ratio and the dark conductivity are also discussed. As the light propagates into the crystal i t decays exponentially with distance travelled due to absorption. Therefore the rate of build up of the hologram is different at different depths in the crystal. The model proposed in Chapter 4 is extended to allow for this effect. It is shown that overlooking this complication leads to an over prediction of the calculated diffraction efficiency. The hologram becomes nonuniform through the thickness of the crystal. The effect is more significant when the ratio of the dark to the photoinduced conductivity is large (>.01) and, of course, for large absorption constants. Staebler and Amodei (1972b) were the f i r s t to point out the implications arising from beam coupling during hologram writing. The hologram being written interacts with the light pattern which is writing i t. This interaction causes energy transfer between the two writing beams and thus modifies the light interference pattern. In Chapter 4 a dynamic model for hologram writing is proposed. The model allows simultaneously, for the f i r s t time, for both the feedback effect of the space charge field and the effect of the hologram in modifying the light pattern which is writing i t, as well as the effect of light absorption. It is shown that neither the amplitude nor the spatial phase shift of the index modulation is uniform through the grating thickness. The effects of the applied and virtual fields, the ab- sorption constant and the dark conductivity on the hologram writing process are also investigated. Hologram writing with nonuniform illumination would produce a large scale space charge field associated with the envelope of the light pattern in addition to the sinusoidal and higher harmonic components. Cornish et al.

19 6. (1976a) have shown that this large scale field affects greatly the hologram writing process. In Chapter 5, a model for hologram writing with a onedimensional Gaussian beam incident on a finite crystal under constant applied voltage is developed. It is shown that an increase in the ratio of the crystal length to the Gaussian beam width (i.e. increase in the fractional illumination of the crystal) improves the hologram writing process. The ratio of the dark conductivity to the photoconductivity is shown to have an important effect on the process. Holograms stored by the photorefractive effect may be optically erased by exposure to uniform illumination. However, as Staebler et al. (1972b) have shown, i f the erasing beam is incident at the Bragg angle of the hologram, the light beam interacts with the hologram and the inter ference pattern of the erasing (destructive reading) beam and the diffrac ted beam writes a new hologram which may add to or subtract from the hologram to be erased. This, of course, is in addition to the optical erasure caused by the uniform part of the pattern. In Chapter 6 a model for optical erasure both on and off the Bragg angle is developed. The special feature of this model is that i t allows for the feedback effect of the space charge field and the effect of the absorption on reducing the light intensity, the erasure due to the uniform part of the illumination, and the new hologram written by the interference pattern of the reading and diffracted beams (for the case of incidence at the Bragg angle). It is shown that the model reproduces a l l the reported types of erasure characteristics. Measurements of the build-up of the diffraction efficiency with time have been used by several authors to test various physical models for the photorefractive effect. Usually a prototype hologram is written by the interference of two plane waves. It is then necessary to relate the ob-

20 7. served diffraction efficiency to the predicted refractive index modulation. Cornish et al. (1975) have shown that neglecting the effects of multiple reflection can cause serious errors. The problem is exacerbated by the fact that these errors are not typically constant during an experiment since significant changes in the optical thickness of the crystal can occur due to heating by the light beam. In Chapter 7, an experimental method is described in which a normally incident ancillary light beam of different wavelength than that used to write the hologram allows the diffraction efficiency to be determined without errors due to multiple internal reflections. Time permitted only a restricted experimental investigation of the hologram storage process in lithium niobate. Results are reported in Chapter 8. Photocurrent measurements were made to investigate the bulk photovoltaic effect. In the hologram writing experiments, very high diffraction efficiencies were measured (almost 100%). According to the hologram writing model in Chapter 4, the drift field equivalent to the bulk photovoltaic effect should be about 50 kv/cm to achieve such high efficiencies. This value is in very good agreement with the value obtained from the photocurrent measurements. The equivalent diffusion field was 1 kv/cm. That i s, for the crystal used, diffusion was negligible. However, energy transfer between the two writing beams of up to 70% was observed whereas the model predicted about 5% energy transfer. This discrepancy might be accounted for i f the transport length of the free electrons in this experiment was not short compared to the grating spacing. For this case drift alone might produce large enough spatial phase shift between the light pattern and the index modulation (Young et a l. 1974) to cause the observed energy transfer.

21 8. CHAPTER II THEORY OF LIGHT DIFFRACTION BY PERIODIC PHASE GRATINGS 2.1 Introduction In recent years, extensive research has been done on the topic of light diffraction by periodic phase gratings, both holographically and acoustically produced. Examples are Klein and Cook (1967), Burckhardt (1966) Kogelnik (1969), Chu and Tamir (1970) and Magnusson and Gaylord (1977). There is general agreement that i t is convenient to define two regimes in which phase gratings operate. In the Raman-Nath regime, several diffracted waves are produced. In the Bragg regime, essentially only one diffracted wave is produced and that only for near Bragg incidence. It has been customary to refer to gratings which operate in the Raman-Nath and the Bragg regimes as "thin" and "thick" gratings respectively. Phase gratings operating in the Bragg regimes have numerous potential applications based on their properties of high diffraction efficiency, wavelength and angular selectivity (Kogelnik 1969 and Forshaw 1974). Examples are narrow band spectral f i l t e r s (Crawford 1954), deflectors and modulators (Hammer 1971). Volume holograms are of interest due to their potential use in high capacity information storage (Heerden 1963), color holography (Pennington et al. 1965), and in white light holograms (Stroke et al. 1966). It would seem to be useful and convenient to have a reliable criterion to determine whether a grating is operating in the Raman-Nath or the Bragg regime of diffraction. Klein and Cook (1967) proposed a parameter 2 Q defined as 2irA L/A n, where o o \ o is the light wavelength, L is the grating thickness, A is the grating spacing and n is the medium refractive index,

22 9. as the criterion for deciding which diffraction regime w i l l apply. Q is a normalized measure of the grating thickness. Small values of Q < 1, i.e. thin gratings were believed to give Raman-Nath operation. Values of Q > 10. i.e. thick gratings, were believed to give Bragg regime operation. Although the parameter Q has been extensively used to distinguish between the two regimes, i t is not generally realized that i t is not always a reliable c r i terion but requires, as Klein and Cook noted at the end of their paper, a limitation on the grating strength im^l/a cos6 (where 0 is the angle diffraction of the incident wave, and n^ is the amplitude of the index modulator (assumed purely sinusoidal)). Klein and Cook have indicated that, in order for Q to predict the diffraction regime, the grating strength must be less than three. Recently, Magnusson and Gaylord (1977) have shown theoretically that for large modulation, higher order waves become important (Raman-Nath regime) even for large Q. Conversely, for small modulation, only a single wave is diffracted (Bragg regime) -in spite of small values of Q. Kaspar (1973) came to a similar conclusion when he compared his theory with the coupled, wave theory of Kogelnik (1969). Experimentally, higher order diffraction (Raman-Nath regime) has been observed from holographic gratings formed in lithium niobate and dichromated gelatin where the value of the parameter Q predicted Bragg regime of diffraction i.e. only one diffracted wave. (Wood ettal. 1975, Magnusson et al. 1977, Moharam and Young 1978a and Alferness 1976). In this chapter, a condensed version of the theory of light diffraction by periodic phase gratings is presented. A parameter p defined 2 2 as A q/a n Q n^, is proposed as a reliable replacement for Q as the criterion to distinguish between the two diffraction regimes. The effects of the higher order modulation of the index (smaller grating spacings) on the

23 10. diffraction problem are also considered. 2.2 Theory of Light Diffraction by Phase Gratings The system configuration of the diffraction problem under consideration is shown in Fig A monochromatic plane wave $ with propagation vector ~o is incident, at an angle 0 to the surface normal, on a periodic o grating described by refractive index j. v / 17 ~ \ n = n + E, n cos(pk-r) (2.1) o p=l p where n^ is the mean refractive index, n^ is the amplitude of the p t n Fourier component of the index modulation, K = (2TT/A) [x cos 'ip + z sin ip] is the grating vector, i\> is the slant angle of the grating and A is the grating wavelength. For aholographically produced grating A = A /2 sin 6 w w here A is the recording light wavelength and 0 is one-half the angle w w between the two recording beams. For acoustically produced gratings, A is the ultrasonic wavelength in the grating medium. Reflections at the grating surfaces are neglected. The scalar wave equation is [V 2 + (6 2 - jab)] E (x,z) = 0 (2.2) where E^ is the complex amplitude of the electric field, a is the intensity absorption constant, 3 = is the propagation constant and A q is the light wavelength in air. Eqs. 2.1 and 2.2 may be solved by resolving the electric field into its Fourier expansion which may be written as where a = d ~ qk. q o ^ 00 E (x,z) = E^. <j> exp (-ja -r) (2.3) * q= Combining 2.1, 2.2 and 2.3 and assuming that 3» a and XIQ» n p (p>0) i;e. a weakly modulated medium, we arrive at an infinite set of coupled wave

24 L - > Fig. 2.1 System configuration

25 equations. 3<j> 2 CqT7~ + " J (7TX O^A n o ) q^ q " 2 A s i n ^V^q 00 -J p l ( V >I *q-p + W (2 ' 4) where C q = cos 0 - q cos ihx /An ), and 6 is the angle of refraction of the H & o o zero order mode (q=0) in the medium. Second order derivatives of <f>^ with respect to z are neglected as in previous work. Chu and Tamir (1970) and Kong (1977) showed that these simplified f i r s t order coupled wave equations (Eq. 2.4) may be accurately applied to weakly modulated medium (n <0.01). P. To gain insight into the phenomenon, Eq. 2.4 w i l l be simplified, without significant loss of generality by considering only unslanted gratings (^=90 ) and assuming that the index modulation is purely sinusoidal i.e. n = 0 for p>l. Eq. 2.4 may then be rewritten as: 9<j> 3.= T JPq V * q + J l V l + V l ] ( 2 ' 5 ) 2 2 where p = X /Ann.., E,' c ',= v(z/l) and v = TTn.,L/X cos 0. L is the grating o o l 1 o thickness, B = 2A sin 0/X q. q o B q is equal to one for light B incidence satis- fying the q t n Bragg angle. The absorption constant a has been neglected, i t can be allowed for by the substitution = <j> exp (-az/cos 0). 2.3 Discussion Examination of Eq. 2.5 shows the q mode is coupled to itself and to the two adjacent modes ( q - l t n and q+l t n modes). Effective energy 2 transfer between modes require> essentially that the factor pq (1 - B^) be relatively small since, i f this factor is much larger than 1, a l l the energy w i l l be coupled back to the q mode. This can be easily seen from Eq. 2.5 by neglecting the second term of the right hand side. Therefore, i f p<l, appreciable energy may be transferred successfully to higher order

26 modes up to some value of q provided the magnitude of B^ Is appropriately limited. The number of observable higher order modes depends on the factor 2 pq (i.e. the smaller p i s, the larger the number of diffracted waves). If p=0 Eq. 2.5 gives the well-known solution in terms of Bessel functions < >q = j^j^(2v). This solution is often obtained by Fourier expansion of the transmitted wave with spatially sinusoidal phase modulation. A similar solution was obtained by Klein and Cook (1967) for their parameter 2 Q = n Q = 0. However, as the thickness L of the grating goes to zero (Q;-> 0) the modulation of the refractive index must go to infinity to retain the finite phase shift (n^l) and as ny+ <*> } p 0. Clearly, the case where Q or p = 0 is a non-physical situation. Thus for p-cl the diffraction pro- 2 cess is in the Raman-Nath regime. If p»l the factor pq (1 - B^) is much larger than one for a l l modes except the zero order mode (q=0) and the mode with q such that B -'l for given 0 i.e. the Bragg condition holds or nearly holds. Therefore, for p>>l appreciable energy may be interchanged between the zero order mode and the mode for which B^~ 1. The coupling constant between these two modes is proportional to n^. Thus, for purely sinusoidal gratings, energy may be transferred only to the f i r s t order mode i f the angle of incidence satisfies the f i r s t Bragg angle (i.e. B^ = 1). If the modulation of the refractive index contains harmonics in addition to the fundamental component, energy may be transferred only to the q fc^ order provided that n^ ^ 0 and the angle of incidence satisfies the q Bragg angle (i.e. B - 1). Therefore, for p>>l only one diffracted wave is produced even i f the grating is nonsinusoidal and thus the diffraction process is in the Bragg regime. Also as p becomes larger, the grating becomes more selec- 2 tive i.e. B q must be closer to one so that the product pq (1 - B ) is q small enough that the energy may be transferred to the q t n mode.

27 The above qualitative analysis was confirmed by solving Eq. 2.5 using the University of British Columbia computer centre, fourth order Runge-Kutta routine with error control with the boundary conditions <f> (0) = 1.0 and <f> (0) = 0.0 for q 4 0. The relative intensity I of the q P q wave is defined as <J>^ (v) <j>^ (v) (the absolute intensity is 1^/2^ where Q, is the characteristic impedance). The number of modes included (i.e. the number of equations) was increased for each case until no further s i g n i f i cant effect occurred for a given value of p, e.g. for p = 50, six modes were included and for p = 0.1, 24 modes were needed. Fig. 2.2 shows the relative intensities of several diffracted modes f o r p = 1 as a f u n c t i o n o f t h e g r a t i n g s t r e n g t h v. Exact Bragg i n cidence for the f i r s t mode was assumed (i.e. B^ = 1). In fact, the intensities of the diffracted waves are plotted vs. the grating thickness L since n^ must be kept constant to keep p constant. By implication, the horizontal axis represents the normalized thickness parameter Q of Klein and Cook (1966) since Q = 2vp. Fig. 2.2 shows that even for values of v corresponding to a thick grating and large values of Q (values of Q>10"were believed to give Bragg regime operation), the energy is transferred successively into a large number of modes and the Raman-Nath regime applies. Fig. 2.3 shows a plot of the intensities of the zero and f i r s t order modes against the grating strength v (and by implication Q) for p ^ 50 and Bragg incidence for the f i r s t order wave ( B -j= 1) The energy is transferred to and fro between only the zero and f i r s t order modes. It has been shown by Phariseau (1956) that the intensities of the higher order modes are 2 of the order of 1/p or less of the zero order mode (provided that p>>l). A p l o t i d e n t i c a l t o F i g. 2.3 was obtained when higher order harmonics of the modulation of the refractive index were included in the

28 GRATING STRENGTH (V) Fig. 2.2 The intensity, of the first:four 'diffracted-modes vs.'.the I-gr-a ting strength v for p= 1-and B = 1/q (Bragg incidence for the f i r s t order mode). -The sum of the intensitiel of the four modes is almost unity. A l l the other 1 modes are negligible.

29 16. GRATING STRENGTH (V) Fig. 2.3 The intensity of the zero and the first order modes vs. the grating strength v for p= 50 and B = 1/q. All the other modes are negligible.

30 calculations, n^ was taken to be equal to n^ for a l l p>l, and as in Fig. 2.3, p was 50 and = 1. It is clearly evident from Fig. 2.3 that for P>>1 only one diffracted wave is produced (i.e. Bragg regime operation) regardless of the value Q (small values of Q (Q<1) were believed to indicate Raman-Nath regime). As i t has been shown above for p>>l and B^ = 1, energy may be interchanged only between the zero and f i r s t order modes (i.e. <j> =0 q>l) q Eq. 2.5 may be rewritten as 9<f> r j *, (2.6) d ' J r l 8 = J4> J T (2.7) o Solving the above two equations with the boundary condition <j> 0 (0) = 1.0 and < >, (0) = 0.0 and defining the diffraction efficiency DE as 1 q DE = <f)*(v) A (v)/<f,*(v):<f>*(v) (2.8) - q q q o o we arrive at the well known expression DE = cos 2 (v) 2 DE 1 = sin (v) (2.9) Phariseau (1956) and Kogelnik (1969) and Chu and Tamir (1970) have obtained similar expressions to Eq These expressions were obtained assuming purely sinusoidal modulation of the refractive index. However, based on the above discussion, these expressions are valid also for nonsinusoidal gratings provided that p>>l ahd B^ = 1. An interesting observation is that, if the change of the refrac tive index is produced holographically (i.e. i t develops with time), i n i t i a l l y p w i l l be very large, since n^ is very small, therefore, the

31 diffraction process w i l l be in the Bragg regime. As n^ increases, p decreases and, unless n^ saturates at a value such that p is s t i l l large enough (>10), the diffraction process w i l l eventually move into the Raman- Nath regime. A further point is that the grating thickness L (or the normalized thickness Q) is irrelevant in i t s e l f, since i t does not enter p. This may be explained as follows: To obtain a significant amount of diffraction, the grating strength v has to be large enough in some sense. Forvv to be large, then, i f L is large, n^ may be either large or small, so that p would be small or large respectively, and we could be in either regime. On the other hand, i f L is small, n^ would have to be large to obtain a sufficiently large v, in which case p would be small, and the Raman-Nath regime would hold. Therefore, the distinction between "thick" and "thin" gratings or holograms as determined by the value L (for given X, n Q and A) in 2 Q = 2-nX^L/A n Q is invalid as a description of whether a single diffracted beam w i l l be produced or whether many diffracted beams w i l l be produced. 2.4 Summary It has been shown for the f i r s t time that a parameter p is always a reliable criterion for deciding whether the Raman-Nath or the Bragg regime w i l l be observed with a given phase grating. The parameter Q of Klein and Cook (1967) which has been extensively used for this purpose was shown to be unreliable. A large p favours the Bragg regime. The relative light intensity going into higher order modes (other than the mode for which the 2 Bragg condition holds) is of the order 1/p so that a value of p>10 indicates more or less ideal Bragg behaviour. It has been shown also that the parameter p w i l l work whether the grating modulation is sinusoidal or nonsinusoidal. It has also been shown that the distinction between thick and

32 thin gratings i s, strictly speaking, invalid as a description of which diffraction regime is operating.

33 CHAPTER III MECHANISMS OF THE PHOTOREFRACTIVE EFFECT 3.1 Introduction As was mentioned in Chapter 1, a number of models have been proposed to explain the photorefractive effect. The development of these models is outlined and their merits are discussed. 3.2 The Electro-optic Nature of the Photorefractive Effect Chen, LaMacchia and Frazer (1968) found that the polarization of the writing beams was not c r i t i c a l during hologram writing in lithium niobate. However, reconstruction of the hologram was only about 1/10 as efficient for ordinary ray illumination as for extraordinary ray illumination. To explain this observation, they suggested that the photorefractive process responsible for hologram storage involves the electro-optic effect in the crystal. The diffraction efficiency at the i n i t i a l stages of hologram formation is proportional to the square of the refractive index modulation n^ (Kogelnik 1967). This implies that, for Chen et al.'s observa- O 6 tion, n^/n^ = 0.3, (neglecting such complications as reflection, absorption, anisotropy, etc.). For a space charge field along the x^ (c-axis) of the crystal, the ratio of the change in the ordinary refractive index to the change in the extraordinary refractive index (for X = mm) is (Appendix B). That i s, Chen et al.'s observation is consistent with a refractive index modulation by a space charge field along the c-axis of the crystal. See Appendix B for an outline of the electro-optic effect in lithium niobate.

34 3.3 Chen's Internal Field Model Chen (1969) measured the changes in birefringence induced with a single laser beam - in lithium niobate using an adjustable compensator method. Fig. 3.1 shows the optically induced changes in birefringence along the b- and c-axis of the crystal (Chen 1969). The birefringence along the c-axis reverses sign and that along the b-axis does not. Assuming, an electro- optic effect, Chen concluded that drift, not diffusion causes the effect along the c-axis. To explain this observation, Chen proposed a model in which there are two types of traps before light illumination. Traps of the f i r s t type are i n i t i a l l y f i l l e d and neutral, and they can provide electrons by photoexcitation. Traps of the second type are i n i t i a l l y empty and can capture electrons. Chen also postulated that there is an internal electric field directed from the positive end of the spontaneous polarization of the crystal to the negative end. This field would cause the photoexcited electrons to drift along the c-axis toward the positive c-axis end, leaving behind positive charges of ionized trap centres. Chen claimed that "the photo-excited electrons w i l l be retrapped and re-excited out of the traps until they eventually drift out of the illuminated region and are finally retrapped there. Since there is no photo-excitation outside the illuminated region and for deep traps, the thermal excitation is too weak to re-excite charges out of the traps, the negative charges stay trapped there. The space charge field thus created between the trapped electrons and the positive ionized centres causes the observed spatial variation of the indices of refraction via the electro-optic effect" Chen (1969). Since lithium niobate exhibits a linear electro-optic effect, the variation A(n e ~n Q ) is linearly related to the spatial variation of the electric field. Chen estimated that the space charge field required to produce the maximum observed value of

35 L:22. BEAM DIAMETER (a) J J b (b) Fig. 3.1 (a) The solid line ( ) shows the change in birefringence along the c-axis and the dashed line (- - -) shows the change along the b-axis due to a beam of circular symmetry, (b) Chen's postulated space charge field distribution which causes the observed change in A(n - n ). e o

36 A(n -n ) was 67 kv/cm. e o Although no origin was given for this built-in internal field, Chen's observations of short-circuit photocurrent in lithium niobate led him to claim that this internal field exists. The direction of the photocurrent was consistent with a field opposite to P. Chynoweth (1956) had also observed photocurrent in BaTiO^ in the absence of applied fields. Chen (1969) has shown that fields of pyroelectric origin due to the nonuniform heating of the crystal by the light beam could not account for the observed short-circuit photocurrent since dp/dt<0 and, therefore, the field would be in the wrong direction for his observation. Amodei and Staebler (1972b) suggested that this built-in field was of pyroelectric origin developed when the crystal was cooled from a high temperature. The development of such a field may be explained in the following way. A ferroelectric crystal with no free charges and no net space charges would have a field corresponding to a polarization charge P r e m per unit area on faces normal to the c-axis. This field would, in fact, be above normal dielectric breakdown. In prac--' tice the crystal would have been cooled from some high temperature at which appreciable conductivity existed, sufficient to cancel the field. Excess charges would accumulate close to each c-face. As cooling progressed, the conductivity would freeze out while the remanent polarization P continued rem to change. Finally, an uncompensated component of T^ would exist giving a built-in field of magnitude 1 j; 1 _^em e t ' 3T o where T q and T^ are the temperature at which the conductivity disappears and the temperature of the experiment respectively, and e is the permittivity.

37 However, Cornish, Moharam and Young (1976) have measured the short-circuit photocurrent in a Fe-doped lithium niobate crystal after cooling i t (a) with and (b) without short-circuit applied to the electrodes on the faces at the ends of the c-axis. They found that the photocurrent was independent of the electrical conditions during cooling. The photocurrents were measured after the pyroelectric current due to the light had decayed. Their experiments showed that these short-circuit photocurrents are produced in the absence of any built-in field of pyroelectric origin. Glass, von der Linde and Negran (1974) have reported that, after 20 hours of continuous illumination, the short-circuit photocurrents which they measured remained constant. Cornish (1976) has reported similar measurements for 43 hours of continuous illumination. Glass et al. have claimed that this photoconductivity would relax any internal fields and a decay of the photocurrent (if i t is due to such fields) would be noticeable. 3.4 Johnston's Polarization Model To remove the need to assume a built-in field of unknown origin, Johnston (1970) proposed an alternative model in which photoinduced variations in the macroscopic polarization caused the photorefractive effect. He claimed that illumination of the crystal would excite the trapped elec^ trons to the conduction band resulting in a change in the density of f i l l e d traps in the region of illumination. This in turn would cause a local change in the polarization. The divergence of the polarization produces a field. This field causes the electrons to drift before retrapping and thus produces permanent change in the polarization. After the light is turned off, there remains a change in the macroscopic polarization which induces a change in the refractive indices of the crystal. Using this model, Johnston was able to account qualitatively for

38 the spatially dependent features of Chen's observations (Fig. 3.1). However, Amodei (1971a) and Amodei and Staebler (1972b) have shown that there are a number of difficulties with this mechanism. They concluded that too large a density of electrons would be required to enter the conduction band to generate the large fields necessary to account for the observed effect. The same effect could result from space charge fields created through simple diffusion and retrapping processes. The number of electrons involved would 3 be less by a factor 10 than would be required in Johnston's model. One might add that this model cannot explain a steady state short-circuit photocurrent with the crystal uniformly illuminated. 3.5 Defect Sites and Impurities The photorefractive effect is most efficient when light of the wavelengths 400 to 500 nm is used,(serreze and Goldner 1973). Clark et al. (1973) suggested that excitation occurs from traps within the band gap. It is believed that both impurities and defects related to the non-stoichiometry of the crystal act as defect sites. Phillips et al. (1972) have shown that gamma irradiation of undoped LlNbO^ increases the photorefractive sens i t i v i t y by increasing the concentration of lattice defects which act as electron traps. Impurity doping with elements such as iron, manganese, copper, rhodium, chromium and uranium (Phillips et al. 1972, Peterson et al. 1971, 1973; Mikami et al. 1973, Glass et al. 1974a) also improves the photorefractive sensitivity of the crystal. Phillips, Amodei and Staebler (1972) have shown that iron i s, so far, the best dopant that has been found. Peterson et a l. (1971) and Clark et al. (1973) have suggested that iron replaces a lithium ion in the crystal lattice. Mossbauer-effect study of iron impurities in LiNbO^ led Keune 3+ et al. (1975) to suggest that the most likely site of the Fe ion is the