UBC Theses and Dissertations

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UBC Theses and Dissertations

Hologram storage by the photorefractive effect Moharam, M. Gamal 1978

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HOLOGRAM STORAGE BY THE PHOTOREFRACTIVE EFFECT by M. Gamal Moharam B.Sc. (Hon), A l e x a n d r i a U n i v e r s i t y ( E g y p t ) , 1971 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department o f E l e c t r i c a l E n g i n e e r i n g ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA Ju n e , 1978 © M. Gamal Moharam In p r e s e n t i n g t h i s t h e s i s in partial f u l f i l m e n t o f the requ i rement s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I ag ree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s tudy . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rpo se s may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Department o f E l e c t r i c a l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 Date June % (? , 1Q7R ABSTRACT Expo s u r e o f some i n s u l a t i n g c r y s t a l s s u c h as l i t h i u m n i o b a t e t o l i g h t o f a p p r o p r i a t e w a v e l e n g t h i n d u c e s s m a l l changes i n t h e r e f r a c t i v e i n -dex. T h i s e f f e c t has been named t h e p h o t o r e f r a c t i v e e f f e c t . I t a l l o w s phase holograms t o be s t o r e d i n t h e s e c r y s t a l s . The work t o be d e s c r i b e d was under-t a k e n i n o r d e r t o o b t a i n a b e t t e r u n d e r s t a n d i n g o f t h e h o l o g r a m s t o r a g e p r o -c e s s w h i c h i s b e l i e v e d t o i n v o l v e t h e s p a t i a l r e d i s t r i b u t i o n of p h o t o e x c i t e d e l e c t r o n s among t r a p s . T h i s c a u s e s a space c h a r g e f i e l d t o d e v e l o p w h i c h m o d u l a t e s t h e r e f r a c t i v e i n d e x v i a t h e l i n e a r e l e c t r o - o p t i c e f f e c t . A new r e l i a b l e c r i t e r i o n f o r d e c i d i n g whether t h e Raman-Nath o r t h e B r a g g r e g i m e o f d i f f r a c t i o n w i l l be o b s e r v e d w i t h a g i v e n h o l o g r a m was p r o p o s e d . I t i s shown t h a t t h e d i s t i n c t i o n between " t h i n " and " t h i c k " h o l o -grams i s i n v a l i d as a c r i t e r i o n f o r w h i c h r e g i m e o p e r a t e s . The new b u l k p h o t o v o l t a i c e f f e c t p r o p o s e d by G l a s s e t a l . i s an i m p o r t a n t mechanism i n t h e p h o t o r e f r a c t i v e e f f e c t i n f e r r o e l e c t r i c c r y s t a l s . I t i s shown t h a t as f o r m u l a t e d by G l a s s e t a l . i t i s f o r m a l l y e q u i v a l e n t t o a f i c t i o n a l " v i r t u a l f i e l d " a c t i n g on t h e p h o t o - l i b e r a t e d e l e c t r o n s p r o v i d e d t h a t t h e i r m i g r a t i o n l e n g t h i s s h o r t compared t o t h e g r a t i n g s p a c i n g . Hologram w r i t i n g by t h e p h o t o r e f r a c t i v e e f f e c t was m o d e l l e d i n p r o -g r e s s i v e s t a g e s o f c o m p l e x i t y . A l l t h e models were based on t h e a s s u m p t i o n t h a t t h e t r a n s p o r t l e n g t h of t h e f r e e e l e c t r o n s i s s h o r t compared t o t h e g r a t i n g s p a c i n g . T h i s appeared t o be a g e n e r a l l y a c c e p t e d a s s u m p t i o n . The f i r s t t r e a t m e n t a l l o w e d f o r t h e f e e d b a c k e f f e c t o f t h e space c h a r g e f i e l d and f o r t h e d a r k c o n d u c t i v i t y . I t was f o r u n i f o r m i l l u m i n a t i o n and c o n s t a n t 1 a p p l i e d v o l t a g e . The e f f e c t s of t h e m o d u l a t i o n r a t i o and t h e a p p l i e d f i e l d were i n v e s t i g a t e d . T h i s t r e a t m e n t was t h e n m o d i f i e d t o a l l o w f o r t h e e f f e c t of t h e a b s o r p t i o n c o n s t a n t i n r e d u c i n g t h e i n t e n s i t y of t h e l i g h t as i t i i p r o p a g a t e s t h r o u g h t h e c r y s t a l . I t was shown t h a t t h e h o l o g r a m becomes n o n u n i f o r m t h r o u g h t h e c r y s t a l t h i c k n e s s as a r e s u l t o f t h i s e f f e c t . H o l o -gram w r i t i n g w i t h o n e - d i m e n s i o n a l G a u s s i a n beams was m o d e l l e d a l l o w i n g f o r t h e f e e d b a c k e f f e c t o f t h e space c h a r g e f i e l d . A l a r g e s c a l e s p a c e c h a r g e f i e l d a s s o c i a t e d w i t h t h e e n v e l o p e o f t h e l i g h t p a t t e r n was shown t o a f f e c t t h e w r i t i n g p r o c e s s . I t was f o u n d t h a t an i n c r e a s e i n t h e f r a c t i o n a l i l l u m i -n a t i o n o f t h e c r y s t a l i mproves t h e w r i t i n g p r o c e s s . The d a r k c o n d u c t i v i t y i s shown t o have an i m p o r t a n t e f f e c t on t h e p r o c e s s . The f i n a l model was a g a i n f o r u n i f o r m i l l u m i n a t i o n and a l l o w e d n o t o n l y f o r t h e f e e d b a c k e f f e c t o f t h e p h o t o i n d u c e d f i e l d and t h e e f f e c t o f t h e d a r k c o n d u c t i v i t y and a b s o r p -t i o n b u t a l s o f o r t h e i n t e r a c t i o n between t h e h o l o g r a m b e i n g w r i t t e n and t h e l i g h t p a t t e r n w h i c h i s w r i t i n g i t . T h i s c a u s e s energy t r a n s f e r between t h e two w r i t i n g beams, t h u s m o d i f y i n g t h e l i g h t p a t t e r n . O p t i c a l e r a s u r e o f holograms w i t h t h e l i g h t i n c i d e n t e i t h e r on and o f f t h e Bragg a n g l e was m o d e l l e d . The t r e a t m e n t a l l o w s f o r t h e f e e d b a c k e f f e c t o f t h e s p a c e c h a r g e f i e l d s and f o r t h e e f f e c t o f t h e a b s o r p t i o n i n r e d u c i n g t h e l i g h t i n t e n s i t y . The model a l l o w e d f o r t h e i n t e r a c t i o n between t h e d i f f r a c t e d and t h e r e a d i n g beams f o r t h e c a s e o f i n c i d e n c e a t t h e Bragg a n g l e . The r e s u l t i n g i n t e r f e r e n c e p a t t e r n w r i t e s a new h o l o g r a m w h i c h may add t o o r s u b t r a c t f r o m t h e h o l o g r a m t o be e r a s e d . An e x p e r i m e n t a l method i s d e s c r i b e d i n w h i c h a n o r m a l l y i n c i d e n t a n c i l l a r y l i g h t beam o f d i f f e r e n t w a v e l e n g t h t h a n t h a t used t o w r i t e t h e h o l o g r a m a l l o w s t h e d i f f r a c t i o n e f f i c i e n c y t o be d e t e r m i n e d w i t h o u t e r r o r s due t o m u l t i p l e i n t e r n a l r e f l e c t i o n s . A l i m i t e d e x p e r i m e n t a l i n v e s t i g a t i o n was made of hologram s t o r a g e i n LiNbO^. P h o t o c u r r e n t and o p t i c a l measurement were c a r r i e d o u t on t h e same c r y s t a l . A l m o s t 100% r e l a t i v e d i f f r a c t i o n e f f i c i e n c y was o b s e r v e d . i i i The v a l u e o f t h e v i r t u a l f i e l d o b t a i n e d f r o m h o l o g r a p h i c measurement was fo u n d t o a g r e e w i t h i n 10% w i t h t h e v a l u e o b t a i n e d f r o m p h o t o c u r r e n t measure-ments. D u r i n g h o l o g r a m w r i t i n g , energy t r a n s f e r o f up t o 70% between t h e two w r i t i n g beams was o b s e r v e d . However, s i n c e t h e " v i r t u a l " f i e l d i n t h e s e e x p e r i m e n t s was much l a r g e r t h a n t h e d i f f u s i o n e q u i v a l e n t f i e l d , t h e model p r e d i c t e d o n l y about 5% energy t r a n s f e r . I t i s , t h e r e f o r e , s u g g e s t e d t h a t t h e t r a n s p o r t l e n g t h o f t h e p h o t o e x c i t e d e l e c t r o n s i n t h e c r y s t a l u s e d , was n o t s h o r t compared t o t h e g r a t i n g s p a c i n g . • I t i s a l s o shown t h a t l i g h t i n d u c e d s c a t t e r i n g can cause s e r i o u s e r r o r i n m e a s u r i n g t h e d i f f r a c t i o n e f f i c i e n c y e x p e c i a l l y d u r i n g o p t i c a l e r a s u r e . i v 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 8 2.1 I n t r o d u c t i o n 8 2.2 T h e o r e t i c a l A n a l y s i s . . . . 10 2.3 D i s c u s s i o n 12 2.4 Summary 18 3. MECHANISMS OF THE PHOTOREFRACTIVE EFFECT 20 3.1 I n t r o d u c t i o n 20 3.2 The E l e c t r o - o p t i c N a t u r e o f t h e P h o t o r e f r a c t i v e E f f e c t . 20 3.3 Chen's I n t e r n a l F i e l d M o del 21 3.4 J o h n s t o n ' s P o l a r i z a t i o n M o d e l . . 24 3.5 D e f e c t S i t e s and I m p u r i t i e s 25 3.6 The T r a n s p o r t L e n g t h 27 3.6.1 I n t r o d u c t i o n 27 3.6.2 Amodei's Model f o r S h o r t T r a n s p o r t L e n g t h . . . . 27 3.6.3 Young e t a l . ' s M o d e l w i t h A r b i t r a r y 29 T r a n s p o r t L e n g t h 31 3.6.4 D i s c u s s i o n 32 3.7 The B u l k P h o t o v o l t a i c E f f e c t 32 3.8 D i s c u s s i o n 34 3.8.1 The B u l k P h o t o v o l t a i c E f f e c t 34 3.8.2 D i f f u s i o n 37 4. THEORY OF HOLOGRAM WRITING BY THE PHOTOREFRACTIVE EFFECT . . . 41 4.1 I n t r o d u c t i o n 41 4.2 The Feedback E f f i c t o f t h e Space Charge F i e l d 42 v Page 4.2.1 I n t r o d u c t i o n . . 42 4.2.2 M o d e l '42 4.2.3 C a l c u l a t e d R e s u l t s and D i s c u s s i o n 45 4.2.3.1 The Time Development o f t h e Space Charge F i e l d 45 4.2.3.2 The E f f e c t o f t h e M o d u l a t i o n R a t i o 48 4.2.3.3 The D i f f r a c t i o n E f f i c i e n c y 53 4.2.4 Summary 56 4.3 The E f f e c t o f L i g h t L o s s due t o A b s o r p t i o n 58 4.3.1 Model 58 4.3.2 D i s c u s s i o n 60 4.3.3 Summary 61 4.4 The E f f e c t s o f Beam C o u p l i n g D u r i n g Hologram W r i t i n g . . 64 4.4.1 I n t r o d u c t i o n . . . 64 4.4.2 M o d e l 67 4.4.3 A l g o r i t h m 74 4.4.4 C a l c u l a t e d R e s u l t s and D i s c u s s i o n 76 4.4.4.1 I n t r o d u c t i o n 76 4.4.4.2 The N o n u n i f o r m i t y o f t h e G r a t i n g 77 4.4.4.3 The D i f f r a c t i o n E f f i c i e n c y 81 4.4.5 Summary 84 5. HOLOGRAM WRITING WITH GAUSSIAN BEAMS 87 5.1 I n t r o d u c t i o n 87 5.2 M o d e l 8 8 5.3 C a l c u l a t e d R e s u l t s and D i s c u s s i o n 91 5.3.1 I n t r o d u c t i o n 91 5.3.2 The E f f e c t o f t h e R a t i o o f C r y s t a l L e n g t h t o Beam W i d t h 92 5.3.3 The E f f e c t o f t h e Dark C o n d u c t i v i t y . . . . . . . 98 5.4 Summary 103 6. READING AND OPTICAL ERASURE OF HOLOGRAMS STORED BY THE PHOTOREFRACTIVE EFFECT 1 0 4 6.1 I n t r o d u c t i o n 1 ^ 6.2 M o d e l 1 0 5 6.2.1 O f f Brag g A n g l e I n c i d e n c e 107 6.2.2 Bragg A n g l e I n c i d e n c e 108 6.3 C a l c u l a t e d R e s u l t s and D i s c u s s i o n 110 6.4 Summary 118 v i Page 7. EXPERIMENTAL CONSIDERATIONS. FOR HOLOGRAM FORMATION 119 7.1 Introduction . 119 7.2 The Optical System 122 7.3 Influence of Multiple Internal Reflections 130 8. PHOTOCURRENT AND HOLOGRAPHIC MEASUREMENTS 138 8.1 Introduction 138 8.2 Photocurrent Measurements 138 8.2.1 Experimental Procedure . . . . . . 138 8.2.2 Results and Discussion 139 8.3 Holographic Measurements . 142 8.3.1 Experimental Procedure . . . 142 8.3.2 Results and Discussion 143 9. CONCLUSIONS 149 9.1 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 LiNb0 3 158 APPENDIX B THE ELECTRO-OPTIC EFFECT IN LITHIUM NIOBATE 159 APPENDIX C READ-WRITE HOLOGRAPHIC OPTICAL MEMORY SYSTEM . . . . . 163 v i i LIST OF ILLUSTRATIONS F i g u r e Page 2.1 System c o n f i g u r a t i o n f o r l i g h t d i f f r a c t i o n by p e r i o d i c phase g r a t i n g s 11 2.2 The i n t e n s i t y o f t h e f i r s t f o u r d i f f r a c t e d modes v s . t h e g r a t i n g s t r e n g t h i n t h e Raman-Nath regime 15 2.3 The i n t e n s i t y o f t h e z e r o and f i r s t o r d e r modes v s . t h e g r a t i n g s t r e n g t h i n t h e Bragg regime 16 3.1 O p t i c a l l y i n d u c e d b i r e f r i n g e n c e change caused by a c i r c u l a r beam 22 3.2 C o n f i g u r a t i o n f o r h o l o g r a m r e c o r d i n g 28 3.3 R e l a t i o n o f t h e c r y s t a l axes and t h e two w r i t i n g beams f o r d i f f e r e n t c o n f i g u r a t i o n s o f f o r m i n g holograms 39 4.1 C a l c u l a t e d t i m e development o f t h e f u n d a m e n t a l and f i r s t f o u r h armonic components of t h e p h o t o -i n d u c e d s p a c e c h a r g e f i e l d 46 4.2 C a l c u l a t e d t i m e development o f t h e f u n d a m e n t a l component of t h e f i e l d f o r d i f f e r e n t v a l u e s o f t h e e f f e c t i v e m o d u l a t i o n r a t i o 49 4.3 The dependence of t h e s t e a d y s t a t e s p a c e c h a r g e f i e l d on t h e e f f e c t i v e m o d u l a t i o n r a t i o 50 4.4 The dependence o f t h e s t e a d y s t a t e f u n d a m e n t a l component o f t h e f i e l d on t h e d a r k c o n d u c t i v i t y 52 4.5 C a l c u l a t e d t i m e development o f t h e d i f f r a c t i o n e f f i c i e n c y d u r i n g h o l o g r a m w r i t i n g f o r d i f f e r e n t v a l u e s o f t h e a p p l i e d f i e l d 55 4.6 C a l c u l a t e d t i m e development of t h e d i f f r a c t i o n e f f i c i e n c y f o r d i f f e r e n t v a l u e s o f t h e a b s o r p t i o n c o n s t a n t 62 4.7 The e f f e c t o f t h e a b s o r p t i o n c o n s t a n t on t h e t i m e development of t h e d i f f r a c t i o n e f f i c i e n c y 63 4.8 C o n f i g u r a t i o n f o r h o l o g r a m w r i t i n g w i t h c o n s t a n t v o l t a g e a p p l i e d t o t h e c - f a c e s of t h e c r y s t a l 68 4.9 The t i m e development of t h e a m p l i t u d e of t h e f u n d a m e n t a l component o f t h e change i n t h e r e f r a c t i v e i n d e x a t d i f f e r e n t d e p t h s i n t h e c r y s t a l 78 v i i i F i g u r e Page 4.10 The t i m e development o f t h e s p a t i a l phase s h i f t o f t h e change i n t h e r e f r a c t i v e i n d e x a t d i f f e r e n t d e p t h s i n t h e c r y s t a l 80 4.11 The t i m e development of t h e d i f f r a c t i o n e f f i c i e n c y d u r i n g h o l o g r a m w r i t i n g f o r d i f f e r e n t v a l u e s o f t o t a l a p p l i e d f i e l d 82 4.12 The dependence o f t h e t i m e development o f the d i f f r a c t i o n e f f i c i e n c y d u r i n g w r i t i n g on t h e a b s o r p t i o n c o n s t a n t 83 4.13 The t i m e development o f t h e d i f f r a c t i o n . ' . e f f i c i e n c y d u r i n g w r i t i n g on t h e d a r k c o n d u c t i v i t y 85 5.1 S p a t i a l d i s t r i b u t i o n o f t h e F o u r i e r compo-n e n t o f t h e p h o t o i n d u c e d s p a c e c h a r g e f i e l d 94 5.2 The dependence o f t h e l o c a l dc component o f the space c h a r g e f i e l d on t h e d a r k c o n d u c t i v i t y f o r d i f f e r e n t v a l u e s o f t h e c r y s t a l f r a c t i o n a l i l l u m i n a t i o n 95 5.3 The Fundamental F o u r i e r component o f t h e f i e l d due t o d r i f t v s . t h e r a t i o o f l i g h t t o d a r k c a r r i e r c o n c e n t r a t i o n f o r d i f f e r e n t v a l u e s o f th e c r y s t a l f r a c t i o n i l l u m i n a t i o n . . . . 96 5.4 The dependence o f t h e f i r s t h a r monic component of t h e f i e l d due t o d r i f t on t h e d a r k c o n d u c t i v i t y f o r d i f f e r e n t v a l u e s o f t h e c r y s t a l f r a c t i o n i l l u m i n a t i o n . . . . . . . . 97 5.5 E x p e r i m e n t a l o b s e r v a t i o n o f t h e dependence of t h e p h o t o i n d u c e d space c h a r g e f i e l d on t h e c r y s t a l f r a c t i o n a l i l l u m i n a t i o n and t h e l i g h t i n t e n s i t y ( C o r n i s h e t a l . ) 99 5.6 The dependence of t h e f u n d a m e n t a l and f i r s t h a r m onic component o f t h e p h o t o i n d u c e d f i e l d on t h e d a r k c o n d u c t i v i t y 101 6.1 A s c h e m a t i c r e p r e s e n t a t i o n o f a q u a l i t a t i v e model f o r t h e o p t i c a l e r a s u r e p r o c e s s I l l 6.2 The t i m e development o f t h e a b s o l u t e d i f f r a c t i o n e f f i c i e n c y d u r i n g h o l o g r a m w r i t e - e r a s e c y c l e 113 6.3 The t i m e development o f t h e a b s o l u t e d i f f r a c t i o n e f f i c i e n c y and t h e f u n d a m e n t a l component o f t h e p h o t o i n d u c e d f i e l d a t d i f f e r e n t d e p t h s i n t h e c r y s t a l 116 i x F i g u r e Page 6.4 O p t i c a l e r a s u r e c h a r a c t e r i s t i c s o f f and on the Bragg a n g l e 117 7.1 I n t e r f e r e n c e p a t t e r n o f two p l a n e waves 120 7.2 D i f f r a c t i o n o f t h e r e f e r e n c e beam by t h e hologram. 121 7.3 E x p e r i m e n t a l arrangement f o r m e a s u r i n g t h e d i f f r a c t i o n e f f i c i e n c y 123 7.4 A l t e r n a t i v e arrangement f o r m e a s u r i n g t h e d i f f r a c t i o n e f f i c i e n c y 123 7.5 The i n t e n s i t y o f t h e w r i t i n g beams w i t h and w i t h o u t t h e p l e x i g l a s s t a b l e c o v e r . . . . 127 7.6 An i n t e n s i t y s c a n a c r o s s t h e d i a m e t e r o f t h e s p a t i a l l y f i l t e r e d c o l l i m a t e d beam 129 7.7 The dependence o f t h e m u l t i p l e i n t e r n a l r e f l e c t i o n c o r r e c t i o n f a c t o r on t h e o p t i c a l t h i c k n e s s and on t h e i n t r i n s i c d i f f r a c t i o n e f f i c i e n c y 132 7.8 E x p e r i m e n t a l arrangement used t o d e t e r m i n e t h e i n t r i n s i c d i f f r a c t i o n e f f i c i e n c y u s i n g an a n c i l l a r y t h i r d beam 135 7.9 E x p e r i m e n t a l r e s u l t s on t h e e f f e c t i v e and i n t r i n s i c d i f f r a c t i o n e f f i c i e n c y 136 8.1 The dependence o f p h o t o c u r r e n t s i n Fe-doped l i t h i u m n i o b a t e on t h e l i g h t i n t e n s i t y and t h e a p p l i e d v o l t a g e 140 8.2 Measured t i m e development of t h e r e l a t i v e d i f f r a c t i o n e f f i c i e n c y o f holograms s t o r e d i n l i t h i u m n i o b a t e c r y s t a l 144 8.3 Observed r e l a t i v e i n t e n s i t i e s o f t h e t r a n s -m i t t e d and d i f f r a c t e d beams d u r i n g o p t i c a l e r a s u r e on t h e Bragg a n g l e 146 8.3 The t r a n s m i t t e d and d i f f r a c t e d beams' i n t e n -s i t i e s d u r i n g d e s t r u c t i v e r e a d i n g 147 G . l A s c h e m a t i c o f r e a d - w r i t e - e r a s e h o l o g r a p h i c o p t i c a l memory 164 x ACKNOWLEDGEMENTS I am most grateful to my supervisor, Dr. L. Young for his encourage-ment 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 1976-1977) and The University of British Columbia (graduate fellowship awarded 1975-1976) are gratefully acknowledged for their financial support. x i To Gehan x i i 1. CHAPTER I INTRODUCTION I n Fe-doped l i t h i u m n i o b a t e and s i m i l a r c r y s t a l s , e x p o s u r e t o l i g h t o f t h e a p p r o p r i a t e w a v e l e n g t h i n d u c e s s m a l l changes i n t h e r e f r a c t i v e i n d e x . T h i s phenomenon i s sometimes c a l l e d t h e p h o t o r e f r a c t i v e e f f e c t . I t i s b e l i e v e d t h a t t h e mechanism o f t h e p h o t o r e f r a c t i v e e f f e c t i n t h e s e c r y s -t a l s i s b r o a d l y as f o l l o w s . The i n c i d e n t l i g h t c a u s e s p h o t o e x c i t a t i o n o f e l e c t r o n s f r o m t r a p s . The e l e c t r o n s d r i f t and d i f f u s e and a r e s u b s e q u e n t l y r e t r a p p e d . I n f e r r o e l e c t r i c c r y s t a l s a new t y p e o f p h o t o e f f e c t i s a l s o i n v o l v e d , as w i l l be d i s c u s s e d l a t e r . The s p a t i a l r e d i s t r i b u t i o n o f e l e c -t r o n s among t r a p s s e t s up a spac e c h a r g e f i e l d w h i c h m o d u l a t e s t h e r e f r a c -t i v e i n d e x o f t h e c r y s t a l v i a t h e e l e c t r o - o p t i c - e f f e c t . The o p t i c a l l y i n d u c e d changes i n r e f r a c t i v e i n d e x may be removed e i t h e r by u n i f o r m i l l u m i -n a t i o n o r by h e a t i n g . These t r e a t m e n t s cause the e l e c t r o n s t o be o p t i c a l l y o r t h e r m a l l y e x c i t e d f r o m t r a p s and u n i f o r m l y r e d i s t r i b u t e d , so t h a t t h e m o d u l a t i o n i n t h e r e f r a c t i v e i n d e x i s removed. The p h o t o r e f r a c t i v e e f f e c t was f i r s t e n c o u n t e r e d i n e l e c t r o o p t i c m o d u l a t o r s and f r e q u e n c y d o u b l e r s where t h e o p t i c a l l y i n d u c e d inhomogenei-t i e s c aused s c a t t e r i n g and d e c o l l i m a t i o n o f t h e l i g h t , t h u s d e g r a d i n g t h e p e r f o r m a n c e o f t h e d e v i c e s ( A s h k i n e t a l . 1966). I t was l a t e r r e c o g n i z e d t h a t t h i s p h o t o r e f r a c t i v e e f f e c t c o u l d be u s e f u l as a new means of g e n e r a -t i n g phase h o l o g r a m s . Volume phase holograms have been s t o r e d by t h e p h o t o -r e f r a c t i v e e f f e c t i n t h e f e r r o e l e c t r i c c r y s t a l s l i t h i u m n i o b a t e (Chen e t a l . 1 9 68), s t r o n t i u m b a r i u m n i o b a t e (SBN) ( T h a x t e r 1 9 6 9 ) , b a r i u m t i t a n a t e (Townsend e t a l . 1 9 7 0 ) , b a r i u m sodium n i o b a t e (Amodei e t a l . 1971c) and 2. l e a d z i r c o n a t e t i t a n a t e (PLZT) ( M i c h e r o n e t a l . 1974). P h o t o r e f r a c t i v e m a t e r i a l s a r e p o t e n t i a l c a n d i d a t e s f o r t h e s t o r a g e media i n h o l o g r a p h i c memory syste m s . They do n o t r e q u i r e development or b l e a c h i n g p r o c e s s e s and, t h e r e f o r e , can be used i n r e a l t i m e , i n c o n t r a s t t o p h o t o g r a p h i c e m u l s i o n s and t h e r m o - p l a s t i c s . They can be used f o r r e a d -w r i t e a p p l i c a t i o n s b e c a u s e t h e h o l o g r a m can be o p t i c a l l y o r t h e r m a l l y e r a s e d and a new h o l o g r a m w r i t t e n . D i f f r a c t i o n e f f i c i e n c i e s a p p r o a c h i n g 100% a r e t h e o r e t i c a l l y p o s s i b l e and have been o b s e r v e d e x p e r i m e n t a l l y i n l i t h i u m n i o b a t e ( C h a p t e r 8 ) . (For d e f i n i t i o n s o f d i f f r a c t i o n e f f i c i e n c y see Chap-t e r 7 ) . A t p r e s e n t , p h o t o r e f r a c t i v e media have l o w e r s e n s i t i v i t y t h a n m i g h t be d e s i r e d b u t improvement i s p o s s i b l e and s t e p s have been t a k e n i n t h a t d i r e c t i o n , e s p e c i a l l y w i t h l i t h i u m n i o b a t e ( P h i l l i p s e t a l . 1974). Volume h o l o g r a p h i c s t o r a g e media have, i n t h e o r y , t h e a t t r a c t i v e c h a r a c t e r i s t i c o f v e r y h i g h d a t a p a c k i n g d e n s i t y . I t was shown by v a n Heerden (1963) t h a t t h e t h e o r e t i c a l u l t i m a t e s t o r a g e c a p a c i t y o f a volume 3 h o l o g r a m i s V/X b i t s where. V i s t h e volume and/A. i s t h e wave l e n g t h of 12 l i g h t . T h i s means t h a t t h e o r e t i c a l l y more t h a n 10 b i t s c a n be s t o r e d i n 3 a 1 cm c r y s t a l . However, t h e p r a c t i c a l l i m i t s e t by o t h e r o p t i c a l p a r a -meters of t h e s t o r a g e s y s t e m i s l o w e r t h a n t h i s (van der L u g t 1973). The f e a s i b i l i t y o f r e a d - i n and r e a d - o u t r e q u i r i n g no m e c h a n i c a l l y moving p a r t s and t h e h i g h p a c k i n g d e n s i t y of o p t i c a l systems p r o m i s e t o o f f e r l a r g e c a p a c i t y w i t h r e l a t i v e l y f a s t a c c e s s compared t o o t h e r l a r g e c a p a c i t y s y s -tems. A number o f r e v i e w a r t i c l e s a r e a v a i l a b l e w h i c h d i s c u s s t h e advan-t a g e s and l i m i t a t i o n s o f o p t i c a l memories (Rajchman 1970, K i n g 1972, A n d e r s o n 1972, H i l l 1972, K i e m l e 1974, Chen and Zook 1975). The t o p i c i s a l s o d i s c u s s e d i n A p p e n d i x C. The work d e s c r i b e d h e r e was u n d e r t a k e n i n o r d e r t o o b t a i n a b e t t e r 3. u n d e r s t a n d i n g o f h o l o g r a m s t o r a g e by t h e p h o t o r e f r a c t i v e e f f e c t i n connec-t i o n w i t h p o s s i b l e e n g i n e e r i n g a p p l i c a t i o n s . A l t h o u g h e x t e n s i v e e x p e r i -m e n t a l i n v e s t i g a t i o n o f t h e h o l o g r a m s t o r a g e p r o c e s s has been c a r r i e d o u t , a l l t h e o r e t i c a l a n a l y s i s so f a r has been l i m i t e d i n a p p l i c a b i l i t y by r e a s o n s of c e r t a i n s i m p l i f i c a t i o n s . A more r i g o r o u s model of h o l o g r a m w r i t i n g and e r a s u r e p r o c e s s e s was needed t o i n v e s t i g a t e t h e mechanisms o f t h e p h o t o -r e f r a c t i v e e f f e c t by c o m p a r i s o n w i t h e x p e r i m e n t a l r e s u l t s . Such a model s h o u l d a l l o w one t o o b t a i n t h e p h y s i c a l p a r a m e t e r s o f t h e c r y s t a l and t o d e t e r m i n e t h e u s e f u l n e s s of a c r y s t a l i n a s p e c i f i c a p p l i c a t i o n . The a b i l i -t y t o p r e d i c t t h e d i f f r a c t i o n e f f i c i e n c y under a g i v e n o p t i c a l e x p o s u r e i s a l s o i m p o r t a n t f o r t h e d e s i g n o f t h e s y s t e m and t h e p r e d i c t i o n o f t h e s y s t e m p e r f o r m a n c e . The work was c a r r i e d o u t w i t h s p e c i a l i n t e r e s t i n Fe-doped l i t h i u m n i o b a t e because i t seemed t h e most p r o m i s i n g m a t e r i a l f o r a p p l i c a -t i o n s and b e c a u s e h i g h q u a l i t y c r y s t a l s a r e r e a d i l y a v a i l a b l e . However, t h e t h e o r e t i c a l a n a l y s i s s h o u l d a p p l y t o o t h e r f e r r o l e c t r i c p h o t o r e f r a c t i v e m a t e r i a l s w h i c h e x h i b i t t h e new b u l k p h o t o v o l t a i c e f f e c t o b s e r v e d i n l i t h i u m n i o b a t e ( G l a s s e t . a l . 1974). The a n a l y s i s s h o u l d a l s o a p p l y t o p a r a e l e c t r i c m a t e r i a l s , s u c h as p o t a s s i u m t a n t a l a t e n i o b a t e (KTN) w h i c h e x h i b i t a com-p a r a b l e p h o t o r e f r a c t i v e e f f e c t i f an e l e c t r i c f i e l d i s a p p l i e d t o t h e c r y s -t a l (Chen 1967). F o r holograms s t o r e d by t h e p h o t o r e f r a c t i v e e f f e c t t o p l a y a major r o l e i n h o l o g r a p h i c memory s y s t e m s , t h e y s h o u l d e x h i b i t , among o t h e r t h i n g s , h i g h d i f f r a c t i o n e f f i c i e n c y and a n g u l a r and w a v e l e n g t h s e l e c t i v i t i e s . That i s , t h e y s h o u l d o p e r a t e i n t h e Bragg r e g i m e of d i f f r a c t i o n . I n C h a p t e r 2, t h e t h e o r y o f l i g h t d i f f r a c t i o n by phase g r a t i n g s (holograms) i s o u t l i n e d and a new r e l i a b l e c r i t e r i o n t o d e t e r m i n e whether t h e phase h o l o g r a m i s o p e r a t i n g i n t h e Bragg o r t h e Raman-Nath r e g i m e s o f d i f f r a c t i o n i s p r o p o s e d . 4. I t i s shown t h a t t h i c k n e s s o f t h e g r a t i n g does n o t e n t e r i n t o t h e c r i t e r i o n and, t h e r e f o r e , t h e d i s t i n c t i o n between " t h i c k " and " t h i n " g r a t i n g s o r h o l o -grams i s i n v a l i d as a d e s c r i p t i o n of whether t h e g r a t i n g i s i n t h e B r a g g o r t h e Raman-Nath regime o f d i f f r a c t i o n . I t i s g e n e r a l l y r e c o g n i z e d t h a t d r i f t and d i f f u s i o n a r e i n v o l v e d i n t h e t r a n s p o r t mechanism o f t h e p h o t o r e f r a c t i v e e f f e c t (Amodei 1971a, 1971b). G l a s s , v o n \ d e r L i n d e and Negran (1974, 1975) have p r o p o s e d t h a t t h e p h o t o r e f r a c t i v e e f f e c t i n v o l v e s a l s o an e n t i r e l y new t r a n s p o r t mechanism w h i c h t h e y have l a b e l l e d t h e " b u l k p h o t o v o l t a i c e f f e c t " . T h i s mechanism i s t h o u g h t t o be r e s p o n s i b l e f o r t h e p h o t o c u r r e n t s w h i c h were p r e v i o u s l y a t -t r i b u t e d t o i n t e r n a l f i e l d s of p y r o e l e c t r i c o r i g i n . These mechanisms a r e o u t l i n e d i n C h a p t e r 3 and i t i s shown t h a t t h i s new p h o t o v o l t a i c e f f e c t , as r e p r e s e n t e d by a c u r r e n t d e n s i t y p r o p o r t i o n a l t o t h e l i g h t i n t e n s i t y , i s f o r m a l l y e q u i v a l e n t t o a f i c t i o n a l " v i r t u a l f i e l d " a c t i n g on t h e p h o t o -r e l e a s e d e l e c t r o n s p r o v i d e d t h a t t h e i r m i g r a t i o n l e n g t h i s s h o r t compared t o t h e g r a t i n g s p a c i n g . I n C h a p t e r 8, some e x p e r i m e n t a l r e s u l t s w h i c h b e a r upon th e p h o t o v o l t a i c e f f e c t a r e d i s c u s s e d . The f e e d b a c k e f f e c t o f t h e p h o t o i n d u c e d s p a c e c h a r g e f i e l d on t h e r e d i s t r i b u t i o n o f e l e c t r o n s d u r i n g h o l o g r a m w r i t i n g i s c o n s i d e r e d i n C h a p t e r 4. A new model f o r h o l o g r a m w r i t i n g a p p l i c a b l e o v e r t h e e n t i r e r a n g e o f e x p o s u r e a l l o w i n g f o r t h i s f e e d b a c k e f f e c t as w e l l as t h e d a r k c o n d u c t i v i t y i s d e v e l o p e d f o r u n i f o r m l y i l l u m i n a t e d c r y s t a l under c o n s t a n t a p p l i e d v o l -t a g e . S h o r t t r a n s p o r t l e n g t h of t h e f r e e e l e c t r o n s i s assumed. I t i s shown t h a t b o t h t h e d r i f t and d i f f u s i o n p r o d u c e d components o f t h e p h o t o i n d u c e d s p a c e c h a r g e f i e l d e v o l v e w i t h t h e same t i m e c o n s t a n t and t h u s t h e s p a t i a l phase s h i f t o f t h e i n d e x m o d u l a t i o n f r o m t h e i n c i d e n t i n t e n s i t y p a t t e r n i s c o n s t a n t i n t i m e and i t s v a l u e i s d e t e r m i n e d by t h e r e l a t i v e c o n t r i b u t i o n o f 5. d r i f t and d i f f u s i o n . The dependence o f t h e s p a c e c h a r g e f i e l d on t h e modu-l a t i o n r a t i o and t h e d a r k c o n d u c t i v i t y a r e a l s o d i s c u s s e d . As t h e l i g h t p r o p a g a t e s i n t o t h e c r y s t a l i t decays e x p o n e n t i a l l y w i t h d i s t a n c e t r a v e l l e d due t o a b s o r p t i o n . T h e r e f o r e t h e r a t e o f b u i l d up o f t h e h o l o g r a m i s d i f f e r e n t a t d i f f e r e n t d e p t h s i n t h e c r y s t a l . The model p r o p o s e d i n C h a p t e r 4 i s e x t e n d e d t o a l l o w f o r t h i s e f f e c t . I t i s shown t h a t o v e r l o o k i n g t h i s c o m p l i c a t i o n l e a d s t o an o v e r p r e d i c t i o n o f t h e c a l -c u l a t e d d i f f r a c t i o n e f f i c i e n c y . The h o l o g r a m becomes n o n u n i f o r m t h r o u g h t h e t h i c k n e s s o f t h e c r y s t a l . The e f f e c t i s more s i g n i f i c a n t when t h e r a t i o o f t h e d a r k t o t h e p h o t o i n d u c e d c o n d u c t i v i t y i s l a r g e (> .01) and, o f c o u r s e , f o r l a r g e a b s o r p t i o n c o n s t a n t s . S t a e b l e r and Amodei (1972b) were t h e f i r s t t o p o i n t o u t t h e i m p l i -c a t i o n s a r i s i n g f r o m beam c o u p l i n g d u r i n g h o l o g r a m w r i t i n g . The h o l o g r a m b e i n g w r i t t e n i n t e r a c t s w i t h t h e l i g h t p a t t e r n w h i c h i s w r i t i n g i t . T h i s i n t e r a c t i o n c auses e n e r g y t r a n s f e r between t h e two w r i t i n g beams and t h u s m o d i f i e s t h e l i g h t i n t e r f e r e n c e p a t t e r n . I n C h a p t e r 4 a dynamic model f o r h o l o g r a m w r i t i n g i s p r o p o s e d . The model a l l o w s s i m u l t a n e o u s l y , f o r t h e f i r s t t i m e , f o r b o t h t h e f e e d b a c k e f f e c t o f t h e space c h a r g e f i e l d and t h e e f f e c t of t h e h o l o g r a m i n m o d i f y i n g t h e l i g h t p a t t e r n w h i c h i s w r i t i n g i t , as w e l l as t h e e f f e c t o f l i g h t a b s o r p t i o n . I t i s shown t h a t n e i t h e r t h e a m p l i t u d e n o r t h e s p a t i a l phase s h i f t o f t h e i n d e x m o d u l a t i o n i s u n i f o r m t h r o u g h t h e g r a t i n g t h i c k n e s s . The e f f e c t s of t h e a p p l i e d and v i r t u a l f i e l d s , t h e ab- • s o r p t i o n c o n s t a n t and t h e d a r k c o n d u c t i v i t y on t h e h o l o g r a m w r i t i n g p r o c e s s a r e a l s o i n v e s t i g a t e d . Hologram w r i t i n g w i t h n o n u n i f o r m i l l u m i n a t i o n w o u l d p r o d u c e a l a r g e s c a l e s p a c e c h a r g e f i e l d a s s o c i a t e d w i t h t h e e n v e l o p e o f t h e l i g h t p a t t e r n i n a d d i t i o n t o t h e s i n u s o i d a l and h i g h e r h armonic components. C o r n i s h e t a l . 6. (1976a) have shown t h a t t h i s l a r g e s c a l e f i e l d a f f e c t s g r e a t l y t h e ho l o g r a m w r i t i n g p r o c e s s . I n C h a p t e r 5, a model f o r h o l o g r a m w r i t i n g w i t h a one-d i m e n s i o n a l G a u s s i a n beam i n c i d e n t on a f i n i t e c r y s t a l under c o n s t a n t app-l i e d v o l t a g e i s d e v e l o p e d . I t i s shown t h a t an i n c r e a s e i n t h e r a t i o o f t h e c r y s t a l l e n g t h t o t h e G a u s s i a n beam w i d t h ( i . e . i n c r e a s e i n t h e f r a c -t i o n a l i l l u m i n a t i o n o f t h e c r y s t a l ) i m p r o v e s t h e ho l o g r a m w r i t i n g p r o c e s s . The r a t i o o f t h e d a r k c o n d u c t i v i t y t o t h e p h o t o c o n d u c t i v i t y i s shown t o have an i m p o r t a n t e f f e c t on t h e p r o c e s s . Holograms s t o r e d by t h e p h o t o r e f r a c t i v e e f f e c t may be o p t i c a l l y e r a s e d by e x p o s u r e t o u n i f o r m i l l u m i n a t i o n . However, as S t a e b l e r e t a l . (1972b) have shown, i f t h e e r a s i n g beam i s i n c i d e n t a t t h e Bragg a n g l e o f the h o l o g r a m , t h e l i g h t beam i n t e r a c t s w i t h t h e ho l o g r a m and t h e i n t e r -f e r e n c e p a t t e r n o f t h e e r a s i n g ( d e s t r u c t i v e r e a d i n g ) beam and t h e d i f f r a c -t e d beam w r i t e s a new h o l o g r a m w h i c h may add t o o r s u b t r a c t f r o m t h e h o l o -gram t o be e r a s e d . T h i s , o f c o u r s e , i s i n a d d i t i o n t o t h e o p t i c a l e r a s u r e caused by t h e u n i f o r m p a r t o f t h e p a t t e r n . I n C h a p t e r 6 a model f o r o p t i -c a l e r a s u r e b o t h on and o f f t h e Bragg a n g l e i s d e v e l o p e d . The s p e c i a l f e a t u r e o f t h i s model i s t h a t i t a l l o w s f o r t h e f e e d b a c k e f f e c t o f t h e spac e c h a r g e f i e l d and t h e e f f e c t o f t h e a b s o r p t i o n on r e d u c i n g t h e l i g h t i n t e n -s i t y , t h e e r a s u r e due t o t h e u n i f o r m p a r t o f t h e i l l u m i n a t i o n , and t h e new h o l o g r a m w r i t t e n by t h e i n t e r f e r e n c e p a t t e r n o f t h e r e a d i n g and d i f f r a c t e d beams ( f o r t h e case o f i n c i d e n c e a t t h e Bragg a n g l e ) . I t i s shown t h a t t h e model r e p r o d u c e s a l l t h e r e p o r t e d t y p e s o f e r a s u r e c h a r a c t e r i s t i c s . Measurements o f t h e b u i l d - u p o f t h e d i f f r a c t i o n e f f i c i e n c y w i t h t i m e have been used by s e v e r a l a u t h o r s t o t e s t v a r i o u s p h y s i c a l models f o r the p h o t o r e f r a c t i v e e f f e c t . U s u a l l y a p r o t o t y p e h o l o g r a m i s w r i t t e n by t h e i n t e r f e r e n c e o f two p l a n e waves. I t i s t h e n n e c e s s a r y t o r e l a t e t h e ob-7. s e r v e d d i f f r a c t i o n e f f i c i e n c y t o t h e p r e d i c t e d r e f r a c t i v e i n d e x m o d u l a t i o n . C o r n i s h e t a l . (1975) have shown t h a t n e g l e c t i n g t h e e f f e c t s o f m u l t i p l e r e f l e c t i o n can cause s e r i o u s e r r o r s . The p r o b l e m i s e x a c e r b a t e d by t h e f a c t t h a t t h e s e e r r o r s a r e n o t t y p i c a l l y c o n s t a n t d u r i n g an e x p e r i m e n t s i n c e s i g n i f i c a n t changes i n t h e o p t i c a l t h i c k n e s s o f t h e c r y s t a l can o c c u r due t o h e a t i n g by t h e l i g h t beam. I n C h a p t e r 7, an e x p e r i m e n t a l method i s d e s -c r i b e d i n w h i c h a n o r m a l l y i n c i d e n t a n c i l l a r y l i g h t beam o f d i f f e r e n t wave-l e n g t h t h a n t h a t used t o w r i t e t h e h o l o g r a m a l l o w s t h e d i f f r a c t i o n e f f i c i e n c y t o be d e t e r m i n e d w i t h o u t e r r o r s due t o m u l t i p l e i n t e r n a l r e f l e c t i o n s . Time p e r m i t t e d o n l y a r e s t r i c t e d e x p e r i m e n t a l i n v e s t i g a t i o n o f t h e ho l o g r a m s t o r a g e p r o c e s s i n l i t h i u m n i o b a t e . R e s u l t s a r e r e p o r t e d i n Chap-t e r 8. P h o t o c u r r e n t measurements were made t o i n v e s t i g a t e t h e b u l k p h o t o -v o l t a i c e f f e c t . I n t h e h o l o g r a m w r i t i n g e x p e r i m e n t s , v e r y h i g h d i f f r a c t i o n e f f i c i e n c i e s were measured ( a l m o s t 1 0 0 % ) . A c c o r d i n g t o t h e ho l o g r a m w r i t i n g model i n C h a p t e r 4, t h e d r i f t f i e l d e q u i v a l e n t t o t h e b u l k p h o t o v o l t a i c e f -f e c t s h o u l d be about 50 kV/cm t o a c h i e v e s u c h h i g h e f f i c i e n c i e s . T h i s v a l u e i s i n v e r y good agreement w i t h t h e v a l u e o b t a i n e d from t h e p h o t o c u r r e n t mea-surements. The e q u i v a l e n t d i f f u s i o n f i e l d was 1 kV/cm. That i s , f o r t h e c r y s t a l u s e d , d i f f u s i o n was n e g l i g i b l e . However, energy t r a n s f e r between t h e two w r i t i n g beams of up t o 70% was o b s e r v e d whereas t h e model p r e d i c t e d about 5% energy t r a n s f e r . T h i s d i s c r e p a n c y m i g h t be a c c o u n t e d f o r i f t h e t r a n s p o r t l e n g t h o f t h e f r e e e l e c t r o n s i n t h i s e x p e r i m e n t was n o t s h o r t com-p a r e d t o t h e g r a t i n g s p a c i n g . F o r t h i s c a s e d r i f t a l o n e m i g h t p r o d u c e l a r g e enough s p a t i a l phase s h i f t between t h e l i g h t p a t t e r n and t h e i n d e x modula-t i o n (Young e t a l . 1974) t o cause t h e o b s e r v e d energy t r a n s f e r . 8. CHAPTER I I THEORY OF LIGHT DIFFRACTION BY PERIODIC PHASE GRATINGS 2.1 I n t r o d u c t i o n I n r e c e n t y e a r s , e x t e n s i v e r e s e a r c h has been done on t h e t o p i c o f l i g h t d i f f r a c t i o n by p e r i o d i c phase g r a t i n g s , b o t h h o l o g r a p h i c a l l y and a c o u s t i c a l l y p r o d u c e d . Examples a r e K l e i n and Cook ( 1 9 6 7 ) , B u r c k h a r d t (1966) K o g e l n i k ( 1 9 6 9 ) , Chu and Tamir (1970) and Magnusson and G a y l o r d ( 1 9 7 7 ) . There i s g e n e r a l agreement t h a t i t i s c o n v e n i e n t t o d e f i n e two r e g i m e s i n w h i c h phase g r a t i n g s o p e r a t e . I n t h e Raman-Nath r e g i m e , s e v e r a l d i f f r a c t e d waves a r e p r o d u c e d . I n t h e Bragg r e g i m e , e s s e n t i a l l y o n l y one d i f f r a c t e d wave i s pr o d u c e d and t h a t o n l y f o r n e a r Bragg i n c i d e n c e . I t has been c u s -tomary t o r e f e r t o g r a t i n g s w h i c h o p e r a t e i n t h e Raman-Nath and t h e Bragg r e g i m e s as " t h i n " and " t h i c k " g r a t i n g s r e s p e c t i v e l y . Phase g r a t i n g s o p e r a -t i n g i n the Bragg r e g i m e s have numerous p o t e n t i a l a p p l i c a t i o n s based on t h e i r p r o p e r t i e s o f h i g h d i f f r a c t i o n e f f i c i e n c y , w a v e l e n g t h and a n g u l a r s e l e c t i v i t y ( K o g e l n i k 1969 and Forshaw 1974). Examples a r e n a r r o w band s p e c t r a l f i l t e r s ( C r a w f o r d 1 9 5 4 ) , d e f l e c t o r s and m o d u l a t o r s (Hammer 1971). Volume holograms a r e o f i n t e r e s t due t o t h e i r p o t e n t i a l use i n h i g h c a p a c i t y i n f o r m a t i o n s t o r a g e (Heerden 1 9 6 3 ) , c o l o r h o l o g r a p h y ( P e n n i n g t o n e t a l . 1965), and i n w h i t e l i g h t h olograms ( S t r o k e e t a l . 1966). I t w o u l d seem t o be u s e f u l and c o n v e n i e n t t o have a r e l i a b l e c r i t e r i o n t o d e t e r m i n e whether a g r a t i n g i s o p e r a t i n g i n t h e Raman-Nath o r the B r a g g regime o f d i f f r a c t i o n . K l e i n and Cook (1967) p r o p o s e d a parameter 2 Q d e f i n e d as 2irA L/A n , where \ i s t h e l i g h t w a v e l e n g t h , L i s t h e g r a t i n g o o o t h i c k n e s s , A i s t h e g r a t i n g s p a c i n g and n i s t h e medium r e f r a c t i v e i n d e x , 9. as t h e c r i t e r i o n f o r d e c i d i n g w h i c h d i f f r a c t i o n r e g i m e w i l l a p p l y . Q i s a n o r m a l i z e d measure o f t h e g r a t i n g t h i c k n e s s . S m a l l v a l u e s o f Q < 1, i . e . t h i n g r a t i n g s were b e l i e v e d t o g i v e Raman-Nath o p e r a t i o n . V a l u e s o f Q > 10. i . e . t h i c k g r a t i n g s , were b e l i e v e d t o g i v e Bragg regime o p e r a t i o n . A l t h o u g h the p a r a m e t e r Q has been e x t e n s i v e l y used t o d i s t i n g u i s h between t h e two r e g i m e s , i t i s n o t g e n e r a l l y r e a l i z e d t h a t i t i s n o t a l w a y s a r e l i a b l e c r i -t e r i o n b u t r e q u i r e s , as K l e i n and Cook n o t e d a t t h e end o f t h e i r p a p e r , a l i m i t a t i o n on t h e g r a t i n g s t r e n g t h im^L/A cos6 (where 0 i s t h e a n g l e d i f -f r a c t i o n o f t h e i n c i d e n t wave, and n^ i s t h e a m p l i t u d e o f t h e i n d e x modu-l a t o r (assumed p u r e l y s i n u s o i d a l ) ) . K l e i n and Cook have i n d i c a t e d t h a t , i n o r d e r f o r Q t o p r e d i c t t h e d i f f r a c t i o n r e g i m e , t h e g r a t i n g s t r e n g t h must be l e s s t h a n t h r e e . R e c e n t l y , Magnusson and G a y l o r d (1977) have shown t h e o -r e t i c a l l y t h a t f o r l a r g e m o d u l a t i o n , h i g h e r o r d e r waves become i m p o r t a n t (Raman-Nath regime) even f o r l a r g e Q. C o n v e r s e l y , f o r s m a l l m o d u l a t i o n , o n l y a s i n g l e wave i s d i f f r a c t e d (Bragg regime) - i n s p i t e of s m a l l v a l u e s o f Q. K a s p a r (1973) came t o a s i m i l a r c o n c l u s i o n when he compared h i s t h e o r y w i t h t h e c o u p l e d , wave t h e o r y of K o g e l n i k ( 1 9 6 9 ) . E x p e r i m e n t a l l y , h i g h e r o r d e r d i f -f r a c t i o n (Raman-Nath re g i m e ) has been o b s e r v e d f r o m h o l o g r a p h i c g r a t i n g s formed i n l i t h i u m n i o b a t e and d i c h r o m a t e d g e l a t i n where t h e v a l u e o f t h e p a r a m e t e r Q p r e d i c t e d B r a g g r e g i m e o f d i f f r a c t i o n i . e . o n l y one d i f f r a c t e d wave . (Wood e t t a l . 1975, Magnusson e t a l . 1977, Moharam and Young 1978a and A l f e r n e s s 1976). I n t h i s c h a p t e r , a condensed v e r s i o n of t h e t h e o r y o f l i g h t d i f f r a c t i o n by p e r i o d i c phase g r a t i n g s i s p r e s e n t e d . A p a r ameter p d e f i n e d 2 2 as A q / a n Q n ^ , i s p r o p o s e d as a r e l i a b l e r e p l a c e m e n t f o r Q as t h e c r i t e r i o n t o d i s t i n g u i s h between t h e two d i f f r a c t i o n r e g i m e s . The e f f e c t s o f t h e h i g h e r o r d e r m o d u l a t i o n of t h e i n d e x ( s m a l l e r g r a t i n g s p a c i n g s ) on t h e 10. d i f f r a c t i o n p r o b l e m a r e a l s o c o n s i d e r e d . 2.2 The o r y o f L i g h t D i f f r a c t i o n by Phase G r a t i n g s The s y s t e m c o n f i g u r a t i o n o f t h e d i f f r a c t i o n p r o b l e m under c o n -s i d e r a t i o n i s shown i n F i g . 2.1. A monochromatic p l a n e wave $ w i t h p r o p a -g a t i o n v e c t o r ~o i s i n c i d e n t , a t an a n g l e 0 t o t h e s u r f a c e n o r m a l , on a p e r i o d i c o g r a t i n g d e s c r i b e d by r e f r a c t i v e i n d e x j . v / 17 ~ \ (2.1) n = n + E, n c o s ( p K - r ) o p = l p where n^ i s t h e mean r e f r a c t i v e i n d e x , n^ i s t h e a m p l i t u d e of t h e p t n F o u r i e r component of t h e i n d e x m o d u l a t i o n , K = (2TT/A) [x cos 'ip + z s i n ip] i s t h e g r a t i n g v e c t o r , i\> i s t h e s l a n t a n g l e o f t h e g r a t i n g and A i s t h e g r a t i n g w a v e l e n g t h . F o r a h o l o g r a p h i c a l l y p r o d u c e d g r a t i n g A = A /2 s i n 6 w w h e r e A i s t h e r e c o r d i n g l i g h t w a v e l e n g t h and 0 i s o n e - h a l f t h e a n g l e w w between the two r e c o r d i n g beams. F o r a c o u s t i c a l l y p r o d u c e d g r a t i n g s , A i s t h e u l t r a s o n i c w a v e l e n g t h i n t h e g r a t i n g medium. R e f l e c t i o n s a t t h e g r a t i n g s u r f a c e s a r e n e g l e c t e d . The s c a l a r wave e q u a t i o n i s [ V 2 + ( 6 2 - j a B ) ] E ( x , z ) = 0 (2.2) where E^ i s t h e complex a m p l i t u d e o f t h e e l e c t r i c f i e l d , a i s t h e i n t e n s i t y a b s o r p t i o n c o n s t a n t , 3 = i s t h e p r o p a g a t i o n c o n s t a n t and A q i s t h e l i g h t w a v e l e n g t h i n a i r . Eqs. 2.1 and 2.2 may be s o l v e d by r e s o l v i n g t h e e l e c t r i c f i e l d i n t o i t s F o u r i e r e x p a n s i o n w h i c h may be w r i t t e n as 00 — E ( x , z ) = E^. <j> exp (-ja - r ) (2.3) * q=—°° where a„ = d ~ qK. q o ^ Combining 2.1, 2.2 and 2.3 and assuming t h a t 3 » a and XIQ » n p (p>0) i ; e . a w e a k l y m o d u l a t e d medium, we a r r i v e a t an i n f i n i t e s e t of c o u p l e d wave L -—> F i g . 2.1 System c o n f i g u r a t i o n e q u a t i o n s . 3<j> 2  C q T 7 ~ + " J ( 7 T XO^ 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 = cos 0 - q cos iHX /An ) , and 6 i s t h e a n g l e o f r e f r a c t i o n o f t h e q H • o o & z e r o o r d e r mode (q=0) i n t h e medium. Second o r d e r d e r i v a t i v e s o f <f>^  w i t h r e s p e c t t o z a r e n e g l e c t e d as i n p r e v i o u s work. Chu and Tamir (1970) and Kong (1977) showed t h a t t h e s e s i m p l i f i e d f i r s t o r d e r c o u p l e d wave e q u a t i o n s (Eq. 2.4) may be a c c u r a t e l y a p p l i e d t o w e a k l y modulated medium (n <0.01). •P. To g a i n i n s i g h t i n t o t h e phenomenon, Eq. 2.4 w i l l be s i m p l i f i e d , w i t h o u t s i g n i f i c a n t l o s s o f g e n e r a l i t y by c o n s i d e r i n g o n l y u n s l a n t e d g r a t i n g s (^=90°) and assuming t h a t t h e i n d e x m o d u l a t i o n i s p u r e l y s i n u s o i d a l i . e . n = 0 f o r p > l . Eq. 2.4 may t h e n be r e w r i t t e n a s : 9<j> 3.= • 2 8 5 T J P q V * q + J l V l + V l ] ( 2 ' 5 ) 2 2 where p = X / A n n . . , E,'c',= v ( z / L ) and v = TTn.,L/X cos 0. L i s t h e g r a t i n g o o l 1 o t h i c k n e s s , B = 2A s i n 0/X q. B i s e q u a l t o one f o r l i g h t i n c i d e n c e s a t i s -q o q B f y i n g t h e q t n Bragg a n g l e . The a b s o r p t i o n c o n s t a n t a has been n e g l e c t e d , i t can be a l l o w e d f o r by t h e s u b s t i t u t i o n = <j> exp (-az/cos 0 ) . 2.3 D i s c u s s i o n E x a m i n a t i o n o f Eq. 2.5 shows t h e q mode i s c o u p l e d t o i t s e l f and t o t h e two a d j a c e n t modes ( q - l t n and q + l t n modes). E f f e c t i v e e nergy 2 t r a n s f e r between modes r e q u i r e > e s s e n t i a l l y t h a t t h e f a c t o r pq (1 - B^) be r e l a t i v e l y s m a l l s i n c e , i f t h i s f a c t o r i s much l a r g e r t h a n 1, a l l t h e energy w i l l be c o u p l e d b a c k t o t h e q mode. T h i s c a n be e a s i l y seen f r o m Eq. 2.5 by n e g l e c t i n g t h e second t e r m o f t h e r i g h t hand s i d e . T h e r e f o r e , i f p < l , a p p r e c i a b l e energy may be t r a n s f e r r e d s u c c e s s f u l l y t o h i g h e r o r d e r modes up t o some v a l u e o f q p r o v i d e d t h e magnitude o f B^ I s a p p r o p r i a t e l y l i m i t e d . The number o f o b s e r v a b l e h i g h e r o r d e r modes depends on t h e f a c t o r 2 pq ( i . e . t h e s m a l l e r p i s , t h e l a r g e r t h e number of d i f f r a c t e d w a v e s ) . I f p=0 Eq. 2.5 g i v e s t h e w e l l - k n o w n s o l u t i o n i n terms o f B e s s e l f u n c t i o n s <|>q = j ^ J ^ ( 2 v ) . T h i s s o l u t i o n i s o f t e n o b t a i n e d by F o u r i e r e x p a n s i o n of t h e t r a n s m i t t e d wave w i t h s p a t i a l l y s i n u s o i d a l phase m o d u l a t i o n . A s i m i l a r s o l u t i o n was o b t a i n e d by K l e i n and Cook (1967) f o r t h e i r p a r a m e t e r 2 Q = n Q= 0. However, as t h e t h i c k n e s s L of t h e g r a t i n g goes t o z e r o (Q;-> 0) t h e m o d u l a t i o n o f t h e r e f r a c t i v e i n d e x must go t o i n f i n i t y t o r e t a i n t h e f i n i t e phase s h i f t (n^L) and as ny+ <*>} p 0. C l e a r l y , t h e c a s e where Q o r p = 0 i s a n o n - p h y s i c a l s i t u a t i o n . Thus f o r p-cl t h e d i f f r a c t i o n p r o -2 c e s s i s i n the Raman-Nath r e g i m e . I f p » l t h e f a c t o r pq (1 - B^) i s much l a r g e r t h a n one f o r a l l modes e x c e p t t h e z e r o o r d e r mode (q=0) and t h e mode w i t h q s u c h t h a t B — - ' l f o r g i v e n 0 i . e . t h e Bragg c o n d i t i o n h o l d s o r n e a r l y h o l d s . T h e r e f o r e , f o r p>>l a p p r e c i a b l e energy may be i n t e r c h a n g e d between th e z e r o o r d e r mode and t h e mode f o r w h i c h B^~ 1. The c o u p l i n g c o n s t a n t between t h e s e two modes i s p r o p o r t i o n a l t o n^. Thus, f o r p u r e l y s i n u s o i d a l g r a t i n g s , e n e r g y may be t r a n s f e r r e d o n l y t o t h e f i r s t o r d e r mode i f t h e a n g l e o f i n c i d e n c e s a t i s f i e s t h e f i r s t B r a g g a n g l e ( i . e . B^ = 1 ) . I f t h e m o d u l a t i o n o f t h e r e f r a c t i v e i n d e x c o n t a i n s h a r m o n i c s i n a d d i t i o n t o t h e f u n d a m e n t a l component, e n e r g y may be t r a n s f e r r e d o n l y t o t h e q f c ^ o r d e r p r o -v i d e d t h a t n^ ^ 0 and t h e a n g l e o f i n c i d e n c e s a t i s f i e s t h e q Bragg a n g l e ( i . e . B - 1 ) . T h e r e f o r e , f o r p>>l o n l y one d i f f r a c t e d wave i s p r o d u c e d even i f t h e g r a t i n g i s n o n s i n u s o i d a l and t h u s t h e d i f f r a c t i o n p r o c e s s i s i n t h e B r a g g r e g i m e . A l s o as p becomes l a r g e r , t h e g r a t i n g becomes more s e l e c -2 t i v e i . e . B must be c l o s e r t o one so t h a t t h e p r o d u c t pq (1 - B ) i s s m a l l q q enough t h a t t h e energy may be t r a n s f e r r e d t o t h e q t n mode. The above q u a l i t a t i v e a n a l y s i s was c o n f i r m e d by s o l v i n g Eq. 2.5 u s i n g t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a computer c e n t r e , f o u r t h o r d e r R u n g e - K u t t a r o u t i n e w i t h e r r o r c o n t r o l w i t h t h e boundary c o n d i t i o n s <f> (0) = 1.0 and <f> (0) = 0.0 f o r q 4 0. The r e l a t i v e i n t e n s i t y I o f t h e ° q P q wave i s d e f i n e d as <J>^  (v) <j>^  (v) ( t h e a b s o l u t e i n t e n s i t y i s 1^/2^ where Q, i s t h e c h a r a c t e r i s t i c i mpedance). The number o f modes i n c l u d e d ( i . e . t h e number o f e q u a t i o n s ) was i n c r e a s e d f o r each c a s e u n t i l no f u r t h e r s i g n i f i -c a n t e f f e c t o c c u r r e d f o r a g i v e n v a l u e o f p, e.g. f o r p = 50, s i x modes were i n c l u d e d and f o r p = 0.1, 24 modes were needed. F i g . 2.2 shows t h e r e l a t i v e i n t e n s i t i e s o f s e v e r a l d i f f r a c t e d 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. E x a c t Bragg i n -c i d e n c e f o r t h e f i r s t mode was assumed ( i . e . B^ = 1 ) . I n f a c t , t h e i n t e n -s i t i e s o f t h e d i f f r a c t e d waves are p l o t t e d v s . t h e g r a t i n g t h i c k n e s s L s i n c e n^ must be k e p t c o n s t a n t t o keep p c o n s t a n t . By i m p l i c a t i o n , t h e h o r i z o n t a l a x i s r e p r e s e n t s t h e n o r m a l i z e d t h i c k n e s s p a r a meter Q o f K l e i n and Cook (1966) s i n c e Q = 2vp. F i g . 2.2 shows t h a t even f o r v a l u e s o f v c o r r e s p o n d i n g t o a t h i c k g r a t i n g and l a r g e v a l u e s o f Q ( v a l u e s o f Q>10"were b e l i e v e d t o g i v e Bragg r e g i m e o p e r a t i o n ) , t h e energy i s t r a n s f e r r e d s u c c e s s i v e l y i n t o a l a r g e number o f modes and t h e Raman-Nath regime a p p l i e s . F i g . 2.3 shows a p l o t o f t h e i n t e n s i t i e s o f t h e z e r o and f i r s t o r d e r modes a g a i n s t t h e g r a t i n g s t r e n g t h v (and by i m p l i c a t i o n Q) f o r p ^ 50 and Br a g g i n c i d e n c e f o r t h e f i r s t o r d e r wave ( B-j= 1) • The energy i s t r a n s -f e r r e d t o and f r o between o n l y t h e z e r o and f i r s t o r d e r modes. I t has been shown by P h a r i s e a u (1956) t h a t t h e i n t e n s i t i e s o f t h e h i g h e r o r d e r modes a r e 2 of t h e o r d e r o f 1/p o r l e s s o f t h e z e r o o r d e r mode ( p r o v i d e d t h a t p>>l). A p l o t i d e n t i c a l t o F i g . 2.3 was o b t a i n e d when h i g h e r o r d e r h a r -monics o f t h e m o d u l a t i o n o f t h e r e f r a c t i v e i n d e x were i n c l u d e d i n t h e 15. 10 15 20 GRATING STRENGTH ( V ) F i g . 2.2 The i n t e n s i t y , o f t h e f i r s t : f o u r ' d i f f r a c t e d - m o d e s vs.'.the I-gr-a t i n g s t r e n g t h v f o r p= 1-and B = 1/q ( B r a g g i n c i d e n c e f o r the f i r s t o r d e r mode). -The sum of the i n t e n s i t i e l o f the f o u r modes i s a l m o s t u n i t y . A l l t h e o t h e r modes a r e n e g l i g i b l e . 1 16. G R A T I N G S T R E N G T H ( V ) F i g . 2.3 The in t e n s i t y of the zero and the f i r s t order modes vs. the grating strength v for p= 50 and B = 1/q. A l l the other modes are n e g l i g i b l e . c a l c u l a t i o n s , n^ was t a k e n t o be e q u a l t o n^ f o r a l l p > l , and as i n F i g . 2.3, p was 50 and = 1. I t i s c l e a r l y e v i d e n t f r o m F i g . 2.3 t h a t f o r P>>1 o n l y one d i f f r a c t e d wave i s p r o d u c e d ( i . e . Bragg regime o p e r a t i o n ) r e g a r d l e s s o f t h e v a l u e Q ( s m a l l v a l u e s o f Q (Q<1) were b e l i e v e d t o i n d i -c a t e Raman-Nath r e g i m e ) . As i t has been shown above f o r p>>l and B^ = 1, energy may be i n t e r c h a n g e d o n l y between the z e r o and f i r s t o r d e r modes ( i . e . <j> = 0 q>l) q Eq. 2.5 may be r e w r i t t e n as 9<f> ° r j * , (2.6) d £ '• J r l = J4> (2.7) 8£ J T o S o l v i n g t h e above two e q u a t i o n s w i t h t h e boundary c o n d i t i o n <j>0(0) = 1.0 and <|>, (0) = 0.0 and d e f i n i n g t h e d i f f r a c t i o n e f f i c i e n c y DE as 1 q DE = <f)*(v) A (v)/<f,*(v):<f>*(v) (2.8) - q q q o o we a r r i v e a t t h e w e l l known e x p r e s s i o n DE = c o s 2 ( v ) (2.9) 2 D E 1 = s i n (v) P h a r i s e a u (1956) and K o g e l n i k (1969) and Chu and Tamir (1970) have o b t a i n e d s i m i l a r e x p r e s s i o n s t o Eq. 2.9. These e x p r e s s i o n s were o b t a i n e d assuming p u r e l y s i n u s o i d a l m o d u l a t i o n of t h e r e f r a c t i v e i n d e x . However, bas e d on t h e above d i s c u s s i o n , t h e s e e x p r e s s i o n s a r e v a l i d a l s o f o r non-s i n u s o i d a l g r a t i n g s p r o v i d e d t h a t p>>l ahd B^ = 1. An i n t e r e s t i n g o b s e r v a t i o n i s t h a t , i f t h e change o f t h e r e f r a c -t i v e i n d e x i s p r o d u c e d h o l o g r a p h i c a l l y ( i . e . i t d e v e l o p s w i t h t i m e ) , i n i -t i a l l y p w i l l be v e r y l a r g e , s i n c e n^ i s v e r y s m a l l , t h e r e f o r e , t h e d i f f r a c t i o n p r o c e s s w i l l be i n t h e B r a g g r e g i m e . As n^ i n c r e a s e s , p de-c r e a s e s and, u n l e s s n ^ s a t u r a t e s a t a v a l u e s u c h t h a t p i s s t i l l l a r g e enough (>10), t h e d i f f r a c t i o n p r o c e s s w i l l e v e n t u a l l y move i n t o t h e Raman-Na t h r e g i m e . A f u r t h e r p o i n t i s t h a t t h e g r a t i n g t h i c k n e s s L ( o r t h e n o r m a l i z e d t h i c k n e s s Q) i s i r r e l e v a n t i n i t s e l f , s i n c e i t does n o t e n t e r p . T h i s may be e x p l a i n e d as f o l l o w s : To o b t a i n a s i g n i f i c a n t amount of d i f f r a c t i o n , t h e g r a t i n g s t r e n g t h v has t o be l a r g e enough i n some s e n s e . F o r v v t o be l a r g e , t h e n , i f L i s l a r g e , n ^ may be e i t h e r l a r g e o r s m a l l , so t h a t p w o u l d be s m a l l o r l a r g e r e s p e c t i v e l y , and we c o u l d be i n e i t h e r r e g i m e . On t h e o t h e r hand, i f L i s s m a l l , n^ w o u l d have t o be l a r g e t o o b t a i n a s u f f i c i e n t l y l a r g e v, i n w h i c h case p w o u l d be s m a l l , and t h e Raman-Nath regime would h o l d . T h e r e f o r e , t h e d i s t i n c t i o n between " t h i c k " and " t h i n " g r a t i n g s o r holograms as d e t e r m i n e d by t h e v a l u e L ( f o r g i v e n X , n Q and A) i n 2 Q = 2-nX^L/A n Q i s i n v a l i d as a d e s c r i p t i o n o f whether a s i n g l e d i f f r a c t e d beam w i l l be p r o d u c e d o r w h e t h e r many d i f f r a c t e d beams w i l l be p r o d u c e d . 2.4 Summary I t has been shown f o r t h e f i r s t t i m e t h a t a p a r a m e t e r p i s a l w a y s a r e l i a b l e c r i t e r i o n f o r d e c i d i n g whether t h e Raman-Nath o r t h e B r a g g r e g i m e w i l l be o b s e r v e d w i t h a g i v e n phase g r a t i n g . The parameter Q o f K l e i n and Cook (1967) w h i c h has been e x t e n s i v e l y used f o r t h i s p u r p o s e was shown t o be u n r e l i a b l e . A l a r g e p f a v o u r s t h e B r a g g r e g i m e . The r e l a t i v e l i g h t i n t e n s i t y g o i n g i n t o h i g h e r o r d e r modes ( o t h e r t h a n t h e mode f o r w h i c h t h e 2 Bragg c o n d i t i o n h o l d s ) i s o f t h e o r d e r 1/p so t h a t a v a l u e of p>10 i n d i -c a t e s more o r l e s s i d e a l B r a g g b e h a v i o u r . I t has been shown a l s o t h a t t h e p a r a m e t e r p w i l l work whether t h e g r a t i n g m o d u l a t i o n i s s i n u s o i d a l o r non-s i n u s o i d a l . I t has a l s o been shown t h a t t h e d i s t i n c t i o n between t h i c k and t h i n g r a t i n g s i s , s t r i c t l y s p e a k i n g , i n v a l i d as a d e s c r i p t i o n o f w h i c h d i f f r a c t i o n r e g i m e i s o p e r a t i n g . CHAPTER I I I MECHANISMS OF THE PHOTOREFRACTIVE EFFECT 3.1 I n t r o d u c t i o n As was m e n t i o n e d i n C h a p t e r 1, a number o f models have been p r o -posed t o e x p l a i n t h e p h o t o r e f r a c t i v e e f f e c t . The development o f t h e s e models i s o u t l i n e d and t h e i r m e r i t s a r e d i s c u s s e d . 3.2 The E l e c t r o - o p t i c N a t u r e of t h e P h o t o r e f r a c t i v e E f f e c t Chen, L a M a c c h i a and F r a z e r (1968) found t h a t t h e p o l a r i z a t i o n of t h e w r i t i n g beams was n o t c r i t i c a l d u r i n g h ologram w r i t i n g i n l i t h i u m n i o -b a t e . However, r e c o n s t r u c t i o n o f t h e h o l o g r a m was o n l y about 1/10 as e f f i c i e n t f o r o r d i n a r y r a y i l l u m i n a t i o n as f o r e x t r a o r d i n a r y r a y i l l u m i n a -t i o n . To e x p l a i n t h i s o b s e r v a t i o n , t h e y s u g g e s t e d t h a t t h e p h o t o r e f r a c t i v e p r o c e s s r e s p o n s i b l e f o r h o l o g r a m s t o r a g e i n v o l v e s t h e e l e c t r o - o p t i c e f f e c t i n the c r y s t a l . The d i f f r a c t i o n e f f i c i e n c y a t t h e i n i t i a l s t a g e s of h o l o -gram f o r m a t i o n i s p r o p o r t i o n a l t o t h e s q u a r e o f t h e r e f r a c t i v e i n d e x modu-l a t i o n n^ ( K o g e l n i k 1967). T h i s i m p l i e s t h a t , f o r Chen e t a l . ' s o b s e r v a -O 6 • t i o n , n^/n^ = 0.3, ( n e g l e c t i n g s u c h c o m p l i c a t i o n s as r e f l e c t i o n , a b s o r p t i o n , a n i s o t r o p y , e t c . ) . F o r a space c h a r g e f i e l d a l o n g t h e x^ ( c - a x i s ) o f t h e c r y s t a l , t h e r a t i o of t h e change i n t h e o r d i n a r y r e f r a c t i v e i n d e x t o t h e change i n t h e e x t r a o r d i n a r y r e f r a c t i v e i n d e x ( f o r X = 632.8 mm) i s 0.315 (Appendix B ) . That i s , Chen e t a l . ' s o b s e r v a t i o n i s c o n s i s t e n t w i t h a r e -f r a c t i v e i n d e x m o d u l a t i o n by a s p a c e charge f i e l d a l o n g t h e c - a x i s o f t h e c r y s t a l . See A p p e n d i x B f o r an o u t l i n e o f t h e e l e c t r o - o p t i c e f f e c t i n l i t h i u m n i o b a t e . 3.3 Chen's I n t e r n a l F i e l d M o d e l Chen (1969) measured t h e changes i n b i r e f r i n g e n c e i n d u c e d w i t h a s i n g l e l a s e r b e a m - i n l i t h i u m n i o b a t e u s i n g an a d j u s t a b l e compensator method. F i g . 3.1 shows t h e o p t i c a l l y i n d u c e d changes i n b i r e f r i n g e n c e a l o n g t h e b-and c - a x i s o f t h e c r y s t a l (Chen 1969). The b i r e f r i n g e n c e a l o n g t h e c - a x i s r e v e r s e s s i g n and t h a t a l o n g t h e b - a x i s does n o t . Assuming, an e l e c t r o -o p t i c e f f e c t , Chen c o n c l u d e d t h a t d r i f t , n o t d i f f u s i o n c auses t h e e f f e c t a l o n g t h e c - a x i s . To e x p l a i n t h i s o b s e r v a t i o n , Chen p r o p o s e d a model i n w h i c h t h e r e a r e two t y p e s o f t r a p s b e f o r e l i g h t i l l u m i n a t i o n . T r a p s o f t h e f i r s t t y p e a r e i n i t i a l l y f i l l e d and n e u t r a l , and t h e y can p r o v i d e e l e c t r o n s by p h o t o e x c i t a t i o n . T r a p s o f t h e second t y p e a r e i n i t i a l l y empty and c a n c a p t u r e e l e c t r o n s . Chen a l s o p o s t u l a t e d t h a t t h e r e i s an i n t e r n a l e l e c t r i c f i e l d d i r e c t e d f r o m t h e p o s i t i v e end o f t h e spontaneous p o l a r i z a t i o n o f t h e c r y s t a l t o t h e n e g a t i v e end. T h i s f i e l d w o u l d cause t h e p h o t o e x c i t e d e l e c t r o n s t o d r i f t a l o n g t h e c - a x i s toward t h e p o s i t i v e c - a x i s end, l e a v i n g b e h i n d p o s i t i v e c h a r g e s o f i o n i z e d t r a p c e n t r e s . Chen c l a i m e d t h a t " t h e p h o t o - e x c i t e d e l e c t r o n s w i l l be r e t r a p p e d and r e - e x c i t e d o u t o f t h e t r a p s u n t i l t h e y e v e n t u a l l y d r i f t o u t o f t h e i l l u m i n a t e d r e g i o n and a r e f i n a l l y r e t r a p p e d t h e r e . S i n c e t h e r e i s no p h o t o - e x c i t a t i o n o u t s i d e t h e i l l u m i n a t e d r e g i o n and f o r deep t r a p s , t h e t h e r m a l e x c i t a t i o n i s too weak t o r e - e x c i t e c h a r g e s o u t o f t h e t r a p s , t h e n e g a t i v e c h a r g e s s t a y t r a p p e d t h e r e . The space c h a r g e f i e l d t h u s c r e a t e d between t h e t r a p p e d e l e c t r o n s and t h e p o s i -t i v e i o n i z e d c e n t r e s c a u s e s t h e o b s e r v e d s p a t i a l v a r i a t i o n o f t h e i n d i c e s o f r e f r a c t i o n v i a t h e e l e c t r o - o p t i c e f f e c t " Chen (1969). S i n c e l i t h i u m n i o b a t e e x h i b i t s a l i n e a r e l e c t r o - o p t i c e f f e c t , t h e v a r i a t i o n A ( n e ~ n Q ) i s l i n e a r l y r e l a t e d t o t h e s p a t i a l v a r i a t i o n o f t h e e l e c t r i c f i e l d . Chen e s t i m a t e d t h a t t h e s p a c e c h a r g e f i e l d r e q u i r e d t o p r o d u c e t h e maximum o b s e r v e d v a l u e o f L:22. BEAM DIAMETER (a) J J b (b) F i g . 3.1 (a) The s o l i d l i n e (— ) shows t h e change i n b i r e f r i n g e n c e a l o n g the c - a x i s and the dashed l i n e (- -•-) shows the change a l o n g t h e b - a x i s due t o a beam o f c i r c u l a r symmetry, (b) Chen's p o s t u l a t e d space charge f i e l d d i s t r i b u t i o n w h i c h causes the o b s e r v e d change i n A(n - n ). e o A(n -n ) was 67 kV/cm. e o A l t h o u g h no o r i g i n was g i v e n f o r t h i s b u i l t - i n i n t e r n a l f i e l d , Chen's o b s e r v a t i o n s o f s h o r t - c i r c u i t p h o t o c u r r e n t i n l i t h i u m n i o b a t e l e d him t o c l a i m t h a t t h i s i n t e r n a l f i e l d e x i s t s . The d i r e c t i o n o f t h e p h o t o -c u r r e n t was c o n s i s t e n t w i t h a f i e l d o p p o s i t e t o P . Chynoweth (1956) had a l s o o b s e r v e d p h o t o c u r r e n t i n BaTiO^ i n t h e absence o f a p p l i e d f i e l d s . Chen (1969) has shown t h a t f i e l d s o f p y r o e l e c t r i c o r i g i n due t o t h e n o n u n i f o r m h e a t i n g o f t h e c r y s t a l by t h e l i g h t beam c o u l d n o t a c c o u n t f o r t h e o b s e r v e d s h o r t - c i r c u i t p h o t o c u r r e n t s i n c e dP/dT<0 and, t h e r e f o r e , t h e f i e l d w o u l d be i n t h e wrong d i r e c t i o n f o r h i s o b s e r v a t i o n . Amodei and S t a e b l e r (1972b) s u g g e s t e d t h a t t h i s b u i l t - i n f i e l d was o f p y r o e l e c t r i c o r i g i n d e v e l o p e d when th e c r y s t a l was c o o l e d f r o m a h i g h t e m p e r a t u r e . The development o f s u c h a f i e l d may be e x p l a i n e d i n t h e f o l l o w i n g way. A f e r r o e l e c t r i c c r y s t a l w i t h no f r e e c h a r g e s and no n e t spac e c h a r g e s would have a f i e l d c o r r e s p o n d i n g t o a p o l a r i z a t i o n c h a r g e P r e m p e r u n i t a r e a on f a c e s n o r m a l t o t h e c - a x i s . T h i s f i e l d w o u l d , i n f a c t , be above n o r m a l d i e l e c t r i c breakdown. I n prac--' t i c e t h e c r y s t a l w o u l d have been c o o l e d f r o m some h i g h t e m p e r a t u r e a t w h i c h a p p r e c i a b l e c o n d u c t i v i t y e x i s t e d , s u f f i c i e n t t o c a n c e l t h e f i e l d . E x c e s s c h a r g e s w o u l d a c c u m u l a t e c l o s e t o each c - f a c e . As c o o l i n g p r o g r e s s e d , t h e c o n d u c t i v i t y w o u l d f r e e z e o u t w h i l e t h e remanent p o l a r i z a t i o n P c o n t i n u e d rem t o change. F i n a l l y , an uncompensated component o f T^ w o u l d e x i s t g i v i n g a b u i l t - i n f i e l d o f magnitude 1 j; 1 _ ^ e m e t ' 3T o where T q and T^ a r e t h e t e m p e r a t u r e a t w h i c h t h e c o n d u c t i v i t y d i s a p p e a r s and t h e t e m p e r a t u r e o f t h e e x p e r i m e n t r e s p e c t i v e l y , and e i s t h e p e r m i t t i -v i t y . However, C o r n i s h , Moharam and Young (1976) have measured t h e s h o r t - c i r c u i t p h o t o c u r r e n t i n a Fe-doped l i t h i u m n i o b a t e c r y s t a l a f t e r c o o l -i n g i t (a) w i t h and (b) w i t h o u t s h o r t - c i r c u i t a p p l i e d t o t h e e l e c t r o d e s on t h e f a c e s a t t h e ends o f t h e c - a x i s . They f o u n d t h a t t h e p h o t o c u r r e n t was i n d e p e n d e n t o f t h e e l e c t r i c a l c o n d i t i o n s d u r i n g c o o l i n g . The p h o t o c u r r e n t s were measured a f t e r t h e p y r o e l e c t r i c c u r r e n t due t o t h e l i g h t had decayed. T h e i r e x p e r i m e n t s showed t h a t t h e s e s h o r t - c i r c u i t p h o t o c u r r e n t s a r e p r o d u c e d i n t h e absence o f any b u i l t - i n f i e l d o f p y r o e l e c t r i c o r i g i n . G l a s s , von d e r L i n d e and Negran (1974) have r e p o r t e d t h a t , a f t e r 20 h o u r s of c o n t i n u o u s i l -l u m i n a t i o n , t h e s h o r t - c i r c u i t p h o t o c u r r e n t s w h i c h t h e y measured remained c o n s t a n t . C o r n i s h (1976) has r e p o r t e d s i m i l a r measurements f o r 43 h o u r s o f c o n t i n u o u s i l l u m i n a t i o n . G l a s s e t a l . have c l a i m e d t h a t t h i s p h o t o c o n d u c t i -v i t y w o u l d r e l a x any i n t e r n a l f i e l d s and a decay of t h e p h o t o c u r r e n t ( i f i t i s due t o s u c h f i e l d s ) w o u l d be n o t i c e a b l e . 3.4 J o h n s t o n ' s P o l a r i z a t i o n M o d e l To remove t h e need t o assume a b u i l t - i n f i e l d o f unknown o r i g i n , J o h n s t o n (1970) p r o p o s e d an a l t e r n a t i v e model i n w h i c h p h o t o i n d u c e d v a r i a -t i o n s i n t h e m a c r o s c o p i c p o l a r i z a t i o n caused t h e p h o t o r e f r a c t i v e e f f e c t . He c l a i m e d t h a t i l l u m i n a t i o n of t h e c r y s t a l would e x c i t e t h e t r a p p e d e l e c ^ t r o n s t o t h e c o n d u c t i o n band r e s u l t i n g i n a change i n t h e d e n s i t y o f f i l l e d t r a p s i n t h e r e g i o n o f i l l u m i n a t i o n . T h i s i n t u r n w o u l d c a u s e a l o c a l change i n t h e p o l a r i z a t i o n . The d i v e r g e n c e o f t h e p o l a r i z a t i o n p r o d u c e s a f i e l d . T h i s f i e l d c a uses t h e e l e c t r o n s t o d r i f t b e f o r e r e t r a p p i n g and t h u s p r o d u c e s permanent change i n t h e p o l a r i z a t i o n . A f t e r t h e l i g h t i s t u r n e d o f f , t h e r e r e m a i n s a change i n t h e m a c r o s c o p i c p o l a r i z a t i o n w h i c h i n d u c e s a change i n t h e r e f r a c t i v e i n d i c e s o f t h e c r y s t a l . U s i n g t h i s m o del, J o h n s t o n was a b l e t o a c c o u n t q u a l i t a t i v e l y f o r t h e s p a t i a l l y dependent f e a t u r e s o f Chen's o b s e r v a t i o n s ( F i g . 3.1). How-e v e r , Amodei (1971a) and Amodei and S t a e b l e r (1972b) have shown t h a t t h e r e a r e a number o f d i f f i c u l t i e s w i t h t h i s mechanism. They c o n c l u d e d t h a t t o o l a r g e a d e n s i t y o f e l e c t r o n s w o u l d be r e q u i r e d t o e n t e r t h e c o n d u c t i o n band t o g e n e r a t e t h e l a r g e f i e l d s n e c e s s a r y t o a c c o u n t f o r t h e o b s e r v e d e f f e c t . The same e f f e c t c o u l d r e s u l t f r o m s p a c e c h a r g e f i e l d s c r e a t e d t h r o u g h s i m p l e d i f f u s i o n and r e t r a p p i n g p r o c e s s e s . The number of e l e c t r o n s i n v o l v e d w o u l d 3 be l e s s by a f a c t o r 10 t h a n w o u l d be r e q u i r e d i n J o h n s t o n ' s model. One mig h t add t h a t t h i s model cannot e x p l a i n a s t e a d y s t a t e s h o r t - c i r c u i t p h o t o -c u r r e n t w i t h t h e c r y s t a l u n i f o r m l y i l l u m i n a t e d . 3.5 D e f e c t S i t e s and I m p u r i t i e s The p h o t o r e f r a c t i v e e f f e c t i s most e f f i c i e n t when l i g h t o f t h e w a v e l e n g t h s 400 t o 500 nm i s u s e d , ( S e r r e z e and G o l d n e r 1 9 7 3 ) . C l a r k e t a l . (1973) s u g g e s t e d t h a t e x c i t a t i o n o c c u r s f r o m t r a p s w i t h i n t h e band gap. I t i s b e l i e v e d t h a t b o t h i m p u r i t i e s and d e f e c t s r e l a t e d t o t h e n o n - s t o i c h i o m e -t r y o f t h e c r y s t a l a c t as d e f e c t s i t e s . P h i l l i p s e t a l . (1972) have shown t h a t gamma i r r a d i a t i o n o f undoped LlNbO^ i n c r e a s e s t h e p h o t o r e f r a c t i v e s e n -s i t i v i t y by i n c r e a s i n g t h e c o n c e n t r a t i o n o f l a t t i c e d e f e c t s w h i c h a c t as e l e c t r o n t r a p s . I m p u r i t y d o p i n g w i t h e l e m e n t s s u c h as i r o n , manganese, c o p p e r , r h o d i u m , chromium and u r a n i u m ( P h i l l i p s e t a l . 1972, P e t e r s o n e t a l . 1971, 1973; M i k a m i e t a l . 1973, G l a s s e t a l . 1974a) a l s o i m p r o v e s t h e p h o t o -r e f r a c t i v e s e n s i t i v i t y o f t h e c r y s t a l . P h i l l i p s , Amodei and S t a e b l e r (1972) have shown t h a t i r o n i s , so f a r , t h e b e s t dopant t h a t has been f o u n d . P e t e r s o n e t a l . (1971) and C l a r k e t a l . (1973) have s u g g e s t e d t h a t i r o n r e p l a c e s a l i t h i u m i o n i n t h e c r y s -t a l l a t t i c e . M o s s b a u e r - e f f e c t s t u d y o f i r o n i m p u r i t i e s i n LiNbO^ l e d Keune 3+ e t a l . (1975) t o s u g g e s t t h a t t h e most l i k e l y s i t e o f t h e Fe i o n i s t h e 2+ Nb s i t e . They were n o t a b l e t o i d e n t i f y t h e most l i k e l y s i t e o f t h e Fe 2+ i o n s ; - D i s c h l e r and Rauber (1975) have s u g g e s t e d t h a t t h e e f f i c i e n t Fe i o n s a r e a s s o c i a t e d w i t h oxygen v a c a n c i e s . 3+ I r o n i m p u r i t i e s i n t h e t r i v a l e n t s t a t e Fe a r e t h e empty traps-, ( P e t e r s o n e t a l . 1971). The o c c u p i e d t r a p s can be c o n s i d e r e d as t h e same 2+ 2+ i o n i n a r e d u c e d v a l e n c e s t a t e Fe . The o c c u p i e d t r a p (Fe i o n ) i n t r o -duce o p t i c a l a b s o r p t i o n due t o p h o t o e x c i t a t i o n o f t h e t r a p p e d e l e c t r o n i n t o t h e c o n d u c t i o n band. The amount o f c o l o r a t i o n f o r a g i v e n c r y s t a l depends on t h e f r a c t i o n o f t r a p s t h a t a r e o c c u p i e d w h i c h i n t u r n depends on t h e o x i d a t i o n / r e d u c t i o n s t a t e o f t h e c r y s t a l ( P e t e r s o n e t a l . 1971, 1973; P h i l l i p s e t a l . 1972, 1974). S t a e b l e r and P h i l l i p s (1974) have shown t h a t t h e p h o t o r e f r a c t i v e s e n s i t i v i t y depends on t h e o x i d a t i o n / r e d u c t i o n s t a t e of t h e i r o n i m p u r i t i e s . The o x i d a t i o n / r e d u c t i o n s t a t e can be a d j u s t e d by c h e m i c a l t r e a t m e n t s . O x i -2+ 3+ d a t i o n o f Fe i o n t o t h e t r i v a l e n t s t a t e Fe can be a c h i e v e d by a n n e a l i n g t h e c r y s t a l i n a i r o r oxygen a t 600°C ( P e t e r s o n e t a l . 1 9 7 1 ) . S m i t h e t a l . (1968) found t h a t f i e l d a n n e a l i n g l i t h i u m n i o b a t e made t h e c r y s t a l a l m o s t i n s e n s i t i v e . The a p p l i e d f i e l d caused t h e i r o n i o n t o m i g r a t e towards t h e n e g a t i v e e l e c t r o d e s . A y e l l o w - b r o w n d e p o s i t e v e n t u a l l y appeared on t h e n e g a t i v e e l e c t r o d e as i r o n came r i g h t o u t of t h e c r y s t a l . C l a r k e t a l . (1973) r e p o r t e d t h a t a n n e a l i n g i n a i r o r oxygen c o n v e r t e d about 96% of t h e t , 3+ l r o n t o Fe 3+ 2+ Methods t o r e d u c e Fe i o n s t o t h e d i v a l e n t s t a t e Fe i n c l u d e h e a t i n g t h e c r y s t a l s i n an a r g o n atmosphere a t around 1000°C and h e a t i n g t h e c r y s t a l s i n a i r w h i l e packed i n a l i t h i u m s a l t s u c h as I^CO-j a t around 500°C f o r 40 h o u r s ( P h i l l i p s and S t a e b l e r 1 9 7 4 ) . R e d u c t i o n o f more t h a n 90% o f i r o n i m p u r i t y i o n s u s i n g t h e s e two methods was a c h i e v e d . ( S t a e b l e r and P h i l l i p s 1974). 3.6 The T r a n s p o r t L e n g t h 37-6.1 I n t r o d u c t i o n To i n v e s t i g a t e t h e phenomenon of h o l o g r a m s t o r a g e i n l i t h i u m n i o -b a t e , i t i s c o n v e n i e n t t o a n a l y s e t h e f o r m a t i o n o f p r o t o t y p e holograms formed by t h e i n t e r f e r e n c e p a t t e r n o f two monochromatic, c o h e r e n t , i n f i n i t e p l a n e waves. The two waves a r e i n c i d e n t s y m m e t r i c a l l y on t h e c r y s t a l and a r e i n t e r s e c t i n g a t an a n g l e 29. The two waves a r e i n t h e x - p l a n e ( w h i c h a l s o c o n t a i n s t h e c - a x i s ) as shown i n F i g . 3.2. The two p l a n e waves a r e commonly c a l l e d t h e r e f e r e n c e wave R and t h e s u b j e c t o r s i g n a l wave S. They may be r e p r e s e n t e d by R = f exp ( - j a ' r ) exp ( j u t ) (3.1) "S-= 3 exp (-jp»r) exp ( j w t ) (3.2) where ? and s a r e t h e a m p l i t u d e s o f t h e two waves. a and p a r e t h e p r o p a -g a t i o n v e c t o r s o f t h e r e f e r e n c e and s u b j e c t waves, r e s p e c t i v e l y , OJ i s t h e f r e q u e n c y of t h e l i g h t wave and r = ( x , y , z ) . The i n t e n s i t y o f t h e i n t e r f e r e n c e p a t t e r n o f t h e two waves may be w r i t t e n as I = |R + "S | 2 (3.3) S u b s t i t u t i n g Eqs. 3.1 and 3.2 i n Eq. 3.3 and c o n s i d e r i n g t h e c o o r d i n a t e s y s t e m i n F i g . 3.2, we a r r i v e a t I ( x ) = 1 (1 + m c o s ( K x ) ) (3.4) o 2 2 where I =|r| +|s| i s t h e av e r a g e l i g h t i n t e n s i t y and m = 2 f ' s / l o i s t h e m o d u l a t i o n r a t i o . K = 2ir/A where A = A, / ( 2 s i n 8 ) . A i s t h e g r a t i n g wave-o l e n g t h , and A q i s t h e l i g h t w a v e l e n g t h . I t i s n o t c o n c l u s i v e l y known how f a r t h e p h o t o e x c i t e d e l e c t r o n s move b e f o r e r e t r a p p i n g . Chen e t a l . (1968) found t h a t t h e y c o u l d s t o r e 28. X A A T A _L D F i g . 3.2 C o n f i g u r a t i o n f o r h o l o g r a m r e c o r d i n g . The two p l a n e waves R and S i n t e r f e r e t o produce a s i n u s i o d a l l i g h t i n t e n s i t y p a t t e r n w i t h a p e r i o d A. holograms w i t h a r e s o l u t i o n o f g r e a t e r t h a n 1600 line/mm. T h i s l e d them t o assume t h a t t h e d i s p l a c e m e n t of e l e c t r o n s r e l a t i v e t o t h e i l l u m i n a t i o n must be a f r a c t i o n o f a m i c r o n t o be a b l e t o r e c o r d t h e v a r i a t i o n i n i n t e n s i t y . G l a s s e t a l . (1974) and K r a t z i g e t a l . ( 1 9 7 7 a ) , e s t i m a t e d f r o m p h o t o c u r r e n t o measurement t h a t t h e t r a n s p o r t l e n g t h i s of t h e o r d e r 0.5 and 2.4A r e s p e c -t i v e l y . They assumed u n i t y quantum e f f i c i e n c y and p o i n t e d o u t t h a t , i f t h e quantum e f f i c i e n c y i s l e s s t h a n u n i t y , t h e d i s p l a c e m e n t w o u l d i n c r e a s e . I f t h e s e e s t i m a t e s a r e c o r r e c t , i t w o u l d be r e a s o n a b l e t o r e s t r i c t t h e t r a n s p o r t l e n g t h t o a v e r y s m a l l f r a c t i o n o f t h e g r a t i n g w a v e l e n g t h A ( f o r most p r a c t i -c a l c a s e s A r a n g e s f r o m 0.5 t o 5 ym). S t a e b l e r and Amodei (1972b) have o b s e r v e d energy t r a n s f e r between th e r e f e r e n c e R and t h e s u b j e c t wave S d u r i n g h o l o g r a m w r i t i n g . T h e i r a n a l y s i s , u s i n g a model d e v e l o p e d by Amodei (1972a) assuming s h o r t t r a n s -p o r t l e n g t h , l e d them t o c o n c l u d e t h a t d i f f u s i o n i s an i m p o r t a n t mechanism i n h o l o g r a m f o r m a t i o n . Young e t a l . (1974) d e v e l o p e d a model w i t h o u t t h e r e -s t r i c t i o n on t h e t r a n s p o r t l e n g t h . T h e i r model can a c c o u n t f o r t h e energy t r a n s f e r between t h e two beams, even when d r i f t i s t h e dominant mechanism. S t u d i e s o f t h e dependence o f t h e d i f f r a c t i o n e f f i c i e n c y o f holograms s t o r e d i n l i t h i u m n i o b a t e on t h e g r a t i n g space ( A l p h o n s e e t a l . 1975) were t a k e n as t e n d i n g t o s u p p o r t Young e t a l . (1974) s u g g e s t i n g t h a t t h e t r a n s p o r t l e n g t h s h o u l d n o t be assumed s h o r t compared t o t h e g r a t i n g s p a c i n g under a l l c i r c u m s t a n c e s . 3.6.2 Amodei's Model f o r S h o r t T r a n s p o r t L e n g t h Amodei (1971a) assumed t h a t t h e r a t e of g e n e r a t i o n of p h o t o e x c i -t e d e l e c t r o n s i s p r o p o r t i o n a l t o t h e l i g h t i n t e n s i t y (Eq. 3.4). The assump-t i o n t h a t t h e t r a n s p o r t l e n g t h o f t h e f r e e e l e c t r o n i s s u b s t a n t i a l l y s h o r t e r t h a n t h e g r a t i n g wave l e n g t h A, i m p l i e s t h a t t h e f r e e c a r r i e r c o n c e n t r a t i o n w o u l d r e m a i n a f a i t h f u l r e p l i c a o f t h e g e n e r a t i o n r a t e and wou l d be g i v e n by n ( x ) = g Q T ( l + m c o s Kx) ( 3 . 5 ) where t i s t h e l i f e t i m e o f t h e c a r r i e r s and i s assumed t o be i n d e p e n d e n t o f th e c a r r i e r c o n c e n t r a t i o n . g Q i s t h e a v e r a g e g e n e r a t i o n r a t e and i s p r o p o r -t i o n a l t o I q , t h e a v e r a g e l i g h t i n t e n s i t y (Eq. 3 . 4 ) . The s p a t i a l d i s t r i b u -t i o n of t h e c u r r e n t d e n s i t y was t a k e n as t h e sum of t h e d r i f t and d i f f u s i o n components J ( x ) = qD + q u n ( x ) E ( x ) ( 3 . 6 ) where q i s t h e e l e c t r o n i c c h a r g e , D i s t h e d i f f u s i o n c o n s t a n t , u i s t h e m o b i l i t y and E ( x ) i s t h e t o t a l e l e c t r i c f i e l d . The r a t e a t w h i c h c h a r g e d e n s i t y p a c c u m u l a t e s a t any p o i n t i s g i v e n by t h e c o n t i n u i t y e q u a t i o n s c d p s c = _ d J ( x ) ( 3 . 7 ) d t dx The space c h a r g e f i e l d s u p p o r t e d by t h e spa c e c h a r g e d e n s i t y i s P s c E (x) = / — dx (3.8) SC £ Amodei assumed t h a t , i n t h e l i n e a r i n i t i a l s t a g e of h o l o g r a m f o r m a t i o n , t h e spa c e c h a r g e f i e l d E g c may be n e g l e c t e d i n t h e t r a n s p o r t e q u a t i o n (Eq. 3 . 6 ) . He c o n s i d e r e d (a) t r a n s p o r t due t o d r i f t o n l y J = q u n E ^ E ^ i s t h e d r i f t f i e l d ) ; and (b) t r a n s p o r t due t o d i f f u s i o n o n l y dn J = 9 D S o l u t i o n o f Eqs. 3.7 and 3.8 y i e l d f o r d r i f t o n l y E = - (quTg /e)E tm cos Kx ( 3 . 9 ) SC o o and f o r d i f f u s i o n o n l y , E = (qyTg o/e)E Dtm s i n Kx ( 3 . 1 0 ) where E^ = kTK/q i s t h e e q u i v a l e n t d i f f u s i o n f i e l d , k i s Boltzmann's c o n - .. s t a n t and T i s t h e t e m p e r a t u r e . Thus, w i t h t h e a s s u m p t i o n of s h o r t t r a n s p o r t l e n g t h , t h e spa c e 31. c h a r g e f i e l d i s o b t a i n e d w i t h a d i f f e r e n c e o f IT/2 i n phase s h i f t a c c o r d i n g t o w h e t h e r d i f f u s i o n o r d r i f t i s o p e r a t i v e . 3.6.3 Young e t a l . ' s M o d e l w i t h A r b i t r a r y T r a n s p o r t L e n g t h Young, Wong, Th e w a l t and C o r n i s h (1974) removed t h e need f o r t h e a s s u m p t i o n t h a t t h e t r a n s p o r t l e n g t h be s h o r t compared t o t h e g r a t i n g s p a c -i n g by u s i n g t h e c o n t i n u i t y e q u a t i o n | a . g o ( i + , „ , K K ) - i + } H (3 .1D I n t h e i n i t i a l l i n e a r s t a g e o f ho l o g r a m f o r m a t i o n , t h e r a t e o f change of t h e c o n c e n t r a t i o n o f f r e e c a r r i e r s i n t h e c o n d u c t i o n i s z e r o a t c o n s t a n t l i g h t i n t e n s i t y so t h a t 0 = g (1 + m cos Kx) - - + - ^ (3.12) °o T q 8x Amodei's (1971a) a s s u m p t i o n t h a t n = T g Q ( l + m cos Kx) {Eq. 3.5) c o r r e s p o n d s 1 9 J t o d r o p p i n g t h e te r m — — i n Eq. 3.12. Young e t a l . o b t a i n e d a s o l u t i o n f o r t h e space c h a r g e f i e l d i n t h e i n i t i a l s t a g e o f ho l o g r a m f o r m a t i o n f o r (a) t r a n s p o r t due t o d r i f t o n l y J = q y n E Q ; and (b) t r a n s p o r t due t o d i f f u -s i o n o n l y J = qD — . S o l u t i o n o f Eqs. 3.7,3.8 and 3.12 y i e l d s f o r d r i f t o n l y mE t E = - ( q y r g /e) ~-r [ c o s Kx + KL s i n Kx] (3.12) S C ° 1 + K 2 L 2 and f o r d i f f u s i o n o n l y mE Dt E = (qyxg /e) x—„ s i n Kx (3.14) s c o 1 + k 2 l . 2 L = V T E i s t h e d r i f t t r a n s p o r t l e n g t h and L' = (DT) i s t h e d i f f u s i o n t r a n s -o p o r t l e n g t h . 32. 3.6.4 D i s c u s s i o n Eq. 3.13 s h o w s , " f o r space c h a r g e f i e l d s d e v e l o p e d by d r i f t , t h e phase s h i f t between t h e i n c i d e n t p e r i o d i c l i g h t i n t e n s i t y p a t t e r n and t h e p e r i o d i c space c h a r g e f i e l d depends on t h e d r i f t l e n g t h . I f KL<<1, Eq. 3.13 i s r e d u c e d t o Amodei's e x p r e s s i o n (Eq. 3.10) and t h e space c h a r g e f i e l d i s i n phase w i t h t h e l i g h t p a t t e r n . I f KL>>1 Eq. 3.13 r e d u c e s t o E ^ c = ( q g o m t / K e ) s i n Kx. Here t h e phase s h i f t i s TT/2, and t h e f i e l d E g c i s in d e p e n d e n t o f E q ( t h e d r i f t f i e l d ) . Young e t a l . s u g g e s t e d t h a t t h i s de-pendence o f t h e phase s h i f t on t h e d r i f t l e n g t h c o u l d a c c o u n t f o r t h e o b s e r -v a t i o n o f energy t r a n s f e r between t h e two w r i t i n g beams. I n c a s e o f d i f f u s i o n - f o r m e d space c h a r g e f i e l d , i f KL'>>1, t h e same l i m i t i n g c a s e i s o b t a i n e d as f o r d r i f t o n l y w i t h KL>>1. F o r s h o r t d i f f u s i o n (KL'<<1) Eq. 3.14 r e d u c e s t o Amodei's e x p r e s s i o n (Eq. 3.11). However, w h a t e v e r t h e magnitude o f L'K, t h e phase s h i f t r e m a i n s t h e same because t h e f r e e e l e c t r o n s have e q u a l p r o b a b i l i t y o f moving i n e i t h e r d i r e c -t i o n . As we m e n t i o n e d b e f o r e , . - e s t i m a t i o n s o f t h e magnitude o f t h e d r i f t l e n g t h r a n g e s f r o m 0.5 t o 2.4 & assuming u n i t y quantum e f f i c i e n c y ( G l a s s e t a l . 1974 and K r a t z i g e t a l . 1977a). T h e r e f o r e , assuming g r a t i n g s p a c i n g A = lum t h e upper l i m i t o f KL = 3.14x10 where E, i s t h e quantum e f f i -c i e n c y . T h e r e f o r e , t h e d r i f t l e n g t h may be assumed s h o r t compared t o t h e g r a t i n g s p a c i n g , i f t h e quantum e f f i c i e n c y i s g r e a t e r t h a n 1%. 3.7 The B u l k P h o t o v o l t a i c E f f e c t G l a s s , von. d e r L i n d e and Negran (1974, 1975a) have p r o p o s e d t h a t t h e p h o t o r e f r a c t i v e e f f e c t i s caused by an e n t i r e l y new t r a n s p o r t mechanism w h i c h t h e y have l a b e l l e d t h e " b u l k p h o t o v o l t a i c e f f e c t " . O b s e r v a t i o n o f s t e a d y s t a t e s h o r t c i r c u i t p h o t o c u r r e n t a f t e r 20 h r s . of c o n t i n u o u s i l l u m i n a t i o n l e d G l a s s e t a l . t o b e l i e v e t h a t t h i s p h o t o -c u r r e n t i s due t o a b u l k p h o t o v o l t a i c e f f e c t and n o t b u i l t - i n i n t e r n a l f i e l d s (Chen 1 9 6 9 ) , s i n c e p h o t o c o n d u c t i v i t y w o u l d r e l a x any i n t e r n a l f i e l d and a decay o f t h e p h o t o c u r r e n t would be n o t i c e a b l e . G l a s s e t a l . have shown t h a t t h i s new b u l k p h o t o v o l t a i c e f f e c t c o u l d a c c o u n t q u a n t i t a t i v e l y f o r t h e o p t i c a l i n d u c e d change i n t h e r e f r a c t i v e i n d e x and i t does n o t have th e drawbacks of o t h e r models (See 3.3 and 3.4). A l t h o u g h t h e p h y s i c s o f the new phenomenon has n o t been f u l l y e s t a b l i s h e d , t h e r e i s a g e n e r a l a g r e e -ment t h a t t h i s new phenomenon e x i s t s . To a c c o u n t f o r t h e b u l k p h o t o v o l t a i c e f f e c t G l a s s e t a l . have p o s t u l a t e d t h a t e l e c t r o n s c o n t r i b u t i n g t o t h e p h o t o c u r r e n t and t h e p h o t o -2+ r e f r a c t i v e p r o c e s s r e s i d e i n asymmetric p o t e n t i a l w e l l s (Fe i o n s ) . Upon e x c i t a t i o n , t h e p r o b a b i l i t i e s o f movement a l o n g the ± c - a x i s w i l l d i f f e r , +2 s i n c e t h e Nb-Fe d i s t a n c e s i n t h e ± c - a x i s a r e d i f f e r e n t . Because of t h e p r e f e r r e d d i r e c t i o n t h e r e i s a n e t e l e c t r o n c u r r e n t F o l l o w i n g t h e e x c i t a t i o n o f na e l e c t r o n , t h e i o n i z e d i m p u r i t y i s d i s p l a c e d a l o n g t h e p o l a r a x i s o f t h e c r y s t a l due t o Franck-Condon r e l a x a t i o n . T h i s g i v e s r i s e t o a d i s p l a c e m e n t c u r r e n t J £ 2 * A f t e r a c e r t a i n t i m e t h e o r i g i n a l e l e c t r o n i c momentum w i l l be r a n d o m i z e d and no l o n g e r c o n t r i b u t e t o t h e c u r r e n t d e n s i t y . G l a s s e t a l . a l s o s u g g e s t t h a t t h e r e t r a p p i n g p r o c e s s may be as y m m e t r i c . The p r o b a b i l i t i e s o f r e t r a p p i n g f r o m e l e c t r o n s a p p r o a c h i n g a t r a p f r o m t h e ± c - d i f e c t i o n a r e d i f f e r e n t and a n e t r e t r a p p i n g c u r r e n t J a r i s e s . The s t e a d y s t a t e c o n t r i b u t i o n t o t h e t o t a l c u r r e n t d e n s i t y i s J = J 1 + J 0 - J = i c o l (3.15) p e l e2 r where a i s t h e a b s o r p t i o n c o n s t a n t and I i s t h e i n t e n s i t y o f l i g h t . The p a r a m e t e r K i s dependent on t h e n a t u r e o f t h e a b s o r p t i o n c e n t r e ( t h e d i r e c t i o n a l p r o b a b i l i t i e s and mean f r e e p a t h s o f e l e c t r o n m o t i o n on ex-c i t a t i o n and r e c o m b i n a t i o n and t h e p h o t o n e n e r g y ) . 3.8 D i s c u s s i o n A t t h e p r e s e n t t i m e , t h e r e i s a g e n e r a l agreement t h a t t h e c h a r g e t r a n s p o r t i n v o l v e d i n t h e p h o t o r e f r a c t i v e e f f e c t may be d e s c r i b e d by J = KOL + qD I 2 - + qyn(V/L + E.) (3.16) where V i s t h e a p p l i e d v o l t a g e ( i f any) t o t h e e l e c t r o d e s a c r o s s t h e c r y s t a l o f a l e n g t h L. E^ i s t h e p h o t o i n d u c e d space c h a r g e f i e l d . C o n t r i b u t i o n s o f t h e d r i f t , d i f f u s i o n and b u l k p h o t o v o l t a i c e f f e c t depend upon t h e e x p e r i -m e n t a l s e t u p and t h e p h y s i c a l p r o p e r t i e s o f t h e c r y s t a l u s e d . These w i l l be d i s c u s s e d i n t h e f o l l o w i n g two s e c t i o n s . The space c h a r g e f i e l d E^ may be c a l c u l a t e d u s i n g Eqs. 3.7, 3.8 and 3.16 and t h e change i n t h e i n d i c e s o f r e f r a c t i o n may be f o u n d f r o m t h e e l e c t r o - o p t i c t e n s o r o f t h e c r y s t a l ( a p p e n d i x B ) . 3.8.1 The B u l k P h o t o v o l t a i c E f f e c t T a k i n g t h e t e r m K a i i n t h e c u r r e n t e q u a t i o n Eq. 3.16 as t h e c o r -r e c t d e s c r i p t i o n of t h e new b u l k p h o t o v o l t a i c e f f e c t , Moharam and Young (1976b and 1977) have shown t h a t t h i s e f f e c t can a l s o be m a t h e m a t i c a l l y d e s c r i b e d as though an i m a g i n a r y b u i l t - i n f i e l d , w h i c h t h e y named t h e " v i r t u a l f i e l d " E^, a c t e d on t h e p h o t o r e l e a s e d e l e c t r o n s , n o t on t h o s e p r e s e n t i n t h e d a r k ( p r o v i d e d t h a t t h e t r a n s p o r t l e n g t h i s s h o r t ) ) . T h i s v i r t u a l f i e l d i s p u r e l y a f i c t i o n a l o r s y m b o l i c q u a n t i t y . F o r s h o r t t r a n s -p o r t l e n g t h , t h e e x t r a e l e c t r o n c o n c e n t r a t i o n p r o d u c e d by t h e l i g h t n = gx (Eq. 3.5) where T i s t h e e l e c t r o n s ' l i f e t i m e and g i s t h e . l o c a l r a t e of g e n e r a t i o n of f r e e e l e c t r o n (g = a£l/hv where a i s t h e a b s o r p t i o n con-s t a n t , £ i s t h e quantum e f f i c i e n c y , hv t h e p h o t o n energy o f l i g h t and I i s t h e i n t e n s i t y o f l i g h t ) . T h e r e f o r e , t h e p h o t o c u r r e n t p r oduced by a " v i r t u a l f i e l d " a c t i n g on p h o t o i n d u c e d c a r r i e r c o n c e n t r a t i o n n i s qunE^. E q u a t i n g t h i s p h o t o c u r r e n t by Kai,'we f i n d t h a t E^ = Khv/qpx£. T h e r e f o r e , f o r s h o r t t r a n s p o r t l e n g t h , we may a l l o w f o r t h e new b u l k p h o t o v o l t a i c e f f e c t by c o n s i d e r i n g a " v i r t u a l f i e l d " t o be added on t o w h a t e v e r f i e l d s a r e p r e -s e n t , due t o spac e c h a r g e o r e x t e r n a l a p p l i c a t i o n ( " r e a l " e l e c t r o s t a t i c f i e l d s w i l l a c t a l s o on t h e f r e e c a r r i e r c o n c e n t r a t i o n i n t h e d a r k ) . F o r a r b i t r a r y t r a n s p o r t l e n g t h , we use t h e c o n t i n u i t y e q u a t i o n (Eq.3.12) and f o r J = K a i , we a r r i v e a t where = Khv/q£. F o r g r a t i n g s p a c i n g A = l y m, X'= 0.5 ym and K = 3x10 cm.A/W ( G l a s s e t a l . 1 974), KL^ = ( 4 . 6 8 / c j ) x l 0 ~ 4 . T h e r e f o r e , i f t h e quantum e f f i c i e n c y E, i s g r e a t e r t h a n 1%, K L ^ « 1 and Eq. 3.17 r e d u c e s t o t h e case o f s h o r t t r a n s p o r t l e n g t h . T h e r e f o r e , t h e b u l k p h o t o v o l t a i c e f f e c t may be r e -p r e s e n t e d by a " v i r t u a l f i e l d " E^ = Khv/qy£x a c t i n g on t h e p h o t o i n d u c e d e l e c t r o n s i f t h e quantum e f f i c i e n c y £ i s g r e a t e r t h a n a few %. The v i r t u a l f i e l d E^ i s i n d e p e n d e n t o f t h e l i g h t i n t e n s i t y u n l e s s T i s so dependent. I t s h o u l d be d i s t i n g u i s h e d f r o m t h e a p p l i e d f i e l d n e c e s -s a r y t o r e d u c e t h e p h o t o c u r r e n t t o z e r o , w h i c h G l a s s e t a l . (1974) found t o be i n t e n s i t y dependent. T h i s f i e l d s h o u l d be i n t e n s i t y dependent (even i f E^ i s n o t ) u n l e s s t h e d a r k c o n d u c t i v i t y i s n e g l i g i b l e . Thus, w i t h J = qynE + K a i , i n w h i c h we a s s u n e u n i f o r m i l l u m i n a t i o n and, hence, z e r o c o n -c e n t r a t i o n g r a d i e n t , and where n = n^ + n^, i n w h i c h n^ i s t h e c o n c e n t r a t i o n of e l e c t r o n s i n t h e d a r k and n^ i s t h e e x t r a c o n c e n t r a t i o n o f e l e c t r o n s due t o i l l u m i n a t i o n . F o r J = 0, we o b t a i n E = E ^ ( l + n^/n^) \ w h i c h i s i n t e n -s i t y dependent. Kim e t a l . (1976b) i n t r o d u c e d an " e q u i v a l e n t d r i f t f i e l d " w h i c h combines b o t h an i n t e n s i t y dependent component and a component w h i c h was n o t so dependent. They d i d n o t r e l a t e t h i s c o n c e p t t o t h e v i r t u a l f i e l d . n ( x ) = Tg I I + m(cos Kx - KL, s i n K x ) ] (3.17) ,-9 However, t h e agreement between t h e i r t h e o r y (Kim e t a l . 1976a) and e x p e r i -ments does n o t e s t a b l i s h t h a t s u c h i n t e n s i t y dependent d r i f t f i e l d s e x i s t s i n c e t h e r e a r e s e v e r a l d i f f i c u l t i e s w i t h t h e i r t h e o r e t i c a l a n a l y s i s and t h e e x p e r i m e n t a l c o n d i t i o n s were n o t c o n t r o l l e d ( t h e y d i d n o t s t a t e whether th e c r y s t a l s i n t h e i r e x p e r i m e n t s were f u l l y o r p a r t i a l l y i l l u m i n a t e d o r whether t h e y were h e l d a t c o n s t a n t v o l t a g e o r an open c i r c u i t ) . By a l l o w -i n g f o r t h e d a r k c o n d u c t i v i t y i n t h e t r a n s p o r t e q u a t i o n , as was shown above, one may a c c o u n t f o r Kim e t a l . ' s o b s e r v a t i o n s w i t h o u t t h e need t o p o s t u l a t e such an i n t e n s i t y dependent e q u i v a l e n t d r i f t f i e l d . The magnitude o f t h e c i r t u a l f i e l d = Khv/qyx^ depends on t h e d i f f e r e n t p r o p e r t i e s o f t h e c r y s t a l . G l a s s e t a l . (1974) c l a i m e d t h a t K i s a c o n s t a n t c h a r a c t e r i s t i c o f t h e i r o n dopant b u t K r a t z i g and K u r t z (1979b) have r e p o r t e d some change o f K w i t h t h e d o p i n g c o n c e n t r a t i o n . E^ would i n -c r e a s e w i t h i n c r e a s e i n t h e c o n c e n t r a t i o n o f t h e empty t r a p s , s i n c e i s i n v e r s e l y p r o p o r t i o n a l t o T. The v a l u e o f E^, e s t i m a t e d f r o m t h e e x p e r i m e n t a l d a t a r e p o r t e d by G l a s s e t a l . 1974b, C o r n i s h e t a l . 1976, K i m e t a l . 1976, Shah e t a l . 1976 and K r a t z i g e t a l . 1977, r a n g e s f r o m 0.5 - 50 kV/cm. T h i s l a r g e v a r i a t i o n i s due t o t h e d i f f e r e n t c r y s t a l p r o p e r t i e s and p r e p a r a t i o n . To summarize, t h e b u l k p h o t o v o l t a i c e f f e c t may be d e s c r i b e d mathe-m a t i c a l l y by a v i r t u a l f i e l d E a c t i n g on t h e p h o t o i n d u c e d e l e c t r o n s . T h i s i m i g i n a r y b u i l t - i n f i e l d may be added t o w h a t e v e r r e a l f i e l d s a r e p r e s e n t s u c h as f i e l d s due t o s p a c e c h a r g e o r t o e x t e r n a l l y a p p l i e d v o l t a g e s . T h i s i s p r o v i d e d t h a t t h e t r a n s p o r t l e n g t h i s s h o r t ( o r by i m p l i c a t i o n t h a t t h e quantum e f f i c i e n c y £ i s g r e a t e r t h a n a few % ) . T h i s v i r t u a l f i e l d i s i n t e n -s i t y i n d e p e n d e n t . I t s m a g n i t u d e , w h i c h depends on t h e p h y s i c a l p a r a m e t e r s of t h e c r y s t a l , can be f o u n d f r o m e i t h e r p h o t o c u r r e n t measurements o r h o l o -g r a p h i c e x p e r i m e n t s . 3.8.2 D i f f u s i o n E x a m i n a t i o n o f Eqs. 3.9 and 3.10 shows t h a t t h e r e l a t i v e i mpor-t a n c e of d r i f t and d i f f u s i o n i n t h e f o r m a t i o n o f t h e space c h a r g e f i e l d s depends on t h e r e l a t i v e m a gnitude o f and Ep. The t o t a l d r i f t f i e l d E Q may c o n t a i n any a p p l i e d f i e l d and t h e v i r t u a l f i e l d . The magnitude of t h e e q u i v a l e n t d i f f u s i o n f i e l d E^ = kTK/q i s 1600 V/cm f o r g r a t i n g s p a c i n g A = lum. Such a f i e l d w o u l d be more t h a n enough t o p r o d u c e 100% d i f f r a c t i o n e f f i c i e n c y f o r a 1 cm t h i c k c r y s t a l (Amodei 1971b). T h e r e f o r e , u n l e s s E Q i s much g r e a t e r t h a n 1600 V/cm, d i f f u s i o n s h o u l d n o t be n e g l e c t e d i n t h e t r a n s p o r t e q u a t i o n . D i f f u s i o n c o u l d be dominant i f t h e d r i f t f i e l d i s o f t h e o r d e r o f 200 V/cm. D i f f u s i o n c o u l d be t h e o n l y o p e r a t i v e mechanism i f th e a p p l i e d f i e l d , o r t h e l a r g e s c a l e s p a c e c h a r g e f i e l d , c a n c e l l e d t h e e f f e c t o f t h e p h o t o v o l t a i c e f f e c t , as r e p r e s e n t e d by t h e v i r t u a l f i e l d . B o t h C o r n i s h e t a l . (1976) and S p h i n h i r n e e t a l . (1977) c l a i m e d t o have s t o r e d holograms by d i f f u s i o n o n l y by a p p l y i n g an e x t e r n a l f i e l d t o c a n c e l t h e v i r t u a l f i e l d . Some o f t h e c o n f u s i o n o v e r t h e p r e c i s e r o l e o f d i f f u s i o n has a r i s e n b e c a u s e holograms a r e n o t r e a d i l y s t o r e d when t h e c - a x i s o f t h e c r y -s t a l i s p e r p e n d i c u l a r t o t h e p l a n e of t h e two w r i t i n g beams. F o r example, von d e r L i n d e and G l a s s (1975) c l a i m t h a t t h e d i f f u s i o n model a l s o f a i l s t o e x p l a i n t h e s t r o n g a n i s o t r o p y o f h o l o g r a m r e c o r d i n g s i n c e o n l y v e r y weak holograms can be r e c o r d e d w i t h p l a n e s p a r a l l e l t o t h e c - a x i s and t h e e f f i -c i e n c y has a maximum when t h e g r a t i n g p l a n e s a r e n o r m a l t o t h e a x i s . How-e v e r , t h i s s t r o n g a n i s o t r o p y , as i t w i l l be shown, would a r i s e , even f o r no d r i f t , f r o m t h e e l e c t r o - o p t i c p r o p e r t i e s o f l i t h i u m n i o b a t e and h o t because th e e f f e c t o f d i f f u s i o n i s n e g l i g i b l e . E x a m i n a t i o n of t h e e l e c t r o - o p t i c t e n s o r o f l i t h i u m n i o b a t e i l l u s t r a t e s why holograms may n o t be s t o r e d i n some c o n f i g u r a t i o n s as f o l l o w s i The e q u a t i o n f o r t h e o p t i c a l i n d i c a t r i x i s ( 1 2 ~ r 2 2 E 2 + r 1 3 E 3 ) x l + ("T + r 2 2 E 2 + r 2 3 E 3 ) x 2 n n o o + ("T + r 3 3 E 3 3 ) x 3 " 2 r 6 1 E l X l X 2 n e + 2 r 4 2 E 2 x 2 x 3 + 2 r 5 1 E 1 x 3 x 1 = 1 (3.18) where E^, E 2 and E^ a r e t h e e l e c t r i c f i e l d components i n t h e x^, x 2 and x^ d i r e c t i o n r e s p e c t i v e l y ; n Q and n g a r e t h e o r d i n a r y and e x t r a o r d i n a r y i n -d i c e s o f r e f r a c t i o n . I n t h e u s u a l c o n f i g u r a t i o n f o r s t o r i n g h o l o g r a m s , t h e c - a x i s (x^) i s i n t h e p l a n e o f i n c i d e n c e and n o r m a l t o t h e b i s e c t o r o f t h e two beams as shown i n F i g . 3 . 3 ( a ) . T h i s c r e a t e s an e l e c t r i c f i e l d E^. From Eq. 3.18 t h e major e f f e c t s a r e changes i n n g p r o p o r t i o n a l t o (30.8 x 10 "^cm/V) and a change i n n^ p r o p o r t i o n a l t o r ^ (8.6 x 10 ^cm/V) (Appendix B ) . The change i n n Q o c c u r s whether t h e l i g h t i s p r o p a g a t i n g i n t h e x^ o r x 2 d i r e c t i o n s . I f t h e c r y s t a l i s t u r p e d t h r o u g h 90° so t h a t t h e c - a x i s i s n o r m a l to t h e p l a n e o f i n c i d e n c e , t h e f i e l d component t h a t i s c r e a t e d depends upon t h e d i r e c t i o n i n w h i c h t h e l i g h t i s p r o p a g a t i n g . F i g . 3.3(b) shows t h e case where t h e g r a t i n g v e c t o r i s i n t h e x^ d i r e c t i o n . T h i s c r e a t e s a f i e l d E,. I n t h i s c a s e t h e r e i s no d i r e c t e f f e c t on e i t h e r n o r n , t h e o n l y 1 e o J change i n t h e i n d i c a t r i x b e i n g a s m a l l r o t a t i o n due t o t h e c r o s s term. However, t h i s has a v e r y s m a l l e f f e c t . F i g . 3.3(c) shows t h e case where t h e g r a t i n g v e c t o r i s i n t h e x 2 d i r e c t i o n . T h i s c r e a t e s a f i e l d E 2 . A change i n n Q p r o p o r t i o n a l t o 39. F i g . 3.3 R e l a t i o n o f the c r y s t a l axes and the two w r i t i n g beams f o r d i f f e r e n t c o n f i g u r a t i o n o f f o r m i n g h o l o g r a m s . 40. r 2 2 ^ " ^ X ±KJcm/V) w i l l r e s u l t . The o n l y change i n n^ w i l l be v e r y s m a l l s i n c e i t i s due t o s m a l l r o t a t i o n o f t h e p r i n c i p a l a x i s o f t h e i n d i c a t r i x . To r e a d holograms i n t h i s c o n f i g u r a t i o n , t h e e l e c t r i c v e c t o r o f t h e l i g h t w o u l d have t o be p o l a r i z e d i n t h e x 2 d i r e c t i o n . The e f f i c i e n c y w o u l d be about 100 t i m e s l e s s t h a n f o r holograms s t o r e d i n t h e c o n f i g u r a t i o n o f F i g . 3 . 3 ( a ) . I t i s e v i d e n t t h a t , f r o m t h e e l e c t r o - o p t i c p r o p e r t i e s o f l i t h i u m n i o b a t e , t h e component o f t h e spac e c h a r g e f i e l d has t h e g r e a t e s t e f f e c t on t h e i n d i c e s o f r e f r a c t i o n . V e r y weak holograms c o u l d be s t o r e d w i t h t h e g r a t i n g v e c t o r n o r m a l t o t h e c - a x i s under c e r t a i n c o n d i t i o n s , namely, l i g h t must p r o p a g a t e i n t h e d i r e c t i o n o f t h e a - a x i s ( x ^ ) and l i g h t s h o u l d be p o l a r i z e d p a r a l l e l t o t h e b - a x i s ( x 2 ) . T h e r e f o r e , t h e s t r o n g a n i s o t r o p y o f ho l o g r a m r e c o r d i n g i s n o t a v a l i d argument a g a i n s t d i f f u s i o n as a t r a n s p o r t mechanism. I n c o n c l u s i o n , d i f f u s i o n may n o t be t h e dominant mechanism o f t h e p h o t o r e f r a c t i v e e f f e c t b u t i s a c o n t r i b u t i n g f a c t o r t o t h e p r o c e s s . I n some e x p e r i m e n t a l c o n d i t i o n s i t c o u l d be t h e o n l y o p e r a t i n g mechanism. D i f f u s i o n s h o u l d n o t be n e g l e c t e d u n l e s s t h e d r i f t f i e l d s ( a p p l i e d and v i r t u a l ) a r e much g r e a t e r t h a n 1600 V/cm. CHAPTER IV THEORY OF HOLOGRAM WRITING BY THE PHOTOREFRACTIVE EFFECT 4.1 I n t r o d u c t i o n I n t h i s c h a p t e r , h o l o g r a m w r i t i n g i s m o d e l l e d o v e r t h e e n t i r e range o f e x p o s u r e f o r u n i f o r m i l l u m i n a t i o n . I n C h a p t e r 3,, t h e models o f Amodei (1971a) and Young e t a l . (1974) were d i s c u s s e d . These models a r e a p p l i c a b l e o n l y i n t h e l i n e a r i n i -t i a l s t a g e o f h o l o g r a m f o r m a t i o n . One r e a s o n f o r t h i s l i m i t a t i o n i s t h a t th e f e e d b a c k e f f e c t o f t h e p h o t o i n d u c e d space c h a r g e f i e l d on t h e r e d i s t r i -b u t i o n o f t h e p h o t o e x c i t e d e l e c t r o n s was n e g l e c t e d . S e v e r a l a u t h o r s have a t t e m p t e d t o a l l o w f o r t h i s f e e d b a c k e f f e c t o f t h e spac e c h a r g e f i e l d ( A l p h o n s e e t a l . 1975, Su and G a y l o r d 1975 and Kim e t a l . 1976a). These models a r e d i s c u s s e d b r i e f l y and some d i f f i c u l t i e s c o n c e r n i n g t h e i r a p p l i c a -b i l i t y t o r e a l p h y s i c a l s i t u a t i o n s a r e n o t e d . An improved model i s t h e n p r o p o s e d , f o r h o l g r a m w r i t i n g under c o n s t a n t a p p l i e d v o l t a g e w h i c h i s of i n t e r e s t i n p r a c t i c a l a p p l i c a t i o n s . The model i s f o r a u n i f o r m l y i l l u m i n a -t e d c r y s t a l w i t h f i n i t e d a r k c o n d u c t i v i t y . I n Sec. 4.3 t h e model i s ex-tended t o a l l o w f o r t h e e f f e c t o f a b s o r p t i o n i n r e d u c i n g t h e l i g h t i n t e n s i t y as i t t r a v e l s t h r o u g h t h e c r y s t a l . I n Sec. 4.4 a more dynamic model f o r hologram w r i t i n g i s p r o p o s e d . The model a l l o w s s i m u l t a n e o u s l y , f o r t h e f i r s t t i m e , f o r b o t h t h e f e e d b a c k e f f e c t o f t h e space c h a r g e f i e l d and t h e e f f e c t o f t h e h o l o g r a m b e i n g w r i t -t e n i n m o d i f y i n g t h e l i g h t p a t t e r n w h i c h i s w r i t i n g i t . The e f f e c t o f ab-s o r p t i o n and t h e d a r k c o n d u c t i v i t y a r e a l s o a l l o w e d f o r . I t i s shown t h a t t h i s model r e p r o d u c e s a l l t h e r e p o r t e d forms o f t h e t i m e development o f t h e d i f f r a c t i o n e f f i c i e n c y . The e f f e c t s of s e v e r a l p a r a m e t e r s , i n c l u d i n g t h e m o d u l a t i o n r a t i o , t h e d a r k c o n d u c t i v i t y , t h e a b s o r p t i o n , ; : t h e a p p l i e d v o l t a g e and t h e v i r t u a l f i e l d , on t h e w r i t i n g p r o c e s s , a r e . a l s o d i s c u s s e d . 4.2 The Feedback E f f e c t of t h e Space Charge F i e l d 4.2.1 I n t r o d u c t i o n As w i t h most p r e v i o u s work, a p r o t o t y p e h o l o g r a m i s c o n s i d e r e d , t h a t i s , one w h i c h i s formed by t h e i n t e r f e r e n c e o f two i n f i n i t e p l a n e waves. The c r y s t a l i s c o n s i d e r e d t o be f u l l y and u n i f o r m l y i l l u m i n a t e d . I t i s assumed t h a t t h e m i g r a t i o n l e n g t h o f t h e f r e e e l e c t r o n s i s s h o r t com-p a r e d t o t h e g r a t i n g w a v e l e n g t h . M u l t i p l e i n t e r n a l r e f l e c t i o n i s n e g l e c t e d . T h i s model d i f f e r s f r o m p r e v i o u s t r e a t m e n t s i n t h a t t h e c r y s t a l i s assumed t o have a f i n i t e d a r k c o n d u c t i v i t y and t h e most f r e q u e n t l y e n c o u n t e r e d r e a l p h y s i c a l s i t u a t i o n o f c o n s t a n t v o l t a g e a p p l i e d a c r o s s t h e c r y s t a l i s con-s i d e r e d . I n p r e v i o u s work, t h e c r y s t a l was assumed t o be under c o n s t a n t a p p l i e d f i e l d w h i c h i s n o t an easy c o n d i t i o n t o r e a l i z e e x p e r i m e n t a l l y . The h o l o g r a m i s assumed t o be w r i t t e n by d r i f t , d i f f u s i o n and t h e p h o t o -v o l t a i c e f f e c t . . . . 4.2.2 M o d e l The i n t e n s i t y o f t h e i n t e r f e r e n c e p a t t e r n formed by two i n f i n i t e p l a n e waves (See 3.6) may be w r i t t e n as I ( x ) = 1 (1 + m c o s Kx) (4.1) o where I i s t h e a v e r a g e l i g h t i n t e n s i t y , m i s t h e m o d u l a t i o n r a t i o , K = 2ir/A and A i s t h e g r a t i n g s p a c i n g . Under t h e a s s u m p t i o n o f s h o r t t r a n s -p o r t l e n g t h , t h e f r e e c a r r i e r c o n c e n t r a t i o n i n t h e c o n d u c t i o n band may be w r i t t e n as n ( x ) = n^ + + m cQs Kx) (4.2) where n_ i s t h e f r e e c a r r i e r c o n c e n t r a t i o n i n t h e d a r k and n° = atcJI^/hv, a i s t h e a b s o r p t i o n c o n s t a n t , x i s t h e f r e e c a r r i e r l i f e t i m e , E, i s t h e quan-tum e f f i c i e n c y , I i s t h e a v e r a g e i n c i d e n t l i g h t i n t e n s i t y and hv i s t h e p h o t o n energy. The s p a t i a l d i s t r i b u t i o n o f t h e c u r r e n t d e n s i t y i s g i v e n by J ( x , t ) = q u n ( x ) k j T + E ± ( x , t ) ] + q D ~ + K a l ( x ) (4.3) w h i c h i n c l u d e s d i f f u s i o n , d r i f t and t h e p h o t o v o l t a i c e f f e c t . V i s t h e a p p l i e d v o l t a g e , L i s t h e c r y s t a l l e n g t h and E ^ ( x , t ) i s t h e d e v i a t i o n o f t h e V e l e c t r i c f i e l d f r o m i t s a v e r a g e v a l u e —, due t o t h e p h o t o i n d u c e d space c h a r g e f i e l d . I t d e s c r i b e s t h e e f f e c t o f i n t e r e s t . The c o n t i n u i t y e q u a t i o n f o r t h e t r a p p e d c h a r g e d e n s i t y p due t o SO t h e m o t i o n o f t h e p h o t o r e l e a s e d e l e c t r o n s (assuming t h a t t h e e l e c t r o n con-c e n t r a t i o n i n t h e c o n d u c t i o n band does n o t v a r y w i t h t i m e ) i s s c _ 9J ( 4 > 4 ) 9t 9x P o i s s o n ' s e q u a t i o n i s 8 E T p s c — i = -S£- (4.5) 9x e where E^ i s t h e t o t a l e l e c t r o s t a t i c f i e l d ( s p a c e c h a r g e and a p p l i e d ) and e i s t h e p e r m i t t i v i t y . C ombining Eqs. 4.4 and 4 .5, we o b t a i n 3 E e + J = G ( t ) (4.6) o t where G ( t ) i s d e t e r m i n e d by t h e boundary c o n d i t i o n s , d e p e n d i n g on t h e ex-t e r n a l c i r c u i t o f t h e c r y s t a l . There a r e two most f r e q u e n t l y e n c o u n t e r e d p h y s i c a l s i t u a t i o n s : e i t h e r t h e c r y s t a l i s open c i r c u i t e d and, hence, G ( t ) = 0 a t a l l t i m e s , o r t h e c r y s t a l i s under c o n s t a n t a p p l i e d v o l t a g e ( i n c l u d i n g s h o r t c i r c u i t V=0) where G ( t ) i s d e t e r m i n e d by t h e c o n s t r a i n t t h a t L _ L ; E T d x = V 2 P r e v i o u s models f o r ho l o g r a m w r i t i n g ( A l p h o n s e e t a l . 1975, Su e t a l . 1975 and Kim e t a l . 1976) do n o t c o r r e s p o n d t o e i t h e r o f t h e s e . T h i s was n o t e x p l i c i t l y s t a t e d b u t i t was i m p l i e d by c h o o s i n g G ( t ) = c o n s t a n t (not z e r o b u t d i f f e r e n t i n each t r e a t m e n t ) . T h e r e f o r e , t h e s e t r e a t m e n t s may l e a d t o e r r o r s i n i n t e r p r e t i n g t h e e x p e r i m e n t a l o b s e r v a t i o n . R e c e n t l y , B l o t e k j a e r (1977) has t r e a t e d t h e case o f open c i r c u i t c r y s t a l . S u b s t i t u t i n g Eqs. 4.2 and 4.3 i n Eq. 4.6, we o b t a i n -T -7~ + [l+m'cosKx]E. = - (E + f ) (1+m'cosKx)+ E m ' s i n K x + G ' ( t ) (4.8) O a t " i V Lt u where T q = e/qu(n^ + Hq) and E^ = kTK/q i s t h e e q u i v a l e n t d r i f t f i e l d and E i s t h e v i r t u a l f i e l d r e p r e s e n t i n g t h e p h o t o v o l t a i c e f f e c t , G' = GT /e and v o m' = m / [ l + ( " j j / 1 1 ^ ) ] i s a n e f f e c t i v e m o d u l a t i o n r a t i o r e s u l t f r o m a l l o w i n g f o r t h e d a r k c o n d u c t i v i t y i n t h e model. G ' ( t ) i s d e t e r m i n e d by t h e boun-d a r y c o n d i t i o n s . The c o n s t r a i n t o f c o n s t a n t a p p l i e d v o l t a g e (Eq. 4.7) im-p l i e s L 1f2 E . ( x , t ) d x =0 (4.9) Li 1 " 2 S o l u t i o n o f Eq. 4.8 under t h e c o n s t r a i n t g i v e n by Eq. 4.9 w i t h t h e i n i t i a l c o n d i t i o n E^(x,0) = 0 i s m ' E s i n K x u E i = [ 1 W c o s K x ] [ l - e x p [ - ( l + m ' c o s K x ) | ] ] - ( E v + ^ ) [ l - I o ( ^ ) e x p ( - f ) o o o 2 - i -_ 1-m' rT I (m'u)exp(-u)du (4.10) 1+m'cosKx o ° ° ,2 ±-+ L , 1 m ' T r -(1-m'cosKx)] / T o I n ( m ' u ) e x p ( - u ) e x p [ ( l - h i i , c o s K x ) ( u - T^-)]du] 1+m cosKx o 0 l The f i r s t t e r m i n t h e e x p r e s s i o n f o r t h e s p a c e c h a r g e f i e l d i s due t o d i f -f u s i o n . The r e s t i s due t o d r i f t i n t h e a p p l i e d v o l t a g e and t h e v i r t u a l f i e l d . I Q i s t h e m o d i f i e d z e r o o r d e r B e s s e l f u n c t i o n . E x p a n d i n g Eq. 4.10 2 i n powers o f t / T Q and n e g l e c t i n g powers o f o r d e r ( t / T Q ) and h i g h e r we o b t a i n E ± ( x , t ) = j- [ E D s i n Kx - (fy + ^ ) c o s Kx] (4.11) o the above e x p r e s s i o n (Eq. 4.11) i s i d e n t i c a l t o t h e two e x p r e s s i o n s o b t a i n e d by Amodei (1971c) f o r t h e space c h a r g e f i e l d due t o d i f f u s i o n and d r i f t i n the i n i t i a l l i n e a r s t a g e s . S o l u t i o n o f Eq. 4.10 as t-W y i e l d s t h e f o l l o w i n g e x p r e s s i o n f o r the s t e a d y s t a t e space c h a r g e f i e l d . m'E s i n K x ,2 E , ( x ) = — ^ — - (E_ + ~ ) [ 1 - * 7 m v ] (4.12) i 1+m cosKx V L 1+m cosKx 4.2.3 C a l c u l a t e d R e s u l t s and D i s c u s s i o n 4.2.3.1 The Time Development o f t h e Space Charge F i e l d E x a m i n a t i o n o f Eq. 4.11 i n d i c a t e s t h a t d u r i n g t h e i n i t i a l l i n e a r s t a g e s (no f e e d b a c k e f f e c t o f t h e f i e l d ) t h e space c h a r g e f i e l d i s p u r e l y s i n u s o i d a l w i t h a s p a t i a l f r e q u e n c y e q u a l t o t h a t o f t h e l i g h t i n -t e n s i t y p a t t e r n . The phase s h i f t between t h e l i g h t p a t t e r n and t h e space c h a r g e f i e l d p a t t e r n i s d e t e r m i n e d by t h e r e l a t i v e magnitude o f E^ and ( E v + ^) . The phase s h i f t <j> = t a n 1 [ E E ) / ( ^ + E v')]« However, as t h e expo-s u r e i n c r e a s e s and t h e f e e d b a c k e f f e c t o f t h e p h o t o i n d u c e d s p a c e c h a r g e f i e l d becomes i m p o r t a n t , t h e p r o c e s s becomes n o n l i n e a r and t h e s p a c e c h a r g e f i e l d d e v e l o p s h i g h e r o r d e r harmonic components i n a d d i t i o n t o f u n d a m e n t a l components. F i g . 4.1 shows a p l o t o f t h e t i m e development of t h e fundamen-t a l component and of t h e f i r s t f o u r h i g h e r o r d e r h a r m o n i c s o f t h e d r i f t and d i f f u s i o n c o n t r i b u t i o n s t o t h e s p a c e c h a r g e f i e l d v s . t h e n o r m a l i z e d 2.0 F i g . 4.1 C a l c u l a t e d time development o f t h e f u n d a m e n t a l and t h e f i r s t f o u r h a rmonic components o f t h e p h o t o i n d u c e d space charge f i e l d ( i n u n i t s o f E ) w i t h m'=0.99. e t i m e ( t / T Q ) . The f u n d a m e n t a l component b u i l d s up a t a much f a s t e r r a t e and s a t u r a t e s a t a l a r g e r v a l u e t h a n t h e h i g h e r o r d e r h a r m o n i c s . N u m e r i c a l c a l c u l a t i o n s a l s o show t h a t b o t h t h e d r i f t and d i f f u s i o n p r o d u c e d components of t h e space c h a r g e f i e l d d e v e l o p w i t h t h e same "time c o n s t a n t " and t h e i r r e l a t i v e magnitude a t any t i m e i s d e t e r m i n e d by t h e r e l a t i v e magnitude of and ( E ^ + j-) . T h e r e f o r e , t h e phase s h i f t <f> between t h e l i g h t p a t t e r n and t h e p h o t o i n d u c e d s p a c e c h a r g e f i e l d r e m a i n s c o n s t a n t a t a l l t i m e s , and r e t a i n s t h e v a l u e t a n ^"[E^/ + E^) ] . S t a e b l e r and Amodei (1972b) have shown t h a t t h i s phase s h i f t d e t e r m i n e s t h e e x t e n t and t h e d i r e c t i o n o f t h e energy t r a n s f e r between t h e two w r i t i n g beams. They p o i n t e d o u t t h a t , i f <!> = 0, t h e r e w o u l d be no e n e r g y t r a n s f e r and t h e maximum en e r g y t r a n s f e r w o u l d o c c u r when <J> = 90. T h e r e f o r e , s i n c e t h e phase s h i f t <j> i s c o n s t a n t a t a l l t i m e s , t h e energy t r a n s f e r a t any t i m e w i l l depend on t h e r e l a t i v e mag-n i t u d e o f E and ~ ( E + —) . The f i r s t t erm i n Eq. 4.10 r e p r e s e n t s t h e dif-? u V Li f u s i o n p r o d u c e d component o f t h e s p a c e c h a r g e f i e l d . m'E^sinKx E . S ( x , t ) = 1 + m . c o s K x [ l - - e x p [ - j - a + m ' c o s K x ) ] ] (4.13) o E x p a n d i n g E^ ( x , t ) i n Eq. 4.13 i n i t s F o u r i e r s e r i e s we o b t a i n Q 00 t " /T E ^ f c , * ) " m ' E D p Z 1 s i n ( p K x ) o / t / o exp (-u) [ I ^ ^ u m ' ) ) - I n + 1 ( u m ' ) ]du ! (4.14) By i m p l i c a t i o n , t h e d r i f t component o f t h e s p a c e c h a r g e f i e l d can be expan-ded i n i t s F o u r i e r s e r i e s i n t h e f o r m r V oo fc/T E. ( x , t ) = -m' (E_+f) I L c o s ( p K x ) / ' o e x p ( - u ) [ I _ (um' ) - I (m'u) ]du l V L p = l o n-1 n+1 (4.15) Here I n(um') a r e t h e m o d i f i e d n t n o r d e r B e s s e l f u n c t i o n s . Combining Eqs. 4.14 and 4.15 we a r r i v e a t t h e s p a c e c h a r g e due t o b o t h d r i f t and d i f f u s i o n . t/T CO r\ E . ( x , t ) = -m'E E 1 cos(pKx+<|)) / e x p ( - u ) [ I .(um')-I ...(umMldu (4.16) 1 e p = l ^ Y o n-1 n+1 where E = [E„ + (— + E„) l" 5 a t t h e s t e a d y s t a t e t/T -**> and Eq. 4.16 can be e D L V J J o ^ i n t e g r a t e d t o o b t a i n E . ( x ) = 2E X c o s ( p K x + ( ( ) ) [ ( - ~ - 1 ) ^ - - ^ r - ] p (4.17) l e p = l ,I ,1 r m m 4.2.3.2 The E f f e c t of t h e M o d u l a t i o n R a t i o I t was shown t h a t , a l l o w i n g f o r t h e d a r k c o n d u c t i v i t y i n t h e model i s m a t h e m a t i c a l l y e q u i v a l e n t t o r e d u c i n g t h e m o d u l a t i o n r a t i o m of t h e l i g h t p a t t e r n t o an e f f e c t i v e m o d u l a t i o n r a t i o m'= m / [ l + n /ri°]. I n most O Lj p r a c t i c a l s i t u a t i o n s TL^IX^ i s v e r y s m a l l b u t t h i s does n o t mean t h a t i t can be n e g l e c t e d , s i n c e , as w i l l be shown l a t e r , i n some c a s e s , t h e f i e l d b u i l d -up and i t s s a t u r a t i o n v a l u e a r e v e r y s e n s i t i v e t o t h e v a l u e o f t h e e f f e c t i v e m o d u l a t i o n r a t i o m'. I n F i g . 4.2 t h e t i m e development o f t h e f u n d a m e n t a l component o f t h e s p a c e c h a r g e f i e l d i s p l o t t e d v s . t h e n o r m a l i z e d t i m e t / T Q f o r d i f f e r e n t v a l u e of m'. F i g . 4.2 shows t h a t t h e a m p l i t u d e of t h e f u n d a -m e n t a l component of t h e f i e l d i n c r e a s e s w i t h i n c r e a s e i n t h e e f f e c t i v e m o d u l a t i o n r a t i o m'. However, t h e r e l a t i o n i s n o n l i n e a r . A t t h e s t e a d y s t a t e , a s m a l l change i n m' causes a l a r g e change i n t h e f i e l d . To i n v e s t i -g a t e t h i s p o i n t f u r t h e r , t h e s t e a d y s t a t e a m p l i t u d e s o f t h e f u n d a m e n t a l and t h e f i r s t f o u r h i g h e r o r d e r h a r m o n i c s components of t h e space c h a r g e f i e l d a r e p l o t t e d v s . t h e e f f e c t i v e m o d u l a t i o n r a t i o m' i n F i g . 4.3 (m' i s a l w a y s l e s s t h a n one even i f m = 1 ) . I t i s e v i d e n t f r o m t h e d i a g r a m t h a t t h e de-pendence o f t h e f i e l d on t h e m o d u l a t i o n r a t i o i s s t r o n g l y n o n l i n e a r i n m' s i n c e r e l a t i v e l y l a r g e changes i n m', f o r moderate and s m a l l v a l u e s o f m', cause r e l a t i v e l y s m a l l changes i n t h e f i e l d w h i l e s m a l l changes i n m', when m' i s c l o s e t o one, cause v e r y s i g n i f i c a n t changes i n t h e f i e l d . T h i s F i g . 4.2 C a l c u l a t e d time development of t h e f u n d a m e n t a l component of the p h o t o i n d u c e d f i e l d ( i n u n i t s o f E ) f o r d i f f e r e n t v a l u e s o f the e f f e c t i v e m o d u l a t i o n r a t i o m'. 50. F i g . 4.3 The s t e a d y s t a t e f u n d a m e n t a l and f i r s t f o u r h armonic components o f the p h o t o i n d u c e d f i e l d ( i n u n i t s of E ) v s . t h e e f f e c t i v e m o d u l a t i o n r a t i o m'. may be qualitatively explained in terms of the relative intensity of the two incident writing beams. From Eq.- 3.3 the modulation ratio of the incident light pattern m = 2r/(l+r 2) where r i s the ratio of the amplitude of the two writing beams. Unity modulation ratio (m = 1) is obtained with two equal writing beams. Since allowing for the dark conductivity, results in an effective modulation ratio m', i t also results in effective beam amplitude ratio (4.18) Substituting Eq. 4.18 in Eq. 4.17 we arrive at CO T\ E.(x) = 2E E. (-r')pcos(pKx + (j>) (4.19) x e p=l Eqs. 4.19 shows that the steady state amplitude of the fundamental component of the space charge f i e l d i s linear with the effective beam amplitude ratio r'. The amplitude of the p t n order harmonic component of the f i e l d i s pro-portional to r , P +"'". The amplitude of the various Fourier component could be equal (a nonphysical situation) i f r 1 = 1.. However, r' is always less than unity, even i f the two incident beams are equal, due to allowing for the dark conductivity. To study the effect of the dark conductivity on the effective beam amplitude ratio r', and by implication, i t s effect on the space charge f i e l d , Fig. 4.4 shows a plot of r 1 vs. the ratio f° r different values of the incident beams amplitude ratio r. The factor n^/n^ represents either the effect of the dark conductivity i f the incident light intensity i s kept constant or i t represents the effect of the incident light intensity. Fig. 4.4 shows that especially when r 1 is close to one, the dark conductivity of the crystal i s more important. Although most of the ferroelectric crystals that exhibit, the photorefractive effect are very good insulators, e.g. the 52. Fig. 4.4 The steady state fundamental component of the photoinduced f i e l d (in units of E ) vs. the ratio n ^ / n j ) f° r different values of the incident beams amplitude ratio r. r e s i s t i v i t y o f l i t h i u m n i o b a t e r a n g e s f r o m 10"^ - 1 0 ^ ohm cm, t h e e f f e c t s o f t h e d a r k c o n d u c t i v i t y on t h e w r i t i n g s h o u l d n o t be n e g l e c t e d . The im-p o r t a n t p a r a m e t e r h e r e i s t h e r a t i o n ^/ 1^* T h i s f a c t o r may be r e w r i t t e n as t h e r a t i o o f t h e two c o n d u c t i v i t i e s a /a . S m a l l v a l u e s o f a T / a n can o c c u r i n s p i t e o f v e r y s m a l l a b s o l u t e d a r k c o n d u c t i v i t y . We may e s t i m a t e o * T / a n as f o l l o w s . The c u r r e n t d e n s i t y term K a i i n t r o d u c e d by G l a s s e t a l . (1974) may be r e p r e s e n t e d by a term a E ( v i r t u a l f i e l d a c t i n g on a l i g h t i n d u c e d Jli v c o n d u c t i v i t y ) . T h i s g i v e s a = K a l / E where K = 3x10 Acm/w f o r i r o n doped J_j V l i t h i u m n i o b a t e ( G l a s s e t a l . 1974). The v a l u e s o f E^, c a l c u l a t e d f r o m t h e r e p o r t e d e x p e r i m e n t a l d a t a ( C h a p t e r 3 ) , r a n g e s f r o m a few kV/cm t o a few t e n s of kV/cm. Hence, f o r a = 1.0 cm \ o^/a^ r a n g e s from 101 t o 10^1 where I i s 2 t h e l i g h t i n t e n s i t y i n W/cm . G l a s s e t a l . have r e p o r t e d t h a t a / a n was 1001 i n one o f t h e i r e x p e r i m e n t s . T h e r e f o r e , even f o r l i g h t i n t e n s i t y as 2 h i g h as a few W/cm , a / a _ c o u l d be s m a l l enough t o a f f e c t t h e s p a c e c h a r g e Li JJ f i e l d development p r o c e s s . 4.2.3.3 The D i f f r a c t i o n E f f i c i e n c y As was p o i n t e d o u t i n Sec. 4.3.4.1, i n t h e i n i t i a l l i n e a r s t a g e o f h o l o g r a m f o r m a t i o n , t h e p h o t o i n d u c e d s p a c e c h a r g e i s p u r e l y s i n u -s o i d a l w i t h a s p a t i a l f r e q u e n c y e q u a l t o t h a t o f t h e i n c i d e n t l i g h t p a t t e r n . As t h e f e e d b a c k e f f e c t of t h e p h o t o i n d u c e d f i e l d becomes more s i g n i f i c a n t h i g h e r o r d e r h a r m o n i c s s t a r t t o d e v e l o p . S i n c e t h e s e h i g h e r o r d e r components have h i g h e r s p a t i a l f r e q u e n c y , t h e y v i o l a t e s t r o n g l y t h e Bragg c o n d i t i o n when t h e r e a d i n g beam i s i n c i d e n t a t an a n g l e s u c h t h a t i t s a t i s f i e s t h e Bragg c o n d i t i o n f o r t h e f u n d a m e n t a l component o f t h e g r a t i n g . T h e r e f o r e , t h e s e h i g h e r o r d e r g r a t i n g s do n o t d i f f r a c t t h e r e a d i n g beam. However, t h i s i s t r u e o n l y as l o n g as t h e d i f f r a c t i o n p r o b l e m i s i n t h e B r a g g d i f f r a c t i o n r e g i m e . I n Ch. 2, i t was shown t h a t f o r t h e Bragg r e g i m e t o h o l d 2 2 p = X M n Q n 2 must be g r e a t e r t h a n 10 ( X q i s t h e l i g h t w a v e l e n g t h , A i s t h e g r a t i n g w a v e l e n g t h , n Q i s t h e b u l k r e f r a c t i v e i n d e x o f t h e medium and n^ i s t h e a m p l i t u d e o f t h e f u n d a m e n t a l component o f t h e i n d e x m o d u l a t i o n ) . F o r X = 500 nm, n^ = 2.24 and a n g l e of i n c i d e n c e o f 10°, t h e upper l i m i t on n^ i s 0.005 i n o r d e r t h a t p r e m a i n s g r e a t e r t h a n 10. F o r an e l e c t r i c f i e l d a l o n g t h e c - a x i s and l i g h t p o l a r i z e d p a r a l l e l t o t h e x - a x i s Eq. 3.17 shows t h a t t h e change i n t h e r e f r a c t i v e i n d e x i s 2 n An = - y r 3 3 E f (4.20) where r 0 0 = 30.8 x 10~ 1 0cm/V and n = 2 . 2 4 f o r X = 500 nm. T h e r e f o r e , t h e 33 e o upper l i m i t o f t h e a m p l i t u d e of t h e f u n d a m e n t a l component o f t h e s p a c e c h a r g e f i e l d E^ f o r t h e B r a g g r e g i m e t o h o l d i s 300 kV/cm. Such l a r g e s p a c e c h a r g e f i e l d s c annot be p h o t o i n d u c e d by t h e p h o t o v o l t a i c e f f e c t . T h e r e f o r e , t h e h i g h e r o r d e r h a r m o n i c s c a n be n e g l e c t e d as f a r as t h e d i f f r a c t i o n p r o b l e m i s c o n c e r n e d , however, t h e i r f e e d b a c k e f f e c t on t h e b u i l d - u p o f t h e s p a c e c h a r g e f i e l d c a n n o t be i g n o r e d . S i n c e t h e s p a c e c h a r g e f i e l d ( t h e g r a t i n g ) i s u n i f o r m t h r o u g h t h e c r y s t a l t h i c k n e s s , t h e K o g e l n i k f o r m u l a f o r t h e d i f f r a c t i o n e f f i c i e n c y w o u l d a p p l y , t h a t i s 2 n = s i n (irAn D/X cos 0,) (4.21) q o 1 where 8^ i s t h e a n g l e o f r e f r a c t i o n i n t h e c r y s t a l , and D i s t h e c r y s t a l t h i c k n e s s and 3 n An. = - ± r E_ (4.22) q 2 q3 f q i n d i c a t e s t h e a x i s p a r a l l e l t o t h e p o l a r i z a t i o n o f t h e l i g h t . E^ i s t h e a m p l i t u d e o f t h e f u n d a m e n t a l F o u r i e r component of the s p a c e c h a r g e f i e l d due t o d i f f u s i o n and d r i f t . F i g . 4.5 shows a p l o t of t h e d i f f r a c t i o n e f -f i c i e n c y o f t h e h o l o g r a m v s . t h e n o r m a l i z e d t i m e c o n s t a n t t/T f o r d i f f e r e n t 55. F i g . 4.5 The time development of the relative d i f f r a c t i o n e f f i c i e n c y f o r d i f f e r e n t v a l u e s o f E w i t h m'= 0.99. v a l u e s o f E^. The l i g h t i s assumed t o be p o l a r i z e d p e r p e n d i c u l a r t o t h e c - a x i s ( r i s t h e o p e r a t i o n a l e l e c t r o - o p t i c c o e f f i c i e n t ) and t h e c r y s t a l t h i c k n e s s D i s 1 cm. F i g . 4.5 shows t h a t t h e d i f f r a c t i o n e f f i c i e n c y s t a r t s w i t h z e r o i n i t i a l s l o p e and t h e n i n c r e a s e s i n a p a r a b o l i c f a s h i o n , t h e c h a r a c t e r i s t i c s d epending on t h e v a l u e o f E . F o r s m a l l v a l u e s of E , t h e e e d i f f r a c t i o n e f f i c i e n c y s a t u r a t e s as t h e s p a c e charge s a t u r a t e s . F o r l a r g e v a l u e s o f E^ t h e d i f f r a c t i o n e f f i c i e n c y r e a c h e s a maximum and t h e n d e c r e a s e s and o s c i l l a t e s b u t e v e n t u a l l y s a t u r a t e s . The e x t e n t and t h e shape o f t h e o s c i l l a t i o n depends on how l a r g e i s E g . The d i f f r a c t i o n e f f i c i e n c y 2 ncc. s i n (aEg) where a depends on t h e p a r a m e t e r s o f t h e c r y s t a l and i n c r e a s e s w i t h e x p o s u r e . T h e r e f o r e , f o r t h e e f f i c i e n c y t o s a t u r a t e w i t h o u t o s c i l l a t i o n E^ must be l e s s t h a n u/2a. I f E^ i s g r e a t e r t h a n i\/2a, t h e e f f i c i e n c y r e a c h e s a maximum o f 1 and t h e n d e c r e a s e s and f i n a l l y s a t u r a t e s . Thus, l a r g e enough E^ may r e s u l t i n a r e d u c e d s a t u r a t i o n d i f f r a c t i o n e f f i c i e n c y . T h e r e -f o r e , t o maximize t h e d i f f r a c t i o n e f f i c i e n c y , t h e a p p l i e d f i e l d s h o u l d be chosen i n a c c o r d a n c e w i t h o t h e r p a r a m e t e r s of t h e • c r y s t a l s . 4.2.4 Summary I n summary, a new model f o r t h e d e s c r i p t i o n of h o l o g r a m f o r m a -by t h e p h o t o r e f r a c t i v e e f f e c t w h i c h i s a p p l i c a b l e o v e r t h e e n t i r e r ange of e x p o s u r e i s p r o p o s e d . The model i s f o r a..uniformly i l l u m i n a t e d c r y s t a l under c o n s t a n t a p p l i e d v o l t a g e . The model a l l o w s f o r t h e f e e d b a c k e f f e c t o f t h e s p a c e c h a r g e f i e l d and f o r t h e d a r k c o n d u c t i v i t y . The p h o t o i n d u c e d space c h a r g e f i e l d i s s i n u s o i d a l i n t h e l i n e a r i n i t i a l s t a g e of h o l o g r a m f o r m a t i o n . As t h e p r o c e s s p r o g r e s s e s , h i g h e r o r d e r h a r m o n i c s d e v e l o p due t o t h e n o n l i n e a r n a t u r e of t h e f e e d b a c k e f f e c t o f t h e s p a c e c h a r g e . The a m p l i t u d e s o f t h e d i f f e r e n t h a r m o n i c components a r e a l -ways l e s s t h a n t h a t of t h e f u n d a m e n t a l component. However, t h e s e h a r m o n i c s do n o t a f f e c t t h e d i f f r a c t i o n e f f i c i e n c y o f t h e h o l o g r a m s i n c e t h e y s t r o n g l y v i o l a t e t h e Bragg c o n d i t i o n . Each h i g h e r o r d e r g r a t i n g ( f i e l d component) w i l l d i f f r a c t t h e r e a d i n g beam i f t h e c r y s t a l i s r o t a t e d s u c h t h a t t h e Bragg c o n d i t i o n i s s a t i s f i e d f o r t h a t o r d e r . I t was a l s o shown t h a t t h e f i e l d components due t o d r i f t and d i f -f u s i o n a r e d e v e l o p e d w i t h t h e same " t i m e c o n s t a n t " and t h e r e l a t i v e phase of t h e f u n d a m e n t a l component of t h e f i e l d w i t h r e s p e c t t o t h e i n c i d e n t l i g h t p a t t e r n r e m a i n s c o n s t a n t o v e r t h e e n t i r e r ange of e x p o s u r e and i s d e t e r m i n e d by t h e r e l a t i v e m a gnitudes o f t h e d r i f t and d i f f u s i o n f i e l d s . A l t h o u g h t h e a m p l i t u d e of t h e s p a c e c h a r g e f i e l d i s p r o p o r t i o n a l t o t h e m o d u l a t i o n r a t i o a t t h e i n i t i a l l i n e a r s t a g e s , t h i s dependence changes g r a d u a l l y as t h e e x p o s u r e i n c r e a s e s t o a dependence on t h e e f f e c t i v e beams a m p l i t u d e r a t i o r ' . A t s t e a d y s t a t e , t h e a m p l i t u d e of t h e f u n d a m e n t a l component i s d i r e c t l y p r o p o r t i o n a l t o r ' , and h i g h e r o r d e r h a r m o n i c s a r e p r o p o r t i o n a l t o h i g h e r o r d e r powers o f r ' . T h i s dependence on r ' a c c o u n t s f o r t h e s t r o n g l y n o n l i n e a r dependence on t h e m o d u l a t i o n r a t i o . The d a r k c o n d u c t i v i t y i s shown t o have a v e r y i m p o r t a n t e f f e c t on t h e development o f t h e space c h a r g e f i e l d and,.- a l t h o u g h i t s a b s o l u t e v a l u e i s v e r y s m a l l ( 1 0 1 0 "^(ohm cm) "*"), i t s h o u l d n o t be i m m e d i a t e l y d i s m i s -s e d . I t has been shown t h a t t h e r a t i o n^/n£ (not t h e a b s o l u t e v a l u e of n^) has a s i g n i f i c a n t . e f f e c t on r ' even f o r v e r y s m a l l v a l u e s o f n^/n^. T h i s e f f e c t i s m a g n i f i e d when r ' i s c l o s e t o 1 . The s e n s i t i v i t y o f h o l o g r a m f o r m a t i o n and t h e s a t u r a t i o n v a l u e of t h e space charge f i e l d s c o u l d be improved by i n c r e a s i n g t h e d r i f t f i e l d ( a p p l i e d and v i r t u a l ) and t h e d i f f u s i o n f i e l d . However, as t h e d i f f u s i o n and d r i f t f i e l d s become l a r g e enough, t h e d i f f r a c t i o n e f f i c i e n c y o s c i l l a t e s w i t h i n c r e a s e d e x p o s u r e and may s a t u r a t e a t a l o w e r v a l u e t h a n t h a t w h i c h w o u l d be o b t a i n e d w i t h s m a l l e r d r i f t and d i f f u s i o n f i e l d s (depending on t h e c r y s t a l p r o p e r t i e s , e s p e c i a l l y c r y s t a l t h i c k n e s s and l i g h t p o l a r i z a t i o n ) . The main drawback o f t h e r e l a t i v e l y s i m p l i f i e d model i s t h a t i t does n o t a l l o w f o r e i t h e r t h e e f f e c t o f t h e n o n u n i f o r m i t y o f t h e s p a c e c h a r g e f i e l d t h r o u g h t h e c r y s t a l t h i c k n e s s due t o t h e a b s o r p t i o n o f l i g h t , o r f o r t h e more i m p o r t a n t e f f e c t s o f t h e beam c o u p l i n g and energy t r a n s f e r between t h e two w r i t i n g beams. These two e f f e c t s w i l l be c o n s i d e r e d i n Sec. 4.3 and 4.4 r e s p e c t i v e l y . However, t h e model i s u s e f u l i n p r e d i c t i n g , a t l e a s t q u a l i t a t i v e l y , t h e t i m e development o f t h e h o l o g r a m f o r m a t i o n . I t a l l o w s t h e v a r i o u s p a r a m e t e r s i n v o l v e d t o be f ound and, hence, shows how t o o p t i m i z e t h e w r i t i n g p r o c e s s , e.g. how t o improve t h e s e n s i t i v i t y and t h e maximum d i f f r a c t i o n e f f i c i e n c y . 4.3 The E f f e c t o f L i g h t L o s s Due t o A b s o r p t i o n 4.3.1 M o d e l I n the model p r e s e n t e d i n Sec. 4.2, i t was assumed t h a t t h e l i g h t i n t e r f e r e n c e p a t t e r n , and c o n s e q u e n t l y t h e f r e e c a r r i e r c o n c e n t r a t i o n i n t h e c o n d u c t i o n band, i s u n i f o r m t h r o u g h t h e t h i c k n e s s o f t h e c r y s t a l . The r e -d u c t i o n i n the l i g h t i n t e n s i t y due t o a b s o r p t i o n was n e g l e c t e d . The model may be m o d i f i e d t o a c c o u n t f o r t h e e f f e c t o f t h e a b s o r p t i o n by i n s e r t i n g an e x p o n e n t i a l d e c a y i n g term i n t h e e x p r e s s i o n o f t h e i n t e n s i t y o f t h e l i g h t i n t e r f e r e n c e p a t t e r n (Eq. 4.1) I ( x , z ) = I e x p ( - a z / c o s 0 ) ( 1 + m cos Kx) (4.23) and under t h e a s s u m p t i o n of s h o r t t r a n s p o r t l e n g t h , t h e f r e e c a r r i e r c o ncen-t r a t i o n may be w r i t t e n as n ( x , z ) = n^ + n£ exp (-az/cos 9 ) ( 1 + m cos Kx) (4.24) The c u r r e n t d e n s i t y now has components i n t h e x and z d i r e c t i o n s . I t may be w r i t t e n as J ( x , z , t ) = I q u n ( x , z ) ^ + i c a l ( x , z ) ] x + qDVn(x,z) + q u n ( x , z ) E \ ( x , z , t ) ( 4 . 2 5 ) f r o m Gauss's l a w _ 9E. _ _ £ — - + J = G ( 4 . 2 6 ) o t where V*G = 0 and may be d e t e r m i n e d f r o m t h e boundary c o n d i t i o n . The open c i r c u i t c o n d i t i o n i n t h e z d i r e c t i o n i m p l i e s t h a t G = 0 . G may be d e t e r -Z X mined f r o m t h e c o n s t r a i n t of c o n s t a n t a p p l i e d v o l t a g e g i v e n by L 2 x . / E . ( x , z , t ) d x = 0.0. S u b s t i t u t i n g Eq. 4.25. i n t o Eq. 4.26, we a r r i v e a t L i 2 3 E ? T ( z ) - ~ + ( l + m ' c o s K x ) E Z = — — ~ ( l + m ' ( z ) c o s K x ) ( 4 . 2 7 ) 3 t l q cos8-^ 9 E X T ( z ) - r - ^ + (1+m'(z)coskx)Ev = - ( E +V/L)(1+m*(z)cosKx) at I V +E ,m'(z)sinkx+G'(+) ( 4 . 2 8 ) D X E_^  and E_^  a r e t h e components o f t h e f i e l d i n t h e x and z d i r e c t i o n r e s p e c -o ~ct z o t i v e l y . T ( z ) = e / q y t n ^ + n L e x P ( ~ c o s 9 ) ] and m' ( z ) = m / [ l + ( n I ) / n L ) e x p ( a z / c o s 0 ) ] The o t h e r p a r a m e t e r s have t h e i r p r e v i o u s d e f i n i t i o n s . The s o l u t i o n o f Eq. z 4.27 f o r the i n i t i a l c o n d i t i o n of E ^ ( x , z , o ) = 0.0 i s E Z ( x , z , t ) = - ^ L _ _ [ l - e x p [ - ( l + m ' ( z ) c o s K x ) t / T ( z ) ] ] ( 4 . 2 9 ) x q c o s 0 ^ o S i n c e t h e upper l i m i t on a f o r any u s e f u l p r a c t i c a l a p p l i c a t i o n i s about —1 kT ct 30 cm t h e n t h e upper l i m i t on i s about 75 V/cm. T h e r e f o r e , t h e r r q cos6 space c h a r g e f i e l d i n t h e z d i r e c t i o n may be n e g l e c t e d s i n c e i t i s v e r y s m a l l compared t o t h e f i e l d s on t h e x d i r e c t i o n ( s e v e r a l kV/cm). Eq. 4.28 i s s i m i l a r t o Eq. 4.8 e x c e p t t h a t T q and m' a r e r e p l a c e d by T Q ( Z ) and m ' ( z ) . T h e r e f o r e , s o l u t i o n s of Eq. 4.8 (Eqs. 4.10 t o 4.19) a r e t h e s o l u t i o n s f o r Eq. 4.28 when T ( z ) and m'(z) a r e s u b s t i t u t e d f o r T Q and m' r e s p e c t i v e l y . 4.3.2 D i s c u s s i o n The p a r a m e t e r T ( z ) i s a measure o f t h e t i m e c o n s t a n t w h i c h d e t e r -mines t h e r a t e of t h e b u i l d - u p of t h e space c h a r g e f i e l d . I t d e c r e a s e s t h r o u g h t h e c r y s t a l t h i c k n e s s . The e f f e c t i v e m o d u l a t i o n r a t i o m'(z) ( w h i c h d e t e r m i n e s t h e a m p l i t u d e o f t h e s p a c e c h a r g e f i e l d ) a l s o d e c r e a s e s t h r o u g h t h e t h i c k n e s s o f t h e g r a t i n g . T h i s d e c r e a s e depends on t h e a b s o r p t i o n con-s t a n t o f t h e c r y s t a l i n a c o m p l i c a t e d way. F o r example, m'(z) = m/(1 + ( n ^ / n ^ ) e x p ( a z / c o s 6 ^ ) ] , i . e . t h e e f f e c t o f a l l o w i n g f o r ab-s o r p t i o n i n t h e model i s t h a t t h e r a t i o n^/n^ i n c r e a s e s t h r o u g h t h e c r y s t a l t h i c k n e s s . I t was shown i n Sec. 4.2.3 t h a t , i f m i s c l o s e t o 1, t h e a m p l i -t u d e of t h e space c h a r g e f i e l d i s v e r y s e n s i t i v e t o n^/n^. I n t h i s c a s e , one e x p e c t s t h e a b s o r p t i o n t o have a s i g n i f i c a n t e f f e c t on t h e f i e l d . On t h e o t h e r hand, i f m i s n o t c l o s e t o 1, t h e f i e l d i s n o t t o o s e n s i t i v e t o V v The a m p l i t u d e of t h e f u n d a m e n t a l F o u r i e r component of t h e s p a c e c h a r g e f i e l d due t o d r i f t and d i f f u s i o n (Eqs. 4.14, 4.15 w i t h T Q ( z ) and m'(z) r e p l a c i n g T q and m') i s t/T ( z ) E f ( z , t ) = m ' ( z ) E / e x p ( - u ) [ l o ( u m ' ( z ) ) - I 2 ( u m ' ( z ) ) ] d u (4.30) where I and I ~ a r e t h e m o d i f i e d B e s s e l f u n c t i o n s o f t h e z e r o and second o 2 o r d e r r e s p e c t i v e l y . The a m p l i t u d e o f t h e f u n d a m e n t a l F o u r i e r component o f t h e s i n u s o i d a l change i n the r e f r a c t i v e i n d e x may be c a l c u l a t e d u s i n g Eq. 4.22. S i n c e t h e change i n t h e r e f r a c t i v e i n d e x An^ i s n o t u n i f o r m t h r o u g h t h e g r a t i n g t h i c k n e s s , t h e K o g e l n i k f o r m u l a f o r t h e d i f f r a c t i o n efficiency of such grating is n = sin2[(TTAcos6..) /DAn. (z,t)dz] (4.31) ± o q If An i s constant, i.e. uniform grating Eq. 4.31 reduces to Kogelnik's formula (Eq. 4.21). To investigate the effect of allowing for the light absorption in the model, the normalized diffraction efficiency of the hologram i s plotted vs. the normalized time constant t/T^, for different values of the absorp-tion constant a (Fig. 4.6). The diffraction efficiency i s calculated for m = 0.99 and n D / n ° = 0.01 and the crystal thickness D = 1.0 cm. The effec-tive f i e l d E g = 3 kV/cm and the light was taken as polarized parallel to the c-axis (or = 9.6 kV/cm and light polarized normal to the c-axis). Both Fig. 4.6 and 4.7 show that neglecting the effects of the absorption of light as i t propagates into the crystal (a = 0.0) results in over estima-tion of the diffraction efficiency of the grating and, of course, the error increases with increase in the value of the absorption constant. As for the influence of the n^/n^ (the dark conductivity) on the effects of absorp-tion, Fig. 4.7 shows a similar plot to Fig. 4.6 but for the ratio n /h° =0.1 (i.e. one tenth of i t s value in Fig. 4.6). Fig. 4.7 shows that errors due to neglecting the effects, the absorption of light on the diffraction e f f i -ciency become more significant as the ratio VL^/TL^ increases (weak light intensity or large dark conductivity). Therefore, the effects of the absorption of light should not, in general, be neglected unless the ratio iLp/n^ i s very small (less than 0.001). 4.3.3 Summary In conclusion, the model proposed in Sec. 4.2 for hologram writing by the interference pattern of two plane waves allowing for the feedback Fig. 4.6 The time development of the relative diffraction efficiency for different values of the absorption constant a with m'= 0.99 and n /n = 0.01. F i g . 4.7 Same as F i g . 4.6 b u t w i t h n /n°= 0.1. e f f e c t o f t h e p h o t o i n d u c e d s p a c e c h a r g e f i e l d and t h e d a r k c o n d u c t i v i t y has been extended t o a l l o w f o r t h e d e c r e a s e o f t h e l i g h t i n t e n s i t y as i t p r o p a -g a t e s i n t h e c r y s t a l due t o t h e a b s o r p t i o n o f l i g h t . I n t h i s e x t e n d e d model the p h o t o i n d u c e d space c h a r g e f i e l d i s n o t u n i f o r m t h r o u g h t h e t h i c k n e s s o f the c r y s t a l . A l l o w i n g f o r t h e d e c r e a s e o f t h e l i g h t i n t e n s i t y due t o a b s o r p -t i o n , r e s u l t s a l s o i n r e d u c e d b u i l d - u p r a t e and a m p l i t u d e o f t h e s p a c e c h a r g e f i e l d t h a n i n t h e o r i g i n a l model. The e f f e c t o f t h e a b s o r p t i o n c o n s t a n t i s more s i g n i f i c a n t when t h e m o d u l a t i o n r a t i o i s c l o s e t o 1 and, o f c o u r s e , when a i s l a r g e . A l t h o u g h t h i s e x t e n d e d model wou l d g i v e more a c c u r a t e q u a n t i t i v e p r e d i c t i o n t h a t t h e p r e v i o u s model (Sec. 4.2) i t s t i l l has t h e drawback of n e g l e c t i n g beam c o u p l i n g . T h i s i m p o r t a n t e f f e c t w i l l be a l l o w e d f o r i n the g e n e r a l i z e d model p r o p o s e d i n t h e n e x t s e c t i o n . 4.4 The E f f e c t s o f Beam C o u p l i n g D u r i n g Hologram W r i t i n g 4.4.1 I n t r o d u c t i o n S t a e b l e r and Amodei (1972b) were t h e f i r s t t o c o n s i d e r t h e i m p l i -c a t i o n s o f beam c o u p l i n g d u r i n g r e a d i n g and w r i t i n g o f holograms. They con-s i d e r e d t h e c a s e where two c o h e r e n t beams R and S a r e s y m m e t r i c a l l y i n c i d e n t a t an a n g l e 6 r e l a t i v e t o t h e z - a x i s on a r e g i o n w i t h p e r i o d i c v a r i a t i o n o f the r e f r a c t i v e i n d e x An = n1 cos (Kx) (4.32) t h a t e x t e n d s f r o m z = 0 t o z = D as shown i n F i g . 3.2. B o t h waves a r e p o l a r i z e d n o r m a l t o t h e p l a n e o f i n c i d e n c e . K = 2TT/A and A = X Q s i n 0 i s t h e g r a t i n g s p a c i n g and X q i s t h e o p t i c a l w a v e l e n g t h . The g r a t i n g i s assumed u n i f o r m t h r o u g h t h e t h i c k n e s s o f t h e medium. N e g l e c t i n g r e f l e c t i o n s and assuming t h a t t h e Bragg c o n d i t i o n s p r e v a i l , t h e two waves R and S can be w r i t t e n i n th e f o r m R = r X z ) e x p [ (-j (2irnzcos6/X. ) + Kx/2] (4.33) S = s ( z ) e x p j ( - j ( 2 i r n z c o s e / X o ) - Kx/2] (4.34) K o g e l n i k (1969) u s i n g t h e coupled-wave a n a l y s i s has shown t h a t i n t h e r e g i o n of a s i m p l e phase g r a t i n g and f o r p e r f e c t B r a g g c o n d i t i o n s % ^ - = - J K S ( Z ) (4.35) d z d s ( z ) . / \ — — = - j K r ( z ) dz where K = Trn^/X co s 8 , 1 o 1 S o l u t i o n o f t h e c o u p l e d wave e q u a t i o n s Eqs. 4.35 f o r t h e g e n e r a l boundary c o n d i t i o n s r ( 0 ) = 1 and s ( 0 ) = A exp(-j<j>) i s r ( z ) = COS(KZ) - j A e x p ( - j ( f > ) s i n ( K z ) (4.36) s ( z ) = Aexp(j<J>)cos(Kz) - j s i n (KZ) 2 2 2 I R = |r(z)-.| / = cos (KZ) + A s i n ( K z ) - A sin(sKz)siri<f> ,2 2 2 2 ( 4 - 3 7 ) I = | s ( z ) | = s i n (KZ) + A cos (KZ) + A sin(2Kz)sin<j> I f t h e beams have e q u a l i n c i d e n t a m p l i t u d e , t h e n A = 1 and Eq. 4.37 g i v e s 1 = 1 - s i n ( K z ) sin<j> K (4.38) I g = 1 + sin(Kz) /'sin<j) S t a e b l e r and Amodei (1972b) have shown t h a t Eq. 4.37 and 4.38 a l s o d e s c r i b e t h e s i t u a t i o n when t h e two i n c i d e n t waves a r e i n phase b u t t h e phase g r a t i n g i s moveable a l o n g t h e x - a x i s w i t h An = n± c o s ( K x +cj)) (4.39) However, f o r t h i s c a s e , Eq. 4.36 becomes r ( z ) = COS(KZ) - j A sin(Kz)exp(-j<J>) (4.40) s ( z ) = A COS(KZ) - j sin(Kz)exp(+j<f>) Eq. 4.37 and 4.38 show,- t h a t t h e l i g h t i n t e r a c t i o n w i t h t h e g r a t i n g r e s u l t s , i n an e n e r g y t r a n s f e r between t h e two beams. The magnitude and d i r e c t i o n o f t h e e nergy t r a n s f e r depends on t h e a m p l i t u d e and s p a t i a l phase o f t h e g r a -t i n g . T h i s phase f a c t o r <j> r e p r e s e n t s t h e phase s h i f t between t h e l i g h t i n -t e n s i t y p a t t e r n and t h e i n d e x m o d u l a t i o n i t p r o d u c e s . S t a e b l e r and Amodei (1972b) have p o i n t e d o u t t h a t t h i s c o u p l e d wave a n a l y s i s i s a p o w e r f u l t o o l t o a s s e s s t h e r e l a t i v e i m p o r t a n c e o f t h e d i f f e r e n t mechanisms p r o p o s e d f o r s t o r i n g phase holograms by t h e p h o t o r e f r a c t i v e e f f e c t . F o r example, t h e y have f o u n d t h a t , d u r i n g h o l o g r a m f o r m a t i o n , t h e r e i s e nergy t r a n s f e r be-tween t h e two beams. S i n c e t h e g r a t i n g p r o d u c e d by d r i f t o n l y (Sec. 3.3 and 4.3) has 0 o r ir phase s h i f t and f o r t h i s c a s e , Eq. 4.28 p r e d i c t s no energy t r a n s f e r , t h e y s u g g e s t e d t h e holograms were s t o r e d by d i f f u s i o n . The e f f e c t o f the changes o f t h e l i g h t p a t t e r n on t h e w r i t i n g p r o c e s s was n o t a d d r e s s e d . The i n t e r f e r e n c e p a t t e r n o f t h e two i n c i d e n t R and S wave i n t h e g r a t i n g i s I ( x ) = | R + S| 2 (4.41) S u b s t i t u t i n g Eqs. 4.34 and 4.40 i n Eq. 4.41, we a r r i v e a t 2 2 I ( x , z ) = I Q [ 1 + m [ c o s (KX) + s i n (KZ)COS(KX + 2(|>)]] + ( A 2 - l ) s i n ( 2 K z ) s i n ( K z + < j > ) (4.42) 2 2 where I = 1 + A and m = 2A/1+A . The l i g h t p a t t e r n i n t h e medium b e f o r e t h e g r a t i n g i s formed i s I ( x ) = 1 ( 1 + m c o s ( K x ) ) (4.43) o Thus, i n t h e i n i t i a l l i n e a r s t a g e s o f h o l o g r a m f o r m a t i o n , an i n c i d e n t l i g h t p a t t e r n (Eq. 4.43) p r o d u c e s a u n i f o r m s i n u s o i d a l phase g r a t i n g (Eq. 4.39) h a v i n g a p o s i t i o n r e l a t i v e t o t h e l i g h t p a t t e r n , t h a t depends on t h e w r i t i n g mechanisms, t h i s phase w i l l change t h e o r i g i n a l l i g h t p a t t e r n i n t o a l i g h t p a t t e r n (Eq. 4-42) w h i c h i s n o t u n i f o r m t h r o u g h t h e g r a t i n g t h i c k n e s s and i s s h i f t e d i n s p a t i a l phase r e l a t i v e t o t h e o r i g i n a l p a t t e r n . The new p a t t e r n w i l l p r o d u c e an a d d i t i o n a l phase g r a t i n g w i t h w h i c h i t i s n o t u n i f o r m t h r o u g h t h i c k n e s s and i s a g a i n s h i f t e d i n phase r e l a t i v e t o t h e p r e v i o u s l y r e c o r d e d g r a t i n g . T h i s c u m u l a t i v e change i n r e f r a c t i v e i n d e x w i l l t h e n m o d i f y t h e l i g h t and so on. To d e a l w i t h t h i s e f f e c t a "dynamic" c o u p l e d wave t h e o r y must be used. Such a dynamic t h e o r y was f i r s t d e v e l o p e d by N i n o m i y a ( 1 9 7 3 ) . S e v e r a l a u t h o r s have p r o p o s e d models f o r t h e t h e o r y of h o l o g r a m w r i t i n g by t h e p h o t o r e f r a c t i v e e f f e c t a l l o w i n g f o r t h e change i n t h e l i g h t p a t t e r n due t o t h e i n d e x m o d u l a t i o n i t p r o d u c e s . N i n o m i y a (1973) and Magnusson and G a y l o r d (1976) t o o k t h e i n d e x change as d e v e l o p i n g l i n e a r l y w i t h e x p o s u r e . I n o t h e r words, t h e y n e g l e c t e d t h e e f f e c t o f t h e space c h a r g e on t h e r e d i s t r i b u t i o n o f e l e c t r o n s w h i c h becomes s i g n i f i c a n t a t about t h e same t i m e s t a g e as does t h e need f o r a dynamic t h e o r y . Vahey (1975) assumed t h a t t h e i n d e x change i s p r o p o r t i o n a l t o t h e w r i t i n g i n t e n s i t y m u l t i p l i e d by an a r b i t r a r y time-dependent f u n c t i o n r a t h e r t h a n s o l v e t h e e l e c t r o n t r a n s -p o r t e q u a t i o n . A model f o r h o l o g r a m w r i t i n g i s now o u t l i n e d w h i c h i s t h e f i r s t t o a l l o w s i m u l t a n e o u s l y f o r t h e f e e d b a c k e f f e c t o f t h e space c h a r g e f i e l d on t h e r e d i s t r i b u t i o n o f e l e c t r o n s , f o r t h e e f f e c t o f t h e h o l o g r a m i n m o d i f y i n g t h e l i g h t p a t t e r n w h i c h n i s w r i t i n g i t and f o r t h e e f f e c t o f t h e d a r k c o n d u c t i v i t y . T h i s model r e p r o d u c e s a l l t h e r e p o r t e d forms of t i m e development of t h e d i f -f r a c t i o n e f f i c i e n c y . However, t h e e x p e r i m e n t s d e s c r i b e d i n c h a p t e r 8 i n d i -c a t e t h a t t h e a s s u m p t i o n of s h o r t t r a n s p o r t l e n g t h may n o t a l w a y s be v a l i d . 4.4.2 M o d e l (Moharam and Young 1977) The s y s t e m c o n f i g u r a t i o n f o r h o l o g r a m r e c o r d i n g and c o o r d i n a t e s y s t e m i s shown i n F i g . 4.8. The h o l o g r a m i s produced by t h e i n t e r f e r e n c e p a t t e r n of two c o h e r e n t monochromatic i n f i n i t e p l a n e waves, t h e r e f e r e n c e 68. F i g . 4.8 C o n f i g u r a t i o n f o r h o l o g r a m r e c o r d i n g . The R and S beams are a l l o c a t e d i n r e l a t i o n t o t h e c o o r d i n a t e s y s t e m w h i c h a l s o s e r v e s t o d e f i n e the s i g n of the a p p l i e d v o l t a g e . The c - a x i s o f the c r y s t a l may be p a r a l l e l o r a n t i p a r a l l e l w i t h t h e x - a x i s . wave R and t h e s u b j e c t wave S. They a r e i n c i d e n t s y m m e t r i c a l l y oh t h e r e -c o r d i n g medium w i t h an a n g l e 0. T h e r e f o r e , t h e g r a t i n g f r i n g e s a r e no r m a l t o t h e c r y s t a l s u r f a c e ( u n s l a n t e d ) . The e x t e n s i o n t o s l a n t e d g r a t i n g i s s t r a i g h t f o r w a r d b u t l i t t l e a d d i t i o n a l i n s i g h t i s g a i n e d . F o r c o n v e n i e n c e , i t i s assumed t h a t t h e same a v e r a g e r e f r a c t i v e i n d e x e x i s t s i n s i d e and o u t -s i d e t h e g r a t i n g and t h u s , t h e r e i s no r e f l e c t i o n . R e a d i n g i s done by b l o c k i n g one o f t h e two beams m o m e n t a r i l y . I t i s assumed t h a t t h e r e a d i n g i s c a r r i e d on f o r a s u f f i c i e n t l y s h o r t t i m e t h a t t h e i n t e r u p t i o n does n o t d i s t u r b t h e r e c o r d e d h o l o g r a m . The r e a d i n g p r o c e s s does n o t depend on w h i c h beam i s b l o c k e d as l o n g as t h e above a s s u m p t i o n i s v a l i d . As shown i n Sec. 4.2, t h e w r i t i n g p r o c e s s i s n o n l i n e a r due t o t h e s p a c e - c h a r g e f i e l d f e e d b a c k e f f e c t s . T h e r e f o r e , t h e change i n t h e r e f r a c -t i v e i n d e x c o n t a i n s n o t o n l y f u n d a m e n t a l components b u t a l s o h i g h e r s p a t i a l h a r m o n i c s c o r r e s p o n d i n g t o s m a l l e r g r a t i n g s p a c i n g s . The two beams i n v o l v e d i n w r i t i n g t h e h o l o g r a m a r e c o u p l e d by t h e f u n d a m e n t a l g r a t i n g ; t h e harmonic components v i o l a t e s t r o n g l y t h e B r a g g - a n g l e c o n d i t i o n and can be i g n o r e d i n c o n s i d e r i n g t h e e f f e c t o f t h e h o l o g r a m i n c o u p l i n g t h e two w r i t i n g beams. N a t u r a l l y , t h i s does n o t mean t h a t t h e h a r m o n i c s of t h e spac e c h a r g e f i e l d can be i g n o r e d i n c o n s i d e r i n g t h e r e d i s t r i b u t i o n o f t h e p h o t o r e l e a s e d e l e c t r o n . As was f i r s t n o t e d by S t a e b l e r and Amodei (19 7 2 b ) , t h e c o u p l i n g between t h e w r i t i n g beams causes t h e p l a n e s o f c o n s t a n t i n d e x i n t h e h o l o g r a m t o be s l i g h t l y b e n t . However, as w i l l be shown, t h e a n g l e o f b e n d i n g i s so s l i g h t t h a t i t can be i g n o r e d i n c o n n e c t i o n w i t h t h e B r a g g - a n g l e c o n d i t i o n . The a p p r o p r i a t e c o u p l e d wave e q u a t i o n s a r e d e v e l o p e d f o r l i g h t p o l a r i z e d n o r m a l t o t h e p l a n e o f i n c i d e n c e . E x t e n s i o n t o t h e p a r a l l e l p o l a r i -z a t i o n c a s e w i l l be g i v e n a l s o . F o l l o w i n g K o g e l n i k (1969) and N i n o m i y a ( 1 9 7 3 ) , t h e s c a l a r wave e q u a t i o n i s V 2 E + T 2 E = 0 (4.44) 2 2 where r = 6 - j a g (4.45) and E ( x , z , t ) i s t h e complex a m p l i t u d e o f t h e y component o f t h e f i e l d , a i s t h e i n t e n s i t y a b s o r p t i o n c o n s t a n t and g = 2TTNA , where N i s t h e r e f r a c t i v e o i n d e x and A q i s t h e l i g h t w a v e l e n g t h . N e g l e c t i n g t h e m o d u l a t i o n o f t h e a b s o r p t i o n c o n s t a n t , t h e r e f r a c t i v e i n d e x o f t h e medium f o r t h e p u r p o s e o f c o n s i d e r i n g beam c o u p l i n g ( i . e . , n e g l e c t i n g h a r m o n i c s as d i s c u s s e d above) i s N ( x , z , t ) = N + A N ( z , t ) c o s ( K * r + <j>(z,r)) o where AN = ( N 2 + N 2 ) 2 (4.45) c s where N i s t h e unmodulated v a l u e o f t h e r e f r a c t i v e i n d e x and N and N a r e o c s t h e s p a t i a l c o s i n e and s i n e f u n d a m e n t a l F o u r i e r components r e s p e c t i v e l y , o f t h e change i n t h e r e f r a c t i v e i n d e x due t o e x p o s u r e t o t h e l i g h t . K i s t h e g r a t i n g v e c t o r and K = 2h/h where A i s t h e g r a t i n g s p a c i n g . S u b s t i t u t i n g Eqs. 4.46 i n Eq. 4.45, and assuming t h a t 3>>a and 2 terms i n AN a r e n e g l i g i b l e , we a r r i v e a t T 2 = 3 2 + 4 g o ( i r A N ( z , t ) / X o ) c o s ( K - r + <f> ( z , t ) ) - j a B (4.47) and 3 Q = 2TTN^/\^ i s t h e p r o p a g a t i o n c o n s t a n t . The t o t a l o p t i c a l e l e c t r i c f i e l d i n t h e medium may be w r i t t e n as E ( x , z , t ) = e x p ( - a z / 2 c o s 0 ) [ R ( z , t ) e x p ( — j p * r ) + S ( z , t ) e x p ( - j o • r ) ] ( 4 . 4 7 ) where R ( z , t ) and S ( z , t ) a r e t h e complex a m p l i t u d e s o f t h e r e f e r e n c e and s u b j e c t waves w h i c h v a r y a l o n g z and i n t i m e as a r e s u l t o f t h e i n t e r a c t i o n w i t h t h e h o l o g r a m . The e x a c t Bragg r e l a t i o n i s a = p - K (4.48) where p and a a r e t h e p r o p a g a t i o n v e c t o r of t h e r e f e r e n c e beam R and t h e 71. F i g . 4.8(b) The r e l a t i o n between t h e p r o p a g a t i o n v e c t o r p and a and t h e g r a t i n g v e c t o r K f o r e x a c t Bragg i n c i d e n c e . s u b j e c t beam S, r e s p e c t i v e l y . |a| = | p | = B and fr o m t h e Bragg r e l a t i o n (Eq. 4 . 4 8 ) °"z = P z = 6 Q cos 6. S u b s t i t u t i n g Eqs. 4 . 4 6 and 4 . 4 7 i n Eq. 4 . 4 4 and comparing terms i n e x p ( - j p - r ) and e x p ( - j o ' r ) ( t h e o n l y two a l l o w e d waves) we o b t a i n t h e c o u p l e d wave e q u a t i o n s : d R i l f t ) = - j C ( z , t ) exp(-j<Kz,t)) S ( z , t ) , d z ( 4 . 4 9 ) 5 S ^ ? t ) = - j C ( z , t ) exp(+j<j,(z,t)) R ( z , t ) where C ( z , t ) = TTAN(Z,t) A QCOS9. The wave g e n e r a t e d i n t h e d i r e c t i o n s p + K and a — K v i o l a t e t h e Bragg c o n d i t i o n and a r e , t h e r e f o r e , n e g l e c t e d . The change i n R ( z , t ) and S ( z , t ) w i t h r e s p e c t t o z i s assumed t o be v e r y s l o w , so t h e second d e r i v a t i v e s o f R ( z , t ) and S ( z , t ) w i t h r e s p e c t t o z a r e n e g l e c -t e d as i n p r e v i o u s work. T h i s a s s u m p t i o n i s v a l i d when C ( z , t ) i s l e s s t h a n 0 . 0 1 w h i c h i s much l a r g e r t h a n t h a t w h i c h c o u l d be o b t a i n e d by t h e p h o t o -r e f r a c t i v e e f f e c t . These c o u p l e d wave e q u a t i o n s may be extended t o a p p l y f o r t h e c a s e o f l i g h t p o l a r i z e d p a r a l l e l t o t h e p l a n e o f i n c i d e n c e when C ( z , t ) i s r e p l a c e d by C ' ( z , t ) = C ( z , t ) cos 26. S o l u t i o n o f t h e c o u p l e d wave e q u a t i o n s y i e l d s R ( z , t ) and S ( z , t ) . The i n t e n s i t y o f t h e l i g h t i n t e r f e r e n c e p a t t e r n formed by t h e R and S waves i s p r o p o r t i o n a l t o |E| and may be w r i t t e n as I ( x , z , t ) = I Qexp(-az/cos9) [ 1 + m ( z , t ) c o s ( K - r + i|»(z,t))] where I q = % g [ | R ( z , t ) | 2 + | s ( z , t ) | 2 ] m ( z , t ) = 2 R ( z , t ) S ( z , t ) / t | R | 2 + |s| 2] ( 4 . 5 0 ) iKz,t) = a r c t a n [ ( S R * - S*R)/(SR* + S*R) ] where g i s t h e i n v e r s e o f t h e c h a r a c t e r i s t i c impedence o f t h e g r a t i n g medium. However, i n o r d e r t o s o l v e t h e c o u p l e d wave e q u a t i o n s , t h e a m p l i -tude and phase o f t h e f u n d a m e n t a l g r a t i n g must be d e t e r m i n e d f i r s t . T h e r e -f o r e , an e x p r e s s i o n f o r t h e p h o t o i n d u c e d space charge f i e l d i s needed. The t o t a l c a r r i e r c o n c e n t r a t i o n i n t h e c o n d u c t i o n band (assuming s h o r t t r a n s -p o r t l e n g t h as b e f o r e ) i s n(x,z,t) = % + \ [ I ( x , z , t ) / l o ] (4.51) and t h e c u r r e n t d e n s i t y i s g i v e n by (Eq. 4.25) 7 ( x , z , t ) = qDVn + qunE. (x,z,t) + [ q y n ^ + K c t l ( x , z , t ) ] x (4.52) X .Li The p h o t o i n d u c e d space c h a r g e f i e l d E^(x,z,t) has two v e c t o r i a l components. The f i r s t component i s a l o n g t h e x - a x i s , t h e second component i s a l o n g t h e z-axis. However, as was shown i n Sec. 4.3, t h e magnitude o f t h e z component of t h e f i e l d i s a t most about a few t e n s V/cm, i . e . , v e r y much s m a l l e r t h a n t h e f i e l d component a l o n g t h e x - a x i s w h i c h i s i n t h e o r d e r o f a few kV/cm. T h e r e f o r e , t h e z component o f t h e f i e l d may be n e g l e c t e d w i t h o u t any s i g n i -f i c a n t e f f e c t s on t h e h o l o g r a m w r i t i n g p r o c e s s . Combining P o i s s o n ' s and t h e c o n t i n u i t y e q u a t i o n s (Eqs. 4.4 and 4.5) and i n t e g r a t i n g w i t h r e s p e c t , t o t f r o m t = t t o t and w i t h r e s p e c t t o x under t h e c o n s t r a i n t o f c o n s t a n t o a p p l i e d v o l t a g e g i v e n by Eq. 4.9, we a r r i v e a t t -E. (x,z,t) = E. (x,z,t ) - - / J ( x , z , t ) d t + \ y f 2 ^ J ( x , z , t ) d t dx (4.53) 1 _ X O - £ t £Li Jj t o -j o where E ^ ( x , z , t Q ) i s t h e i n i t i a l v a l u e o f t h e space c h a r g e f i e l d . Combining Eqs. 4.50, 4.51 and 4.52 and s u b s t i t u t i n g t h e r e s u l t i n g e q u a t i o n i n Eq. 4.53 we o b t a i n -1 t V E i ( x , z , t ) = ^ y t/ {m' ( z , t ) [ ( E y + £)cos(Kx+iKz,t)) - E Dsin(Kx+^(z,t)) ] o + m'(z,t) cos(Kx+iKz,t)) E ± ( x , z , t ) } d t + —— ~ j S 2 f11 m' (z,t)cos(Kx+iKz,t))E. (x,z,t) d t dx T(z) L-f fco + E i ( x , z , t Q ) where i s t h e v i r t u a l f i e l d r e p r e s e n t i n g t h e b u l k p h o t o v o l t a i c e f f e c t . T(z) = e/qu[n D+n£ exp(-az/cos8)] and m'(z,t)=m(z,t)/[l+(n^/n£)exp(az/cos8)]. The space c h a r g e f i e l d E^ i s o b t a i n e d by s o l v i n g t h i s d i f f e r e n t i a l i n t e g r a l e q u a t i o n (Eq. 4.54). By r e s o l v i n g t h e spac e c h a r g e f i e l d i n t o i t s F o u r i e r components, t h e a m p l i t u d e AN and phase <j> o f t h e change i n t h e r e f r a c t i v e i n d e x , w h i c h i s p r o d u c e d v i a t h e e l e c t r o - o p t i c e f f e c t , i s AN = 0.5 N r „(F 2 + F 2 ) ^ (4.55) q q3 s c (4.55) <f> = a r c t a n (-F /F ) s c where F g and F c a r e t h e f u n d a m e n t a l s i n e and c o s i n e components o f t h e space ch a r g e f i e l d E_p N^ and r ^ a r e t h e p r o p e r r e f r a c t i v e i n d e x and e l e c t r o -o p t i c c o e f f i c i e n t (Eq. 4.21). 4.4.3 A l g o r i t h m Eqs. 4.50 t o 4.55 were s o l v e d u s i n g n u m e r i c a l methods. S i n c e t h e v a r i a b l e s i n t h e e q u a t i o n s a r e f u n c t i o n s o f x,z and t and t h e y a r e s l o w l y v a r y i n g i n z, t h e c r y s t a l i s d i v i d e d n o r m a l t o t h e z - a x i s i n t o a l a r g e num-b e r o f segments, each w i t h s u f f i c i e n t l y s m a l l t h i c k n e s s Az. I t i s assumed t h a t t h e l i g h t p a t t e r n i s c o n s t a n t i n z w i t h i n any segment, t h e r e f o r e , t h e space c h a r g e f i e l d E^, AN and <j> a r e a l s o c o n s t a n t w i t h i n t h a t segment. F o r example, i f a segment i s d e f i n e d by z = Z q and z = z^ where z^ = Z q + Az, and AN arid c|) a r e c o n s t a n t w i t h i n t h i s segment and e q u a l t o t h e i r v a l u e a t z = z o > t h e c o u p l e d wave e q u a t i o n s (Eq. 4.50) may be s o l v e d w i t h i n t h e s e g -ment s u b j e c t t o t h e boundary c o n d i t i o n R ( z Q , t ) and S ( z Q , t ) we a r r i v e a t t h e v a l u e s o f R(z.j,,t) and S(z^,t) as f o l l o w s R(z , t ) = R(z , t ) c ( z , t ) ~ jS(z ,t)s(z,t)exp(-j<j)(z . t ) ) l o o o o ( 4 > 5 6 ) S ( Z ; L , t ) = S ( z o , t ) c ( z d , t ) - jR(z o,t)s(z,t)exp(+jcf)(z^,t)) where c ( z , t ) = cos ( C ( z , t ) A z ) o o s ( z Q , t ) = s i n ( C ( z Q , t ) A z ) The v a l u e s o f R ( z ^ , t ) and S ( z ^ , t ) t h u s o b t a i n e d p r o v i d e t h e boundary c o n d i -t i o n s on the n e x t segment d e f i n e d by z = z^ t o 2.^ where = + Az. They a l s o d e t e r m i n e t h e l i g h t i n t e r f e r e n c e p a t t e r n i n t h e n e x t segment a t any tim e (Eq. 4.51). T h i s l i g h t p a t t e r n , w h i c h i s assumed c o n s t a n t w i t h i n t h i s segment, t h e n i s used t o d e t e r m i n e A N ( z ^ , t ) and <f>(z^,t) ( t h r o u g h Eqs. 4.54 and 4.55 f o r z=z^) f o r t h i s segment a t any t i m e . Eq. 4.56 may be used a g a i n t o c a l c u l a t e R ( s 2 5 t ) and S(z2»t) u s i n g t h e p r e v i o u s l y o b t a i n e d v a l u e s o f A N ( z ^ , t ) , <j>(z^,t) and R ( z ^ , t ) . By r e p e a t i n g t h i s a l g o r i t h m , t h e v a l u e s o f R(D , t ) and S ( D , t ) ( t h e v a l u e o f R and S a t e x i t ) a r e f i n a l l y o b t a i n e d where D i s t h e t h i c k n e s s o f t h e c r y s t a l . F o r t h e h o l o g r a m w r i t i n g p h a s e , t h e i n p u t v a l u e s o f R ( 0 , t ) and S ( 0 , t ) a r e d e t e r m i n e d by t h e i n c i d e n t l i g h t p a t t e r n . T h e i r r e l a t i v e v a l u e s depend on t h e m o d u l a t i o n r a t i o m(0,t) o f t h e i n c i d e n t l i g h t , e.g. i f m(0,t) = M, t h e n [ S ( 0 , t ) / R ( 0 , t ) ] = (1/M) [1 - (1 - M 2 ) * 2 ] , i . e . , i f M = 1 t h e n S/R = 1. F o r t h e r e s u l t s g i v e n b e l o w , i t i s assumed, f o r c o n v e n i e n c e , t h a t 2 2 I R ( 0 , t ) I + | s ( 0 , t ) | = 2. T h i s a s s u m p t i o n does n o t a f f e c t t h e r e s u l t s i n c e any p r o p o r t i o n a l i t y c o n s t a n t may be a b s o r b e d i n T ( z ) . F o r r e a d i n g t h e h o l o g r a m , t h e c o u p l e d wave e q u a t i o n s (Eq. 4.50) a r e s o l v e d w i t h t h e i n i t i a l boundary c o n d i t i o n s R ( 0 , t ) = 1 and S ( 0 , t ) = 0.0 and t h e v a l u e s A N ( z , t ) and cf>(z,t) o b t a i n e d f r o m t h e w r i t i n g phase. The a b s o l u t e d i f f r a c t i o n e f f i c i e n c y , w h i c h i s d e f i n e d as t h e r a t i o between t h e d i f f r a c t e d and i n c i d e n t beam i n t e n s i t i e s i s g i v e n by n ( t ) = e x p ( - c d V c o s 6 ) | s ( D , t ) | 2 / | E ( 0 , t ) | 2 (4.60) To s o l v e Eq. 4.54 f o r t h e s p a c e c h a r g e f i e l d E^, t h r e e d i f f e r e n t methods were t r i e d , t h e E u l e r method, second o r d e r and f o u r t h o r d e r Runge-K u t t a methods. The second o r d e r R u n g e - K u t t a method was f o u n d t o be t h e most s u i t a b l e and t h e f o u r t h o r d e r method d i d n o t make s i g n i f i c a n t improvement on t h e a c c u r a c y b u t t h e e x e c u t i o n t i m e o f t h e program was much worse s i n c e Eq. 4.54 i s an i n t e g r a l d i f f e r e n t i a l e q u a t i o n and t h e h i g h e r t h e o r d e r o f t h e method, t h e l a r g e r t h e number of i n t e g r a l s t o be e v a l u a t e d . A l t h o u g h t h e E u l e r method d i d n o t r e q u i r e any i n t e g r a l e v a l u a t i o n , a v e r y s m a l l t i m e s t e p was r e q u i r e d t o a c h i e v e t h e same a c c u r a c y as t h e R u n g e - K u t t a method. The F o u r i e r a n a l y s i s t o o b t a i n t h e f u n d a m e n t a l components o f t h e s p a c e c h a r g e f i e l d was c a r r i e d o u t u s i n g t h e t r a p e z o i d a l r u l e . The a c c u r a c y o f t h e s o l u t i o n was t e s t e d by i n c r e a s i n g t h e number o f segments Az and r e d u c i n g t h e t i m e s t e p s i z e u n t i l improvements i n c o n v e r g e n c e were m i n i m a l . 4.4.4 C a l c u l a t e d R e s u l t s and D i s c u s s i o n 4.4.4.1 I n t r o d u c t i o n The c a l c u l a t e d r e s u l t s o b t a i n e d f r o m t h e model w i l l be p r e -s e n t e d i n terms o f t h e dependence on t h r e e p a r a m e t e r s w h i c h we now d i s c u s s . They a r e (a) t h e ( i n t e n s i t y ) a b s o r p t i o n c o e f f i c i e n t a, (b) t h e r a t i o n^/n^ and (c) what may be c a l l e d t h e e f f e c t i v e t o t a l f i e l d (V/L + E ) . W i t h i r o n doped c r y s t a l s , t h e a b s o r p t i o n c o e f f i c i e n t depends on t h e c o n c e n t r a t i o n o f 2+ Fe . We assume no o t h e r a b s o r p t i o n p r o c e s s a t t h e w a v e l e n g t h of i n t e r e s t . F o r a g i v e n c r y s t a l a c o u l d be a l t e r e d by c h a n g i n g t h e s t a t e o f o x i d a t i o n 2+ 3+ i . e . t h e r a t i o Fe /Fe . T h i s w o u l d , however, a l t e r t h e l i f e t i m e T (and hence t h e t r a n s p o r t l e n g t h s ) s i n c e t h e mean d i s t a n c e f r o m a f i l l e d t r a p 2+ (Fe ) t o an empty t r a p w o u l d change. R e d u c t i o n m i g h t a l s o i n t r o d u c e 3+ s h a l l o w t r a p s i n a d d i t i o n t o t h e Fe c e n t r e s ( C o r n i s h , Moharam and Young 1976b). T h i s c o m p l i c a t i o n has n o t been i n t r o d u c e d i n t o our model. R e d u c t i o n m i g h t a l s o change t h e d a r k c o n d u c t i v i t y . The second p a r a m e t e r n ^ / n ^ - i n c r e a s e s w i t h i n c r e a s i n g l i g h t i n t e n s i t y ( t h r o u g h n£) and d e c r e a s e s w i t h i n c r e a s i n g d a r k c o n d u c t i v i t y . I t s h o u l d be n o t e d t h a t t h e d a r k conduc-t i v i t y c a n n o t be c o n s i d e r e d as s i m p l y c a u s i n g a r e d u c t i o n i n t h e m o d u l a t i o n r a t i o , s i n c e t h e m o d u l a t i o n r a t i o i s n o t c o n s t a n t i n t i m e o r space. F i n a l l y , t h e e f f e c t i v e t o t a l f i e l d can be s o - c a l l e d o n l y because we a r e assuming u n i f o r m , c o m p l e t e i l l u m i n a t i o n o f t h e c r y s t a l under c o n d i t i o n s o f c o n s t a n t a p p l i e d v o l t a g e , s i n c e (e.g.) n o n - u n i f o r m i l l u m i n a t i o n w o u l d i n t r o d u c e a l a r g e s c a l e f i e l d a s s o c i a t e d w i t h t h e e n v e l o p e o f t h e l i g h t . The o r i e n t a -t i o n o f t h e c r y s t a l ( F i g . 4 . 8 ) , as r e g a r d s r e v e r s a l o f t h e d i r e c t i o n o f t h e c + a x i s w i t h r e s p e c t t o t h e r e f e r e n c e a x e s , a f f e c t s t h e r e l a t i v e s i g n s o f and V/L and a l s o t h e s i g n o f t h e e l e c t r o - o p t i c e f f e c t . The c a l c u l a t e d r e s u l t s a r e f o r holograms s t o r e d i n a LiNbO^ c r y s t a l o f t h i c k n e s s D = 1.0 cm. B o t h t h e w r i t i n g and r e a d i n g beams a r e of w a v e l e n g t h X = 514.5 nm and a r e p o l a r i z e d p e r p e n d i c u l a r t o t h e p l a n e o f i n c i d e n c e and t o t h e c - a x i s w h i c h i s i n t h i s p l a n e . F o r t h i s c a s e , t h e o r d i n a r y r e f r a c t i v e i n d e x a p p l i e s , N q = 2.33. The a n g l e o f i n c i d e n c e i n a i r i s 15°. The two i n c i d e n t w r i t i n g beams a r e t a k e n t o be e q u a l . R ( 0 , t ) = S ( 0 , t ) = 1.0, i . e . u n i t y i n c i d e n t m o d u l a t i o n r a t i o m(0,t) = 1.0. The t i m e s c a l e i s t/T where t i s t h e ti m e i n seconds and T = e/qun°, i . e . o o L t h e t i m e i s e x p r e s s e d i n u n i t s o f T q w h i c h i s a measure o f t h e d i e l e c t r i c r e l a x a t i o n t i m e a s s o c i a t e d w i t h t h e p h o t o i n d u c e d c a r r i e r s . 4.4.4.2 The N o n u n i f o r i i i i t y o f t h e G r a t i n g Because of t h e energy t r a n s f e r between t h e two w r i t i n g beams due t o t h e d e v e l o p i n g h o l o g r a m , t h e l i g h t p a t t e r n i s c h a n g i n g w i t h t i m e and w i t h t h e g r a t i n g t h i c k n e s s . F i g . 4.9 shows t h e t i m e development o f t h e a m p l i t u d e AN o f t h e c u m u l a t i v e change i n t h e f u n d a m e n t a l s i n u s o i d a l component 78. Fig. 4.9 The time development of the amplitude AN of the cumulative fundamental component of the change in the refractive index at different planes through the crystal thickness during hologram writing with a= 1.0 (1/cm), n°/n = 10, and Ev+(V/L)= 5 kV/cm. of t h e r e f r a c t i v e i n d e x a t d i f f e r e n t p l a n e s t h r o u g h t h e t h i c k n e s s o f t h e c r y s t a l f o r chosen v a l u e s o f t h e above t h r e e p a r a m e t e r s (a,n^/n° and E ^ + ^ r ) ) . F i g . 4.9 shows t h e a m p l i t u d e AN i n i t i a l l y i n c r e a s e s l i n e a r l y w i t h t i m e t h e n i t m o n o t o n i c a l l y s a t u r a t e s . The a m p l i t u d e AN o f t h e f u n d a m e n t a l g r a t i n g a t t h e b a c k s u r f a c e o f t h e c r y s t a l i s a l m o s t h a l f i t s v a l u e a t t h e f r o n t s u r -f a c e . Q u a l i t a t i v e l y , F i g . 4.9 shows t h a t t h e ho l o g r a m i s as i f i t were w r i t t e n w i t h d i f f e r e n t v a l u e s o f t h e m o d u l a t i o n r a t i o m a t d i f f e r e n t d e p t h s i n t h e c r y s t a l . I t i s i n t e r e s t i n g t o n o t e t h a t , i f t h e two w r i t i n g beams a r e e q u a l i n i t i a l l y (as was assumed i n F i g . 4 . 9 ) , t h e energy t r a n s f e r between th e two beams w i l l c a u s e one o f t h e beams t o i n c r e a s e a t t h e expense o f t h e o t h e r . No m a t t e r w h i c h beam i s g a i n i n g e n e r g y , t h e r e s u l t i s a r e d u c e d modu-l a t i o n r a t i o and weaker h o l o g r a m as t h e l i g h t p r o p a g a t e s i n t o t h e c r y s t a l . B u t , i f the two w r i t i n g beams a r e n o t i n i t i a l l y e q u a l , t h e r e a r e two d i s -t i n c t c a s e s , d e p e n d i n g on t h e d i r e c t i o n o f energy t r a n s f e r . The f i r s t c a s e i s when t h e more i n t e n s e beam i s g a i n i n g e n e r g y r e s u l t i n g i n r e d u c e d m o d u l a t i o n r a t i o and weaker h o l o g r a m as l i g h t t r a v e l s t h r o u g h t h e t h i c k n e s s ( t h i s c a s e i s , i n g e n e r a l , s i m i l a r t o t h e case o f t h e two e q u a l beams). How-e v e r , i f t h e o r i e n t a t i o n o f t h e c - a x i s and t h e s p a t i a l phase s h i f t <J> a r e such t h a t t h e weak beam i s g a i n i n g e n e r g y as i t t r a v e l s t h r o u g h t h e c r y s t a l , t h e m o d u l a t i o n r a t i o and t h e h o l o g r a m s t r e n g t h w i l l i n c r e a s e w i t h d i s t a n c e t r a v e l l e d i n t h e c r y s t a l t h i c k n e s s . F i g 4.10 shows t h e ti m e development a t d i f f e r e n t ' d e p t h s i n t o t h e c r y s t a l o f t h e s p a t i a l phase s h i f t <j). I t i s t h e s p a t i a l phase s h i f t o f t h e f u n d a m e n t a l F o u r i e r component of t h e c u m u l a t i v e change i n t h e r e f r a c t i v e i n -dex w i t h r e s p e c t t o t h e s i n u s o i d a l component o f t h e i n i t i a l l i g h t p a t t e r n . A t a l l d e p t h s i n t o t h e c r y s t a l i n t h e i n i t i a l l i n e a r s t a g e s , and a t a l l t i m e s a t t h e f r o n t s u f r a c e o f t h e c r y s t a l (where t h e c o u p l i n g may be n e g l e c t e d ) 80. (t/r0> Fig. A. 10.The time development of the spatial phase shift <J> of the cumulative change in the refractive index at different planes through the thickness of the crystal. The other parameters are the same as in Fig.-419. (j) i s t h e same as i s found i n t h e o r i e s o f t h e l i n e a r i n i t i a l s t a g e s and i s d e t e r m i n e d by t h e r e l a t i v e c o n t r i b u t i o n s o f t h e t o t a l d r i f t f i e l d ( E v + V/L) and o f t h e " d i f f u s i o n e q u i v a l e n t f i e l d " ( k T K / q ) . F o r t h e chosen p a r a m e t e r s i n F i g . 4.10,' t h e d i f f u s i o n f i e l d was 1.6 kV/cm and ( E v + V/L) was 5.0 kV/cm, r e s u l t i n g i n an i n i t i a l phase s h i f t o f 18°. F i g . 4.10 shows t h a t <{> i n c r e a s e s w i t h t i m e and w i t h d i s t a n c e o f p r o p a g a t i o n i n t o t h e c r y s t a l , i n o t h e r words, as f i r s t n o t e d by S t a e b l e r e t a l . (1972b), t h e p l a n e s o f c o n s t a n t i n d e x change a r e b e n t and t h e i r c u r v a t u r e i n c r e a s e s w i t h t i m e . 4.4.4.3 The D i f f r a c t i o n E f f i c i e n c y I n t h i s s e c t i o n , t h e dependence o f t h e d i f f r a c t i o n e f f i c i e n c y (Eq. 4.60) on t h r e e d i f f e r e n t p a r a m e t e r s , t h e a p p l i e d f i e l d (E + 7") > t h e a b s o r p t i o n c o n s t a n t a and t h e r a t i o I L ^ / I L ^ i s d i s c u s s e d . The t i m e development of t h e d i f f r a c t i o n e f f i c i e n c y f o r d i f f e r e n t v a l u e s o f (E + -f) i s p l o t t e d i n F i g . 4.11 f o r chosen v a l u e s o f t h e o t h e r two p a r a m e t e r s . Here we see a gamut o f b e h a v i o u r o f t h e d i f f r a c t i o n e f f i c i e n c y , f r o m m o n o t o n i c s a t u r a t i o n t o o s c i l l a t i o n f o l l o w e d by s a t u r a t i o n . The l o w e s t c u r v e i n F i g . 4.11 i s f o r E = 0. T h i s means t h a t t h e e x t e r n a l a p p l i e d & v L f i e l d i s j u s t c a n c e l l i n g t h e v i r t u a l f i e l d and t h e h o l o g r a m i s s t o r e d by d i f f u s i o n . As t h e v a l u e o f (E + j-) i n c r e a s e s , d r i f t s t a r t s t o dominate t h e w r i t i n g mechanism. F i g . 4(a) a l s o shows t h a t t h e h i g h e r t h e v a l u e of (E + ^) t h e f a s t e r t h e b u i l d - u p and t h a t t h e r e l a t i o n becomes a l m o s t l i n e a r v L when d r i f t d ominates t h e p r o c e s s . The maximum d i f f r a c t i o n e f f i c i e n c y de-pends on t h e v a l u e o f ( E v+ U P t o a l i m i t beyond w h i c h any f u r t h e r i n c r e a s e does n o t i n c r e a s e t h e e f f i c i e n c y , w h i c h s t a r t s t o o s c i l l a t e . F i g . 4.12 shows t h e t i m e e v o l u t i o n o f t h e d i f f r a c t i o n e f f i c i e n c y f o r d i f f e r e n t v a l u e s o f t h e a b s o r p t i o n c o n s t a n t a and f o r chosen v a l u e s f o r th e o t h e r two p a r a m e t e r s . The t i m e s c a l e i s a d j u s t e d t o a l l o w f o r t h e 82. F i g . 4.11 The time development of the hologram w r i t i n g for dif f e r e n t values (1/cm). absolute d i f f r a c t i o n e f f i c i e n c y during of (Ev+V/L) with n°/nD= 10 and a= 1.0 90 <*/(cm-1) 0.2 (l/T0) -Fig. 4.12 The time development of the absolute diffraction efficiency during holgram writing for different values of the absorption constant a with n /n = 10 and (E +V/L)= 7.5 kV/cm. change i n T as a i s v a r i e d . Here a g a i n we see t h e d i f f e r e n t t y p e s o f be-o h a v i o u r f r o m s a t u r a t i o n t o o s c i l l a t i o n f o l l o w e d by s a t u r a t i o n . F i g . 4.11 shows t h a t t h e s m a l l e r t h e v a l u e o f a t h e s l o w e r t h e b u i l d - u p b u t t h e h i g h e r t h e maximum d i f f r a c t i o n e f f i c i e n c y . However, d e c r e a s e i n a beyond some p o i n t w i l l n o t p r o d u c e h i g h e r e f f i c i e n c y b u t a g a i n r e s u l t s i n o s c i l l a t i o n s . T h e r e -f o r e , t h e l a r g e r t h e v a l u e o f a t h e h i g h e r t h e s e n s i t i v i t y o f t h e w r i t i n g p r o c e s s b u t t h e s m a l l e r t h e maximum d i f f r a c t i o n e f f i c i e n c y and v i c e v e r s a . Thus, t h e r e i s a t r a d e - o f f between t h e s e two d e s i r e d p r o p e r t i e s ( s e n s i t i v i t y and l a r g e e f f i c i e n c y ) when s y n t h e s i z i n g t h e a b s o r p t i o n o f t h e c r y s t a l . S t a e b l e r e t a l . (1972) have shown t h a t t h e b e s t t r a d e - o f f i s o b t a i n e d when the c r y s t a l a b s o r b s 50% o f t h e l i g h t . As f o r t h e e f f e c t o f n^/n^, F i g . 4.13, shows t h e t i m e development of t h e d i f f r a c t i o n e f f i c i e n c y f o r d i f f e r e n t v a l u e s o f n^/n^ and f o r c h o s e n v a l u e s f o r t h e o t h e r two p a r a m e t e r s . F i g . 4.13 shows t h a t t h e h i g h e r t h e v a l u e o f n^/nj) ( i . e . t h e s m a l l e r d a r k c o n d u c t i v i t y o r t h e h i g h e r l i g h t i n -t e n s i t y ) t h e f a s t e r t h e b u i l d - u p and t h e h i g h e r t h e maximum e f f i c i e n c y ( a g a i n up t o c e r t a i n l i m i t s ) . The above b e h a v i o u r may be e x p l a i n e d as f o l l o w s : t h e s m a l l e r t h e d a r k c o n d u c t i v i t y , t h e s m a l l e r t h e r a t e o f t h e r m a l decay d u r i n g h o l o g r a m w r i t i n g . A l s o t h e h i g h e r t h e l i g h t i n t e n s i t y , t h e h i g h e r t h e a v e r a g e l i g h t g e n e r a t e d f r e e c a r r i e r c o n c e n t r a t i o n and t h e s m a l l e r t h e e f f e c t o f t h e f r e e c a r r i e r c o n c e n t r a t i o n i n t h e d a r k . The i n c l u s i o n o f t h e d a r k c o n d u c t i v i t y i n our model r e s u l t s i n an i n t e n s i t y i n d e p e n d e n t be-h a v i o u r o f t h e d i f f r a c t i o n e f f i c i e n c y . 4.4.5 Summary The g e n e r a l i z e d model combines, f o r t h e f i r s t t i m e , t h e f e e d b a c k e f f e c t o f t h e p h o t o i n d u c e d space c h a r g e f i e l d and t h e e f f e c t o f t h e hologram i n m o d i f y i n g t h e l i g h t p a t t e r n ( t h e use of t h e "dynamic t h e o r y " ) . I t 85. Fig. 4.13 The time development of the absolute diffraction efficiency during hologram writing for different values of the ratio n /n with a= 1.0 (1/cm) and (Ev+V/L)= 7.5 kV/cm. p r e d i c t s q u a l i t a t i v e l y a l l t h e r e p o r t e d t y p e s o f t i m e e v o l u t i o n o f t h e d i f -f r a c t i o n e f f i c i e n c y . The p r e v i o u s l y d e v e l o p e d models succeeded i n p r e d i c t i n g some t y p e s of t i m e developments b u t n o t a l l . F o r example, t h e models w h i c h a l l o w o n l y f o r t h e e f f e c t o f t h e f e e d b a c k of t h e space c h a r g e f i e l d ( A l p h o n s e e t a l . ( 1 9 7 5 ) , Su and G a y l o r d ( 1 9 7 5 ) , Kim e t a l . (1976a) and Sec. 4.2) f a i l t o p r e d i c t t h e o s c i l l a t o r y b e h a v i o u r o f t h e d i f f r a c t i o n e f f i c i e n c y r e p o r t e d by Amodei e t a l . (1972) and I s h i d a e t a l . ( 1 9 7 2 ) . CHAPTER V HOLOGRAM WRITING WITH GAUSSIAN BEAMS 5.1 I n t r o d u c t i o n I n C h a p t e r 4, m o d e l l i n g o f t h e h o l o g r a m w r i t i n g w i t h u n i f o r m i l -l u m i n a t i o n was c o n s i d e r e d . However, h o l o g r a m w r i t i n g w i t h n o n u n i f o r m i l l u -m i n a t i o n i s o f a g r e a t p r a c t i c a l i m p o r t a n c e s i n c e i t i s t h e most l i k e l y c on-f i g u r a t i o n i n h o l o g r a p h i c memory systems where a number o f s m a l l holograms 2 (1 mm ) i n the fo r m o f a g r i d w o u l d p r o b a b l y be s t o r e d i n each c r y s t a l . Holograms s t o r e d w i t h n o n u n i f o r m l i g h t p a t t e r n . p r o d u c e a l a r g e s c a l e space c h a r g e f i e l d a s s o c i a t e d w i t h t h e e n v e l o p e o f t h e l i g h t p a t t e r n , i n a d d i t i o n t o t h e f u n d a m e n t a l s i n u s o i d a l and h i g h o r d e r harmonic components o f t h e space c h a r g e f i e l d . A l a r g e s c a l e f i e l d o f t h i s t y p e i s p r o d u c e d by a s i n g l e l i g h t beam i n t h e absence o f t h e i n t e r f e r e n c e p a t t e r n e f f e c t w h i c h p r o d u c e s th e h o l o g r a m when two beams a r e p r e s e n t . Such a f i e l d due t o a s i n g l e beam was, o f c o u r s e , what was o b s e r v e d by Chen (1969) i n h i s o r i g i n a l compensa-t o r e x p e r i m e n t s . The f e e d b a c k e f f e c t o f t h i s l a r g e s c a l e f i e l d on t h e r e -d i s t r i b u t i o n o f p h o t o r e l e a s e d e l e c t r o n s was n e g l e c t e d i n t h e p r e v i o u s models s i n c e u n i f o r m i l l u m i n a t i o n was assumed. E x p e r i m e n t a l i n v e s t i g a t i o n s o f t h e e f f e c t o f t h e a p p l i e d v o l t a g e and of t h e f r a c t i o n a l i l l u m i n a t i o n o f t h e c r y s t a l on h o l o g r a m w r i t i n g were made by C o r n i s h , Moharam and Young (1976a) and i t was r e p o r t e d t h a t t h i s l a r g e s c a l e f i e l d a s s o c i a t e d w i t h t h e l i g h t e n v e l o p e has a s i g n i f i c a n t e f f e c t on t h e ho l o g r a m w r i t i n g . I n t h i s c h a p t e r , a model i s p r o p o s e d f o r h o l o g r a m w r i t i n g w i t h two c o h e r e n t l i g h t beams whose i n t e n s i t i e s v a r y i n G a u s s i a n f a s h i o n i n one d i m e n s i o n . These beams a r e i n c i d e n t on a c r y s t a l t o w h i c h a c o n s t a n t v o l -t age i s a p p l i e d . The e f f e c t o f t h e g r o w i n g h o l o g r a m on m o d i f y i n g t h e l i g h t p a t t e r n (Moharam and Young 1977) i s n o t a l l o w e d f o r } s i n c e i t c o m p l i c a t e s t h e p r o b l e m even w i t h n u m e r i c a l s o l u t i o n s t o an extreme e x t e n t . However, t h e p r e s e n t model as w i l l be shown, g i v e s u s e f u l i n s i g h t i n t o t h e e f f e c t of i n t e r e s t . I t i s a l s o shown t h a t t h e model a c c o u n t s , a t l e a s t q u a l i t i t i v e l y , f o r t h e e x p e r i m e n t a l o b s e r v a t i o n s o f C o r n i s h , Moharam and Young (1976a). 5.2 M o d e l (Moharam and Young 1976b) The i n t e r f e r e n c e p a t t e r n of two c o h e r e n t p l a n e waves w i t h G a u s s i a n a m p l i t u d e d i s t r i b u t i o n i n one d i m e n s i o n ( t h e x - a x i s ) i n c i d e n t s y m m e t r i c a l l y on t h e c r y s t a l w i t h a n g l e 6 and t h e i r p l a n e of i n c i d e n c e i s p a r a l l e l t o t h e x z p l a n e ( n o r m a l t o t h e c - a x i s w h i c h i s a l o n g t h e x - a x i s ) as i n F i g . 4.8 i s I ( x ) = 1 ( 1 + m cos K x ) e x p ( - 2 x 2 c o s 2 0 / a 2 ) (5.1) o where I i s t h e a v e r a g e l i g h t i n t e n s i t y a t t h e c e n t r e o f t h e beam (x = 0) and m i s t h e m o d u l a t i o n r a t i o a l s o a t t h e c e n t r e o f t h e beam. K = 2TT/A where A = A o / 2 s i n 8 i s t h e g r a t i n g w a v e l e n g t h and X q i s t h e l i g h t w a v e l e n g t h . Assuming t h a t t h e m i g r a t i o n l e n g t h o f t h e f r e e e l e c t r o n s i s s u b s t a n t i a l l y s h o r t e r t h a n t h e g r a t i n g s p a c i n g A, t h e f r e e c a r r i e r c o n c e n t r a t i o n i n t h e c o n d u c t i o n band may be w r i t t e n as n ( x ) = + n°[I(x)/I o] ( 5 > 2 ) where i s t h e f r e e c a r r i e r i n t h e d a r k and n£ = a t E ^ / h v . a i s t h e a b s o r p t i o n c o n s t a n t , T i s t h e c a r r i e r l i f e t i m e , £ i s t h e quantum e f f i c i e n c y and hv i s t h e p h o t o n e n e r g y . The e f f e c t o f t h e changes i n t h e t r a p occupancy on g e n e r a t i o n and t r a p p i n g r a t e s w i l l be n e g l e c t e d , s i n c e t h e s e s o u r c e s o f n o n l i n e a r i t y a r e n o r m a l l y l e s s i m p o r t a n t t h a n t h e f e e d b a c k e f f e c t o f t h e s p a c e c h a r g e f i e l d . The c o n d u c t i o n c u r r e n t d e n s i t y i s g i v e n i n t h e u s u a l way as J ( x , t ) = qD ~ + q y n [ f + E , ( x , t ) ] + K a l ( x ) (5.3) OX Li - X w h i c h i n c l u d e s d r i f t , d i f f u s i o n and t h e c u r r e n t due t o t h e p h o t o v o l t a i c e f f e c t . E ^ ( x , t ) i s t h e p h o t o i n d u c e d space charge f i e l d . I t i s t h e d e v i a -t i o n (due t o space charge f i e l d ) o f t h e e l e c t r i c f i e l d f r o m i t s a v e r a g e v a l u e V/L where V i s t h e a p p l i e d v o l t a g e and L t h e l e n g t h o f t h e c r y s t a l . The c o n s t r a i n t o f c o n s t a n t a p p l i e d v o l t a g e i m p l i e s t h a t L l f 1 E T dx = V ( 5 ; 4 ) where x = and x = d e f i n e t h e end o f t h e c r y s t a l w i t h r e s p e c t t o t h e c e n t r e o f t h e beam and L = - I ^ . The c o n t i n u i t y and P o i s s o n ' s e q u a t i o n s a r e 3p ( x , t ) . ,, N  s c = _ 3 J ( x , t ) 3 t 9 t ^ • : > ) and 9 E T ( x , t ) p ( x , t ) r — ( 5 - 6 ) 8x e Combining Eq. 5.5 and 5.6 and- i n t e g r a t i n g w i t h r e s p e c t t o t i m e and space s u b j e c t t o t h e c o n s t r a i n t o f Eq. ( 5 . 4 ) and assuming z e r o i n i t i a l c o n d i t i o n s E i ( x , 0 ) = 0.0, we a r r i v e a t 1 t 1 L l ' E . ( x , t ) = - ± o / t J ( x , t ) d t + — L / Qf J ( x , t ) d t dx ( 5 . 7 ) I n t h e i n i t i a l l i n e a r s t a g e where t << T (T = e/qun°)>E.(x,t) o o M P L J x i s v e r y s m a l l and may be n e g l e c t e d i n t h e t r a n s p o r t e q u a t i o n (Eq. 5 . 3 ) . S u b s t i t u t i n g Eqs. 5.2 and 5.3 i n Eqs. 5.7, we o b t a i n 2 2 2 E _ L ( x , t ) = ( t / T )E e x p ( - 2 x cos 6/a ) [m s i n Kx + (2x cos 2 9 / K a 2 ) ( l + m cos K x ) ] + ( q D t / e L ) [ n ( L ^ ) - n ( L 2 ) ] - ( t / T Q ) [ ( V / L ) + E v ] { e x p ( - 2 x 2 c o s 2 6 / a 2 ) - ( 1 + m cos Kx) L l - ( 1 / L ) T ; 1 ( I ( x ) / l )dx> ( 5 . 8 ) L 2 ° where = kTK/q i s t h e e q u i v a l e n t d i f f u s i o n f i e l d and E^ i s t h e v i r t u a l 2 2 f i e l d due t o t h e p h o t o v o l t a i c e f f e c t . The e x p r e s s i o n 2x cos e/Ka i s n e g l i g i b l y s m a l l i n a l m o s t a l l p r a c t i c a l c a s e s and, t h e r e f o r e , t h e second term i n t h e f i r s t b r a c k e t i n Eq. 5.8 may be n e g l e c t e d f o r a l l p r a c t i c a l p u r p o s e s . The e x p r e s s i o n g i v e n by Eq. 5.8 i s t h e n s i m i l a r t o Amodei's (1971a) e x p r e s s i o n e x c e p t f o r t h e G a u s s i a n e n v e l o p e of t h e space c h a r g e f i e l d . A c l o s e d f o r m s o l u t i o n f o r t h e t i m e development of t h e p h o t o -i n d u c e d s p a c e c h a r g e f i e l d d u r i n g h o l o g r a m w r i t i n g has n o t been o b t a i n e d due t o t h e c o m p l e x i t y o f t h e p r o b l e m . However, an i n f i n i t e s e r i e s s o l u t i o n has been o b t a i n e d as f o l l o w s : CO E . ( x , t ) = E ( - q u t / e ) r - ( l / r ! ) { [ ( V / L ) + E v ] [ C r - C r ] + ( k T / q ) [ D r - D r ] } r - 1 where C = h ( x ) C , D = n ( x ) D , r+1 r r+1 r and = n ( x ) , T>1 = 3n/3x, (5.9) L l - L l C = ( 1 / L ) T / XC dx, T) = ( 1 / L ) T ; X D dx. " 2 2 The f i n a l s a t u r a t i o n s i t u a t i o n i s g i v e n by ^ = — = 0 ( n o t by J = 0, s i n c e t h e c o n d u c t i v i t y i s everwhere f i n i t e ) . F o r t h i s s a t u r a t i o n s t a t e we o b t a i n a s i m p l e s o l u t i o n w h i c h a l l o w s t h e main p o i n t s t o be deduced: E ± ( x ) = { [ ( A / n ( x ) ) - l ] ( V / L + E ^ ) } - ( k T / q ) [ ( 3 n / 9 x ) - B ] / n ( x ) where (5.10) L L A = L/_ / ( l / n ( x ) ) d x , and B = (A/L)_ / [ ( 3 n / 3 x ) / n ( s ) ] d x L 2 ' 2 The f i r s t t e r m i n 5.10 i s due t o d r i f t i n t h e a c t u a l e l e c t r o s t a t i c f i e l d p l u s t h e " v i r t u a l " f i e l d , t h e second t e r m i s due t o d i f f u s i o n . I f t h e i n -c i d e n t l i g h t p a t t e r n i s s y m m e t r i c a l l y p l a c e d w i t h r e s p e c t t o t h e c r y s t a l edges, i . e . L = -L^ = — t h e n = 0 i n Eq. 5.9 and B = 0 i n Eq. 5.10. 5.3 C a l c u l a t e d R e s u l t s and D i s c u s s i o n 5.3.1 I n t r o d u c t i o n E x a m i n a t i o n of Eq. 5.8 shows t h a t , i n t h e i n i t i a l l i n e a r s t a g e s , t h e p h o t o i n d u c e d s p a c e c h a r g e , w h i c h a t t h i s s t a g e c o n t a i n s o n l y f u n d a m e n t a l s i n u s o i d a l components, i s i n d e p e n d e n t of t h e d a r k c o n d u c t i v i t y and of t h e n o n u n i f o r m i t y o f t h e i l l u m i n a t i o n , ( e x c e p t i n t h a t t h i s l a t t e r p r o v i d e s an e n v e l o p e f u n c t i o n on t h e a m p l i t u d e ) s i n c e t h e f e e d b a c k e f f e c t of t h e s p a c e c h a r g e f i e l d i s assumed s m a l l a t t h i s s t a g e . D u r i n g t h e l a t e r development o f t h e h o l o g r a m we have v a r i o u s e f f e c t s due t o t h e f e e d b a c k e f f e c t o f t h e s p a c e c h a r g e f i e l d . T h i s i n -c l u d e s b o t h t h e s i n u s o i d a l components a s s o c i a t e d w i t h t h e i n t e r f e r e n c e of the beams and t h e l a r g e s c a l e f i e l d a s s o c i a t e d w i t h t h e e n v e l o p e o f t h e beam. To a n a l y s e t h e e f f e c t on t h e h o l o g r a m , we assume t h a t t h e G a u s s i a n e n v e l o p e of t h e l i g h t i n t e n s i t y may be t a k e n as s l o w l y v a r y i n g i n d i s t a n c e compared t o t h e s c a l e o f t h e s i n u s o i d a l v a r i a t i o n s o f l i g h t i n t e n s i t y due t o i n t e r f e r e n c e . F o r p r a c t i c a l c a s e s , Ka>>l and t h u s t h e e n v e l o p e may be c o n s i d e r e d as c o n s t a n t o v e r a d i s t a n c e g i v i n g a few p e r i o d s of t h e s i n u -s o i d a l components. T h i s a l l o w s t h e p h o t o i n d u c e d s p a c e c h a r g e f i e l d t o be r e s o l v e d i n t o " l o c a l " F o u r i e r dc p l u s f u n d a m e n t a l and h i g h e r harmonic s i n u -s o i d a l components. The r e s u l t s p r e s e n t e d h e r e a r e l i m i t e d t o t h e s a t u r a t i o n s t a t e of th e s pace c h a r g e f i e l d (Eq. 5.10). A l t h o u g h , t h e t i m e development o f t h e space c h a r g e f i e l d can be c a l c u l a t e d u s i n g Eq. 5.9, i t does n o t p r o v i d e any new i n s i g h t i n t o t h e p r o b l e m t h a t cannot be g a i n e d f r o m s t u d y i n g t h e s a t u r a t i o n s t a t e . 92. 5.3.2 E f f e c t o f t h e R a t i o o f C r y s t a l L e n g t h t o Beam W i d t h E x a m i n a t i o n o f Eq. 5.10 shows t h a t t h e c r y s t a l l e n g t h L e n t e r s i n t o t h e space c h a r g e e x p r e s s i o n t h r o u g h t h e i n t e g r a l s A and B. The i n t e g r a l A d e t e r m i n e s t h e v a l u e o f t h e spac e c h a r g e component due t o d r i f t w h i l e t h e i n t e g r a l B a f f e c t s t h e f i e l d component due t o d i f f u s i o n . I f t h e l i g h t p a t -t e r n i s s y m m e t r i c a l l y p l a c e d on t h e c r y s t a l , t h e v a l u e o f t h e i n t e g r a l B=0, and t h e d i f f u s i o n c o n t r i b u t i o n t o t h e p h o t o i n d u c e d space c h a r g e f i e l d E^ i s in d e p e n d e n t o f t h e c r y s t a l l e n g t h t o t h e beam w i d t h . However, t h e i n t e n s i t y e n v e l o p e o f t h e i l l u m i n a t i o n d e t e r m i n e s t h e e n v e l o p e o f t h e a m p l i t u d e of t h e space c h a r g e f i e l d . The r e l a t i o n between t h e e n v e l o p e o f t h e i n t e n s i t y and the e n v e l o p e o f t h e a m p l i t u d e depends on t h e d a r k c o n d u c t i v i t y i n a c o m p l i -c a t e d manner. The space c h a r g e f i e l d component due t o d i f f u s i o n may be w r i t -t e n as (Eq. 5.10 and B=0) E - E m ' s i n k x i D 1+m'coskx (5.11) m' = m / [ l + ( n D / n ^ ) e x p ( 2 x 2 c o s 2 9 / a 2 ) ] T h e r e f o r e , t h e r a t i o d a r k t o l i g h t c a r r i e r c o n c e n t r a t i o n n ^ / i t ^ ( w h i c h i n -c r e a s e s w i t h i n c r e a s e i n t h e d a r k c o n d u c t i v i t y and d e c r e a s e s w i t h t h e l i g h t i n t e n s i t y ) as w e l l as t h e e n v e l o p e o f t h e l i g h t p a t t e r n a f f e c t s t h e shape of t h e e n v e l o p e o f t h e space c h a r g e f i e l d due t o d i f f u s i o n . I n t h e l i m i t i n g c ase where n^ = 0 t h e space c h a r g e f i e l d due t o d i f f u s i o n i s u n i f o r m o v e r the c r y s t a l l e n g t h . I f t h e l i g h t p a t t e r n i s n o t p l a c e d s y m m e t r i c a l l y on t h e c r y s t a l ( i . e . B^O), an a d d i t i o n a l c o n t r i b u t i o n t o t h e space c h a r g e f i e l d , i n c l u d i n g b o t h l a r g e s c a l e and s i u n s o i d a l components, i s p r o d u c e d . The e x t r a s i n u -s o i d a l component hasa90° phase s h i f t w i t h r e s p e c t t o s i n u s o i d a l components f o r a s y m m e t r i c a l l y p l a c e d beam. T h i s a d d i t i o n a l c o n t r i b u t i o n depends on t h e r a t i o o f t h e c r y s t a l l e n g t h t o t h e beam w i d t h ( t h r o u g h t h e i n t e g r a l B ) . Eq. 5.10 shows t h a t t h e space c h a r g e f i e l d component due t o d r i f t i s p r o p o r t i o n a l t o t h e i n t e g r a l A w h i c h i s d e t e r m i n e d by t h e r a t i o o f t h e c r y s t a l l e n g t h t o t h e beam w i d t h ( t h r o u g h t h e i n t e g r a t i o n l i m i t s ) as w e l l as the l i g h t p a t t e r n e n v e l o p e and t h e r a t i o n^/n^. The l a s t two p a r a m e t e r s a l s o w i l l d e t e r m i n e t h e shape of t h e e n v e l o p e o f t h e f i e l d as i n the d i f f u s i o n c a s e . I n F i g . 5.1 t h e s p a t i a l d i s t r i b u t i o n o f t h e a m p l i t u d e o f t h e l o c a l F o u r i e r dc component of t h e p h o t o i n d u c e d space c h a r g e f i e l d E^ a t s a t u r a t i o n i s p l o t t e d t o g e t h e r w i t h t h e c o n t r i b u t i o n o f (a) d r i f t and (b) d i f f u s i o n t o the f u n d a m e n t a l and f i r s t h armonic s i n u s o i d a l components. T h i s i s f o r sym-m e t r i c a l l y p l a c e d beams and m = 1.0, = 5 and L/2a = 2.5. F i g . 5.1 shows t h a t t h e maximum o f t h e f u n d a m e n t a l s i n u s o i d a l component i s n o t a t t h e c e n t r e o f t h e beam (maximum i n t e n s i t y ) . T h i s i s due t o t h e n e g a t i v e f e e d -back e f f e c t o f t h e l o c a l dc f i e l d component w h i c h i s a maximum a t t h e c e n t r e of t h e beam. The f u n d a m e n t a l component due t o d i f f u s i o n i s shown a l s o t o be g r e a t e r t h a n the f u n d a m e n t a l component due t o d r i f t ( f o r e q u a l e f f e c t i v e d i f -f u s i o n and d r i f t f i e l d s ) . T h i s i s due t o t h e n o n u h i f o r m i t y o f t h e i l l u m i n a -t i o n w h i c h does n o t a f f e c t t h e d i f f u s i o n component b u t a f f e c t s t h e d r i f t component ( t h r o u g h t h e i n t e g r a l A ) . I f t h e beam w i d t h a-*30 ( i . e . u n i f o r m i l l u m i n a t i o n ) t h e two components due t o d r i f t and d i f f u s i o n w o u l d be e q u a l (Sec. 4.2). To i n v e s t i g a t e t h e e f f e c t o f t h e r a t i o o f t h e c r y s t a l l e n g t h t o the beam w i d t h (L/2a) on t h e d r i f t c o n t r i b u t i o n t o t h e spac e c h a r g e f i e l d ( i t has no e f f e c t on t h e d i f f u s i o n c o n t r i b u t i o n f o r s y m m e t r i c a l l y p l a c e d beams), F i g s . 5.2 t o 5.4 show how t h e a m p l i t u d e s a t s a t u r a t i o n o f t h e l o c a l d c , f u n d a m e n t a l and f i r s t h a r m o n i c s components o f E^ ( r e s p e c t i v e l y ) change w i t h t h e r a t i o n?/n_ f o r d i f f e r e n t v a l u e s o f t h e r a t i o o f c r y s t a l l e n g t h t o 94. Ill u« CRYSTAL LENGTH- H DISTANCE FROM BEAM CENTER (UNITS (r) F i g . 5.1 Spa t i a l d i s t r i b u t i o n of some of the Fourier components of the photo-induced f i e l d due to d r i f t (in units of E +V/L) and due to d i f f u s i o n ( i n units of E D) for n°/nD = 0.5, L/2a = 2.5, and mV= 1.0. F i g . 5.2 The locafc dc component of the ghotoinduced f i e l d ( i n units of Ev+V/L) at the center of the beam vs. n^/n^ for d i f f e r e n t values of the cr y s t a l f r a c t i o n a l i l l u m i n a t i o n L/2a. F i g . 5.3 The fundamental Fourier component of E. due to d r i f t ( i n units of Ev+V/L) at the center of the beam vs. f° r d i f f e r e n t values of L /2a. 97. beam w i d t h L/2a. F o r t h e s e diagrams m = 1.0 and t h e l i g h t p a t t e r n i s symme-". t r i c a l l y p l a c e d on the c r y s t a l . F i g . 5.2 shows t h a t the l o c a l dc component of t h e p h o t o i n d u c e d f i e l d ( a t t h e c e n t r e o f t h e beam) i n c r e a s e s w i t h i n c r e a s e i n L/2a f o r a l l v a l u e s o f n^/n^. That i s , t h e s m a l l e r t h e f r a c t i o n a l i l l u m i -n a t i o n , t h e l a r g e r t h e l a r g e s c a l e f i e l d . F o r example, when t h e beam d i a -meter i s about f o u r t i m e s t h e s i z e of t h e c r y s t a l , t h e l o c a l dc f i e l d i s about 0.01 of t h e t o t a l a p p l i e d f i e l d . T h e r e f o r e , a l t h o u g h u n i f o r m i l l u m i -n a t i o n i s an i d e a l i z e d s i t u a t i o n , n o t s t r i c t l y r e a l i z a b l e , t h e v a l u e o f L/2a<0.5 would g i v e an e x c e l l e n t a p p r o x i m a t i o n t o t h e i d e a l c a s e of a u n i -f o r m l y i l l u m i n a t e d c r y s t a l . F i g . 5.3 and F i g . 5 . 4 show t h a t b o t h t h e f u n d a -m e n t a l and f i r s t h a r monic s i n u s o i d a l components of t h e d r i f t c o n t r i b u t i o n t o t h e space c h a r g e f i e l d ( a t t h e c e n t r e o f t h e beam) d e c r e a s e w i t h i n -c r e a s e i n t h e r a t i o L/2a f o r a l l v a l u e s o f V^/^' T h i s b e h a v i o u r i s , as e x p e c t e d , i n c o n t r a s t t o t h e b e h a v i o u r o f t h e l o c a l dc component w h i c h has a n e g a t i v e f e e d b a c k e f f e c t on t h e development o f t h e o t h e r components o f t h e space c h a r g e f i e l d . E x p e r i m e n t a l o b s e r v a t i o n s by C o r n i s h e t a l . c o n f i r m t h i s dependence o f t h e f u n d a m e n t a l component o f t h e f i e l d on t h e f r a c t i o n a l i l l u m i n a t i o n o f t h e c r y s t a l . T h e i r e x p e r i m e n t a l d a t a ( F i g . 5.5) shows t h a t t h e l a r g e r t h e r a t i o o f beam d i a m e t e r t o c r y s t a l s i z e , t h e l a r g e r t h e p h o t o -i n d u c e d space c h a r g e f i e l d and t h e l a r g e r t h e d i f f r a c t i o n e f f i c i e n c y of t h e g r a t i n g . T h i s w o u l d l i m i t t h e p o t e n t i a l o f t h e p h o t o r e f r a c t i v e m a t e r i a l i n a p p l i c a t i o n s where o n l y a s m a l l f r a c t i o n of t h e c r y s t a l i s i l l u m i n a t e d . 5.3.3 The E f f e c t o f t h e D a r k C o n d u c t i v i t y As was shown i n Sec. 4.2, i n t h e i l l u m i n a t e d r e g i o n , an i n -c r e a s e i n the d a r k c o n d u c t i v i t y i s e q u i v a l e n t t o a d e c r e a s e i n t h e m o d u l a t i o n r a t i o m. Thus, i f n = n^ + n°(1+mcosKx), we may w r i t e t h a t n = (1+m'cosKx), where n' = n^ + n^ and m' = m/(H-n^/n^). However, w i t h a p a r t i a l l y 01 1 1 1 L t I I -3 -2 -/ 0 1 2 3 APPLIED VOLTAGE /kV F i g . 5.5 Experimental observations reported by Cornish et a l . (1976a). Normalized values of arcsinv'n (normalized by dividing by the exposure) are plotted against applied voltage (for electrodes 1 cm apart) for dif f e r e n t configurations of c r y s t a l i l l u m i n a t i o n . The r a l a t i v e area of illumination i s shown by the c i r c l e s and the c r y s t a l face (1 cm square) by the squares. Curves C and D have the same f r a c t i o n a l i l l u m i n a t i o n but the l i g h t i n t e n s i t y used i n curve D was about ten times larger than that used i n curve C. 100. i l l u m i n a t e d c r y s t a l w i t h t o t a l d a r k r e g i o n s , t h e d a r k c o n d u c t i v i t y has an a d d i t i o n a l i m p o r t a n t e f f e c t i n t h a t t h e f i n a l s t e a d y s t a t e c o r r e s p o n d s t o 3J/3x = 0 i f ± 0, as compared t o J = 0 f o r n D = 0. Thus f o r = 0y J = 0 = q y n [ E . ( x ) + E v + L J ] + qD | ^ so t h a t ~ an E (x) = - ( E + h (5.11) 1 v L q n I n the f i n a l s a t u r a t i o n s t a t e , t h e s i n u s o i d a l p a r t s o f t h e s p a c e c h a r g e f i e l d , w h i c h c o n s t i t u t e t h e h o l o g r a m , a r e c r e a t e d by d i f f u s i o n i n b a l a n c e w i t h d r i f t due t o t h e s i n u s o i d a l f i e l d s . I t i s i n t e r e s t i n g t o n o t e t h a t Eq. 5.11 a p p l i e s a l s o t o t h e c a s e of o p e n - c i r c u i t w i t h c o m p l e t e i l l u m i -n a t i o n p r o v i d e d t h a t V i s i n t e r p r e t e d as t h e s p o n t a n e o u s l y g e n e r a t e d v o l t a g e . W i t h G a u s s i a n beams, th e i l l u m i n a t i o n m a t h e m a t i c a l l y e x t e n d s o v e r t h e w h o le specimen so t h a t , even i f ^ 0 t h e c a r r i e r c o n c e n t r a t i o n i s nowhere z e r o and t h e f i n a l s t a t e c o r r e s p o n d s t o 3J/3x = 0. I n t h i s s i t u a t i o n , t h e f i n a l s t e a d y s t a t e v a l u e o f t h e l o c a l dc component o f E^ does n o t r e a c h t h e above maximum v a l u e o f - ( E v + V / L ) , w h i c h w o u l d g i v e a t o t a l e l e c t r o s t a t i c f i e l d j u s t c a n c e l l i n g t h e " v i r t u a l f i e l d " . The l a r g e r n^ and hence t h e l a r g e r s t e a d y s t a t e c u r r e n t , t h e l a r g e r t h e s i n u s o i d a l components and t h e s m a l l e r t h e l o c a l dc component o f E^ a t t h e f i n a l s t e a d y s t a t e ( n e g l e c t i n g t h e change i n t h e m o d u l a t i o n r a t i o m) . F i g . 5.6 shows a p l o t o f t h e f u n d a m e n t a l and f i r s t h a r m o n i c s i n u -s o i d a l components o f t h e d i f f u s i o n c o n t r i b u t i o n o f t h e p h o t o i n d u c e d space c h a r g e f i e l d E^ a t t h e c e n t r e o f t h e beam v s . t h e r a t i o n^/n^. Here t h e i n c i d e n t m o d u l a t i o n r a t i o o f t h e l i g h t p a t t e r n m = 1.0 and t h e l i g h t p a t t e r n i s s y m m e t r i c a l l y p l a c e d on t h e c r y s t a l . F i g . 5.6 shows t h a t b o t h s i n u s o i d a l components o f t h e f i e l d d e c r e a s e w i t h i n c r e a s e i n t h e d a r k c o n d u c t i v i t y ( o r F i g . 5.6 The fundamental and the f i r s t harmonic Fourier component of the photoinduced f i e l d E. due to d i f f u s i o n ( i n units of E_) vs. n /n . d e c r e a s e i n t h e l i g h t i n t e n s i t y ) s i n c e any i n c r e a s e i n r a t i o n /n° w i l l r e -duce t h e m o d u l a t i o n r a t i o m-and d e c r e a s e t h e s i n u s o i d a l components o f t h e p h o t o i n d u c e d f i e l d (Sec. 4.2). As f o r t h e d r i f t c o n t r i b u t i o n t o E^, F i g . 5.2 shows t h a t t h e l o c a l dc component of E^ a t t h e c e n t r e o f t h e beam d e c r e a s e s w i t h d e c r e a s e i n n°/n f o r a l l v a l u e s of L / 2 a . T h i s may be e x p l a i n e d as f o l l o w s : l a r g e Li u v a l u e s of n^ ( s m a l l n^/n^) p r o d u c e s l a r g e s t e a d y s t a t e c u r r e n t s w h i c h t e n d to r e l a x s u c h l a r g e s c a l e f i e l d s . F i g . 5.2 shows a l s o t h a t , f o r l a r g e L / 2 a , t h e e f f e c t o f the, d a r k c o n d u c t i v i t y i s more s i g n i f i c a n t . T h i s i s b ecause f o r l a r g e v a l u e s o f L/2CJ, t h e p h o t o i n d u c e d c a r r i e r c o n c e n t r a t i o n a t t h e edges o f t h e c r y s t a l i s v e r y s m a l l and t h e d a r k c a r r i e r c o n c e n t r a t i o n con-t r o l s t h e s t e a d y s t a t e c u r r e n t . But i f L / 2 a i s s m a l l , a b i g g e r p h o t o i n d u c e d c a r r i e r c o n c e n t r a t i o n a t t h e edges and, hence, a b i g g e r s t e a d y s t a t e c u r r e n t r e g a r d l e s s of t h e v a l u e of n^ i s p r o d u c e d . F i g . 5.3 and 5.4 shows th e dependence of t h e f u n d a m e n t a l and f i r s t h a rmonic s i n u s o i d a l components o f E^ a t t h e c e n t r e o f t h e beam on t h e r a t i o n^/n^. As f a r as t h e s e s i n u s o i d a l components a r e c o n c e r n e d , t h e r a t i o h ^ / n 0 tends t o r e d u c e t h e m o d u l a t i o n r a t i o and hence r e d u c e s t h e a m p l i t u d e of t h e s i n u s o i d a l components. On t h e o t h e r hand, an i n c r e a s e i n n^/n^ t e n d s t o p r o d u c e l a r g e r s t e a d y s t a t e c u r r e n t and hence r e d u c e s t h e l a r g e s c a l e f i e l d and i t s n e g a t i v e f e e d b a c k e f f e c t and t h u s i n c r e a s e s t h e s i n u s o i d a l component F i g . 5.3 and 5.4 show t h a t where L/2a i s s m a l l (where t h e e f f e c t of n^/n° on t h e s t e a d y s t a t e c u r r e n t i s s m a l l ) t h e a m p l i t u d e s of b o t h s i n u s o i d a l components i n c r e a s e w i t h i n c r e a s e n^/n^ ( i . e . i n c r e a s e i n t h e e f f e c t i v e mod-l u a t i o n r a t i o ) . However, i f L / 2 a i s l a r g e , an i n i t i a l i n c r e a s e i n t h e amp-l i t u d e s of t h e f i e l d components o c c u r s due t o t h e i n c r e a s e i n t h e m o d u l a t i o n a f t e r w h i c h t h e e f f e c t o f t h e l a r g e n^/n^ i n r e d u c i n g t h e s t e a d y s t a t e 103. c u r r e n t t a k e s o v e r and p r o d u c e s a l a r g e dc component and a l a r g e n e g a t i v e f e e d b a c k e f f e c t , and hence r e d u c e s t h e a m p l i t u d e s of t h e s i n u s o i d a l compo-n e n t s . The e x p e r i m e n t a l o b s e r v a t i o n s of C o r n i s h e t a l . (1976a) i n F i g . 5.5 ( c u r v e C and D) c l e a r l y show t h e d e c r e a s e i n t h e f u n d a m e n t a l component o f gram s t o r a g e p r o c e s s i s i n v e s t i g a t e d . A model f o r h o l o g r a m w r i t i n g w i t h t h e i n t e r f e r e n c e p a t t e r n o f two p l a n e waves w i t h a G a u s s i a n a m p l i t u d e d i s t r i b u -t i o n i n c i d e n t on a f i n i t e c r y s t a l under c o n s t a n t a p p l i e d v o l t a g e i s p r e s e n t e d . I t i s shown t h a t t h e d a r k c o n d u c t i v i t y has an i m p o r t a n t e f f e c t on t h e w r i t i n g p r o c e s s e s p e c i a l l y when t h e i l l u m i n a t e d r e g i o n o f t h e c r y s t a l i s s m a l l com-p a r e d t o t h e c r y s t a l s i z e . I t i s a l s o shown t h a t t h e f u n d a m e n t a l and h a r -monic s i n u s o i d a l components of t h e space c h a r g e f i e l d w h i c h c o n s t i t u t e t h e h o l o g r a m d e c r e a s e d r a s t i c a l l y w i t h d e c r e a s e i n t h e i l l u m i n a t e d r e g i o n on t h e c r y s t a l . The t h e o r e t i c a l p r e d i c t i o n of t h e model i s shown t o be i n a t l e a s t q u a l i t a t i v e agreement w i t h t h e e x p e r i m e n t a l o b s e r v a t i o n s o f C o r n i s h e t a l . t h e f i e l d when t h e l i g h t i n t e n s i t y i s i n c r e a s e d l a r g e . 5.4 Summary The e f f e c t o f t h e n o n u n i f o r m i t y o f t h e i l l u m i n a t i o n on t h e h o l o -( 1 9 7 6 a ) . CHAPTER"VI READING AND OPTICAL ERASURE OF HOLOGRAMS STORED BY THE PHOTOREFRACTIVE EFFECT 6.1 I n t r o d u c t i o n When holograms s t o r e d by t h e p h o t o r e f r a c t i v e e f f e c t a r e r e a d w i t h t h e o r i g i n a l w a v e l e n g t h o f l i g h t , e l e c t r o n s a r e once more l i b e r a t e d and r e -d i s t r i b u t e d . I f t h e i n c i d e n t l i g h t i s u n i f o r m , t h e produced p h o t o c o n d u c t i -v i t y t e n d s t o r e l a x t h e e x i s t i n g f i e l d and t h u s t o e r a s e t h e hologram. How-e v e r , i f , as S t a e b l e r and Amodei (1972b) have shown, t h e r e c o n s t r u c t e d beam i n t e r a c t s w i t h t h e r e a d i n g beam, t h e r e s u l t i n g l i g h t i n t e r f e r e n c e p a t t e r n w r i t e s a new h o l o g r a m w h i c h may e i t h e r enhance t h e o r i g i n a l h o l o g r a m o r s e l e c t i v e l y e r a s e i t . The d i f f r a c t i o n e f f i c i e n c y i n t h e case o f enhance-ment may i n c r e a s e a t f i r s t w i t h t i m e o r m e r e l y decay more s l o w l y t h a n o t h e r -w i s e . I f t h e l i g h t i s i n c i d e n t o f f t h e Bragg a n g l e ( i . e . no i n t e r a c t i o n w i t h t h e g r a t i n g ) t h e n s i m p l e e r a s u r e o c c u r s w i t h n e i t h e r enhancement n o r d i s -enhancement. I f t h e c r y s t a l i s n o t f u l l y and u n i f o r m l y i l l u m i n a t e d , t h e s i t u a t i o n becomes more complex ( C o r n i s h e t a l . 1976 and Moharam and Young 1976b). s i n c e a l a r g e s c a l e f i e l d a s s o c i a t e d w i t h t h e e n v e l o p e o f t h e beam i s w r i t t e n i n p a r a l l e l w i t h e r a s i n g t h e hologram. T h i s f i e l d may p e r s i s t t h r o u g h t h e r e a d i n g / e r a s u r e phase and a f f e c t subsequent r e w r i t i n g o f h o l o -grams . Amodei (1972a) and l a t e r A l p h o n s e e t a l . (1975) have m o d e l l e d e r a s u r e by u n i f o r m l i g h t i n c i d e n t o f f t h e Bragg a n g l e ( i m p l i e d by n e g l e c t i n g t h e e f f e c t o f t h e i n t e r f e r e n c e o f t h e r e a d i n g and d i f f r a c t e d beams). B o t h a u t h o r s s t a r t e d w i t h a u n i f o r m h o l o g r a m t h r o u g h t h e t h i c k n e s s o f t h e g r a t i n g i . e . t h e y n e g l e c t e d t h e e f f e c t s o f t h e beam c o u p l i n g and l i g h t a b s o r p t i o n 105. d u r i n g h o l o g r a m w r i t i n g . They p r e d i c t e d an e x p o n e n t i a l decay of t h e space c h a r g e f i e l d w h i c h c o n s t i t u t e s t h e h o l o g r a m w i t h a t i m e c o n s t a n t e q u a l t o t h a t of t h e d i e l e c t r i c r e l a x a t i o n t i m e d u r i n g i l l u m i n a t i o n . S t a e b l e r and Amodei (1972b) were t h e f i r s t t o p r e d i c t t h e enhancement and disenhancement, p r o c e s s e s . They p r e s e n t e d a s i m p l i f i e d model f o r t h e s e c o m p l i c a t i o n s . Magnusson and G a y l o r d (1976) used t h e i r dynamic model t o s t u d y t h e enhance-ment, r e a d i n g and o p t i c a l e r a s u r e a t t h e Bragg a n g l e , . b u t t h e y n e g l e c t e d t h e f e e d b a c k e f f e c t o f t h e space cha r g e f i e l d . They a l s o n e g l e c t e d t h e u n i f o r m p a r t o f t h e l i g h t i n t e n s i t y ( w h i c h c a u s e s t h e e r a s u r e ) . I n t h i s c h a p t e r , a model f o r t h e d e s t r u c t i v e r e a d i n g and o p t i c a l e r a s u r e b o t h on and o f f t h e Bragg a n g l e i s p r o p o s e d . The model, f o r t h e f i r s t t i m e , t a k e s i n t o a c c o u n t a l l t h e major c o m p l i c a t i o n s . Thus a b a s i s f o r r e a l i s t i c c o m p a r i s o n w i t h e x p e r i m e n t a l d a t a i s o b t a i n e d . 6.2 M o d e l (Moharam and Young 1978b) The h o l o g r a m t o be e r a s e d i s t a k e n t o be. as i t w o u l d be i n p r a c -t i c e i n t h a t i t i s n o t u n i f o r m t h r o u g h t h e t h i c k n e s s o f t h e c r y s t a l (due t o t h e a b s o r p t i o n o f l i g h t and due t o t h e energy t r a n s f e r between t h e two w r i -t i n g beams). I n f a c t , t h e s t a r t i n g h o l o g r a m i s t h e o u t p u t o f t h e model f o r t h e w r i t i n g p r o c e s s p r o p o s e d i n Sec. 4.4. T h i s w r i t i n g model i s a n a l o g o u s t o t h e p r e s e n t model f o r d e s t r u c t i v e r e a d i n g and e r a s u r e p r o c e s s e s . B o t h models a l l o w f o r t h e e f f e c t o f space cha r g e f i e l d s on t h e r e d i s t r i b u t i o n o f p h o t o r e l e a s e d e l e c t r o n s . The e l e c t r o n s a r e a l l o w e d t o move by d i f f u s i o n and by d r i f t i n e l e c t r o s t a t i c f i e l d s and by t h e p h o t o v o l t a i c e f f e c t . The e f f e c t o f t h e h o l o g r a m i n m o d i f y i n g t h e l i g h t i n t e n s i t y p a t t e r n t h r o u g h t h e c r y s t a l ( i n t h e c a s e o f Bragg a n g l e i n c i d e n c e ) i s t a k e n i n t o a c c o u n t as i s t h e e f f e c t of a b s o r p t i o n . F i n i t e d a r k c o n d u c t i v i t y i s a l s o a l l o w e d f o r . The model i s , however, r e s t r i c t e d t o t h e c a s e of u n i f o r m i l l u m i n a t i o n , c o n s t a n t a p p l i e d 106. v o l t a g e , and s h o r t t r a n s p o r t l e n g t h s of l i b e r a t e d e l e c t r o n s compared t o t h e g r a t i n g s p a c i n g . The s y s t e m c o n f i g u r a t i o n and c o o r d i n a t e s y s t e m i s shown i n F i g . 4.8. From Sec. 4.4.2, t h e c o u p l e d wave e q u a t i o n s f o r r e a d o u t by monochromatic p e r -p e n d i c u l a r l y p o l a r i z e d i n f i n i t e p l a n e wave o f an u n s l a n t e d phase g r a t i n g s a t i s f y i n g t h e Bragg c o n d i t i o n a r e : (Eq. 4.49). 3 Ra( ! ? t ) = - j C ( z , t ) e x p ( - j c j > ( z , t ) ) S ( z , t ) d Z (6.1) 8 Sa( Z ? t ) = - j C ( z , t ) e x p ( - j ( j , ( z , t ) ) R ( z , t ) d Z where C ( z , t ) = 7 r A N ( z , t ) A O cos 8, X q i s t h e l i g h t w a v e l e n g t h and 8 i s t h e a n g l e o f i n c i d e n c e . AN and <(> a r e t h e a m p l i t u d e and r e l a t i v e s p a t i a l phase of t h e f u n d a m e n t a l F o u r i e r component o f t h e m o d u l a t i o n i n t h e r e f r a c t i v e i n d e x . The r e a d i n g beam i s assumed t o be d i f f r a c t e d o n l y by t h e f u n d a m e n t a l g r a t i n g (Sec. 4.2 and 4.3). The above e q u a t i o n a p p l i e s a l s o t o p a r a l l e l p o l a r i z e d l i g h t i f C i s r e p l a c e d by C' = C cos 28. S i n c e t h e change i n t h e 3 r e f r a c t i v e i n d e x i s p r o d u c e d v i a t h e e l e c t r o - o p t i c e f f e c t , t h e n AN = % r N E^. N and r a r e t h e a p p r o p r i a t e r e f r a c t i v e i n d e x . E^ i s t h e a m p l i t u d e of t h e f u n d a m e n t a l F o u r i e r component of t h e space c h a r g e f i e l d , <j> i s t h e s p a t i a l phase s h i f t o f t h i s component. We now d e v e l o p an e x p r e s s i o n f o r t h e p h o t o i n d u c e d space c h a r g e f i e l d d u r i n g t h e e r a s u r e p r o c e s s . Under t h e a s s u m p t i o n o f s h o r t t r a n s p o r t l e n g t h compared t o t h e g r a t i n g s p a c i n g , t h e f r e e c a r r i e r c o n c e n t r a t i o n i n t h e c o n d u c t i o n band may be w r i t t e n as n ^ x . z . t ) = + n£(I(x,z,t)/l o (6.2) where n^ i s t h e f r e e c a r r i e r c o n c e n t r a t i o n i n t h e d a r k , I ( x , z , t ) i s t h e l i g h t i n t e n s i t y p a t t e r n and I i s i t s a v e r a g e i n t e n s i t y , n? = a£rl /hv where a i s O J_i o 107. t h e a b s o r p t i o n c o n s t a n t , £ i s t h e quantum e f f i c i e n c y , T i s t h e c a r r i e r l i g h t t i me and hv i s t h e p h o t o n e n e r g y . The c u r r e n t d e n s i t y e q u a t i o n i s 3. V J ( x , z , t ) = q D - - a + K a I ( x ) z , t ) + q u n ( x , z , t ) [— +. E. ( x , z , t ) ] (6.3) O X Li 1 where E^ i s t h e p h o t o i n d u c e d s p a c e c h a r g e f i e l d and V i s t h e a p p l i e d v o l t a g e and L i s t h e c r y s t a l l e n g t h . The c u r r e n t d e n s i t y component i n t h e z - d i r e c t i o n i s n e g l e c t e d (Sec. 4.3). Combining t h e c u r r e n t d e n s i t y e q u a t i o n (Eq. 6.3) and t h e c o n t i n u i t y and P o i s s o n ' s e q u a t i o n s (Eq. 4.7 and 4.8) under t h e c o n s t r a i n t o f c o n s t a n t a p p l i e d v o l t a g e (Eq. 4.9) we o b t a i n : E ( x , z , t ) = '0=9%{ / f c{[ E + f + E . ( x , z , t ) ] N ( x , z , t ) + — | ^ } d t X G t V J_i x q dx o L ^ T / 2 / t { [E + y- + E ( x , z , t ) ] n ( x , z , t ) + — -^ n-}dt dx i i i i c v L x q <3x " 2 ° + E 1 ( x , z , t o ) (6.4) where E ^ ( x , z , t ^ ) i s t h e i n i t i a l v a l u e o f t h e s p a c e c h a r g e f i e l d . I t r e p r e -s e n t s t h e space c h a r g e f i e l d w h i c h c o n s t i t u t e s t h e ho l o g r a m t o be e r a s e d . E^ i s t h e v i r t u a l f i e l d r e p r e s e n t i n g t h e p h o t o v o l t a i c e f f e c t . F o r any l i g h t i n t e n s i t y p a t t e r n I ( x , z , t ) Eq. 6.3 and 6.4 may be s o l v e d f o r t h e space c h a r g e f i e l d . By r e s o l v i n g t h e f i e l d i n t o F o u r i e r s e r i e s , (and AN) and cf> can be o b t a i n e d and t h e c o u p l e d wave e q u a t i o n s (Eqs. 6.1) may be s o l v e d w i t h t h e a p p r o p r i a t e boundary c o n d i t i o n s . 6.2.1 O f f Bragg A n g l e I n c i d e n c e S i n c e t h e i n c i d e n t e r a s i n g beam w i l l n o t i n t e r a c t w i t h t h e h o l o g r a m ( i . e . no d i f f r a c t e d beam i s p r o d u c e d ) , i t s i n t e n s i t y w i l l r e m a i n i n d e p e n d e n t o f x and t , b u t w i l l d e c r e a s e e x p o n e n t i a l l y w i t h t h e d i s t a n c e t r a v e l l e d i n t o t h e c r y s t a l , i . e . 108.' I ( z ) = I q e x p ( - a z / c o s 0^) (6.5) where 0. i s t h e a n g l e o f r e f r a c t i o n o f t h e e r a s i n g beam. S u b s t i t u t i n g Eqs. 6.2 and 6.5 i n Eq. 6.4 we a r r i v e a t E i ( x , z , t ) = E i ( x , z , t o ) e x p { - ^ (6.6) Eq. 6.6 shows t h a t t h e space c h a r g e f i e l d t o be e r a s e d w i l l decay exponen-t i a l l y b u t t h e " t i m e c o n s t a n t " i s i n c r e a s i n g w i t h t h e d i s t a n c e t r a v e l l e d i n t o t h e c r y s t a l . E ^ ( x , z , t Q ) i s t h e spac e c h a r g e f i e l d a t t h e end of t h e w r i t i n g phase. R e s o l v i n g E^ i n t o i t s F o u r i e r components, we o b t a i n E^ ( o r AN) and the s p a t i a l phase s h i f t ( j i . M o n i t o r i n g t h e h o l o g r a m d u r i n g e r a s u r e i s done u s i n g e i t h e r t h e r e f e r e n c e o r t h e s u b j e c t beam i n c i d e n t a t t h e Bragg a n g l e . However, i t i s assumed t h a t t h e r e a d i n g i s c a r r i e d o u t f o r a s u f f i c i e n t l y s h o r t t i m e t h a t i t does n o t d i s t u r b t h e e r a s u r e p r o c e s s o r i n t e r a c t w i t h t h e hol o g r a m . Under t h i s a s s u m p t i o n , t h e n o n d e s t r u c t i v e r e a d i n g p r o c e s s does n o t depend on w h i c h beam i s u s e d . To c a l c u l a t e t h e d i f f r a c t i o n e f f i c i e n c y t h e c o u p l e d wave e q u a t i o n s (Eqs. 6.1) a r e s o l v e d (knowing A N ( z , t ) and <f>(z,t) f r o m Eq. 6.6) w i t h t h e boundary c o n d i t i o n s R ( 0 , t ) = 1.0 and S ( 0 , t ) = 0.0. 2 The d i f f r a c t i o n e f f i c i e n c y i s d e f i n e d as exp(-aD/cos 0 ) | s ( D , t ) | where D i s the c r y s t a l t h i c k n e s s . 6.2.2 Brag g A n g l e I n c i d e n c e I n t h i s c a s e t h e r e a d i n g beam i s a l s o t h e e r a s i n g beam ( d e s -t r u c t i v e r e a d i n g ) . The l i g h t i n t e r f e r e n c e p a t t e r n o f t h e r e a d i n g beam and t h e r e c o n s t r u c t e d wave may be w r i t t e n as (Eq. 4.50) where I q = J s g [ | R ( z , t ) | 2 . + | S ( z , t ) | 2 ] m ( z , t ) = 2 R ( z , t ) / [ | R ( z , : t ) | 2 + | s ( z , t ) | 2 ] (6.7) 109. iKz,t) = arctan[(SR* - S*R)/(SR* + S*R)] (6.7) and g i s the inverse of the characteristic impedance of the medium. Sub-stituting Eq. 6.2 and 6.7 in Eq. 6.4 we obtain an expression for the space charge which can be desolved into Fourier series to obtain AN(z,t) and <j>(z,t) as the case of off Bragg incidence. To calculate the diffraction efficiency the coupled wave equations (Eqs. 6.1) are solved with the i n i t i a l boundary conditions R(0,t) = 1.41 and S(0,t) =0.0 for reference beam erasure and R(0,t) = 0.0 and S(0,t) = 1.41 for subject beam erasure. The amplitude of the erasing beam is chosen such that the total erasing intensity i s equal to the total intensity of the two beams which are used to write the original holo-gram. The diffraction efficiency i s defined as 0.5|S(D,t)J for reference 2 beam erasure and 0.5|R(D,t)| for subject beam erasure. It is important to note that optical erasure using the reference beam with the c-axis of the crystal pointed i n one direction i s , in general, not equivalent to erasure using the subject beam with the c-axis ;pointed in the other direction, since the photo-voltaic effect i s a unidirectional property and by flipping the crystal around the sign, the current Kai (and of E^) i s changed in addition to the change in the sign of the electro-optic coefficient. The results given in the next section are for light of wavelength X q = 514.4 nm with electric vector parallel to the c-axis. Since the medium is taken to be lithium niobate, the extraordinary refractive index i s taken as 2.24 and the operative electro-optic coefficient r^^ a s 30.8 x 10 "^ cm/V. Holograms are i n i t i a l l y written with two equal beams incident symmetrically on the crystal at an angle of incidence of 15°. The ratio of the i n i t i a l light to dark carrier concentration n^/n^ is 100 for both the writing and the erasure cycle. The calculated results w i l l be presented for a selected set of 1 1 0 . h o l o g r a m p a r a m e t e r s . These p a r a m e t e r s a r e t h e t o t a l f i e l d (7-+ E ) and t h e J _ J V a b s o r p t i o n c o n s t a n t a . The t i m e s c a l e i s t/T o» where T q = e/qun^. T q i s a measure of t h e r e l a x a t i o n t i m e a s s o c i a t e d w i t h t h e p h o t o i n d u c e d c a r r i e r s . These p a r a m e t e r s a , E^, n£ a r e i n t e r d e p e n d e n t , s i n c e a i s p r o p o r t i o n a l t o +2 the c o n c e n t r a t i o n o f f i l l e d t r a p s (Fe ) and t h e l i f e t i m e T, w h i c h e n t e r s b o t h E^ and n£ i s i n v e r s e l y p r o p o r t i o n a l t o t h e c o n c e n t r a t i o n o f empty t r a p s +3 +2 +3 (Fe ) and n^ i s p r o p o r t i o n a l t o f / ( l - f ) ( i . e . Fe /Fe ) , where f i s t h e p r o b a b i l i t y o f t r a p o c c u p a t i o n . Thus n£ i s dependent on t h e s t a t e o f r e d u c -t i o n o r o x i d i z a t i o n o f t h e i r o n c e n t r e s b u t n o t on t h e i r d e n s i t y . The s t a t e o f r e d u c t i o n w i l l a l s o a f f e c t t h e d a r k c o n d u c t i v i t y . I n r e s u l t s t o be p r e s e n t e d i n t h e n e x t s e c t i o n , t h e c o m p l e t e w r i t e -e r a s e c y c l e w i l l be g i v e n , s i n c e , as w i l l be shown, t h e e x t e n t and c h a r a c -t e r i s t i c s o f t h e w r i t i n g phase a f f e c t t h e e r a s u r e phase. 6.3 C a l c u l a t e d R e s u l t s and D i s c u s s i o n As was p o i n t e d o u t b e f o r e , t h e mechanism o f o p t i c a l e r a s u r e i n v o l v e s t h e p h o t o e x c i t a t i o n o f e l e c t r o n s f r o m t r a p s by t h e u n i f o r m i l l u m i n a t i o n and t h e i r s u b sequent r e t r a p p i n g i n such a way as t o r e l a x t h e spac e c h a r g e f i e l d w h i c h c o n s t i t u t e s t h e ho l o g r a m . T h e r e f o r e , t h e o p t i c a l e r a s u r e i s f a s t e r t h e l a r g e r t h e a b s o r p t i o n c o n s t a n t . A l s o , due t o t h e a b s o r p t i o n o f l i g h t , t he h o l o g r a m i s e r a s e d a t a f a s t e r r a t e a t t h e f r o n t s u r f a c e o f t h e c r y s t a l t h a n a t i t s back s u r f a c e . F o r Bragg a n g l e i n c i d e n c e , we have t h e a d d i t i o n a l e f f e c t t h a t a new h o l o g r a m i s w r i t t e n by t h e i n t e r f e r e n c e p a t t e r n o f t h e r e a d -i n g beam and t h e beam d i f f r a c t e d by t h e e x i s t i n g h o l o g ram. F i g . 6.7 shows a s c h e m a t i c r e p r e s e n t a t i o n o f t h e e r a s u r e p r o c e s s . We o r i g i n a l l y hava a space c h a r g e f i e l d (hologram) E^ c o s ( K x + <(>) . The u n i f o r m p a r t o f t h e i l l u m i n a t i o n b o t h f o r on and f o r o f f Bragg a n g l e i l l u m i n a t i o n c a u s e s e r a s u r e o f t h e h o l o -gram AE (Kx + <f>) . The s i n u s o i d a l p a r t o f t h e i l l u m i n a t i o n , w h i c h e x i s t s HOLOGRAM BEFORE ERASURE . _ Ej cos (kx + 0 ) HOLOGRAM ADDED*1 DURING ERASURE DC. ILLUMINATION ____ - AE-, cos(kx + 0) SINUSOIDAL COMPONENTS (AT BRAGG ANGLE ONLY) a - DRIFT J j ;AE a s in(kx+ ' i ) b-DIFFUSION — iAEg 'cos (kx+0 ) OFF BRAGG ANGLE ERASURE — = ERASURE AT BRAGG ANGLE ERASURE R BEAM S BEAM . ENHANCEMENT DISENHANCEMENT F i g . 6 . 1 A schematic representation of a q u a l i t a t i v e model for the o p t i c a l erasure process of holograms stored by the photorefractive e f f e c t . only for Bragg angle erasure, writes a hew hologram. This new hologram w i l l be in phase or 180° out of phase with the original hologram i f i t is written by diffusion i.e. ± AE^ cos (Kx + <j>) . The new hologram w i l l be ± 90° out of phase i f i t is written by d r i f t i.e. ± AE sin (Kx + <}>) . The sign of the new 3. hologram (space charge f i e l d ) , as shown by Staebler and Amodei (1972), de-pends on the orientation of the c-axis of the crystal with respect to the reading beam. Fig. 6.1 illustrates the combined effect of erasure of the hologram by the uniform part of the illumination and vectorial addition of the new hologram (in case of Bragg angle incidence). There are two important cases of optical erasure at the Bragg angle. The f i r s t occurs when the effect of the uniform illumination i s stronger than the effect of the new hologram written during the erasure pro-cess. We would expect an exponential-like decay of the space charge f i e l d (and i n most cases of the diffraction efficiency) for erasure with either of the two beams but with different "time constants", because the result of vectorial addition of the new hologram is different depending on which of the two beams is used (Fig. 6.1). Fig. 6.2 shows the calculated diffraction efficiency of the hologram in a write-erase cycle. The erasure phase i s shown for the three cases of off Bragg angle incidence and for Bragg angle incidence with (a) the reference and (b) the subject beam. Here the crystal thickness D was 0.5 cm, the absorption constant a was 3.0 cm ^ and the total applied f i e l d (E^ + j-) = 5 kV/cm. These parameters were chosen to select the above case (strong absorption and weak new hologram). Fig. 6.2a shows the writing phase of the cycle. The diffraction efficiency reaches a maxi-mum and then starts to decrease. Fig. 6.2b shows the behaviour of the d i f -fraction efficiency during optical erasure when starting from point A in Fig. 6.2a. The diffraction efficiency shows an exponentials-like decay for 113. 25-0 uj 12.5 WRITING F i g . 6.2 The time development of the absolute d i f f r a c t i o n e f f i c i e n c y during hologram write-erase cycle (a) write, (b) erase s t a r t i n g from point A, and (c) esare from B. Erasure i s shown for off the Bragg angle and for destructive reading with (i)the R beam and ( i i ) the S beam. 114. b o t h r e f e r e n c e and s u b j e c t beam e r a s u r e and f o r t h e case o f o f f Bragg a n g l e i n c i d e n c e , b u t w i t h a d i f f e r e n t " t i m e c o n s t a n t " f o r each c a s e as was p r e d i c -t e d above. F i g . 6.2c shows t h e o p t i c a l e r a s u r e c h a r a c t e r i s t i c s when t h e w r i t i n g phase was e x t e n d e d u n t i l t h e d i f f r a c t i o n e f f i c i e n c y has s t a r t e d t o o s c i l l a t e ( p o i n t B i n F i g . 6.2a). The spac e c h a r g e f i e l d decays as b e f o r e i n an e x p o n e n t i a l - l i k e f a s h i o n w i t h a " t i m e c o n s t a n t " depending on i f t h e r e f e r e n c e o r s u b j e c t beam i s used o f i f o f f Bragg a n g l e e r a s u r e i s used. The d i f f r a c t i o n e f f i c i e n c y , on t h e o t h e r hand, shows an i n i t i a l i n c r e a s e and t h e n s t a r t s t o decay i n a manner s i m i l a r t o t h a t o f F i g . 6.2b. The t y p e o f c h a r a c t e r i s t i c s shown i n F i g . 6.2b has been r e p o r t e d i n t h e l i t e r a -t u r e ( S t a e b l e r and Amodei 1972b and S t a e b l e r and P h i l l i p s 1974). The e x p e r i -m e n t a l d a t a r e p o r t e d by I s h i d a e t a l . (1972) f o r o p t i c a l e r a s u r e o f ho l o g r a m s t o r e d i n Rh-doped l i t h i u m n i o b a t e a r e s i m i l a r i n p r i n c i p l e t o t h a t o f F i g . 6.2c e x c e p t t h a t t h e i r e r a s u r e c h a r a c t e r i s t i c s showed two o s c i l l a t i o n s i n t h e d i f f r a c t i o n e f f i c i e n c y b ecause t h e d i f f r a c t i o n e f f i c i e n c y was a l l o w e d t o o s c i l l a t e t w i c e i n t h e w r i t i n g phase. The second c a s e o f i m p o r t a n c e i s when t h e e f f e c t o f t h e new h o l o -gram added d u r i n g r e a d i n g i s s t r o n g e r t h a n t h e e r a s u r e o f t h e o r i g i n a l h o l o -gram due t o t h e u n i f o r m p a r t o f t h e i l l u m i n a t i o n . T h i s c a s e i s a c h i e v e d w i t h s m a l l a b s o r p t i o n , a t h i c k c r y s t a l and t h e h o l o g r a m w r i t t e n by d i f f u s i o n ( i . e . a r e d u c e d l i g h t l y i r o n doped c r y s t a l ) . I t can be deduced f r o m F i g . 6.1 t h a t , i f t h e new h o l o g r a m i s w r i t t e n by d i f f u s i o n and i s s t r o n g e r t h a n t h e e r a s u r e due t o t h e u n i f o r m p a r t o f t h e l i g h t , t h e combined e f f e c t w i l l be c l e a r enhancement o f t h e o r i g i n a l h o l o g r a m f o r r e a d i n g w i t h one beam o r c l e a r disenhancement f o r r e a d i n g w i t h t h e o t h e r beam. However, t h e enhance-ment does n o t c o n t i n u e i n d e f i n i t e l y . The spac e c h a r g e f i e l d i s e r a s e d a t t h e f r o n t s u r f a c e o f t h e h o l o g r a m s i n c e t h e i l l u m i n a t i o n i s u n i f o r m a t t h i s 115. p l a n e . The new h o l o g r a m w r i t t e n d u r i n g r e a d i n g i s a t f i r s t weak as one p r o c e e d s i n t o t h e c r y s t a l away f r o m t h e f r o n t s u r f a c e , and hence does n o t overcome t h e e r a s u r e c a u s e d by t h e u n i f o r m i l l u m i n a t i o n . Thus a k i n d o f " c r e e p i n g e r a s u r e " o c c u r s s t a r t i n g f r o m t h e f r o n t s u r f a c e o f t h e c r y s t a l . T h i s c a u s e s t h e enhancement t o s t o p a f t e r a c e r t a i n t i m e d e p e n d i n g on t h e a b s o r p t i o n c o n s t a n t . The d i f f r a c t i o n e f f i c i e n c y w i l l t h e n decay i n an ex-p o n e n t i a l - l i k e manner. F i g s . 6.3 and 6.4 show t h e c a l c u l a t e d d i f f r a c t i o n e f f i c i e n c y d u r i n g a w r i t e - e r a s e c y c l e . The p a r a m e t e r s o f t h e h o l o g r a m a r e chosen s u c h t h a t t h e above second c a s e i s i n e f f e c t . Here a = 1.0, D = 1.0 V cm and + — = 0.0. The h o l o g r a m i s w r i t t e n by d i f f u s i o n o n l y and t h e s p a t i a l phase s h i f t i s 9 0 ° . F i g . 6.2a shows t h e t i m e development o f t h e d i f f r a c t i o n e f f i c i e n c y d u r i n g t h e w r i t i n g phase. F i g . 6.4a shows t h e be-h a v i o u r o f t h e d i f f r a c t i o n e f f i c i e n c y d u r i n g o p t i c a l e r a s u r e when th e w r i t -i n g phase i s s t o p p e d a t p o i n t A. F o r t h e o f f Bragg a n g l e c a s e , t h e dlff-f r a c t i o n e f f i c i e n c y d ecays i n e x p o n e n t i a l f a s h i o n . F o r t h e Bragg a n g l e c a s e , e r a s u r e w i t h t h e r e f e r e n c e beam causes t h e d i f f r a c t i o n e f f i c i e n c y t o i n c r e a s e a t f i r s t w i t h t i m e , b u t a f t e r some ti m e i t s t a r t s t o decay, as p r e d i c t e d by t h e p r e v i o u s argument. E r a s u r e w i t h t h e s u b j e c t beam i s s i m i l a r i n e f f e c t t o t h a t f o r t h e o f f Bragg a n g l e c a s e b u t a t a f a s t e r r a t e . F i g . 6.4b shows t h e o p t i c a l e r a s u r e c h a r a c t e r i s t i c s when t h e w r i t i n g t i m e i s about t h r e e t i m e s t h a t o f F i g . 6.4c. The w r i t i n g phase was e x t e n d e d t o p o i n t B i n F i g . 6.3a. The e r a s u r e c h a r a c t e r i s t i c s a r e g e n e r a l l y s i m i l a r i n F i g . 6.4a and 6.4b. They d i f f e r i n t h a t , i n F i g . 6.4a, t h e enhancement s t o p s when t h e e r a s u r e t i m e i s about f o u r w r i t i n g t i m e s , w h i l e i n F i g . 6.4b i t s t o p s a f t e r one w r i t i n g t i m e . T h i s i s a r e s u l t of t h e beam c o u p l i n g d u r i n g t h e w r i t i n g phase. F o r l o n g enough w r i t i n g t i m e , a l l t h e i n c i d e n t l i g h t e n ergy i s t r a n s -f e r r e d i n t o one of t h e two w r i t i n g beams a t some d e p t h i n t h e c r y s t a l . T h i s Fig. 6.3 The time development of (a) the absolute diffraction efficiency and (b) the fundamental component of the photoinduced f i e l d at different depths in the crystal during hologram writing. 15.0 (c) ERASURE 3.0 , 6.0 9.0 i F i g . 6.4 Optical erasure for off the Bragg angle and f o r destructive reading (on the Bragg angle) with ( i ) R and ( i i ) S beams s t a r t i n g from points (a) A and (b) B i n F i g . 6.3. causes the hologram at this depth to start being erased. This process w i l l start at the back surface of the crystal. Fig. 6.36 shows the time develop-ment of the fundamental Fourier component of the space charge f i e l d at five equally spaced planes through the thickness of the crystal during the writing phase. The f i e l d at the front surface of the crystal (Z/D = 0.0) is s t i l l increasing but at Z/D = 0.5 i t has almost saturated at a value half that at Z/D = 0.0. For Z/D > 0.5 the f i e l d reaches a saturation and then starts to decrease. Therefore, for this case, the strength of the hologram is mainly near the front surface where i t is erased f i r s t during the erasure phase. This i s not the case for a shorter writing phase where the hologram is more or less uniform throughout the crystal thickness. Experimental data simi-lar to Fig. 6.3c and 6.3d have been reported previously by Staebler et a l . (1974) and by Gaylord et a l . (1973). However, Gaylord et a l . , we believe, aborted the erasure phase before the optical erasure overcame the enhancement. 6.4 Summary A computer model for optical erasure off the Bragg angle and for destructive reading at the Bragg angle of holograms stored by photorefractive effect has been developed. This model allows simultaneously for the feedback effect of the space charge f i e l d , the nonuniformity of the hologram through the crystal thickness and the dark conductivity. It takes account of the new hologram written by interference between the reconstructed and the read-ing waves for the case of destructive reading at the Bragg angle. The model produces a l l the types of erasure characteristics reported in the literature and i s believed to be r e a l i s t i c enough for the determination of the physical parameters of the material by f i t t i n g experimental data to erasure characteri-stics generated by the model. It also can be used to synthesize the crystal parameters and the external experimental conditions to obtain a desired set of erasure characteristics. CHAPTER V I I EXPERIMENTAL CONSIDERATIONS FOR HOLOGRAM FORMATION 7.1 I n t r o d u c t i o n To I n v e s t i g a t e e x p e r i m e n t a l l y t h e p r o c e s s e s o f h o l o g r a m s t o r a g e and e r a s u r e i n LiNbO^, t h e p r o t o t y p e h o l o g r a m formed by t h e i n t e r f e r e n c e p a t t e r n o f two monochromatic c o h e r e n t p l a n e waves was use d . To w r i t e t h e h o l o g r a m , t h e LiN b O ^ c r y s t a l was p l a c e d i n t h e r e g i o n of t h e i n t e r f e r e n c e p a t t e r n ( F i g . 7 . 1 ) . The s p a t i a l v a r i a t i o n o f t h e l i g h t i n t e n s i t y i n d u c e s a s p a t i a l v a r i a t i o n i n t h e r e f r a c t i v e i n d i c e s o f t h e c r y -s t a l (by t h e p h o t o r e f r a c t i v e e f f e c t ) w h i c h c o n s t i t u t e t h e hologram. To r e a d o u t t h e i n f o r m a t i o n s t o r e d i n t h e h o l o g r a m , t h e c r y s t a l was i l l u m i n a t e d w i t h a p l a n e wave ( t h e r e f e r e n c e wave) and t h e volume d i f f r a c t i o n g r a t i n g s c a t t e r s t h e l i g h t i n a manner t h a t r e c o n s t r u c t s t h e s u b j e c t wave used t o fo r m t h e h o l o g r a m ( F i g . 7 . 2 ) . The a b s o l u t e d i f f r a c t i o n e f f i c i e n c y n o f t h e h o l o g r a m i s d e f i n e d as t h e r a t i o o f t h e d i f f r a c t e d i n t e n s i t y t o t h e i n c i d e n t i n t e n s i t y . The r e l a t i v e d i f f r a c t i o n e f f i c i e n c y i s d e f i n e d as t h e r a t i o o f t h e d i f f r a c t e d t o the t r a n s m i t t e d i n t e n s i t i e s . K o g e l n i k ( 1 9 6 9 ) has shown t h a t , f o r phase h o l o -grams o p e r a t i n g i n t h e Bragg regime o f d i f f r a c t i o n ( o n l y one d i f f r a c t e d wave and t h a t a t t h e Bragg a n g l e ) and i f t h e ho l o g r a m i s u n i f o r m t h r o u g h o u t th e t h i c k n e s s o f t h e c r y s t a l , t h e r e l a t i v e d i f f r a c t i o n e f f i c i e n c y i s n = s i n 2 ( v ) ( 7 . 1 ) and t h e a b s o l u t e e f f i c i e n c y i s n = n exp(-aD/cos 6) (7.2) where a i s t h e a b s o r p t i o n c o n s t a n t , 9 i s t h e a n g l e o f i n c i d e n c e of t h e r e a d -i n g wave ( t h e Bragg a n g l e ) , and v = i r A n D / ^ c o s 9 f o r p e r p e n d i c u l a r p o l a r i z e d l i g h t and v = irAnDcos20/X cosO f o r p a r a l l e l p o l a r i z a t i o n . 1 2 0 . X R \ A 1 z h—D—H Y -wvw-F i g . 7.1 Interference pattern of two plane waves. 121 Fig. 7.2 Diffraction of the reference wave by the hologram. 122. Here, X q i s t h e vacuum w a v e l e n g t h , D i s t h e g r a t i n g t h i c k n e s s and An i s t h e a m p l i t u d e o f t h e F o u r i e r f u n d a m e n t a l component o f t h e r e f r a c t i v e i n d e x modu-l a t i o n (assumed u n i f o r m i n t h e g r a t i n g t h i c k n e s s ) . K e r m i s c h (1973) and K i l l a t (1976) have shown t h a t t h e d i f f r a c t i o n e f f i c i e n c y o f n o n u n i f o r m phase holograms ( t h e a m p l i t u d e o f t h e s i n u s o i d a l i n d e x m o d u l a t i o n i s c h a n g i n g w i t h d i s t a n c e t r a v e l l e d i n t o t h e c r y s t a l ) i s g i v e n by t h e K o g e l n i k f o r m u l a (Eq. 7.1) w i t h v = ( i r / X o ) o J " D ^ C O S ^ d z , f o r p e r p e n d i c u l a r p o l a r i z e d l i g h t and v = ( c o s 2 8 / A o ) o / D ^ c o s ^ A n ( z ) d z , f o r p a r a l l e l p o l a r i z e d l i g h t . A n ( z ) i s t h e a m p l i t u d e o f t h e f u n d a m e n t a l component o f t h e i n d e x m o d u l a t i o n . Here v i s t h e a v e r a g e v a l u e o f t h e g r a t i n g s t r e n g t h . I f A n ( z ) i s c o n s t a n t , t h e d i f -f r a c t i o n r e d u c e s t o K o g e l n i k ' s o r i g i n a l f o r m u l a . I f b o t h t h e a m p l i t u d e and phase of t h e f u n d a m e n t a l component o f t h e i n d e x m o d u l a t i o n a r e n o t u n i f o r m t h r o u g h t h e g r a t i n g t h i c k n e s s (bent g r a t i n g ) , no c l o s e d f o r m e x p r e s s i o n f o r t h e d i f f r a c t i o n e f f i c i e n c y i s a v a i l a b l e and t h e p r o p e r c o u p l e d wave e q u a t i o n s a r e s o l v e d n u m e r i c a l l y (Sec. 4.4). Eqs. 7.1 and 7.2 were d e r i v e d n e g l e c t i n g m u l t i p l e i n t e r n a l r e f l e c t i o n between t h e f a c e s o f t h e c r y s t a l . These e f f e c t s a r e d i s c u s s e d i n Sec. 7.3. 7.2 The O p t i c a l System F i g . 7.3 shows a s c h e m a t i c of t h e e x p e r i m e n t a l s e t u p used. Co-h e r e n t l i g h t beam f r o m t h e L a s e r ( S p e c t r a P h y s i c s model 166 a r g o n i o n l a s e r ) was s p l i t w i t h a beam s p l i t t e r i n t o two beams. The t r a n s m i t t a n c e o f t h e beam s p l i t t e r was v a r i a b l e ( i n s t e p s ) w h i c h a l l o w e d a d j u s t m e n t o f t h e r e l a -t i v e beam powers. The p o l a r i z a t i o n o f t h e l i g h t beam was a d j u s t e d by u s i n g a p o l a r i z a t i o n r o t a t o r ( S p e c t r a P h y s i c s M o d e l 310-21). Two f r o n t s u r f a c e l a s e r m i r r o r s were used t o d i r e c t t h e two beams t o cause them t o i n t e r s e c t w i t h i n t h e volume o f t h e c r y s t a l . Two s e t s o f s p a t i a l f i l t e r s and c o l l i m a -t i n g l e n s e s ( S p e c t r a P h y s i c s model 332 and G e r t n e r model R250) were used t o F i g . 7.3. E x p e r i m e n t a l arrangement f o r m e a s u r i n g the d i f f r a c t i o n e f f i c i e n c y o f p l a n e wave h o l o g r a m by i n t e r m i t t e n t l y b l o c k i n g t h e S beam. F i g . 7.4 A l t e r n a t i v e arrangement f o r m e a s u r i n g the d i f f r a c t i o n e f f i c i e n c y by c o n t i n u o u s l y m o n i t o r i n g the a u z i l l a r y He-Ne beam. 124. s p a t i a l l y f i l t e r and expand each o f t h e two beams. The s i z e o f t h e c o l l i -mated beams was 2.5 cm i n d i a m e t e r . A t t e m p t s t o s p a t i a l l y f i l t e r t h e expan-ded l i g h t beam b e f o r e t h e beam s p l i t t e r was abandoned because t h e beam s p l i t t e r i n t r o d u c e d f r i n g e s i n t h e f i l t e r e d expanded beams. The geometry o f t h e s e t u p was s u c h t h a t t h e p a t h l e n g t h s o f t h e two beams f r o m t h e beam s p l i t t e r t o t h e c r y s t a l were w i t h i n 1.0 cm o f b e i n g e q u a l . T h i s e n s u r e d t h a t t h e p a t h d i f f e r e n c e was l e s s t h a n t h e coherence l e n g t h o f t h e l a s e r 12 cm). A s i l i c o n p h o t o v o l t a i c d e t e c t o r ( A l p h a m e t r i c s model dc 1010 w i t h a P1110 broadband p r o b e ) was p l a c e d a f t e r t h e c r y s t a l i n l i n e w i t h t h e s u b j e c t beam. When t h e s h u t t e r was c l o s e d , t h e energy d i f -f r a c t e d f o r t h e r e f e r e n c e beam toward t h e d e t e c t o r c o u l d be measured. A s i m i l a r d e t e c t o r ( A l p h a m e t r i c s model dc 1030 w i t h a P1101 p r o b e ) was p l a c e d a f t e r t h e c r y s t a l t o c o n t i n u o u s l y m o n i t o r t h e r e f e r e n c e ( r e a d i n g ) beam. The s u b j e c t s h u t t e r was c l o s e d m o m e n t a r i l y , from t i m e t o t i m e , t o measure t h e d i f f r a c t e d beam d u r i n g h o l o g r a m w r i t i n g p r o c e s s . D u r i n g h o l o g r a m e r a s u r e ( d e s t r u c t i v e r e a d o u t ) t h e s u b j e c t beam was b l o c k e d c o m p l e t e l y and t h e d i f -f r a c t e d beam-was m o n i t o r e d c o n t i n u o u s l y . The d i f f r a c t i o n e f f i c i e n c y o f t h e h o l o g r a m ( d u r i n g b o t h w r i t i n g and r e a d i n g ) was t a k e n as the r a t i o of the d i f f r a c t e d i n t e n s i t y t o t h e sum of t h e i n t e n s i t i e s of t h e d i f f r a c t e d and t r a n s m i t t e d p a r t s o f the r e f e r e n c e beam. F i g . 7.4 shows a n o t h e r method used t o m o n i t o r t h e d i f f r a c t i o n e f f i c i e n c y . An a n c i l l a r y He-Ne l a s e r was p o s i t i o n e d so t h a t t h e a n g l e o f i n c i d e n c e o f t h e beam s a t i s f i e d t h e Bragg c o n d i t i o n o f t h e phase g r a t i n g p r o d u c e d by t h e h i g h power a r g o n i o n l a s e r . As t h e h o l o g r a m d e v e l o p e d , more and more energy w o u l d be d i f f r a c t e d f r o m t h e i n c i d e n t p a t h o f t h e He-Ne beam a l l o w i n g c o n t i n u o u s m o n i t o r i n g o f t h e d i f f r a c t i o n e f f i c i e n c y . The l ow power He-Ne l a s e r was chosen because t h e p h o t o r e f r a c t i v e e f f e c t i s i n e f f i c i e n t w i t h l i g h t o f w a v e l e n g t h 682.8 nm as compared w i t h l i g h t o f wave-l e n g t h l e s s t h a n 510 nm. Each o f t h e above methods has i t s drawbacks e x p e r i m e n t a l l y . W i t h t h e f i r s t method, t h e h o l o g r a m f o r m a t i o n must be i n t e r r u p t e d t o r e a d t h e d i f f r a c t i o n e f f i c i e n c y . D u r i n g r e a d i n g , t h e r e f e r e n c e beam w i l l c a u s e some o p t i c a l e r a s u r e o f t h e h o l o gram. However, t h e o p t i c a l e r a s u r e on r e a d o u t was r e d u c e d by b l o c k i n g t h e s u b j e c t ( t o r e a d t h e d i f f r a c t i o n e f f i c i e n c y ) f o r v e r y s h o r t p e r i o d s o f t i m e ( t y p i c a l l y l e s s t h a n 1 s e c ) . The second method does n o t e r a s e t h e h o l o g r a m o r n e c e s s i t a t e i n t e r -r u p t i o n o f i t s f o r m a t i o n b u t i t does p r e s e n t o t h e r p r o b l e m s . I f t h e a n g l e of i n c i d e n c e o f t h e He-Ne beam i s n o t v e r y c l o s e t o t h e Bragg a n g l e , t h e n t h e d i f f r a c t i o n e f f i c i e n c y o f t h e h o l o g r a m d e t e r m i n e d w i t h t h i s beam i s much r e d u c e d f r o m i t s t r u e v a l u e . Not o n l y i s t h e a l i g n m e n t v e r y c r i t i c a l b u t d e t e r m i n a t i o n o f t h e a c c u r a c y o f t h e a l i g n m e n t i s n o t an easy t a s k . The f i r s t method does n o t have t h i s a l i g n m e n t p r o b l e m s i n c e t h e r e a d i n g and t h e w r i t i n g beams a r e t h e same. M e c h a n i c a l s t a b i l i t y d u r i n g h o l o g r a m f o r m a t i o n i s a n o t h e r c r i t i c a l e x p e r i m e n t a l c o n s i d e r a t i o n . The h i g h e s t s p a t i a l f r e q u e n c y b e i n g r e c o r d e d d e t e r m i n e s t h e v i b r a t i o n t h a t may be t o l e r a t e d . T h i s i s g e n e r a l l y o f t h e o r d e r o f a w a v e l e n g t h o f t h e l i g h t used f o r r e c o r d i n g . The r e c o r d i n g medium must n o t move more t h a n a f r a c t i o n o f t h i s d i s t a n c e r e l a t i v e t o t h e f r i n g e p a t t e r n b e i n g r e c o r d e d . To keep t h e medium s t e a d y i s n o t a p r o b l e m , b u t t o keep t h e f r i n g e p a t t e r n s t a b l e , s p e c i a l p r e c a u t i o n s a r e n e c e s s a r y . The e s s e n t i a l r e q u i r e m e n t i s t h a t t h e o p t i c a l p a t h s o f t h e r e -f e r e n c e and s u b j e c t beams must r e m a i n c o n s t a n t . T h i s r e q u i r e s t h a t mechani-c a l v i b r a t i o n s , a c o u s t i c a l and t h e r m a l d i s t u r b a n c e s must be m i n i m i z e d . To a c c o m p l i s h t h i s , e x p e r i m e n t s were p e r f o r m e d on an o p t i c a l bench. The o p t i c a l bench had been c o n s t r u c t e d by e p o x y i n g s t e e l s t r i p s t o a m a s s i v e c o n c r e t e b ase (2.15 x 1.75 x 0.15 m). The t a b l e was s u p p o r t e d by two columns o f cement b l o c k s . L a y e r s o f f e l t were used between each row of b l o c k s t o r e -duce t h e e f f e c t o f b u i l d i n g v i b r a t i o n . A l l t h e o p t i c a l components used i n t h e s e t u p were f i r m l y a t t a c h e d t o t h e t a b l e u s i n g h o l d e r s w i t h m a g n e t i c b a s e s . To r e d u c e d t h e r m a l and a c o u s t i c a l d i s t u r b a n c e , a p l e x i g l a s s c o v e r was used t o e n c l o s e t h e components on t h e t a b l e t o p . The l a s e r was l e f t o u t s i d e t h e c o v e r because o f t h e h e a t i t g e n e r a t e d d u r i n g o p e r a t i o n . How-e v e r , t h e l a s e r beam p a t h o u t s i d e t h e c o v e r was c o n f i n e d t o t h i c k p a p e r t u b e s . S t a e b l e r and Amodei (1972b) have shown t h a t t h e energy t r a n s f e r between t h e two w r i t i n g beams depends on t h e r e l a t i v e phase between t h e l i g h t p a t t e r n and t h e i n d e x m o d u l a t i o n . T h e r e f o r e , a s l i g h t r e l a t i v e m o t i o n between t h e f r i n g e p a t t e r n and t h e r e c o r d i n g medium w i l l cause a change i n t h e energy t r a n s f e r between t h e two beams. To check t h e e f f e c t o f t h e m e c h a n i c a l , t h e r m a l and a c o u s t i c a l d i s t u r b a n c e s , t h e t r a n s m i t t e d i n t e n s i t y o f t h e r e f e r e n c e beam was m o n i t o r e d d u r i n g h o l o g r a m w r i t i n g . F i g . 7.5a shows t h e i n t e n s i t y o f t h e r e f e r e n c e beam d u r i n g h o l o g r a m w r i t i n g when t h e p l e x i -g l a s s c o v e r was u s e d . The i n t e n s i t y i n c r e a s e s s m o o t h l y , w i t h t i m e as t h e h o l o g r a m grows and t h e c o u p l i n g i n c r e a s e s , i n d i c a t i n g v e r y s l i g h t o r no r e -l a t i v e s h i f t between the l i g h t p a t t e r n and t h e c r y s t a l . F i g . 7.5b i s s i m i -l a r t o F i g . 7.5a-but f o r h o l o g r a m w r i t i n g when t h e c o v e r was removed. The i n t e n s i t y o f t h e r e f e r e n c e beam changed r a p i d l y i n an u n c o n t r o l l e d manner i n d i c a t i n g t h a t t h e t h e r m a l and a c o u s t i c a l d i s t u r b a n c e s were c a u s i n g r e l a -t i v e m o t i o n between t h e r e c o r d i n g medium and t h e f r i n g e p a t t e r n . The e x p e r i m e n t s were c a r r i e d o u t w i t h c r y s t a l s as u n i f o r m l y LU ( < 0 0 0 2 3 TIME (mih) 2 r > LU LU cm (b) 0 . j 5 0 2 3 TIME (min) F i g . 7.5 The r e l a t i v e i n t e n s i t y o f one o f t h e w r i t i n g beams d u r i n g h o l o g r a m r e c o r d i n g (a) w i t h and (b) w i t h o u t the p l e x i g l a s s c o v e r on. i l l u m i n a t e d as p o s s i b l e . To check t h e u n i f o r m i t y o f t h e expanded beam, i t s i n t e n s i t y was scanned a c r o s s t h e h o r i z o n t a l d i a m e t e r ( F i g . 7.6) ( u s i n g a Gamma S c i e n t i f i c model 2900 s c a n n e r ) . F i g . 7.6 shows t h a t t h e i n t e n s i t y o f t h e beam was r e a s o n a b l y u n i f o r m i n t h e c e n t r e p o r t i o n o f t h e beam. The l i t h i u m n i o b a t e c r y s t a l s used i n t h e e x p e r i m e n t s had a f a c e d i m e n s i o n s o f 1 x 1 cm o r l e s s , t h e r e f o r e , t h e c r y s t a l was f u l l y and u n i f o r m l y i l l u m i n a t e d f o r a l l p r a c t i c a l p u r p o s e s . I n t h e r e a d o u t o f t h e d i f f r a c t i o n e f f i c i e n c y , any u n d e s i r a b l e l i g h t f a l l i n g on t h e d e t e c t o r t e n d s t o p r o d u c e e r r o r s i n t h e measurements. I n Fe-doped l i t h i u m n i o b a t e samples e x h i b i t i n g h i g h s e n s i t i v i t y and c a p a b l e o f h i g h d i f f r a c t i o n e f f i c i e n c y , i n a d d i t i o n t o t h e u s u a l n o i s e due t o s c a t t e r -i n g f r o m i m p e r f e c t i o n s i n t h e c r y s t a l , t h e r e i s o f t e n an o p t i c a l l y i n d u c e d s c a t t e r i n g . T h i s o c c u r s when t h e c r y s t a l i s exposed t o e i t h e r s i n g l e o r two sup e r i m p o s e d c o h e r e n t beams ( P h i l l i p s e t a l . 1972, Magnusson et. a l . 1974 and A l p h o n s e e t a l . 1976). U n i f o r m i n c o h e r e n t l i g h t does n o t g i v e r i s e t o t h i s t y p e o f s c a t t e r i n g . P h i l l i p s e t a l . and E l G u i b a l y (work i n p r o g r e s s h e r e ) have shown t h a t t h e s c a t t e r i n g i s n o t f r o m i n t e r f e r e n c e due t o m u l t i p l e i n t e r n a l r e f l e c t i o n . The f a c t t h a t t h e s c a t t e r i n g a p p e ars o n l y d u r i n g e x-p o s u r e t o c o h e r e n t l i g h t and t h a t i t shows pronounced a n g u l a r s e l e c t i v i t y s u g g e s t s s t r o n g l y t h a t t h e p r o c e s s i s a t l e a s t t r i g g e r e d by p a r a s i t i c h o l o -grams. A l p h o n s e and P h i l l i p s (1976) have s u g g e s t e d t h a t a g r a d u a l b e n d i n g o f t h e l i g h t beam, due t o o p t i c a l l y i n d u c e d i n d e x i n h o m o g e n e i t i e s i s due t o a p o s s i b l e n o n u n i f o r m c r o s s s e c t i o n o f t h e beam, f o l l o w e d by i n t e r f e r e n c e b e -tween t h e be n t and unbent p o r t i o n s o f t h e beam. The i n t e r f e r e n c e p a t t e r n i s r e c o r d e d as a ho l o g r a m . One way t o overcome t h e s c a t t e r i n g p r o b l e m i s by o p t i c a l e r a s u r e . When t h e sample i s r o t a t e d t o a new a n g l e , t h e i n c i d e n t beam t e n d s t o e r a s e t h e s c a t t e r i n g t h a t o c c u r s a t o t h e r a n g l e s . ( P h i l l i p s DISTANCE ( c m ) E i g . 7.6 A s c a n a c r o s s the d i a m e t e r o f one o f t h e t w o i ; s p . a t i a l l y ; f i l t e r e d w r i t i n g beams. The l i g h t i n t e n s i t y i n c i d e n p on t h e c r y s t a l was u n i f o r m w i t h i n 5%. 130. e t a l . 1972) 7.3 ?~.Tnf l u e r i e e o f M u l t i p l e I n t e r n a l R e f l e c t i o n s As was m e n t i o n e d i n Sec. 7.1, t h e K o g e l n i k f o r m u l a (Eq. 8.1) f o r the d i f f r a c t i o n e f f i c i e n c y o f u n i f o r m phase holograms and i t s e x t e n s i o n t o n o n u n i f o r m g r a t i n g n e g l e c t e d m u l t i p l e i n t e r n a l r e f l e c t i o n between t h e f a c e s of t h e c r y s t a l . K o g e l n i k (1967) has shown t h a t n e g l e c t i n g t h e e f f e c t o f m u l t i p l e r e f l e c t i o n c o u l d l e a d t o s e r i o u s e r r o r i n d e t e r m i n i n g t h e i n t r i n s i c e f f i c i e n c y o f t h e g r a t i n g ( n e g l e c t i n g r e f l e c t i o n , Eq. 7.1) from t h e o b s e r v e d d i f f r a c t i o n e f f i c i e n c y . He showed t h a t i f n i s t h e o b s e r v e d d i f f r a c t i o n o e f f i c i e n c y ( i n c l u d i n g r e f l e c t i o n s ) o f a p e r i o d i c phase g r a t i n g whose i n t r i n -s i c d i f f r a c t i o n g i v e n by Eq. 7.1 i s 2 n o = T exp(-aD/cos 8 ) s i n (v) (7.3) and „ „ (1 - R) (1 + 2 R 1 c o s 23D + R x ) (1 - R 2 ) 2 + 4 R 2 ( c o s 2 2 v + c o s 2 2 3 D ) - 4 ^ ( 1 + R 2 ) c o s 2v cos 23D (7.4) where R i s F r e s n e l f o r m u l a f o r t h e power r e f l e c t a n c e f o r p e r p e n d i c u l a r l y 2 2 p o l a r i z e d l i g h t R = s i n ( 8 - - 8 ) / s i n ( 9 O + 9 ) and f o r p a r a l l e l p o l a r i z e d 2 2 l i g h t R = t a n ( 8 - 6 ) / t a n ( 9 Q + 9 ) . 9 i s t h e a n g l e of i n c i d e n c e o f t h e r e a d -i n g wave and 8 i s i t s a n g l e o f r e f r a c t i o n i n t h e medium. R^ = R exp(-aDcos8) 27rncos8 . and 3 = — r : i s t h e p r o p a g a t i o n c o n s t a n t . Here n i s t h e r e f r a c t i v e A o i n d e x o f t h e g r a t i n g medium and A q i s t h e l i g h t w a v e l e n g t h i n vacuum. Eq. 7.4 was o b t a i n e d t h a t $ » a. Eqs. 7.3 and 7.4 may be s o l v e d t o o b t a i n t h e g r a t i n g s t r e n g t h v f o r t h e o b s e r v e d d i f f r a c t i o n e f f i c i e n c y i f t h e cos2gD ( o r t h e o p t i c a l t h i c k -n e s s o f t h e c r y s t a l nDcos8 ) i s known. However, t h e o p t i c a l p a t h cannot be measured t o t h e r e q u i r e d a c c u r a c y s i n c e any s l i g h t e r r o r i n d e t e r m i n i n g t h e o p t i c a l p a t h (as s m a l l as t h e l i g h t w a v e l e n g t h 0.5 um) w o u l d cause a l a r g e 131. change i n cos2gD ( i t s v a l u e c o u l d be changed from -1 t o +1 w i t h s u c h s m a l l change i n t h e o p t i c a l p a t h ) r e s u l t i n g i n a l a r g e e r r o r i n c a l c u l a t i n g t h e p r o p e r v a l u e o f t h e c o r r e c t i o n f a c t o r T. F i g . 7.7 ( o f Moharam e t a l . 1976a) shows a p e r s p e c t i v e p l o t o f t h e c o r r e c t i o n f a c t o r x as a f u n c t i o n o f t h e i n t r i n s i c d i f f r a c t i o n e f f i c i e n c y (Eq. 7.1) and t h e change i n t h e o p t i c a l p a t h A ( n D ) . F i g . 1.1 shows t h a t a change o f t h e o p t i c a l p a t h of about O.lum causes an e r r o r i n c a l c u l a t i n g t h e c o r r e c t i o n f a c t o r x (and t h e d i f f r a c t i o n e f f i c i e n c y ) o f u p * t o 70%. T h i s p r o b l e m i s e x a c e r b a t e d by t h e f a c t t h a t t h e o p t i c a l p a t h i s t y p i c a l l y n o t c o n s t a n t d u r i n g an e x p e r i m e n t s , s m a l l changes i n t e m p e r a t u r e s u c h as t h o s e p r o d u c e d by t h e l a s e r beam used i n w r i t i n g o r r e a d i n g t h e h o l o g r a m , o r f l u c t u a t i o n s i n t h e ambient t e m p e r a t u r e can p r o d u c e changes i n t h e o p t i c a l p a t h l a r g e enough (> 0.1 ym) t o cause s i g n i f i c a n t e r r o r i n c a l c u l a t i n g x ( C o r n i s h e t a l . 1975 and 1976b). The v a l u e o f cos2gD c o u l d be d e t e r m i n e d by m e a s u r i n g t h e t r a n s m i s s i v i t y of t h e c r y s t a l t o t h e r e a d i n g beam b e f o r e s t o r i n g t h e h o l o g r a m and by u s i n g t h e f o r m u l a f o r t r a n s -m i s s i o n w i t h i n t e r n a l r e f l e c t i o n (Heavens 1955) T = e x p ( - q D / c o s 6 ) ( l - R ) 2 ( ? 5 ) •1 - 2R 1cos2gD + R 2 However, as e x p l a i n e d , t h e v a l u e o f cos23D changes d u r i n g t h e w r i t i n g p r o c e s s , and a f t e r t h e h o l o g r a m i s w r i t t e n Eq. 7.5 does n o t r e p r e s e n t t h e t r a n s m i t t e d i n t e n s i t y of t h e r e a d i n g beam, bec a u s e o f t h e i n t e r a c t i o n w i t h t h e h o l o gram. We now d e s c r i b e an e x p e r i m e n t a l method t o d e t e r m i n e cos2f5D a t any t i m e d u r i n g h o l o g r a m w r i t i n g . T h i s e n a b l e s t h e c a l c u l a t i o n o f t h e i n t r i n s i c d i f f r a c t i o n e f f i c i e n c y (and t h e g r a t i n g s t r e n g t h v) from t h e o b s e r v e d e f f i -c i e n c y u s i n g Eqs.. 7.1 t o 7.4. An a n c i l l a r y t h i r d l i g h t beam o f d i f f e r e n t w a v e l e n g t h f r o m t h a t used t o w r i t e t h e h o l o g r a m i s i n c i d e n t on t h e c r y s t a l a t t h e r e g i o n o f t h e i n t e r f e r e n c e p a t t e r n o f t h e two w r i t i n g beams. I t i s 132. F i g . 7.7 Calculated correction factor x:as a fuction of (a) change i n the o p t i c a l thickness A(nd) and (b) the i n t r i n s i c d i f f r a c t i o n e f f i c i e n c y for X = 441.6 nm , d = 0.254 cm and ct= 0.6 (1/cm) . 133. c o n v e n i e n t , b u t n o t n e c e s s a r y , f o r t h e a n c i l l a r y beam t o be n o r m a l l y i n c i -d e nt on the c r y s t a l . T h i s beam i s n o t d i f f r a c t e d by t h e h o l o g r a m , p r o v i d e d i t i s k e p t f a r f r o m i t s Bragg a n g l e and i t does n o t i n t e r a c t w i t h t h e w r i t i n g beams ( s i n c e i t i s o f d i f f e r e n t w a v e l e n g t h ) . The p o l a r i z a t i o n o f t h e beam s h o u l d be s u c h as t o l a u n c h a s i n g l e wave i n t h e c r y s t a l b u t i s o t h e r w i s e u n r e s t r i c t e d . The w a v e l e n g t h o f t h i s a n c i l l a r y beam s h o u l d be d i f f e r e n t and l o n g e r ( p r e f e r a b l y f r o m He-Ne L a s e r A = 632 .8 nm) t h a n t h a t used t o w r i t e the h o l o g r a m . A t t h i s w a v e l e n g t h (632 .8 nm) the p h o t o r e f r a c t i v e e f f e c t i s i n e f f i c i e n t and t h e beam w i l l n o t e r a s e t h e hologram. The t r a n s m i s s i v i t y o f t h e c r y s t a l t o t h e a n c i l l a r y beam i s moni-t o r e d c o n t i n u o u s l y d u r i n g h o l o g r a m w r i t i n g and r e a d i n g . U s i n g Eq. 7.5, w i t h the a p p r o p r i a t e p a r a m e t e r s ( p r i m e d symbols) f o r t h e d i f f e r e n t w a v e l e n g t h A', the v a l u e o f cos2g/T) can be d e t e r m i n e d a t any s t a g e . Here g' = 2Trn'Dcos8' A \ One m i g h t c o n s i d e r c a l c u l a t i n g g o f t h e r e a d i n g beam by m u l t i p l y i n g g' by (A'ncose/An'cosQ'). However, t h i s i s n o t p r a c t i c a b l e s i n c e g' i s v e r y l a r g e (>10^m "*") and any s l i g h t e r r o r i n n, n', 0 o r 6' (even o f O.lum) woul d c a u s e , as was shown, a l a r g e enough e r r o r i n g t o make t h e c a l c u l a t i o n s m e a n i n g l e s s . To overcome t h i s p r o b l e m , t h e v a l u e o f g' a t any s t a g e i s compared w i t h i t s i n i t i a l v a l u e ( b e f o r e h o l o g r a m w r i t i n g ) and t h e change Ag' i s c a l c u l a t e d . T h i s , i n t u r n , g i v e s t h e change i n g by m u l t i p l y i n g by (A ' n c o s 6 / A n ' c o s 6 ' ) . By a d d i n g t h e change i n g t o t h e i n i t i a l v a l u e o f g c a l c u l a t e d u s i n g Eq. 7.5 ( b e f o r e h o l o g r a m w r i t i n g ) t h e v a l u e o f g (and of cos2gD) i s fo u n d a t any s t a g e . Now Eqs. 7.1 t o 7.4 c a n be s o l v e d t o c a l c u l a t e i n t r i n s i c e f f i c i e n c y and t h e g r a t i n g s t r e n g t h can be found a t any s t a g e u s i n g t h e o b s e r v e d d i f -f r a c t i o n e f f i c i e n c y . An e x p e r i m e n t a l i l l u s t r a t i o n o f t h e method was c a r r i e d o u t . A 0.25 cm t h i c k Fe-doped (0.015 mole %) LiNbO^ c r y s t a l was use d . The ho l o g r a m was formed u s i n g a He-Cd l a s e r (A = 441.6 nm) (RCA model LD2186) and t h e a b s o r p t i o n a t t h i s w a v e l e n g t h was 0.43 cm-"'". The l i g h t was p o l a r i z e d p e r -p e n d i c u l a r t o t h e c - a x i s of t h e c r y s t a l . The a n g l e of i n c i d e n c e was 12.2°. The a n c i l l a r y beam was f r o m He-Ne l a s e r ( A ' = 632.8 n m ) ( S p e c t r a P h y s i c s model 132) a l s o p o l a r i z e d p e r p e n d i c u l a r t o t h e c - a x i s . I t was i n c i d e n t n o r m a l l y on t h e c r y s t a l and was c e n t r e d on t h e r e c o r d i n g s p o t . The a b s o r p -t i o n a t A' = 632.8 was n e g l i g i b l e . F i g . 7.8 shows t h e arrangement used. The i n i t i a l t r a n s m i s s i o n o f t h e r e f e r e n c e beam was measured j u s t b e f o r e s t a r t i n g t h e r e c o r d i n g p r o c e s s . D u r i n g t h e h o l o g r a m f o r m a t i o n t h e r e l a t i v e t r a n s m i s s i o n o f t h e a n c i l l a r y beam was m o n i t o r e d c o n t i n u o u s l y . The d i f f r a c -t e d beam i n t e n s i t y was measured by m o m e n t a r i l y e x p o s i n g t h e c r y s t a l t o t h e r e f e r e n c e beam a l o n e f r o m t i m e t o t i m e . F i g . 7.9 shows t h e e x p e r i m e n t a l r e s u l t s on t h e t i m e development of t h e o b s e r v e d (measured) d i f f r a c t i o n e f f i c i e n c y t o g e t h e r w i t h t h e i n t r i n -s i c d i f f r a c t i o n e f f i c i e n c y as d e t e r m i n e d by t h e above methods. A l s o p l o t t e d i s t h e a v e r a g e f u n d a m e n t a l component o f t h e s p a c e charge f i e l d c a l c u l a t e d f r o m t h e i n t r i n s i c d i f f r a c t i o n e f f i c i e n c y (Eq. 7.1). The change i n t h e r e f r a c t i v e e f f e c t was assumed t o be due t o t h e e l e c t r o - o p t i c e f f e c t . An i n i t i a l l i n e a r i t y i n t h e development o f t h e space c h a r g e f i e l d i s , o o f c o u r s e , e x p e c t e d . I n summary, t h e e f f e c t o f m u l t i p l e i n t e r n a l r e f l e c t i o n on t h e r e a d o u t of phase holograms s t o r e d by t h e p h o t o r e f r a c t i v e e f f e c t was c o n -s i d e r e d . An e x p e r i m e n t a l method t o d e t e r m i n e t h e i n t r i n s i c d i f f r a c t i o n e f f i c i e n c y (no m u l t i p l e r e f l e c t i o n ) f r o m t h e o b s e r v e d (measured) d i f f r a c t i o n e f f i c i e n c y was g i v e n . The o n l y l i m i t a t i o n on t h e method i s t h e a s s u m p t i o n t h a t t h e c r y s t a l s u r f a c e s a r e p e r f e c t l y p a r a l l e l and f l a t . I r r e g u l a r i t i e s i n t h e c r y s t a l s u r f a c e s may e i t h e r make m u l t i p l e r e f l e c t i o n u n i m p o r t a n t o r 135. FigJ 7.8 E x p e r i m e n t a l arrangement. 136. Fig. 7.9 Experimental results on the e f f i c t i v e and the i n t r i n s i c diffraction efficiencies as measured by the method described are plotted together with the amplitude of the space charge f i e l d as deduced from the i n t r i n s i c e f f i c i -ency. The correction factor T varied from 0.5 to 0.7. d e s t r o y t h e phase r e l a t i o n between t h e i n d i v i d u a l w a v e l e t s . The e f f e c t s o f m u l t i p l e r e f l e c t i o n d u r i n g t h e w r i t i n g p r o c e s s , a l t h o u g h p r o b a b l y i m p o r t a n t , a r e n o t c o n s i d e r e d s i n c e t h e p r o b l e m i s v e r y complex. The model p r o p o s e d f o r h o l o g r a m w r i t i n g n e g l e c t s t h e m u l t i p l e i n t e r n a l r e f l e c t i o n . I n o r d e r t o t e s t t h i s model c r y s t a l s w i t h a n t i r e f l e c t i o n c o a t i n g must be used. 138. CHAPTER V I I I PHOTOCURRENT AND HOLOGRAPHIC MEASUREMENTS 8.1 I n t r o d u c t i o n I n t h i s c h a p t e r , e x p e r i m e n t a l o b s e r v a t i o n s o f p h o t o c u r r e n t i n Fe-doped l i t h i u m n i o b a t e a r e g i v e n . The e x p e r i m e n t s t o be d e s c r i b e d show t h a t t h e r e l a t i o n between t h e p h o t o c u r r e n t and b o t h t h e r a d i a n t i n t e n s i t y and t h e a p p l i e d v o l t a g e a r e l i n e a r o v e r t h e r a n g e c o n s i d e r e d . The v a l u e s o f t h e p h o t o c o n d u c t i v i t y , t h e p h o t o v o l t a i c e f f e c t c o n s t a n t K and t h e v i r t u a l f i e l d a r e e s t i m a t e d u s i n g t h e s e p h o t o c u r r e n t measurements. Measurements of t h e d i f f r a c t i o n e f f i c i e n c y o f holograms s t o r e d i n l i t h i u m n i o b a t e were c a r r i e d o u t d u r i n g b o t h h o l o g r a m w r i t i n g and o p t i c a l e r a s u r e . The v a l u e o f t h e v i r t u a l f i e l d e s t i m a t e d f r o m t h i s h o l o g r a p h i c measurement i s i n r e a s o n a b l e agreement w i t h v a l u e s o b t a i n e d f r o m t h e p h o t o -c u r r e n t measurement. I t i s shown t h a t t h e o p t i c a l l y i n d u c e d s c a t t e r i n g causes s e r i o u s e r r o r s i n c a l c u l a t i n g t h e d i f f r a c t i o n e f f i c i e n c y o f t h e h o l o -gram. Energy t r a n s f e r between t h e two w r i t i n g beams o f up t o 70% was o b s e r v e d d u r i n g h o l o g r a m w r i t i n g . S i n c e d r i f t was t h e dominant mechanism ( t h e d r i f t f i e l d was about 50 t i m e s t h e d i f f u s i o n f i e l d ) , i t i s s u g g e s t e d t h a t t h e t r a n s -p o r t l e n g t h o f t h e f r e e e l e c t r o n s , i n t h e s e e x p e r i m e n t s , was n o t s h o r t com-p a r e d t o t h e g r a t i n g s p a c i n g , and, t h e r e f o r e , d r i f t a l o n e c o u l d p r o d u c e l a r g e enough s p a t i a l phase s h i f t between t h e l i g h t p a t t e r n and t h e i n d e x m o d u l a t i o n (Young e t a l . 1974) t o a c c o u n t f o r t h e o b s e r v e d energy t r a n s f e r . 8.2 P h o t o c u r r e n t Measurements 8.2.1 E x p e r i m e n t a l P r o c e d u r e The Fe-doped (0.1 mole %) LiN b O ^ c r y s t a l was i l l u m i n a t e d w i t h an a r g o n i o n l a s e r (A = 514.5 nm). The beam was s p a t i a l l y f i l t e r e d and c o l l i -mated. The expanded beam (2.4 cm i n d i a m e t e r ) i l l u m i n a t e d t h e c r y s t a l n e a r l y u n i f o r m l y . Aluminum e l e c t r o d e s were e v a p o r a t e d on t o t h e c - f a c e s o f t h e c r y s t a l . The p h o t o c u r r e n t was measured w i t h a K e i t h l e y 602 e l e c t r o m e t e r . A h i g h v o l t a g e power s u p p l y was c o n n e c t e d i n s e r i e s w i t h t h e c r y s t a l and t h e e l e c t r o m e t e r , t o p r o v i d e t h e a p p l i e d v o l t a g e a l o n g t h e c - a x i s . Measurements were made a f t e r t h e c r y s t a l had t h e r m a l l y e q u i l i b r a t e d w i t h r a d i a t i o n t o a v o i d c o n t r i b u t i o n from t h e p y r o e l e c t r i c e f f e c t t o t h e c u r r e n t . However, p r o l o n g e d i l l u m i n a t i o n c auses s i g n i f i c a n t o p t i c a l i n d u c e d s c a t t e r i n g ( P h i l l i p s e t a l . 1972). To overcome t h i s p r o b l e m , t h e c r y s t a l was r o t a t e d t o a new a n g l e , t h u s t h e i n c i d e n t l i g h t i s n o t d i f f r a c t e d by t h e p a r a s i t i c h o lograms t h a t c a use t h e s c a t t e r i n g . 8.2.2 R e s u l t s . a n d D i s c u s s i o n F i g . 8.1 shows a p l o t o f t h e measured c u r r e n t v s . t h e i n c i d e n t l i g h t i n t e n s i t i e s f o r t h r e e d i f f e r e n t v a l u e s o f t h e a p p l i e d v o l t a g e 0. and ± 3kV. The r e l a t i o n s h i p between t h e c u r r e n t and t h e l i g h t i n t e n s i t y i s c l e a r l y l i n e a r f o r any one v a l u e o f t h e a p p l i e d v o l t a g e . The r e l a t i o n s h i p between t h e c u r r e n t and t h e a p p l i e d v o l t a g e i s a l s o l i n e a r f o r c o n s t a n t l i g h t i n t e n s i t y . The c u r r e n t may be r e p r e s e n t e d by i = a l + bVI + cV (8.1) L e a s t s q u a r e f i t t i n g o f t h e p l o t i n F i g . 8.1 t o Eq. 8.1 gave a = 1118.1 ± 45, b = 2 4 . 8 ± 2 and c = 0.12 ± 0.05 2 where t h e c u r r e n t i i s i n pA and t h e l i g h t i n t e n s i t y I , i s i n W/cm and V i s the a p p l i e d v o l t a g e i n kV. The t h i r d t erm i n Eq. 8.1 i s t h e d a r k c u r r e n t due t o t h e f i n i t e r e s i s t i v i t y o f t h e c r y s t a l h o l d e r and t h e s u r f a c e r e s i s -t a n c e o f t h e c r y s t a l . I t i s t o o l a r g e t o be due t o d a r k c o n d u c t i v i t y o f t h e c r y s t a l . The f i r s t two terms i n Eq. 8.1 a r e t h e p h o t o c u r r e n t s . The f i r s t t e rm i s t h e p h o t o v o l t a i c e f f e c t c o n t r i b u t i o n t o t h e p h o t o c u r r e n t . The F i g . 8.1 P h o t o c u r r e n t s i n Fe-doped l i t h i u m n i o b a t e c r y s t a l v s . the l i g h t i n t e n s i t y f o r t h r e e d i f f e r e n t v a l u e s o f the a p p l i e d v o l t a g e . 1 4 1 . second term i s t h e a d d i t i o n a l c o n t r i b u t i o n t o t h e p h o t o c u r r e n t when a v o l t a g e i s a p p l i e d a c r o s s t h e c r y s t a l . The p h o t o c u r r e n t d e n s i t y a l o n g t h e c - a x i s o f t h e c r y s t a l i n d u c e d by u n i f o r m i l l u m i n a t i o n (no d i f f u s i o n ) may be r e p r e s e n t e d by ( G l a s s e t a l . 1974) J = ral + CTl J (8.2) where K i s t h e p h o t o v o l t a i c e f f e c t c o n s t a n t , a i s t h e a b s o r p t i o n c o n s t a n t , I i s t h e l i g h t i n t e n s i t y and i s d e c a y i n g e x p o n e n t i a l l y w i t h d i s t a n c e t r a v e l -l e d i n t h e c r y s t a l due t o a b s o r p t i o n , V i s t h e a p p l i e d v o l t a g e , L i s t h e c r y s t a l l e n g t h ( a l o n g t h e c - a x i s ) and a = qua -^y where T i s t h e c a r r i e r l i f e t i m e , y i s t h e m o b i l i t y , £ i s t h e quantum e f f i c i e n c y and hv i s t h e l i g h t p h o t o n energy. I f D i s t h e c r y s t a l t h i c k n e s s and H i s t h e l e n g t h o f the e l e c t r o d e s , t h e n t h e p h o t o c u r r e n t i s i = I H [ 1 - e x p ( ^ a D ) ] [K + (8.3) F o r t h e Fe-doped Li N b O ^ c r y s t a l u sed i n t h e measurement, L was 1 cm, D was 0.1 cm. The a b s o r p t i o n c o n s t a n t a was 15 cm (A Cary 14 s p e c t r o p h o t o m e t e r was used and no c o r r e c t i o n s f o r t h e r e f l e c t i o n s were made s i n c e t h e c r y s t a l had a n t i r e f l e c t i o n c o a t i n g s ) . Eq. 8.3 t h e n may be r e -w r i t t e n as i = 0.75 K I + 311.03 yx£IV (8.4) P 2 2 where K i s i n Acm/W. I i s i n W/cm , V i s i n kV and yx£ i s cm /V. Comparing Eq. 8.4 w i t h t h e f i r s t two terms i n Eq. 8.1 shows t h a t K = 1.49 ± 0.06 x 10~ 9A cm/W, yx£ = 8 ± .6 1 0 ~ 1 4 cm 2/V, and E = - ^ = 45±5kV/cm. ' v qyx£ The v i r t u a l f i e l d E i s a l s o e q u a l t o t h e a p p l i e d f i e l d (^) r e q u i r e d t o V 1J c a n c e l t h e p h o t o e l e c t r i c c u r r e n t . The v a l u e o b t a i n e d f o r K i s about one h a l f t h e v a l u e r e p o r t e d by G l a s s e t a l . (197-4) b u t i s v e r y c l o s e t o t h e 142. v a l u e r e p o r t e d by K r a t z i g e t a l . (19 7 7 ) . The v a l u e w h i c h has been o b t a i n e d f o r t h e v i r t u a l f i e l d E i s a t t h e h i g h e r end o f t h e s c a l e o f r e -v p o r t e d v a l u e s i n t h e l i t e r a t u r e . The p h o t o c o n d u c t i v i t y a = quT£a/hv i s Li -13 -1 e s t i m a t e d t o be (4.96 ± 0.7)10 I(ficm) , where I i s t h e l i g h t i n t e n s i t y 2 i n W/cm . Young e t a l . have e s t i m a t e d t h a t t h e m o b i l i t y i s of t h e o r d e r o f 2 10 cm /Vsec f r o m e x t r a p o l a t i o n o f h i g h t e m p e r a t u r e c o n d u c t i v i t y measurements by J o r g e n s e n e t a l . ( 1 9 6 9 ) . Then t h e c a r r i e r l i f e t i m e T i s o f t h e o r d e r o f 8 x 10 s e c . assuming u n i t y quantum e f f i c i e n c y . 8.3 H o l o g r a p h i c Measurements 8.3.1 E x p e r i m e n t a l P r o c e d u r e The holograms were formed i n t h e Fe-doped (0.1 mole%) 0.1 cm t h i c k LiNb0.j c r y s t a l u s i n g two expanded l i g h t beams o r i g i n a t i n g f r o m an a r g o n i o n l a s e r (A. = 51 4 . 5 ) . The a b s o r p t i o n c o n s t a n t a t t h i s w a v e l e n g t h was 15 cm The e x p e r i m e n t a l s e t u p i s shown i n F i g . 7.3. The two beams were s p a t i a l l y f i l t e r e d and c o l l i m a t e d (2.4 c i i n d i a m e t e r ) . The c r y s t a l (1 cm x 1 cm f a c e ) was u n i f o r m l y and f u l l y i l l u m i n a t e d . The i n t e n s i t i e s o f t h e two beams were 2 e q u a l , t h e i n t e n s i t y o f each beam was 35 mW/cm . The a n g l e o f i n c i d e n c e was ±10°. B o t h beams were p o l a r i z e d w i t h t h e e l e c t r i c v e c t o r n o r m a l t o t h e p l a n e of i n c i d e n c e w h i c h a l s o c o n t a i n e d t h e c - a x i s . The c r y s t a l was s h o r t c i r -c u i t e d by j o i n i n g t h e . e l e c t r o d e on t h e c - f a c e s . The d i f f r a c t i o n e f f i c i e n c y , d u r i n g h o l o g r a m r e c o r d i n g , was o b s e r v e d by m o m e n t a r i l y e x p o s i n g t h e c r y s t a l t o t h e r e f e r e n c e beam a l o n e ( i . e . b l o c k i n g t h e s u b j e c t beam), f r o m t i m e t o t i m e . B o t h t h e d i f f r a c t e d and t r a n s m i t t e d beams were measured d u r i n g t h i s n o n d e s t r u c t i v e r e a d o u t . The r e l a t i v e d i f f r a c t i o n e f f i c i e n c y o f the..hologram was d e f i n e d as t h e r a t i o o f t h e d i f f r a c t e d i n t e n s i t y t o t h e sum of t h e i n -t e n s i t y o f t h e d i f f r a c t e d and t r a n s m i t t e d i n t e n s i t i e s . T h i s was u s e d j i n s t e a d of t h e r a t i o of d i f f r a c t e d i n t e n s i t y t o t h e t r a n s m i t t e d beam b e f o r e h o l o g r a m 143. w r i t i n g , t o c o r r e c t f o r t h e l i g h t i n t e n s i t y l o s s due t o s c a t t e r i n g ( P h i l l i p s e t a l . 1 9 7 2 ) . O p t i c a l e r a s u r e o r d e s t r u c t i v e r e a d i n g o f t h e r e c o r d e d h o l o -gram was done by c o n t i n u o u s l y e x p o s i n g t h e c r y s t a l t o t h e r e f e r e n c e beam. B o t h t h e d i f f r a c t e d and t h e t r a n s m i t t e d beams were m o n i t o r e d . B e f o r e t h e s t a r t i n g o f a new w r i t e - e r a s e c y c l e , t h e ho l o g r a m was a l m o s t t o t a l l y e r a s e d (when the e f f i c i e n c y was l e s s t h a n 0.001 o f t h e maximum o b s e r v e d e f f i c i e n c y ) . The c r y s t a l was a l s o i l l u m i n a t e d w i t h a s i n g l e beam, b e f o r e t h e w r i t i n g p hase, a t a:'.different a n g l e o f i n c i d e n c e , f o r about 10 m i n u t e s t o e r a s e t h e p a r a s i t i c holograms c a u s i n g s c a t t e r i n g . 8.3.2 R e s u l t s and D i s c u s s i o n F i g . 8.2 shows a p l o t o f t h e t i m e development o f t h e o b s e r v e d r e l a t i v e d i f f r a c t i o n e f f i c i e n c y . F i g . 8.2 shows t h a t t h e d i f f r a c t i o n e f f i -c i e n c y r e a c h e d a maximum o f 98.5% a f t e r about 250 seconds and t h e n s t a r t e d t o d e c r e a s e t o about 60% a f t e r about 600 sec o n d s . S e v e r a l w r i t i n g c y c l e s were c a r r i e d o u t and t h e o b s e r v e d r e l a t i v e d i f f r a c t i o n e f f i c i e n c i e s were w i t h i n ± 4% f o r t h e maximum e f f i c i e n c y and w i t h i n ± 5% f o r t h e t i m e s c a l e . When b o t h t h e r e f e r e n c e and s u b j e c t beams were m o n i t o r e d d u r i n g h o l o g r a m w r i t i n g , energy t r a n s f e r f r o m t h e s u b j e c t beam t o t h e r e f e r e n c e beam was o b s e r v e d . However, t h e e x t e n t o f t h e en e r g y t r a n s f e r v a r i e d g r e a t l y i n t h e s e v e r a l w r i t i n g c y c l e s . The energy l o s s o f t h e s u b j e c t beam r a n g e d f r o m 15 t o 70%. I n a d d i t i o n , t o o b s e r v e s u c h l a r g e d i f f r a c t i o n e f f i c i e n c y i n 0.1 cm t h i c k c r y s t a l , t h e ave r a g e p h o t o i n d u c e d space c h a r g e f i e l d s h o u l d be i n t h e o r d e r o f 50 kV/cm. F o r l i g h t w a v e l e n g t h o f A q = 514.5 nm and a n g l e kT of i n c i d e n c e o f 10°, t h e e q u i v a l e n t d i f f u s i o n f i e l d = — i s 1.0 kV/cm. T h e r e f o r e , t h e d r i f t f i e l d ( i . e . t h e v i r t u a l f i e l d s i n c e t h e r e i s no a p p l i e d v o l t a g e ) i s g r e a t e r t h a n 50 kV/cm. Thus d r i f t was t h e dominant mechanism. Assuming s h o r t t r a n s p o r t l e n g t h , t h e s p a t i a l phase s h i f t o f t h e spac e c h a r g e 144. TIME ( s ec ) F i g . 8.2. The c i r c l e s (•) show, t h e measured t i m e development of t h e r e l a t i v e d i f f r a c t i o n e f f i c i e n c y d u r i n g h o l o g r a m w r i t i n g i n 0.1 cm Fe-doped l i t h i u m n i o b a t e c r y s t a l . The s o l i d l i n e (—) shows t h e c a l c u l a t e d r e l a t i v e d i f f r a c t i o n e f f i c i e n c y u s i n g t h e model o f Sec. 4.4. 145. f i e l d w i t h r e s p e c t t o t h e i n c i d e n t l i g h t p a t t e r n was i n t h e o r d e r o f 1°. Such a s m a l l phase s h i f t c annot p r o d u c e more t h a n 5% energy t r a n s f e r between t h e two w r i t i n g beams ( S t a e b l e r and Amodei 1972b). However, as Young e t a l . (1974) have shown, i f t h e t r a n s p o r t l e n g t h o f t h e p h o t o - r e l e a s e d e l e c t r o n s i s n o t s h o r t compared t o t h e g r a t i n g s p a c i n g , d r i f t a l o n e c o u l d p r o d u c e a s p a t i a l phase s h i f t l a r g e enough t o a c c o u n t f o r t h e o b s e r v e d energy t r a n s f e r . M e c h a n i c a l v i b r a t i o n , a l s o , c o u l d move t h e r e c o r d i n g medium r e l a t i v e t o t h e i n c i d e n t l i g h t p a t t e r n t h u s p r o d u c i n g a s p a t i a l phase s h i f t . T h i s may a c c o u n t f o r t h e o b s e r v e d v a r i a t i o n s i n t h e e n e r g y t r a n s f e r between t h e two beams. However, h i g h f r e q u e n c y m e c h a n i c a l v i b r a t i o n w o u l d cause r e d u c t i o n i n t h e m o d u l a t i o n r a t i o , b u t i t w o u l d n o t p r o d u c e a s t e a d y s t a t e s p a t i a l phase s h i f t between t h e l i g h t p a t t e r n and t h e r e c o r d i n g medium. The s o l i d l i n e i n F i g . 8.2 i s t h e c a l c u l a t e d r e l a t i v e d i f f r a c t i o n e f f i c i e n c y u s i n g t h e dynamic model o f Sec. 4.4 w i t h = 55 kV/cm, l*-!* 2 o yxEJ = 7.6 x 10 cm /V arid rx^/n^ = 1000. The c a l c u l a t e d and the 1 measured d i f f r a c t i o n e f f i c i e n c y a r e c l o s e and t h e v a l u e s o b t a i n e d f o r E and f o r UT£J v a r e i n agreement w i t h t h e v a l u e s o b t a i n e d from t h e p h o t o c u r r e n t measurements. O n l y i n the l a t e r s t a g e s d i d t h e c a l c u l a t e d e f f i c i e n c y d e v i a t e s l i g h t l y f r o m t h e measured e f f i c i e n c y . T h i s c o u l d be due t o e r r o r s i n e s t i m a t i n g t h e d i f -f r a c t i o n e f f i c i e n c y due t o o p t i c a l l y i n d u c e d s c a t t e r i n g . However, t h e model p r e d i c t e d o n l y 5% energy t r a n s f e r between t h e two w r i t i n g beams r a t h e r t h a n t h e o b s e r v e d 15-70% t r a n s f e r . The f i t between c a l c u l a t e d and o b s e r v e d d i f -f r a c t i o n e f f i c i e n c i e s i s , t h e r e f o r e , i l l u s o r y , s i n c e t h e d i s c r e p a n c y between t h e p r e d i c t e d energy t r a n s f e r (about 5%) and t h a t o b s e r v e d (15-70%) shows t h a t t h e model does n o t a p p l y . The o n l y e x p l a n a t i o n seems t o be t h a t t h e t r a n s p o r t l e n g t h o f t h e e l e c t r o n s was n o t s h o r t i n t h i s c r y s t a l . No o t h e r a n t i r e f l e c t i o n c o a t e d c r y s t a l s were a v a i l a b l e . C l e a r l y more e x p e r i m e n t s F i g . 8.3 Observed r e l a t i v e i n t e n s i t i e s o f the t r a n s m i t t e d and d i f f r a c t e d beams v s . t i m e d u r i n g o p t i c a l e r a s u r e s t a r t i n g from p o i n t A i n F i g . 8 . 2 . 0 10 20 30 TIME (min.) F i g . 8.4 Observed r e l a t i v e I n t e n s i t i e s o f t h e t r a n s m i t t e d and d i f f r a c t e d beams v s . t i m e d u r i n g o p t i c a l e r a s u r e s t a r t i n g f r o m p o i n t B i n F i g . 8.2. 148. a r e needed t o d e t e r m i n e t h e g e n e r a l i t y o f t h i s r e s u l t . As f o r t h e o p t i c a l e r a s u r e and d e s t r u c t i v e r e a d i n g o f t h e hol o g r a m , F i g . 8.3 shows a p l o t o f t h e i n t e n s i t i e s o f b o t h t h e d i f f r a c t e d and t r a n s -m i t t e d beam d u r i n g o p t i c a l e r a s u r e o f t h e hol o g r a m s t o r e d i n F i g . 8.2 when t h e w r i t i n g c y c l e was h a l t e d a f t e r 250 seco n d s . F i g . 8.3 shows t h a t i n i t i a l l y t h e d i f f r a c t e d beam d e c r e a s e d and t h e t r a n s m i t t e d beam i n c r e a s e d by t h e same amount. However, p r o l o n g e d e x p o s u r e d e v e l o p e d o p t i c a l l y i n d u c e d s c a t t e r i n g ( P h i l l i p s e t a l . 1972) and t h e t r a n s m i t t e d beam d e c r e a s e d ( o r d i d n o t i n -c r e a s e by t h e same amount l o s t by t h e d i f f r a c t e d beam). T h i s problem caused an e r r o r i n c a l c u l a t i n g t h e d i f f r a c t i o n e f f i c i e n c y d u r i n g e r a s u r e , e s p e c i a l -l y d u r i n g t h e l a t e r s t a g e s . F i g . 8.4 shows a p l o t o f t h e i n t e n s i t i e s o f t h e d i f f r a c t e d and t r a n s m i t t e d beams d u r i n g o p t i c a l e r a s u r e s t a r t i n g a t t h e end of t h e w r i t i n g phase shown i n F i g . 8.2. Because t h e d i f f r a c t i o n e f f i c i e n c y i n t h e w r i t i n g phase was a l l o w e d t o d e c r e a s e a f t e r i t had r e a c h e d t h e m a x i -mum, o p t i c a l e r a s u r e o f t h e spac e c h a r g e f i e l d was e x p e c t e d t o cause an i n i -t i a l i n c r e a s e i n t h e d i f f r a c t i o n e f f i c i e n c y . F i g . 8.3 shows t h a t t h i s d i d i n d e e d happen. T h i s t y p e o f b e h a v i o u r was p r e d i c t e d by t h e model p r e s e n t e d i n c h a p t e r 6 f o r t h e o p t i c a l e r a s u r e . The t r a n s m i t t e d beam d e c r e a s e d i n i -t i a l l y t o complement t h e d i f f r a c t e d beam. As i n F i g . 8.3, p r o l o n g e d expo-s u r e caused t h e t r a n s m i t t e d beam t o d e c r e a s e due t o o p t i c a l l y i n d u c e d s c a t -t e r i n g . No a t t e m p t was made t o f i t t h e o b s e r v e d d i f f r a c t i o n e f f i c i e n c y t o th e model p r o p o s e d i n c h a p t e r 6 because o f t h e problems caused by t h e l i g h t s c a t t e r i n g . CHAPTER I X CONCLUSIONS The p u r p o s e o f t h i s work was t o o b t a i n a b e t t e r u n d e r s t a n d i n g of the h o l o g r a m s t o r a g e p r o c e s s by t h e p h o t o r e f r a c t i v e e f f e c t . Most o f t h e work was c o n c e r n e d w i t h r i g o r o u s m o d e l l i n g o f h o l o g r a m w r i t i n g and e r a s u r e p r o c e s s e s , r e m o v i n g some o f t h e l i m i t a t i o n s o f p r e v i o u s m o d e l l i n g . P h o t o -c u r r e n t and h o l o g r a p h i c measurements were c a r r i e d o u t t o i n v e s t i g a t e t h e p h o t o r e f r a c t i v e and t h e p h o t o v o l t a i c e f f e c t s and t o t e s t t h e d e v e l o p e d models. The c o n t r i b u t i o n s w h i c h were made t o t h e s u b j e c t may be summarized as f o l l o w s : a) A r e l i a b l e new c r i t e r i o n was p r o p o s e d t o d e t e r m i n e whether a phase g r a t i n g w i l l o p e r a t e i n t h e Raman-Nath o r t h e Bragg r e g i m e . Holograms o p e r a t i n g i n t h e B r a g g r e g i m e a r e d e s i r a b l e f o r most a p p l i c a t i o n s because o f t h e i r h i g h d i f f r a c t i o n e f f i c i e n c y and a n g u l a r s e l e c t i v i t y . b) A t r e a t m e n t f o r t h e t i m e development of p h o t o i n d u c e d space c h a r g e was d e v e l o p e d . T h i s model a l l o w e d f o r t h e f e e d b a c k e f f e c t o f t h e space c h a r g e f i e l d on f u r t h e r development o f t h a t f i e l d . The model i s a p p l i c a b l e o v e r th e e n t i r e range o f e x p o s u r e . I t i s f o r h o l o g r a m w r i t i n g by d r i f t , d i f f u -s i o n and t h e p h o t o v o l t a i c e f f e c t i n a u n i f o r m l y i l l u m i n a t e d c r y s t a l k e p t under c o n s t a n t a p p l i e d v o l t a g e . The t r a n s p o r t l e n g t h o f t h e p h o t o r e l e a s e d e l e c t r o n s was assumed t o be s h o r t . T h i s model was extended t o a l l o w f o r t h e e f f e c t o f t h e d e c r e a s e of t h e i n t e n s i t y o f t h e l i g h t as i t p r o p a g a t e s i n t h e c r y s t a l due t o t h e a b s o r p t i o n . I t was shown t h a t t h e h o l o g r a m became non-u n i f o r m t h r o u g h o u t t h e t h i c k n e s s o f t h e g r a t i n g . I t was a l s o shown t h a t t h e s p a t i a l phase s h i f t o f t h e i n d e x m o d u l a t i o n w o u l d r e m a i n c o n s t a n t o v e r t h e e n t i r e r ange o f e x p o s u r e . c) A new dynamic model f o r t h e d e s c r i p t i o n o f t h e t i m e development o f t h e h o l o g r a m w r i t i n g p r o c e s s was d e v e l o p e d . The model a l l o w e d s i m u l t a n e o u s l y , f o r t h e f i r s t t i m e , f o r t h e f e e d b a c k e f f e c t o f t h e s p a c e c h a r g e f i e l d on t h e r e d i s t r i b u t i o n o f e l e c t r o n s and t h e e f f e c t o f t h e h o l o g r a m i n m o d i f y i n g t h e l i g h t p a t t e r n w h i c h i s w r i t i n g i t as w e l l as t h e e f f e c t o f t h e d a r k conduc-t i v i t y . The decay o f t h e l i g h t i n t e n s i t y w i t h d i s t a n c e t r a v e l l e d i n t h e c r y s t a l due t o a b s o r p t i o n i s a l s o a l l o w e d f o r . The model a p p l i e s t o u n i -f o r m l y i l l u m i n a t e d c r y s t a l s under c o n s t a n t a p p l i e d v o l t a g e . T h i s model r e -p r o d u c e d a l l t h e r e p o r t e d forms o f t i m e development of d i f f r a c t i o n e f f i c i e n c y . d) The p r o b l e m of h o l o g r a m w r i t i n g w i t h o n e - d i m e n s i o n a l G a u s s i a n beams, i n -c i d e n t on a f i n i t e c r y s t a l w i t h f i n i t e d a r k c o n d u c t i v i t y was m o d e l l e d f o r c o n s t a n t a p p l i e d v o l t a g e and s h o r t t r a n s p o r t l e n g t h . I t was shown t h a t t h e r a t i o o f t h e l i g h t t o d a r k c a r r i e r c o n c e n t r a t i o n and t h e r a t i o o f t h e c r y s t a l l e n g t h t o t h e G a u s s i a n beam w i d t h have i m p o r t a n t e f f e c t s on t h e w r i t i n g p r o c e s s . e) A t r e a t m e n t f o r o p t i c a l e r a s u r e w i t h l i g h t i n c i d e n t on and o f f t h e Bragg a n g l e w h i c h a l l o w e d q u a n t i t a t i v e l y , f o r t h e f i r s t t i m e , f o r t h e new h o l o g r a m w r i t t e n by t h e i n t e r f e r e n c e p a t t e r n o f t h e r e a d i n g and d i f f r a c t e d beams and o p t i c a l e r a s u r e due' t o t h e dc p a r t o f t h e l i g h t was d e v e l o p e d . The model a l s o a l l o w e d f o r t h e f e e d b a c k e f f e c t o f t h e c h a r g e f i e l d , t h e e f f e c t o f ab-s o r p t i o n i n r e d u c i n g t h e i n t e n s i t y o f t h e l i g h t as w e l l as t h e e f f e c t of t h e d a r k c o n d u c t i v i t y . The model r e p r o d u c e d a l l t h e r e p o r t e d t y p e s o f e r a s u r e c h a r a c t e r i s t i c s . I t i s b e l i e v e d t o be r e a l i s t i c enough f o r m e a n i n g f u l com-p a r i s o n w i t h e x p e r i m e n t a l d a t a , p r o v i d e d t h a t t h e b a s i c p h y s i c a l a s s u m p t i o n s a p p l y . f ) An e x p e r i m e n t a l method o f d e t e r m i n i n g t h e i n t r i n s i c d i f f r a c t i o n 151. e f f i c i e n c y o f holograms s t o r e d by t h e p h o t o r e f r a c t i v e e f f e c t w i t h o u t e r r o r s due t o m u l t i p l e i n t e r n a l r e f l e c t i o n was d e v e l o p e d . 9.1 S u g g e s t i o n s f o r F u r t h e r R e s e a r c h F u r t h e r i n v e s t i g a t i o n s a r e r e q u i r e d t o c h a r a c t e r i z e more c o m p l e t e -l y t h e mechanisms of t h e p h o t o r e f r a c t i v e e f f e c t . F o r example, t h e p h y s i c s o f the b u l k p h o t o v o l t a i c e f f e c t i s n o t a d e q u a t e l y u n d e r s t o o d . The d e s c r i p -t i o n o f t h e p h o t o v o l t a i c e f f e c t by a p h o t o i n d u c e d c u r r e n t d e n s i t y p r o p o r t i o n a l t o t h e l i g h t i n t e n s i t y r e q u i r e s f u r t h e r c l a r i f i c a t i o n , e s p e c i a l l y when t h e l i g h t p a t t e r n has h i g h s p a t i a l f r e q u e n c i e s . Some of t h e p a r a m e t e r s i n v o l v e d i n t h e t r a n s p o r t , w h i c h a r e n o t p r e s e n t l y known, a r e t h e quantum e f f i c i e n c y o f p h o t o e x c i t a t i o n , t h e l i f e t i m e o f f r e e c a r r i e r s , t h e c a p t u r e c r o s s - s e c t i o n o f t h e t r a p s and t h e m i g r a t i o n l e n g t h o f t h e f r e e e l e c t r o n s . .The models f o r h o l o g r a m w r i t i n g and o p t i c a l e r a s u r e p r e s e n t e d i n t h i s t h e s i s c o u l d be e x t e n d e d t o a p p l y t o a r b i t r a r y m i g r a t i o n l e n g t h o f f r e e e l e c t r o n s . The p r o b l e m o f o p t i c a l l y i n d u c e d s c a t t e r i n g s h o u l d be i n v e s t i g a t e d f u r t h e r t o f i n d t h e mechanisms i n v o l v e d i n t h e s c a t t e r i n g p r o c e s s and t o make t h e c r y s t a l l e s s s u s c e p t i b l e t o t h i s u n d e s i r a b l e phenomenon. 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APPENDIX A PROPERTIES OF LITHIUM NIOBATE A. 1 C r y s t a l Growth S i n g l e c r y s t a l s of l i t h i u m n i o b a t e a r e grown by s e v e r a l t e c h n i q u e s (Nassau e t a l . 1966b). The most common i s t h e C z o c h r a l s k i t e c h n i q u e w i t h an e l e c t r i c f i e l d a p p l i e d d u r i n g g r o wth. The c r y s t a l i s r o t a t e d as i t i s p u l l e d f r o m t h e m e l t . I f no f i e l d i s a p p l i e d d u r i n g g r o w t h a m u l t i - d o m a i n c r y s t a l f o r m s . E i t h e r p o l a r i t y may be a p p l i e d , however, i f p o l a r i t y i s r e v e r s e d d u r i n g g r o w t h , a 180° domain w a l l i s p r o d u c e d . C r y s t a l s w h i c h a r e n o t p o l e d d u r i n g growth may be p o l e d a f t e r w a r d s , b u t o n l y a t t e m p e r a t u r e s above 1000°C. A c r y s t a l grown f r o m a c o n g r u e n t l y m e l t i n g c o m p o s i t i o n (48.6 mole % Li^O) has more homogeneous r e f r a c t i v e i n d i c e s ( R e d f i e l d e t a l . 1974) t h a n t h a t grown f r o m s t o i c h i o m e t r i c m e l t (50 mole % Ll^O). A.2 M i s c e l l a n e o u s P h y s i c a l P r o p e r t i e s The s t r u c t u r e o f t h e f e r r o e l e c t r i c c r y s t a l l i t h i u m n i o b a t e i s r h o m b o h e d r a l , p o i n t group symmetry 3m w i t h a = 0.5499 nm and a = 55° 52' (Nassau e t a l . 1966a). A t room t e m p e r a t u r e , t h e c r y s t a l l i n e s t r u c t u r e con-s i s t s o f p l a n a r s h e e t s o f oxygen atoms i n a p p r o x i m a t e l y h e x a g o n a l c l o s e pack-i n g . The r e s u l t i n g o c t a h e d r a l i n t e r s t i c e s a r e o n e - t h i r d o c c u p i e d by Nb and o n e - t h i r d by L i w i t h t h e r e m a i n d e r v a c a n t (Abrahams e t a l . 1966a, 1966b). A l l t h e oxygen o c t a h e d r a a r e d i s t o r t e d and t h e r e a r e two v a l u e s each f o r L i - 0 and Nb-0 d i s t a n c e s . I t . i s a u n i a x i a l c r y s t a l w i t h o r d i n a r y and e x t r a o r d i n a r y r e f r a c -t i v e i n d i c e s a t 500 nm r e p o r t e d t o be n = 2.34 and n = 2 . 2 4 . The d i e l e c t r i c r o e c o n s t a n t has been measured f o r d i r e c t i o n p e r p e n d i c u l a r t o t h e c - a x i s a s 78 and a l o n g t h e c - a x i s as 32 (Nassau e t a l . 1966a). The p u r e c r y s t a l has 158. v e r y l i t t l e a b s o r p t i o n f r o m 350 nm t o 5 ym. The c u r i e t e m p e r a t u r e i s 1210°C and t h e m e l t i n g p o i n t i s about 1260°C (Nassau e t a l . 1966a). A t room t e m p e r a t u r e , t h e c r y s t a l i s a s t a b l e f e r r o e l e c t r i c . H i g h t e m p e r a t u r e t r a n s p o r t p r o c e s s e s have been measured by J o r g e n s e n and B a r t l e t t (1969) who found t h a t b o t h i o n i c and e l e c t r o n i c c on-2 d u c t i v i t y o c c u r . The e l e c t r o n i c m o b i l i t y was c a l c u l a t e d t o be 1.7 cm /Vsec -2/3 a t 1000°K i n a 50% C0/50% C 0 2 atm and e x h i b i t s a T t e m p e r a t u r e depenr--2 2 dence. The p y r o e l e c t r i c c o e f f i c i e n t i s 10 yC/(m deg) i n t h e r e g i o n 100°C ( R o i t b e r g e t a l . 1970). A.3 Thermal B l e a c h i n g and F i x i n g o f Holograms i n LiNbO^ O p t i c a l e r a s u r e o f holograms w h i c h u s u a l l y o c c u r d u r i n g r e a d o u t can be a v o i d e d i f t h e holograms a r e f i x e d (Amodei e t a l . 1972a and 1972b, S t a e b l e r e t a l . 1972a). When a c r y s t a l i n w h i c h a h o l o g r a m has been r e -co r d e d i s h e a t e d t o 100°C f o r 20 t o 30 m i n u t e s and t h e n c o o l e d , i t i s fo u n d t h a t t h e h o l o g r a m has been b l e a c h e d . The h o l o g r a m may be r e s t o r e d by i l l u m i -n a t i n g t h e c r y s t a l w i t h l i g h t o f w a v e l e n g t h 400 t o 500 nm. The r e s t o r e d h o l o g r a m cannot be o p t i c a l l y e r a s e d . Amodei e t a l . have p r o p o s e d t h a t t h e hol o g r a m i s b l e a c h e d , n o t by t h e r m a l l y a c t i v a t e d e l e c t r o n s w h i c h a r e r e d i s t r i b u t e d u n i f o r m l y , -but by some k i n d o f i o n i c movement w h i c h compensates t h e spac e c h a r g e . When t h e c r y s t a l i s c o o l e d , t h o s e i o n s a r e f r o z e n i n t h e i r new s i t e s i n t h e c r y s t a l l a t t i c e . I t i s s u g g e s t e d ( W i l l i a m s e t a l . 1976) t h a t t h o s e i o n s a r e S i i o n s . When the c r y s t a l i s i l l u m i n a t e d , o p t i c a l l y e x c i t e d e l e c t r o n s r e d i s t r i b u t e u n i -f o r m l y . The h o l o g r a m r e a p p e a r s due t o t h e i o n i c d i s p l a c e m e n t s w h i c h a r e i n s e n s i t i v e t o l i g h t . T h i s e x p l a n a t i o n a c c o u n t s , q u a l i t a t i v e l y , f o r t h e e x p e r i m e n t a l o b s e r v a t i o n s . 159. APPENDIX B THE ELECTRO-OPTIC EFFECT IN LITHIUM NIOBATE The e l e c t r o - o p t i c e f f e c t may he d e f i n e d as t h e change i n t h e r e -f r a c t i v e i n d e x o f a m a t e r i a l when a f i e l d i s a p p l i e d t o i t . The d i e l e c t r i c p o r p e r t i e s o f an a n i s o t r o p i c c r y s t a l a t o p t i c a l f r e q u e n c i e s a r e g i v e n by D. = E e.. E. ( B . l ) where D i s t h e d i s p l a c e m e n t , E i s t h e e l e c t r i c f i e l d , E q i s t h e p e r m i t t i v i t y o f f r e e space and e „ i s t h e p e r m i t t i v i t y t e n s o r o f t h e medium. The p r o p a g a t i o n o f e l e c t r o m a g n e t i c waves i n an a n i s o t r o p i c c r y s t a l i s dependent on t h e p o l a r i z a t i o n and t h e p r o p a g a t i o n d i r e c t i o n o f t h e wave w i t h r e s p e c t t o t h e c r y s t a l a x e s . I t can be shown (Nye 1960) t h a t two waves o f d i f f e r e n t v e l o c i t i e s , i n g e n e r a l , p o r p a g a t e t h r o u g h t h e c r y s t a l f o r a g i v e n wave n o r m a l . The r e f r a c t i v e i n d i c e s o f t h e two waves may be o b t a i n e d by d r a w i n g an e l l i p s o i d known as t h e i n d i c a t r i x . I f x^, a n ^ x 3 a r e t n e p r i n c i p a l d i r e c t i o n o f t h e p e r m i t t i v i t y t e n s o r , t h e i n d i c a t r i x i s d e f i n e d by th e e q u a t i o n 2 2 2 X . X x_ - \ + - f + = 1 (B.2) n l n 2 n 3 where n^ = /s~> = , n^ = v^g~ I f a s t r a i g h t l i n e i s drawn f r o m t h e c e n t r e o f t h e e l l i p s o i d p a r a -l l e l t o t h e wave n o r m a l of t h e p r o p a g a t i n g wave, t h e n an e l l i p s e may be formed by c l e a v i n g t h e e l l i p s o i d t h r o u g h i t s c e n t r e , p e r p e n d i c u l a r t o t h i s l i n e . The semi-axes o f t h i s e l l i p s o i d d e f i n e t h e two d i r e c t i o n s o f p o l a r i -z a t i o n w h i c h may p r o p a g a t e . The i n d i c e s o f r e f r a c t i o n seen by t h e two p r o p a g a t i n g waves a r e t h e n g i v e n by t h e l e n g t h o f t h e s e m i - a x e s . 160. I f an e l e c t r i c f i e l d i s p r e s e n t , t h e r e f r a c t i v e i n d e x o f t h e c r y -s t a l i s a l t e r e d , and t h e g e n e r a l e q u a t i o n o f t h e i n d i c a t r i x t h e n becomes 2 I-i,j,k,£ n T+ zi j k Ek + R i j k £ Ek E* + X i X j " 1 <B'3> where t h e i n d i c e s i,j,k,£ r u n f r o m 1 t o 3. The c o e f f i c i e n t s z . M and R.., XJ K XJ KX. a r e t h e l i n e a r and t h e q u a d r a t i c e l e c t r o - o p t i c c o e f f i c i e n t s . C o n t r a c t i o n s i n t h e i n d i c e s a r e u s u a l l y made as f o l l o w s : f , z,,.>, a n f R R,.. w, n\ J mk ( i j ) k mn (xj)(k£) where m and n r u n fr o m 1 t o 6 and m i s r e l a t e d t o ( i j ) and n t o (Jul) as f o l l o w s : 1 •+ 11, 2 .-»• 22, 3 -> 33, 4 2 3 , 5 -> 13, 6 -> 12. C e r t a i n c r y s t a l s cannot e x h i b i t t h e l i n e a r e l e c t r o - o p t i c e f f e c t ( s u c h as t h o s e w i t h a c e n t r e o f symmetry) w h i l e a l l m a t e r i a l s e x h i b i t t h e q u a t r a t i c e f f e c t . L i t h i u m n i o b a t e e x h i b i t s t h e l i n e a r e l e c t r o - o p t i c e f f e c t . Symmetry c o n s i d e r a t i o n r e q u i r e s t h a t some o f t h e l i n e a r e l e c t r o - o p t i c c o -e f f i c i e n t s a r e e q u a l and t h a t some a r e z e r o as shown by t h e f o l l o w i n g m a t r i x ( c l a s s 3m). - r 0 0 0 0 :42 22 - r 22 :22 0 :42 0 0 "13 :13 :33 0 0 0 -10 ,-10 where ( T u r n e r 1966) r 1 3 = 8.6 x 10 c m / v o l t , = 3.4 x 10 c m / v o l t , r.„ = 28 x 1 0 ~ 1 0 c m / v o l t , r 0 0 = 30.8 x 1 0 ~ 1 0 c m / v o l t . 42 33 A f u r t h e r p r o p e r t y o f LiNbO^ i s t h a t i t i s a u n i a x i a l c r y s t a l w i t h x^ c o n s i d e r e d as t h e p o l a r a x i s . Hence, t h e i n d i c a t r i x i s an e l l i p s o i d o f r e v o l u t i o n and two o f t h e t h r e e p r i n c i p a l semi-axes a r e e q u a l so t h a t n = n_ = n„, n = n_ o 1 2 e 3 The i n d i c a t r i x i s t h u s g i v e n by ( _ 1 2 " r 2 2 E 2 + r 1 3 V X / + ( _ 1 2 + r 2 2 E 2 + r 2 V 3 ^ n n o o + ( - — + r 3 3 E 3 ) x 3 2 + 2 ( - r 2 2 ^ ^ n e + 2 ( r 4 2 E 2 ) e 2 x 3 + 2 ( r 4 2 ^ x± = 1 (B.4) From t h i s e q u a t i o n i t can be s e e n t h a t , i f E 3 i s t h e o n l y f i e l d p r e s e n t , t h e n o n l y an e x t e n s i o n o r c o n t r a c t i o n o f t h e major axes i s p o s s i b l e S i n c e a l l t h e c r o s s terms w o u l d be z e r o , no r o t a t i o n o f t h e p r i n c i p a l axes ; o f t h e i n d i c a t r i x o c c u r s . I f , h o w e v e r , E^ o r E 2 a r e p r e s e n t , t h e n a r o t a -t i o n o c c u r s . F o r h o l o g r a m s t o r a g e i n l i t h i u m n i o b a t e , t h e space c h a r g e f i e l d i s o n l y a l o n g t h e x 3 a x i s ( c - a x i s ) and, f o r t h i s c a s e , t h e i n d i c a t r i x r e -duces t o (~T + r 1 3 E 3 ) x l + ( " T + r 2 3 E 3 ) x 2 + ^2 + r 3 3 E 3 ) x 3 = 1 ( B ' 5 ) n n n o o e The e f f e c t o f E„ i s t o i n t r o d u c e changes, An and An i n t h e two r e f r a c t i v e 3 o e i n d i c e s . F o r a wave p r o p a g a t i n g i n x^ d i r e c t i o n , m a n i p u l a t i n g Eq. B.5, i t can be shown t h a t 3 3 n r nrr„„ An = - ° E, and An = - - ^ r ^ E„ (B.6) o 2 3 e 2 3 The change i n i n d e x i s t h e n p r o p o r t i o n a l t o t h e f i e l d . However, i n h o l o g r a m w r i t i n g and r e a d i n g , t h e l i g h t wave i s i n c i d e n t a t an a n g l e 8 w i t h r e s p e c t t o x^ ( o r x 2 ) . R o t a t i n g t h e p r i n c i p a l axes o f t h e i n d i c a t r i x so t h a t t h e wave n o r m a l c o i n c i d e s w i t h t h e new x^, th e new e q u a t i o n o f t h e i n d i c a t r i x i s ( V r 2 3 ) x 2 2 + ("T + r f 3 ) x 3 2 = 1 (B.7) n n_ o f 2 2 , 1 s i n 8 cos 8 , . 2„ 2 n where — = — 2 1 2 — a r f 3 = r 1 3 S l n r 3 3 C O S n n f o e and t h e change i n t h e i n d i c e s o f r e f r a c t i o n An and An i s 6 o e 3 3 r 7 f n f An = - r 2 3 n o E. and A n £ = ^ E„ (B.8) o — 2 r / J S i n c e f o r l i t h i u m n i o b a t e n > n and r__> r „ t h e n r O J - < r„„ and < n . o e 33 13 3f 33 f e Thus, n e g l e c t i n g t h e e f f e c t o f t h e non-normal i n c i d e n c e o f t h e l i g h t waves, r e s u l t s i n an o v e r e s t i m a t i o n o f t h e change i n e x t r a o r d i n a r y r e f r a c t i v e i n d e x . However, i t does n o t a f f e c t t h e change i n t h e o r d i n a r y i n d e x . 163. APPENDIX C READ-WRITE HOLOGRAPHIC MEMORY SYSTEM R e a d - w r i t e o p t i c a l memories ba s e d on h o l o g r a m s t o r a g e i n an e r a s -a b l e medium a r e of i n t e r e s t b ecause t h e y o f f e r t h e p o s s i b i l i t y o f l a r g e s t o r a g e c a p a c i t y , h i g h - r e s o l u t i o n c a p a b i l i t y , no m e c h a n i c a l m o t i o n , low n o i s e and h i g h speed random a c c e s s i b i l i t y . The b a s i c c o n f i g u r a t i o n o f a h o l o g r a p h i c memory i s shown i n F i g . C . l . I t c o n s i s t s o f a l a s e r s o u r c e , a d e f l e c t i o n s y s t e m , a h o l o l e n s o r a beam s p l i t t e r , a page composer, t h e s t o r a g e medium and a d e t e c t o r a r r a y . The d e f l e c t i o n s y s t e m c o n s i s t s o f two a c o u s t o - o p t i c o r e l e c t r o -o p t i c d e f l e c t o r s a t r i g h t a n g l e s t o each o t h e r w i t h a r e s o l u t i o n of 100 t o 4 16 1000 p o s i t i o n s e a c h ( t o t a l c a p a c i t y 10 t o 10 b i t s ) w i t h a c c e s s t i m e of 2 t o 10 us. The a c o u s t o - o p t i c d e f l e c t o r i s t h e s i m p l e r o f t h e two b u t i t i s s l o w e r . The e l e c t r o - o p t i c d e f l e c t o r i s more c o m p l i c a t e d and r e q u i r e s h i g h o p e r a t i n g v o l t a g e s . B o t h t h e s e d e v i c e s can be made w i t h LiNbO^ c r y s t a l s ( K o r p e l e t a l . 1966, Chen 1970). The " h o l o l e n s " i s a h o l o g r a p h i c o p t i c a l element whose i n p u t i s t h e l i g h t f r o m a d e f l e c t o r and whose o u t p u t c o n s i s t s o f a r e f e r e n c e beam, w h i c h 3 4 i s d i r e c t e d towards t h e s t o r a g e medium, and a c l u s t e r o f 10 t o 10 low-i n t e n s i t y beams d i r e c t e d toward t h e page composer. The._ page composer i s an a r r a y o f e l e c t r o n i c a l l y c o n t r o l l e d l i g h t v a l v e s t h a t m o d u l a t e t h e c l u s t e r o f beams i n a b i n a r y f a s h i o n (ON = " 1 " , OFF = " 0 " ) . The o u t p u t , f r o m t h e page composer, and t h e r e f e r e n c e beam t o a 1-mm s p o t a t t h e computer a d d r e s s e d l o c a t i o n on t h e s t o r a g e medium. Ne m a t i c l i q u i d c r y s t a l s . h a v e been used i n t h e page composer i n e x p e r i m e n t a l h o l o g r a p h i c systems ( S t e w a r t e t a l . 1973, d ' A u r i a e t a l . 1974). However, t h e y r e q u i r e a b u f f e r memory and a r e 164. F i g . C.l A schematic of read, write, erase holographic o p t i c a l memory. 165. i n h e r e n t l y s l o w t o s w i t c h s t a t e ( t y p i c a l l y m i l l i s e c o n d s ) . F e r r o e l e c t r i c c e r a m i c s s u c h as PLZT have a l s o been c o n s i d e r e d f o r use i n page composers. A l t h o u g h . t h e s e d e v i c e s a r e f a s t e r t h a n l i q u i d c r y s t a l , a t p r e s e n t PLZT f a t i g u e s b o t h e l e c t r i c a l l y and o p t i c a l l y . The h o l o g r a m i s w r i t t e n - i n by t h e a p p l i c a t i o n o f t h e r e f e r e n c e beam and t h e o b j e c t beam fr o m t h e page composer, and i t i s r e a d o u t by t h e a p p l i c a t i o n o f t h e r e f e r e n c e beam a l o n e . The o u t p u t l i g h t s i g n a l i s con-v e r t e d t o e l e c t r i c a l s i g n a l by t h e p h o t o d e t e c t o r a r r a y . I n a r e a d - w r i t e s y s t e m , t h e r e a d o u t i s d e s t r u c t i v e and t h e h o l o g r a m must be r e w r i t t e n b e f o r e a new a d d r e s s i s g i v e n t o t h e d e f l e c t o r . W i t h a 1 0 ^ - p o s i t i o n d e f l e c t o r and 4 10 - b i t page composer, t h e c a p a c i t y o f such a page by page d a t a a c c e s s _ memory sys t e m i s 10"^ b i t s . The s t o r a g e medium i s t h e most i m p o r t a n t component o f t h e system. The s t o r a g e medium s h o u l d have h i g h r e s o l u t i o n c a p a b i l i t y o f a t l e a s t 1000 lines/mm. The d a r k s t o r a g e t i m e , depending on usage and on t h e p l a c e o f o p t i c a l memory among t h e h i e r a r c h y o f memories i n u s e , s h o u l d range f r o m a week t o s e v e r a l months. I n f o r m a t i o n w r i t e and e r a s e t i m e s h o u l d be a p p r o x i -m a t e l y e q u a l and be of t h e o r d e r o f a few m i l l i s e c o n d s p e r page ( i . e . few 3 m i c r o s e c o n d s p e r b i t f o r 10 - b i t page composer). The s t o r a g e medium s h o u l d be r e v e r s i b l e i n t h e se n s e t h a t t h e h o l o g r a m s h o u l d be e r a s a b l e upon i l l u m i -n a t i o n . A l t h o u g h s e v e r a l m a t e r i a l s have been s t u d i e d as p o t e n t i a l s t o r a g e media f o r h o l o g r a p h i c memories, i r o n - d o p e d l i t h i u m n i o b a t e i s p a r t i c u l a r l y s u i t a b l e , b ecause i t has t h e above c h a r a c t e r i s t i c s t o a l a r g e e x t e n t . I t s major drawback i s t h a t i t s s e n s i t i v i t y i s s t i l l i n s u f f i c i e n t f o r p r a c t i c a l memory. However, i t i s v e r y p r o m i s i n g because of t h e ease o f p r e p a r a t i o n and because f u r t h e r improvements i n i t s s e n s i t i v i t y a r e p o s s i b l e . I n a t y p i c a l o p t i c a l memory, t h e e f f i c i e n c y o f t h e d e f l e c t i n g s y s t e m (about 20 t o 25%) and t h e t r a n s m i s s i o n and r e f l e c t i o n l o s s e s t h r o u g h t h e v a r i o u s o p t i c a l components a r e s u c h t h a t o n l y about 5% of t h e t o t a l l i g h t r e a c h e s t h e s t o r a g e medium. I f t h e h o l o g r a p h i c d i f f r a c t i o n e f f i c i e n c y i s 1% o n l y 0.05% o f t h e l a s e r power wou l d r e a c h t h e d e t e c t o r a r r a y . I f t h i s 3 -7 was t h e n s h a r e d between 10 d e t e c t o r e l e m e n t s o n l y about 2.5 x 10 o f t h e l a s e r power wou l d r e a c h each d e t e c t o r e l e m e n t . F o r a d e t e c t o r t h a t r e q u i r e s 1 p j o f l i g h t , a r e a d i n g speed o f 4 usee p e r h o l o g r a m w o u l d r e q u i r e a l a s e r w i t h a power 1W. However, s i n c e o n l y 50 mW o f l i g h t a v a i l a b l e f o r w r i t i n g and r e a d i n g - e r a s i n g , about 10% o f t h i s amount ( t h e h o l o l e n s o u t p u t ) i s s p l i t i n t o t h e i n d i v i d u a l beams t o be coded by t h e page composer and t o be s t o r e d 3 h o l o g r a p h i c a l l y i n t h e s t o r a g e medium f o r a page composer of 10 b i t s , t h e 4 r e f e r e n c e - t o - o b j e c t beam r a t i o f o r each b i t i s 10 and t h e m o d u l a t i o n r a t i o i s m = 0.02. T h e r e f o r e , t h e s e n s i t i v i t y o f t h e s t o r a g e medium s h o u l d be h i g h enough t o s t o r e t h e h o l o g r a m i n t h e o r d e r o f aVfew m i l l i s e c o n d s w i t h s u c h s m a l l m o d u l a t i o n r a t i o . C a r l s e n (1974) has p r o p o s e d an a l t e r n a t i v e method f o r r e c o r d -i n g t h e d i g i t a l d a t a . I n h i s sy s t e m , d a t a b i t s w o u l d be s t o r e d s e q u e n t i a l -l y as t h e y a r r i v e d f r o m t h e computer, t h u s a l l e v i a t i n g t h e need f o r a l a r g e page composer. The advan t a g e s o f p a r a l l e l i n p u t a r e l o s t u n l e s s t h e d a t a t o be s t o r e d i n one page i s p r e a r r a n g e d . O t h e r w i s e t h e d a t a s t o r e d i n any one page would be so d i v e r s e t h a t t h e r e w o u l d n o t be much advantage i n r e -t r i e v i n g i t a l l i n a p a r a l l e l o u t p u t mode. W i t h random a c c e s s s e q u e n t i a l i n p u t , the computer c o u l d t a g t h e d a t a f o r s t o r a g e i n a g i v e n page. I n t h i s manner, p a r a l l e l o u t p u t o f r e l a t e d d a t a c o u l d be a c h i e v e d . C a r l s e n p r o p o s e d t h a t e ach b i t i n a page be s t o r e d w i t h a d i f f e r e n t a n g l e w i t h a l l b i t s i n a page super i m p o s e d upon one a n o t h e r t o y i e l d a m u l t i p l e e x p o s u r e h ologram. 167. Each page wou l d be s t o r e d i n a d i f f e r e n t l o c a t i o n i n t h e r e c o r d i n g medium. I t i s of i n t e r e s t t o compare h o l o g r a p h i c memory systems w i t h o t h e r systems t o d e t e r m i n e what r o l e t h e y m i g h t p l a y . K i e m l e (1974) has shown t h a t h o l o g r a p h i c memories u s i n g a s i n g l e d e t e c t o r a r r a y and a s i n g l e h o l o -g gram p l a t e a r e l i m i t e d i n s t o r a g e c a p a c i t y t o about 10 b i t s . New memory t e c h n i q u e s such as c h a r g e - c o u p l e d d e v i c e s and m a g n e t i c b u b b l e domains may be c a p a b l e o f t h i s c a p a c i t y r a n g e and may p r o v i d e cheaper s o l u t i o n s . C o n v e n t i o n a l r e c o r d i n g t e c h n o l o g i e s s u c h as m a g n e t i c drums and d i s k s w i l l p r o b a b l y be improved. Development o f h o l o g r a p h i c memories, t h e r e f o r e , s h o u l d s t r i v e t o complement t h e s e and o t h e r t e c h n o l o g i e s because i t i s u n l i k e l y t h a t t h e y w i l l c o m p l e t e l y r e p l a c e them. I t i s e n v i s i o n e d t h a t h o l o g r a p h i c mamories w i l l be a b l e t o p r o v i d e c a p a c i t i e s comparable t o t h o s e o f m a g n e t i c tape s t o r a g e s y s t e m s , b u t w i t h much s h o r t e r a c c e s s t i m e s . I n t a b l e C . l , a c o m p a r i s o n i s made o f some memory systems ( C o r n i s h 1976) P u b l i c a t i o n s ; M.G. Moharam and L. Young, "Reading and O p t i c a l Erasure of Holograms Stored by the Photorefractive E f f e c t " Appl. Optics, 17, (1978). M.G. Moharam and L. Young, " C r i t e r i o n f o r Bragg and Raman-Nath D i f f r a c t i o n Regimes", Appl. Optics, 17, (1978). M.G. Moharam and L. Young, "Hologram Writing by the Photorefractive Effect",. J . Appl. Phys. 48, 3230 (1977). M.G. Moharam and L. Young, "Hologram Writing by the Photorefractive E f f e c t with Gaussian Beams at Constant Applied Voltage", J . Appl. Phys. 47, 4048 (1976). W.D. Cornish, M.G. Moharam and L. Young, " E l l i p s o m e t r i c I n v e s t i g a t i o n of O p t i c a l Damage i n Lithium Niobate", F e r r o e l e c t r i c s , 10, 153 (1976). W.D. Cornish, M.G. Moharam and L. Young, " E f f e c t of Applied Voltage on Hologram Writing i n Lithium Niobate", J . Appl. Phys., 47, 1479 (1976). M.G. Moharam, W.D. Cornish and L. Yo-. ng, "Experiemntal Method of Determining the I n t r i n s i c D i f f r a c t i o n E f f i c i e n c y of Hologram Stored by the Photorefractive E f f e c t " Appl. Phys. L e t t . , 28, 324 (1976). 

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