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The photorefractive effect in lithium niobate and its applications Elguibaly, Fayez H. F. 1979

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THE PHOTOREFRACTIVE EFFECT IN LITHIUM NIOBATE AND ITS APPLICATIONS by Fayez H. F. e l Guibaly B.Sc. (Hon), Cairo University (Egypt), 1972 B.Sc. (Hon), Ain Shams University (Egypt), 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE STUDIES (Department of El e c t r i c a l Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA . September, 1979 (7)' Fayez H.• F. e l Guibaly > 1979 DOCTOR OF PHILOSOPHY in I n p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r a n advanced degree at the 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 , I a g r e e t h a t 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 a n d s t u d y . 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 p u r p o s e s may be g r a n t e d by the H e a d o f my D e p a r t m e n t o r 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 tha t 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 b e 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 . , E l e c t r i c a l Engineering Department o f ' The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 Sept 21, 1979 ABSTRACT In i r o n - d o p e d l i t h i u m n i o b a t e and o t h e r s i m i l a r c r y s t a l s , exposure t o l i g h t o f a p p r o p r i a t e wavelength i n d u c e s s m a l l changes i n the r e f r a c t i v e i n d e x . T h i s phenomenon i s c a l l e d the 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 to be d e s c r i b e d was under-taken 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 mechanisms o f the p h o t o r e f r a c t i v e e f f e c t and to i n v e s t i g a t e 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 . The p h o t o r e f r a c t i v e e f f e c t i s b e l i e v e d t o i n v o l v e the s p a t i a l r e d i s t r i b u t i o n o f p h o t o e x c i t e d elec-^ t r o n s among t r a p s . T h i s causes a space charge f i e l d t o develop-which modulates the r e f r a c t i v e i n d e x v i a the 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 b u l k p h o t o v o l -t a i c e f f e c t s p e c i a l t o f e r r o e l e c t r i c c r y s t a l s , f i r s t r e c o g n i z e d by G l a s s e t a l . , i s i m p o r t a n t 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 t h e s e . c r y s t a l s . I t i s shown t h a t the f i n i t e e l e c t r o n t r a n s p o r t l e n g t h i n t h i s e f f e c t makes the p h o t o v o l t a i c c u r r -ent d i s t r i b u t i o n s p a t i a l l y s h i f t e d from the l i g h t i n t e n s i t y p a t t e r n t h a t causes i t . Moreover, i t i s shown t h a t the s p a t i a l l y v a r y i n g p h o t o v o l t a i c c u r r e n t comp-onent which i s r e s p o n s i b l e f o r the hologram f o r m a t i o n d e c r e a s e s as the s p a t i a l f r e q u e n c y o f 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 i n c r e a s e s . Hologram w r i t i n g by the p h o t o r e f r a c t i v e e f f e c t i s m o d e l l e d f o r a r b i t r a r y e l e c t r o n t r a n s p o r t l e n g t h . The treatment a l l o w s f o r the feedback e f f e c t o f the space charge f i e l d and f o r the dark 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 model a p p l i e s t o "uniform i l l u m i n a t i o n and c o n s t a n t a p p l i e d v o l t a g e c o n d i t i o n s . I t . i s . shown t h a t e x c e p t - i n . c r y s t a l s where d i f f u s i o n dominates the .hologram i s s p a t i a l l y s h i f t e d from the l i g h t i n t e n s i t y p a t t e r n t h a t caused i t because of the f i n i t e e l e c t r o n t r a n s p o r t l e n g t h a s s o c i a t e d w i t h 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 . E x p e r i m e n t a l r e s u l t s which bear upon the b u l k p h o t o v o l t a i c e f f e c t and t h e a s s o c i a t e d e l e c t r o n t r a n s p o r t l e n g t h a r e r e p o r t e d . Hologram w r i t i n g w i t h an a r b i t r a r y o n e - d i m e n s i o n a l l i g h t i n t e n s i t y i i d i s t r i b u t i o n i s m o d e l l e d a l l o w i n g f o r the feedback e f f e c t o f t h e space charge f i e l d a t a l l w r i t i n g t i m e s , A l a r g e s c a l e space charge f i e l d a s s o c i a t e d w i t h the envelope o f t h e l i g h t i s shown to a f f e c t the w r i t i n g p r o c e s s . I t i s found t h a t f o r any ty p e o f i n t e n s i t y d i s t r i b u t i o n an i n c r e a s e i n the f r a c t i o n o f the c r y s t a l which i s i l l u m i n a t e d improves the e f f i c i e n c y o f the hologram w r i t i n g p r o c e s s . A l s o f o r p a r t i a l l y i l l u m i n a t e d c r y s t a l s t h e s t o r a g e e f f i c i e n c y improves as the p h o t o c o n d u c t i v i t y approaches the dark c o n d u c t i v i t y v a l u e from above. F o r a f u l l y i l l u m i n a t e d c r y s t a l the s t o r a g e e f f i c i e n c y improves as the r a t i o o f the p h o t o c o n d u c t i v i t y t o dark c o n d u c t i v i t y i n c r e a s e s . 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 t h e e f f e c t o f the l a r g e s c a l e f i e l d on hologram s t o r a g e a r e r e p o r t e d . Beam d i s t o r t i o n and 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 a r e two problems encountered w h i l e s t o r i n g holograms i n l i t h i u m h i p b a t e . We r e p o r t e x p e r i -mental o b s e r v a t i o n s and t h e o r e t i c a l models f o r t h e s e phenomena. I t i s shown t h a t beam d i s t o r t i o n i s due to t h e d e f o c u s i n g a c t i o n o f the l a r g e s c a l e r e f -r a c t i v e i n d e x change due to the envelope o f l i g h t . L i g h t s c a t t e r i n g i s sugg-e s t e d t o be due t o the l e n s a c t i o n o f the i n d e x v a r i a t i o n s due to l a s e r s p e c k l e s i n s i d e t h e c r y s t a l . A t h e o r e t i c a l treatment o f the s p a t i a l f i l t e r i n g p r o p e r t i e s o f volume holograms i s p r e s e n t e d . P r a c t i c a l a p p l i c a t i o n s o f volume holograms i n the f i e l d s o f i n t . e r f e r o m e t r i c t e s t i n g and o p t i c a l . communications a r e a l s o d i s c u s s e d . i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF ILLUSTRATIONS i x ACKNOWLEDGEMENTS x i i i 1. INTRODUCTION 1 2. PHYSICAL MODELS OF THE PHOTOREFRACTIVE EFFECT 8 2.1 I n t r o d u c t i o n 8 2.2 The E l e c t r o - O p t i c Nature of the P h o t o r e f r a c t i v e e f f e c t . 8 2.3 The I n t e r n a l F i e l d Model 8 2.4 Joh n s t o n ' s P o l a r i z a t i o n Model 11 2.5 The P y r o e l e c t r i c F i e l d Model . . . . . . 12 2.6 D e f e c t S i t e s and I m p u r i t i e s 13 2.7 F o r m a t i o n o f Holograms by D r i f t o r D i f f u s i o n 15 2.7.1 I n t r o d u c t i o n 15 2.7.2 Amodei's Model f o r Sh o r t T r a n s p o r t l e n g t h . . . . 18 2.7.3 Young e t a l . ' s A r b i t r a r y T r a n s p o r t Length Model . 20 2.7.4 D i s c u s s i o n 21 2.8 E f f e c t s o f Beam C o u p l i n g 22 2.8.1 I n t r o d u c t i o n 22 2.8.2 C o u p l i n g D u r i n g W r i t i n g . . . . 24 2.8.3 C o u p l i n g D u r i n g R e a d o u t . . . 24 2.8.3.A R Beam Read Out 26 2.8.3.B S Beam Read Out . 27 2.8.3.C D i s c u s s i o n 28 2.9 The Bulk P h o t o v o l t a i c E f f e c t 30 2.9.1 I n t r o d u c t i o n . 30 2.9.2 The Asymmetric P h o t o - D e l o c a l i z a t i o n Model ( G l a s s e t a l . 1974) 31 2.9.3 The C o l l e c t i v e Franck-Condon R e l a x a t i o n Model (Chanussot and G l a s s 1976) 34 2.9.4 The P h o t o f l u c t u a t i o n Model 36 2.9.5 The P o l a r i z e d I m p u r i t i e s Model (von B a l t z 1978) . 37 i v Page 3. MATHEMATICAL MODELS FOR THE BULK PHOTOVOLTAIC EFFECT WITH ARBITRARY ELECTRON TRANSPORT LENGTH 39; 3.1 Introdution . 39 3.2 The.-Continuous, S c a t t e r i n g : Model 41 3.2.1 Assumptions . . . 41 3.2.2 M a t h e m a t i c a l A n a l y s i s 41 3.2.3 D i s c u s s i o n 43 3.3 The F i x e d T r a n s p o r t Length Model . 44 3.3.1 Assumptions 44 3.3.2 M a t h e m a t i c a l A n a l y s i s 44 3.3.3 D i s c u s s i o n . . . 44 3.4 The D i s c r e t e S c a t t e r i n g C e n t e r s Model . . 46 3.4.1 Assumptions 46 3.4.2 M a t h e m a t i c a l A n a l y s i s 46 3.4.3 D i s c u s s i o n 48 3.5 Summary and C o n c l u s i o n s 48 4. - HOLOGRAM WRITING IN PHOTOREFRACTIVE CRYSTALS WITH ARBITRARY ELECTRON TRANSPORT LENGTH 49 •'4 ' 4.1 I n t r o d u c t i o n . 49 4.2 P h y s i c a l Model . . . . . . 50 4.3 M a t h e m a t i c a l A n a l y s i s . . . . . . . . . . . . 52 4.4 R e s u l t s . . . . . . ' 54 4.4.1 I n i t i a l Stages 54 4.4.2 S t e a d y - S t a t e L i m i t 56 4.5 D i s c u s s i o n 56 5. HOLOGRAPHIC MEASUREMENTS 58 5.1 I n t r o d u c t i o n 58 5.2 E x p r i m e n t a l P r o c e d u r e s 58 5.3 R e s u l t s 6 1 5.3.1 P h o t o c u r r e n t Measurements (Moharam 1978b). . . . 61 5.3.2 H o l o g r a p h i c Measurements 64 5.4 D i s c u s s i o n 70 v Page 6. HOLOGRAM WRITING WITH NONUNIFORM ILLUMINATION 76 6.1 I n t r o d u c t i o n 76 6.2 Model 77 6.2.1 The Sma l l Time A p p r o x i m a t i o n 82 6.2.2s I n t e r m e d i a t e and S a t u r a t i o n Stages 82 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 83 6.3.1 E f f e c t o f E x t e n t of F r a c t i o n a l I l l u m i n a t i o n . . . 85 6.3.2 E f f e c t o f the Dark C o n d u c t i v i t y . 9 0 7. AN INTERFEROMETRIC METHOD FOR OBSERVING THE SPACE CHARGE FIELD 96 7.1 I n t r o d u c t i o n 96 7.2' F r i n g e . F o rmation Arrangement . 96 7.3 E x p e r i m e n t a l P r o c e d u r e s 98 7.4 E x p e r i m e n t a l R e s u l t s 98 7.5 D i s c u s s i o n 103 7.5.1 E s t i m a t i o n o f the Magnitude o f the Space Charge F i e l d 103 7.5.2 D e t e r m i n i n g the S i g n o f the Index Change . . . . 1 0 3 8. INFLUENCE OF THE ENVELOPE FIELD ON HOLOGRAM STORAGE IN L i N b 0 3 105 8.1 I n t r o d u c t i o n 105 8.2 E x p e r i m e n t a l P r o c e d u r e s and R e s u l t s 105 8.2.1 P h o t o c u r r e n t Measurements 106 8.2.2 R e c t a n g u l a r I l l u m i n a t i o n 106 8.2.3 Q u a s i G a u s s i a n I l l u m i n a t i o n . I l l 8.3 D i s c u s s i o n 113 8.4 C o n c l u s i o n s 114 9. HOLOGRAM FIXING IN LITHIUM NIOBATE 115 9.1 I n t r o d u c t i o n 115 9.2 Two-Photon R e c o r d i n g 115 9.3 Thermal F i x i n g 118 9.3.1 I n t r o d u c t i o n 118 9.3.2 F i x e d Hologram Read Out 119 v i 9.3.3 D i s c u s s i o n . . . . . . . V . . . . . '. . . . . 122 OPTICALLY INDUCED LIGHT SCATTERING AND BEAM DISTORTION IN IRON-DOPED LITHIUM NIOBATE CRYSTALS . . . . . . . . . . . 125 10. . IN  LITHIUM NIOBATE CRYSTALS . . . . . . . 125 10.1 I n t r o d u c t i o n ' . . . . . . . . . . . . . . . . . . . . 125 10.2 Some B a s i c E x p e r i m e n t a l O b s e r v a t i o n s . . . . . .' . . 127 10.2.1 I d e n t i f i c a t i o n o f t h e S c a t t e r i n g as Due to P a r a s i t i c G r a t i n g s . . .. . 130 10.2.2 Lens A c t i o n o f the L a r g e S c a l e P a t t e r n o f O p t i c a l l y Induced R e f r a c t i v e Index Change . 134 10.2.3 Source o f the I n i t i a l S c a t t e r i n g 138 10.2.4 D i s c u s s i o n . . „ . . . . . . . . . . . . . . 144 10.3 O p e r a t i o n o f p a r a s i t i c G r a t i n g s . . . . . . . . . » . 146 11. SPATIAL FILTERING PROPERTIES OF VOLUME HOLOGRAMS . . . . . 148 11.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . 148 11.2 The A n a l y s i s . . . . . . . . . . . . . . . . . . . v 148 11.2.1 The R e c o r d i n g P r o c e s s . . . . . . . 148 11.2.2 Hologram Read Out . . . . . . . . . . . . . . 151 11.3 ' A p p l i c a t i o n s i n H o l o g r a p h i c I n t e r f e r o m e t r y 158 11.4 Volume Holograms as M o d u l a t o r s f o r O p t i c Communications . .< . . . . . . . 162 12. CONCLUSIONS . . . . . . . . . . . . . . . 167 12.1 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 . . . . . . . . . . 170 REFERENCES . . . . . . . . . . . . . 171 APPENDIX A PROPERTIES OF LITHIUM NIOBATE . 180 A . l . C r y s t a l Growth . . . . . . . . . 180 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 '. 181 APPENDIX B ELECTRO-OPTIC EFFECT IN LITHIUM NIOBATE 182 v i i Page APPENDIX C HOLOGRAM RECORDING IN. LiNbO WITH GENERAL GRATING VECTOR ORIENTATION . 186 C . l I n t r o d u c t i o n 186 C.2 The A n a l y s i s . . . . ' 187 APPENDIX D COUPLED WAVE THEORY FOR THICK HOLOGRAMS 192 APPENDIX E SOURCES OF THE LITHIUM NIOBATE CRYSTALS . . . . . . . 197 APPENDIX F DISTORTION PROPERTIES OF LITHIUM NIOBATE 199 F . l I n t r o d u c t i o n 199 F.2 Sources o f D i s t o r t i o n i n L i t h i u m N i o b a t e 199 v i i i LIST OF ILLUSTRATIONS F i g u r e Page 2.1 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 9 2.2 C o n f i g u r a t i o n f o r hologram r e c o r d i n g 16 2.3 E x p e r i m e n t a l a p p a r a t u s f o r r e c o r d i n g s i m p l e phase g r a t i n g s i n LiNbO^ 25 2.4 A schematic 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 the o p t i c a l e r a s u r e p r o c e s s 29 2.5 The asymmetric p h o t o d e l o c a l i z a t i o n model i n LiNbO^ 33 2.6 C o l l e c t i v e Franck-Condon model 35 '2.7 C o o r d i n a t e c o n f i g u r a t i o n f o r the Franck-Condon r e l a x a t i o n 35 3.1 The c o n t i n u o u s s c a t t e r i n g model 42 3.2 The f i x e d t r a n s p o r t l e n g t h model 45 3.3 The d i s c r e t e s c a t t e r i n g c e n t e r s model 47 4.1 C o n f i g u r a t i o n f o r hologram r e c o r d i n g 51 5.1 O p t i c a l arrangement f o r the h o l o g r a p h i c s t o r a g e 60 5.2 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 the l i g h t i n t e n s i t y and the a p p l i e d v o l t a g e 62 5.3 E v o l u t i o n o f the d i f f r a c t i o n e f f i c i e n c y and beam c o u p l i n g d u r i n g hologram r e c o r d i n g and e r a s u r e 65 5.4 D i f f r a c t i o n e f f i c i e n c y and beam c o u p l i n g d u r i n g f i v e c o n s e c u t i v e runs 68 5.5 D i f f r a c t i o n e f f i c i e n c y and beam c o u p l i n g f o r t h r e e runs w i t h the c r y s t a l l e f t f o r s e v e r a l days b e f o r e each run to a l l o w decay o f r e f r a c t i v e i n d e x changes 71 5.6 C a l c u l a t e d beam i n t e n s i t i e s f o r d i f f e r e n t v a l u e s o f the phase s h i f t a s s o c i a t e d w i t h the b u l k p h o t o v o l t a i c e f f e c t 74 6.1 C o n f i g u r a t i o n f o r hologram r e c o r d i n g w i t h a g e n e r a l i z e d l i g h t i n t e n s i t y d i s t r i b u t i o n 78 i x F i g u r e Page '•' 6.2 S p a t i a l d i s t r i b u t i o n o f the F o u r i e r components 1' of the p h o t o i n d u c e d space charge f i e l d . . . . • •. . ... . . V 87 6.3 The dependence o f t h e l o c a l dc component o f t h e • f i e l d on the beam w i d t h 89 6.4 The dependence o f t h e fundamental f i e l d component on the beam w i d t h 89 6.5 The dependence o f the fundamental f i e l d component on the dark c o n d u c t i v i t y under u n i f o r m i l l u m i n 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 the m o d u l a t i o n depth 91 6.6 The dependence o f the l o c a l dc f i e l d on t h e dark c o n d u c t i v i t y f o r r e c t a n g u l a r i l l u m i n a t i o n , , 91 6.7 The dependence o f t h e fundamental f i e l d component on the dark c o n d u c t i v i t y f o r r e c t a n g u l a r i l l u m i n a t i o n c o n d i t i o n s and d i f f e r e n t v a l u e s o f the m o d u l a t i o n depth . 92 6.8 V a r i a t i o n o f t h e fundamental f i e l d components due to • d r i f t and d i f f u s i o n w i t h the dark c o n d u c t i v i t y f o r G a u s s i a n i l l u m i n a t i o n and d i f f e r e n t v a l u e s o f the m o d u l a t i o n depth 92 6.9 S p a t i a l d i s t r i b u t i o n o f the F o u r i e r components o f the p h o t o i n d u c e d space charge f i e l d 94 7.1 I n t e r f e r e n c e o f l i g h t r e f l e c t e d from the f r o n t and back s u r f a c e s o f t h e c r y s t a l 97 7.2 Geometry o f the r e s u l t i n g i n t e r f e r e n c e f r i n g e s 99 7.3" E x p e r i m e n t a l arrangement employed t o i n d u c e an i n d e x change by an argon l a s e r and viewed by a He-Ne l a s e r 99 7.4 F r i n g e s i n a Fe-doped c r y s t a l showing o p t i c a l l y -i n d u c e d changes i n the r e f r a c t i v e i n d i c e s 100 8.1 The.. dependence o f the. 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 ori'-the, l i g n t • • i n t e n s i t y and'the .' , J • -.' a p p l i e d v o l t a g e . . 107 .8.2 v E x p e r i m e n t a l arrangement, f o r • h o l o g r a m s t o r a g e by r e c t a n g u l a r l i g h t i n t e n s i t y d i s t r i b u t i o n 108 8.3 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 and the e n v e l o p e f i e l d v e r s u s exposure . 110 j,'- ' •' x Figure Page 8.4 Effect of an i n i t i a l envelope f i e l d on the diffraction efficiency 110 9.1 Different photoionization processes for the photorefractive effect 116 9.2 Two-photon hologram recording 116 10.1 Development of beam distortion and optically induced scattering 128 10.2 Scattering during hologram reconstruction 132 10.3 Scattering rings appearing after the rotation of the sample 132 10.4 Scattering pattern induced by an argon laser and viewed by a He-Ne laser 132 10.5 The effect of decollimation of a beam in spreading the Ewald sphere into a, spherical shell . . . . . 133 10.6 The assumed refractive indices variations 135 10.7 The coordinate axes chosen and their relation to the crystal major axes 137 10.8 Computed ray paths in the crystal 139 10.9 Ratio of transmitted beam intensity to the i n i t i a l transmitted intensity vs. exposure . . 141 10.10 Ratio of transmitted intensity to the i n i t i a l intensity for a uniformly illuminated crystal . . . . . . . . 142 10.11 Scattering pattern that developes in a f u l l y illuminated crystal 143 11.1 Optical system for studying the spatial f i l t e r i n g properties of volume holograms 149 11.2 Optical Arrangement for the Matched F i l t e r i n g Operation 152 11.3 Set-up for a Holographic Interferometry Scheme 159 xi Figure Page 11.4 Input transparency, the hologram and the r e s u l t i n g interferogram 160 11.5 An o p t i c a l modulator employing a volume hologram 164 C . l Grating vector o r i e n t a t i o n r e l a t i v e to the c r y s t a l major axes 188 C. 2 Relation of the c r y s t a l axes and the two w r i t i n g beams for d i f f e r e n t configurations of forming holograms 190 D. l The r e l a t i o n between the wavevectors of the i n t e r f e r i n g plane waves and the grating vector at exact Bragg incidence 193 x i i ACKNOWLEDGEMENTS I would l i k e t o thank my s u p e r v i s o r , Dr. L. Young f o r h i s encou-ragement and guidance d u r i n g the c o u r s e o f t h i s r e s e a r c h . I am t h a n k f u l f o r the h e l p f u l d i s c u s s i o n s I had w i t h Dr. M. Moharam. I wish t o ex p r e s s my a p p r e c i a t i o n to Mrs. S. Hoy and Miss L. O h r l i n g f o r t y p i n g the t h e s i s , Mr. M.A. e l . S h a r k a w i f o r h i s v a l u a b l e h e l p i n t a k i n g most o f t h e photographs, and t o Mr. J . Stuber f o r h i s a s s i s t a n c e i n the machine shop. The N a t i o n a l Research C o u n c i l o f Canada (Grant No. A3392 and s c h o l a r s h i p awarded 1977-1979) and the U n i v e r s i t y o f B r i t i s h Columbia (graduate f e l l o w s h i p awarded 1976-1977) a r e g r a t e f u l l y acknowledged f o r t h e i r f i n a n c i a l s u p p o r t . x i i i To Mama and Papa xiv' 1. I INTRODUCTION In c e r t a i n i n s u l a t i n g c r y s t a l s such as l i t h i u m niobate, exposure to l i g h t of appropriate wavelength induces small changes i n the r e f r a c t i v e index. This phenomenon i s known as the photorefractive e f f e c t . I t i s believed that the mechanism of the photorefractive e f f e c t i n these c r y s t a l s i s broadly as follows. Exposure to l i g h t excites electrons out of deep traps. The electrons d r i f t and d i f f u s e and are then retrapped forming regions of net space charge, p o s i t i v e where the electrons are depleted, •. . negative where they accumulate. This produces e l e c t r i c f i e l d patterns i n the bulk of the storage medium which, i n turn, s p a t i a l l y modulate the r e -f r a c t i v e index v i a the l i n e a r 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 induced change i n the r e f r a c t i v e index may be removed either by uniform i l l u m i n a t i o n or by heating. These treatments cause the electrons to be o p t i c a l l y or thermally excited from t h e i r traps and uniformly r e d i s t r i b u t e d so that the modulation i n the r e f r a c t i v e index i s removed. The photorefractive e f f e c t , which e a r l i e r was-called " o p t i c a l damage", was f i r s t observed i n c r y s t a l s of LiNbO^ , LiTaO-j and BaTiO-j by Ashkin et a l . (1966) i n the study of o p t i c a l second harmonic generation, .; the damage taking the form of an inhomogeneity i n the index of r e f r a c t i o n causing s c a t t e r i n g and decollimation of l i g h t , thus degrading the perform-ance of the devices. I t was subsequently recognized that this e f f e c t could be useful as a means of s t o r i n g pure phase holograms. Volume phase holograms have been stored by the photorefractive 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 l i t h i u m niobate (Chen et a l . 1968), strontium barium niobate (SBN) (Thaxter 1969), barium t i t a n a t e (Townsend et a l . 1970), barium sodium niobate (Amodei et a l , 1971a) and lead-lanthanum titano-zirconate ( P L Z T ) ( M i c h e r o n et a l . 1974a, 1974b). 2. P h o t o r e f r a c t i v e m a t e r i a l s have become the s u b j e c t of c o n s i d e r a b l e i n t e r e s t on account of t h e i r a b i l i t y to s t o r e volume phase holograms g i v i n g r i s e to v a r i o u s p o t e n t i a l a p p l i c a t i o n s , e.g.; d a t a s t o r a g e , f i l t e r i n g , i n f o r -m a tion p r o c e s s i n g , c o l o r d i s p l a y s , beam s p l i t t e r s , e t c . There a r e c e r t a i n advantages i n u s i n g p h o t o r e f r a c t i v e m a t e r i a l s , as compared to p h o t o g r a p h i c p l a t e s and o t h e r s t o r a g e media, such as e x c e l l e n t r e s o l u t i o n , r e a d out e f f i c i e n c y , r e v e r s i b i l i t y , s t o r a g e c a p a c i t y and s e n s i t i v i t y . In a d d i t i o n , b o t h r e a d / w r i t e and r e a d - o n l y systems a r e f e a s i b l e i n t h e s e m a t e r i a l s . The r e a d / w r i t e o p e r a t i o n i s p a r t i c u l a r l y s i m p l e because the a s - r e c o r d e d hologram can be i m m e d i a t e l y r e a d out w i t h o u t p r o c e s s i n g , and then e r a s e d w i t h the same wavelength used f o r s t o r a g e . F o r r e a d - o n l y p u r p o s e s , the hologram can be r e n d e r e d permanent < by v a r i o u s f i x i n g t e c h n i q u e s , as w i l l be d i s c u s s e d i n Chapter 9. The prime l i m i t a t i o n o f t h e s e m a t e r i a l s a t p r e s e n t i s t h a t most of the h o l o g r a p h i c s t o r a g e p r o p e r t i e s a r e s t r o n g l y i n t e r r e l a t e d ; e . g . , i n enhancing one p r o p e r t y ( e . g . , s e n s i t i v i t y ) by p r o p e r c h o i c e o f m a t e r i a l or m a t e r i a l treatment one f i n d s a t r a d e o f f i n o t h e r s ( e . g . , d i f f r a c t i o n e f f i c i e n c y ) . . The work d e s c r i b e d h e r e was u ndertaken i n o r d e r to 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 hologram s t o r a g e by the p h o t o r e f r a c t i v e e f f e c t and i n v e s t i g a t e 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 . The work was c a r r i e d out w i t h s p e c i a l i n t e r e s t i n i r o n - d o p e d l i t h i u m n i o b a t e because h i g h q u a l i t y c r y s t a l s were r e a d i l y a v a i l a b l e . The t h e o r e t i c a l a n a l y s i s , however, s h o u l d a p p l y to o t h e r p h o t o r e f r a c t i v e m a t e r i a l s as w e l l . A l t h o u g h more was known about the photo-r e f r a c t i v e e f f e c t i n l i t h i u m n i o b a t e than i n o t h e r c r y s t a l s , the b a s i c mechanisms a r e not y e t f u l l y u n d e r s t o o d and the s p e c i f i c a t i o n o f the m a t e r i a l f o r e n g i n e e r i n g a p p l i c a t i o n s was i n c o m p l e t e . 3. 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 p h o t o e x c i t e d e l e c t r o n s i n l i t h i u m n i o b a t e , and i n o t h e r s i m i l a r c r y s t a l s , a r e t r a n s p o r t e d 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 d i f f u s i o n i n c o n c e n t r a t i o n g r a d i e n t s (Amodei 1971b, 1971c). G l a s s , von der L i n d e and Negran (1974, 1975) have proposed t h a t the photo-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 which they c a 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 thought to be r e s p o n s i b l e f o r the p h o t o c u r r e n t s which were p r e v i o u s l y a t t r i b u t e d to i n t e r n a l 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 . The d i s c o v e r y o f t h i s new mechanism opened up a new f i e l d o f r e s e a r c h c a l l e d " p h o t o f e r r o e l e c t r i c i t y " (Chanussot. 1978), which combines the p h o t o e l e c t r i c and the f e r r o e l e c t r i c phenomena. T h i s new mechanism, and the v a r i o u s models a t t e m p t i n g to e x p l a i n i t , a r e o u t l i n e d i n Chapter 2. In Chapter 3, s e v e r a l m a t h e m a t i c a l models f o r the r e s u l t i n g p h o t o c u r r e n t s due t o t h i s p h o t o v o l t a i c e f f e c t a r e d e v e l o p e d f o r v a r i o u s assumed e l e c t r o n s c a t t e r i n g models. I t i s shown t h a t t h i s mechanism produces 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 to the absorbed l i g h t i n t e n s i t y but s p a t i a l l y s h i f t e d from i t by an amount t h a t depends on the t r a n s p o r t l e n g t h of t h e p h o t o e x c i t e d e l e c t r o n s . S t a e b l e r and Amodei (1972a) were the f i r s t to p o i n t out the beam c o u p l i n g e f f e c t t h a t t akes p l a c e as a r e s u l t o f the s p a t i a l phase s h i f t between the hologram and 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 t h a t g e n e r a t e s i t . The hologram 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 the l i g h t p a t t e r n c a u s i n g energy t r a n s f e r between the two w r i t i n g beams and thus m o d i f y 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 . In Chapter 4 a model i s proposed f o r hologram w r i t i n g w i t h a r b i t r a r y e l e c t r o n t r a n s p o r t l e n g t h . I t i s shown t h a t a phase s h i f t i s i n t r o d u c e d between the hologram and the l i g h t p a t t e r n m a i n l y because of the 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 d i f f u s i o n as was p r e v i o u s l y b e l i e v e d . In 4. Chapter 5, some e x p e r i m e n t a l r e s u l t s which bear upon the; bulk p h o t o v o l t a i c e f f e c t and the a s s o c i a t e d e l e c t r o n t r a n s p o r t l e n g t h a r e r e p o r t e d . Hologram w r i t i n g w i t h n onuniform i l l u m i n a t i o n would produce a l a r g e s c a l e space charge f i e l d a s s o c i a t e d w i t h the e n v e l o p e o f l i g h t i n a d d i t i o n to the s i n u s o i d a l components o f the f i e l d c o n s t i t u t i n g the hologram. C o r n i s h e t a l . (1976a) and Moharam and Young (1976a) showed 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 the hologram b e i n g w r i t t e n . In Chapter 6, a model f o r hologram w r i t i n g w i t h an a r b i t r a r y o n e - d i m e n s i o n a l l i g h t i n t e n s i t y 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 d eveloped. T h i s model a l l o w s f o r the feedback e f f e c t of the space charge f i e l d on the 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 a p p l i e s f o r a l l w r i t i n g t i m e s . I t i s shown t h a t an i n c r e a s e i n the f r a c t i o n o f the c r y s t a l which i s i l l u m i n a t e d improves the hologram w r i t i n g p r o c e s s due to the r e d u c t i o n o f t h e l a r g e s c a l 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 of the e f f e c t o f the l a r g e s c a l e f i e l d on hologram s t o r a g e a r e r e p o r t e d i n Chapter 8 and a t e c h n i q u e i s d e s c r i b e d to improve the m u l t i p l e hologram s t o r a g e a b i l i t y o f the p h o t o r e f r a c t i v e m a t e r i a l . A l t h o u g h c r y s t a l s can be p r e p a r e d i n which e r a s u r e takes much l o n g e r time than w r i t i n g ( S t a e b l e r and P h i l l i p s 1974a, Alphonse and P h i l l i p s 1976), c o n t i n u e d r e a d out o f a hologram w i l l e v e n t u a l l y e r a s e i t as does th e r m a l r e l a x a t i o n of the e l e c t r o n i c space charge p a t t e r n . I t i s n o t p o s s i b l e to use f o r r e a d ; o u t a d i f f e r e n t wavelength to which the m a t e r i a l i s i n s e n s i t i v e , s i n c e not a l l f r e q u e n c y components of the hologram w i l l s a t i s f y the Bragg c o n d i t i o n s f o r r e c o n s t r u c t i o n . In Chapter 9, we r e v i e w the f i x i n g t e c h n i q u e s proposed f o r LiNbO^ and d e v e l o p a m a t h e m a t i c a l model which, we b e l i e v e , e x p l a i n 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 o f f i x i n g i n LiNbO^ • T h i s model r e l a t e s the dependence of the r e t r i e v e d hologram a f t e r f i x i n g 5. to m a t e r i a l and r e c o r d i n g parameters such as the \ b u l k p h o t o v o l t a i c e f f e c t , a p p l i e d v o l t a g e , l i g h t i n t e n s i t y l e v e l , i r o n d o p i n g and the 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 dopant i r o n . One problem encountered d u r i n g working w i t h l i t h i u m n i o b a t e i s 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 and beam d i s t o r t i o n ( P h i l l i p s e t a l . 1972). T h i s i s the same phenomenon t h a t l e d to the d i s c o v e r y o f the p h o t o r e f r a c t i v e e f f e c t i n some f e r r o e l e c t r i c m a t e r i a l s as was mentioned e a r l i e r . The s c a t t e r -i n g n o i s e appears b o t h d u r i n g the r e c o r d i n g o f h i g h e f f i c i e n c y holograms and d u r i n g prolonged.-read out.of holograms a t any e f f i c i e n c y . C l e a r l y , t h i s n o i s e l i m i t s the maximum u s e f u l e f f i c i e n c y , i f h i g h e f f i c i e n c y i s d e s i r e d , and the maximum number of r e a d o u t s b e f o r e the s i g n a l - t o - n o i s e r a t i o r e a c h e s u n a c c e p t -a b l e v a l u e s . The i n d u c e d n o i s e s h a r e s common f e a t u r e s w i t h s t o r e d holograms ( e . g . ; a n g u l a r and wavelength s e n s i t i v i t y , dependence on the p o l a r i z a t i o n • . of l i g h t and the need f o r c o h e r e n t r a d i a t i o n to g e n e r a t e i t ) . T h i s i s what l e d P h i l l i p s e t a l . to c o n c l u d e t h a t s c a t t e r i n g was caused by i n t e r f e r e n c e f r i n g e s p r e s e n t i n the l i g h t beam. In Chapter 10, we d i s c u s s our 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 t h e s c a t t e r i n g phenomenon and propose a model to e x p l a i n i t . T h i s model i s based on the l e n s - l i k e a c t i o n o f the i n d u c e d l a r g e s c a l e i n d e x i n h o m g e n i e t i e s due to the n o n u n i f o r m i t y o f the i n c i d e n t i l l u m i n a t i o n . Ray-t r a c i n g c a l c u l a t i o n s o f the r e f r a c t e d l i g h t w i t h i n t h e c r y s t a l c o n f i r m e d t h a t l i g h t i n t e r f e r e s w i t h i n the c r y s t a l g i v i n g r i s e t o p a r a s i t i c holograms (Magnusson and G a y l o r d 1974) which are r e s p o n s i b l e f o r the s c a t t e r e d n o i s e . In a d d i t i o n to the b e n e f i t s of u s i n g p h o t o r e f r a c t i v e m a t e r i a l s as the s t o r a g e media, t h e r e a r e advantages i n u s i n g volume holograms as compared t o t h i n holograms. F i r s t , 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 a r e a t t a i n a b l e t h a t improve the s i g n a l - t o - n o i s e r a t i o a t read{.out. Second,, 6. s e v e r a l holograms can be r e c o r d e d w i t h o u t c r o s s - t a l k between them. T h i r d , volume holograms do n o t produce a c o n j u g a t e image a t the o u t p u t . Because of t h e s e advantages, s e v e r a l a p p l i c a t i o n s were su g g e s t e d and even implemented i n p r o t o t y p e models. By f a r t h e most i n v e s t i g a t e d a p p l i c a t i o n i s f o r a l a r g e c a p a c i t y o p t i c a l r e a d / w r i t e memory. A c c e s s to d a t a would be on a page-by-page b a s i s w i t h I O 4 b i t s ( d ' A u r i a e t a l . 1974, Takeda 1972a) r e a d o r w r i t t e n i n p a r a l l e l u s i n g a p h o t o d e t e c t o r a r r a y o r a page-composer, r e s p e c t i v e l y . A s t o r a g e c a p a c i t y o f 1 0 7 bits/mm 2 i n 5 mm t h i c k c r y s t a l can be p r o j e c t e d f o r such systems (Kurz 1977a). C a r l s e n (1974) has proposed 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 . In h i s system d a t a b i t s would 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 from t h e computer, thus a l l e v i a t i n g the need f o r a page-composer. A number o f r e v i e w a r t i c l e s a r e a v i a l a b l e which d i s c u s s the advantages and l i m i t a t i o n s o f o p t i c a l memories (Rajchman 1970a, 1970b, Anderson 1968, 1972, H i l l 1972, Vander L u g t 1973, K i e m l e 1974, Chen and Zook 1975). L o g i c o p e r a t i o n s by volume h o l o g r a p h y were e x p e r i m e n t a l l y demonstrated by H u i g n a r d e t a l . (1975a, 1976) and were d i s c u s s e d by G a y l o r d e t a l . (1977). The p o t e n t i a l use, however, o f volume h o l o g r a p h y i n t h e f i e l d o f d a t a p r o c e s s i n g , e.g.; matched f i l t e r i n g , has h a r d l y been i n v e s t i g a t e d . T h i s i s i n s p i t e o f t h e known advantages a volume hologram p o s s e s s e s o v e r the w i d e l y used and s t u d i e d t h i n hologram. I b e l i e v e t h a t t h i s i s m a i n l y because no t h e o r e t i c a l t r e a t m e n t f o r t h e use o f volume holograms i n d a t a p r o c e s s i n g has been d e v e l o p e d . In C h a p t e r 11, an a n a l y s i s o f t h e behav^ i o u r o f a volume h o l o g r a m employed i n the Vander Lugt f i l t e r o r s i m i l a r d e v i c e s i s d e r i v e d . From the a n a l y s i s i t i s shown t h a t r e p l a c e m e n t o f • the t h i n hologram (which i s t r a d i t i o n a l l y employed i n the Vander L u g t 7. f i l t e r ) by a volume hologram changes the response of the f i l t e r . Other p o s s i b l e p r a c t i c a l a p p l i c a t i o n s o f volume holograms i n the f i e l d s o f h o l o g r a p h i c i n t e r f e r o m e t r y and o p t i c a l communications a r e a l s o d i s c u s s e d . 8. .. . I l l PHYSICAL MODELS OF THE PHOTOREFRACTIVE EFFECT I I - l I n t r o d u c t i o n A number o f models have been proposed to e x p l a i n the p h o t o r e f r a c -t i v e e f f e c t . The development of t h e s e models i s o u t l i n e d i n t h i s c h a p t e r . I I - 2 The E l e c t r o - O p t i c N a ture of the P h o t o r e f r a c t i v e E f f e c t Chen, Lamacchia and F r a s e r (1968) found t h a t the p o l a r i z a t i o n o f the w r i t i n g beams was not c r i t i c a l d u r i n g hologram 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 of the hologram 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 , they suggested t h a t the 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 hologram s t o r a g e i n v o l v e s the 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 (see Appendix 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 ) . The d i f f r a c t i o n e f f i c i e n c y a t the i n i t i a l s t a g e s of hologram f o r m a t i o n i s p r o p o r t i o n a l to the square o f the 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 An ( K o g e l n i k 1969). T h i s i m p l i e s t h a t , Chen e t a l . ' s o b s e r v a t i o n s , ( A n Q / A n e ) - 0.3. F o r a m o d u l a t i n g space charge f i e l d a l o n g the o p t i c a x i s of the c r y s t a l ( c - a x i s ) , the r a t i o o f the change i n the o r d i n a r y r e f r a c t i v e i n d e x to the change i n the 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 A=632.8 nm) 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 space charge f i e l d a l o n g the c - a x i s o f the c r y s t a l . I I - 3 The I n t e r n a l F i e l d Model Chen (1969) measured the changes i n b i r e f r i n g e n c e ( n e - n Q ) i n d u c e d w i t h a s i n g l e l a s e r beam 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 . 2.1 (a) shows the 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 9 . ( b ) F i g . 2.1 (a) The s o l i d l i n e ( ) shows the 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 t h e dashed l i n e ( - - - - ) shows the change a l o n g the b ~ a x i s due to 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 which causes t h e o b s e r v e d change i n A ( n o _ n „ ) -10. a l o n g the b- and c - a x i s o f the 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 the 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 the b - a x i s does n o t . To e x p l a i n t h i s o b s e r v a t i o n , Chen proposed a model i n which t h e r e a r e two t y p e s of 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 . Traps o f the f i r s t type 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 they 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 . Traps of the second type a r e i n i t i a l l y empty and can 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 was an i n t e r n a l e l e c t r i c f i e l d , E g , i n the d i r e c t i o n from the p o s i t i v e end o f spontaneous p o l a r i z a t i o n o f the c r y s t a l to the n e g a t i v e end, t h a t i s , a n t i p a r a l l e l t o the spontaneous p o l a r i z a t i o n . The d i r e c t i o n o f the f i e l d was determined 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 the c r y s t a l . One d i r e c t i o n o f the f i e l d r e t a r d e d the change i n b i r e f r i n g e n c e w h i l e the o t h e r d i r e c t i o n enhanced the change. The f i e l d would cause the p h o t o e x c i t e d e l e c t r o n s to d r i f t toward the p o s i t i v e end o f the c - a x i s ( p o s i t i v e end o f the c - a x i s i s d e f i n e d as the one which becomes p o s i t i v e on c o o l i n g the c r y s t a l ) l e a v i n g b e h i n d p o s i t i v e charges o f i o n i z e d t r a p c e n t e r s . Chen c l a i m e d t h a t "the 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 out o f the t r a p s u n t i l they e v e n t u a l l y d r i f t out o f the 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 the 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 , the t h e r m a l e x c i t a t i o n i s too weak to r e - e x c i t e charges out o f the t r a p s , the n e g a t i v e charges s t a y t r a p p e d t h e r e . The space charge f i e l d E g thus c r e a t e d between the t r a p p e d e l e c t r o n s and the p o s i t i v e i o n i z e d c e n t e r s causes the 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 the i n d i c e s of r e -f r a c t i o n v i a the e l e c t r o - o p t i c e f f e c t " . ( C h e n 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 , the 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 to the s p a t i a l v a r i a t i o n o f the 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 the space charge f i e l d r e q u i r e d to produce the maximum observed v a l u e o f A ( n e - n Q ) was 67 kVcm 11. A l t h o u g h the o r i g i n o f t h i s d r i f t f i e l d was n o t c o m p l e t e l y under-s t o o d , Chen e t a l . (1968) p o s t u l a t e d t h a t i t i s l i k e l y t o be due t o i n c o m p l e t e compensation of s u r f a c e charges i n t h e s e s t r o n g l y p o l a r i z e d c r y s t a l s . To v e r i f y t h a t t h e r e was an i n t e r n a l f i e l d r e s i d i n g i n the c r y s t a l , Chen (1969) l o o k e d f o r and found a s h o r t - c i r c u i t p h o t o c u r r e n t . 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 g . I t was suggested t h a t the f i e l d might be due to - nonuniform h e a t i n g of the c r y s t a l by a p o r t i o n o f the l i g h t used to form the hologram. However, Chen p o i n t e d out t h a t (dP s/dT < 0) and t h e r e f o r e the f i e l d would be i n the wrong d i r e c t i o n f o r t h i s o b s e r v a t i o n . I I - ^ J o h n s t o n ' s P o l a r i z a t i o n Model To remove the need to 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) proposed an a l t e r n a t i v e model i n which p h o t o i n d u c e d v a r i a -t i o n s i n the m a c r o s c o p i c p o l a r i z a t i o n caused the 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 the c r y s t a l would e x c i t e e l e c t r o n s i n the J 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 the d e n s i t y of f i l l e d t r a p s i n the 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 would cause a l o c a l change i n the 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 the p o l a r i z a t i o n produces a f i e l d . T h i s f i e l d causes the e l e c t r o n s to d r i f t b e f o r e r e t r a p p i n g and thus produces permanent change i n the p o l a r i z a t i o n . A f t e r the l i g h t i s t u r n e d o f f , t h e r e remains a change i n the m a c r o s c o p i c p o l a r i z a t i o n which i n d u c e s a change i n the r e f r a c t i v e i n d i c e s of the c r y s t a l . U s i n g t h i s model, J o h n s t o n was a b l e to q u a l i t a t i v e l y account f o r the 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 . 2.1). However, Amodei (1971b) and Amodei and S t a e b l e r (1972a) have shown t h a t t h e r e a r e a 12. 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 a v e r y l a r g e number of e l e c t r o n s would be r e q u i r e d t o e n t e r the c o n d u c t i o n band to g e n e r a t e the l a r g e f i e l d s n e c e s s a r y to account f o r the observed e f f e c t . The same magnitude of i n d u c e d i n d e x change c o u l d r e s u l t from space charge f i e l d s g e n e r a t e d through 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 3 e l e c t r o n s i n v o l v e d would be l e s s by a f a c t o r 10 than would be r e q u i r e d by Johnston's model. O b s e r v a t i o n , o f m a c r o s c o p i c charge s e p a r a t i o n i n LiNbOg a f t e r o p t i c a l i r r a d i a t i o n , as determined by d e t e c t i n g the accompanying e l e c -t r o s t a t i c p o t e n t i a l , i s i n c o n s i s t e n t w i t h J o h n s t o n ' s i d e a s ( S c h e i n e t a l . 1977). One might 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 the c r y s t a l f u l l y i l l u m i n a t e d . II-?5 The P y r o e l e c t r i c F i e l d Model Amodei and S t a e b l e r (1972a) su g g e s t e d t h a t the b u i l t - i n f i e l d which Chen used to e x p l a i n h i s r e s u l t s was o f p y r o e l e c t r i c o r i g i n and d e v e l o p e d when the c r y s t a l was c o o l e d from a h i g h temperature. The development of such a f i e l d may be e x p l a i n e d i n the 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 charges and no n e t space charges would have a f i e l d c o r r e s p o n d i n g to a p o l a r i z a t i o n charge '|P~S| per u n i t a r e a on f a c e s normal to the c - a x i s . T h i s f i e l d would, i n f a c t , be above normal d i e l e c t r i c breakdown. In p r a c t i c e , the c r y s t a l would have been c o o l e d from some h i g h temperature a t which the i o n i c c o n d u c t i v i t y was h i g h enough f o r the r e s u l t i n g p y r o e l e c t r i c f i e l d s t o r e l a x i n a m a t t e r o f minutes. When the c r y s t a l i s c o o l e d , however, the c o n d u c t i v i t y drops r a p i d l y . E x c e s s charges would accumulate c l o s e t o each c - f a c e as the spontaneous p o l a r i z a -t i o n P s c o n t i n u e d to change. F i n a l l y , an uncompensated component o f P, would 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 — s 1 Fl 8P« 1 s - dT, where T T 3T ' 0 T 0 13. and T 1 are the temperature a t which the c o n d u c t i v i t y d i s a p p e a r s and the i. . temperature of the experiment r e s p e c t i v e l y , and e i s the p e r m i t t i v i t y . However,. C o r n i s h e t a l . (1976a) have measured the 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 a s h o r t - c i r c u i t a p p l i e d to the e l e c t r o d e s on the f a c e s a t the ends of the c - a x i s . They found t h a t the p h o t o c u r r e n t was independent o f the 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 the p y r o e l e c t r i c c u r r e n t due to l i g h t a b s o r p t i o n has decayed. T h e i r experiments showed t h a t these 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 produced i n the absence of any b u i l t - i n f i e l d of p y r o e l e c t r i c o r i g i n . G l a s s , von der 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 hours of c o n t i n u o u s i l l u m i n a t i o n , the 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 which they measured .remained c o n s t a n t . They 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 would r e l a x any i n t e r n a l f i e l d s and a decay of the p h o t o c u r r e n t ( i f i t i s due to such f i e l d s ) would be n o t i c e a b l e . I I - 6 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 wave-l e n g t h 400 to 500 nm i s used ( S e r r e z and G o l d n e r 1973). C l a r k e t a l . (1973) suggested t h a t e x c i t a t i o n o c c u r s from t r a p s w i t h i n the 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 the n o n - s t o i c h i o m e t r y of the 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 n o m i n a l l y undoped LiNbO^ i n c r e a s e s the p h o t o c o n d u 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 the 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 which 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 elements such as i r o n , manganese, copper, rhodium and uranium ( 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, 14. 1973, Mickami e t a l . 1973, G l a s s e t a l . 1974) a l s o improves the 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 the 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 , the b e s t dopant t h a t has been found. 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 suggested t h a t i r o n r e p l a c e s l i t h i u m i n the c r y s t a l l a t t i c e . Mossb.auer 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-j l e d Keune e t a l . (1975) to suggest t h a t the most l i k e l y s i t e o f the 3+ Fe i o n i s the Nb s i t e . They were unable to i d e n t i f y the most l i k e l y s i t e 2+ of the Fe i o n . D i s c h l e r and Rauber (1975) have suggested t h a t the e f f i c i e n t ^ 2+ . , . , 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 the t r i v a l e n t s t a t e (Fe ) a r e the empty t r a p s ( 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 the same 2+ i o n i n a reduced 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 i n t r o d u c e s . . 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 of 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 the 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 which, i n t u r n , depends on the 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 the 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 (1974a,b) have shown t h a t the 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 the 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 the 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 d a -2+ t i o n o f Fe i o n to the t r i v a l e n t s t a t e can be a c h i e v e d by a n n e a l i n g the c r y s t a l i n a i r or oxygen a t 600°C ( P e t e r s o n e t a l . 1971). Smith e t a l . (1968) found t h a t f i e l d a n n e a l i n g of l i t h i u m n i o b a t e made the c r y s t a l almost i n s e n -s i t i v e to l i g h t . The a p p l i e d f i e l d caused the i r o n i o n s to m i g r a t e toward the n e g a t i v e e l e c t r o d e . A yellow-brown d e p o s i t e e v e n t u a l l y appeared on the 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 out o f the c r y s t a l . C l a r k e t a l . (1973) 3+ 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 or oxygen c o n v e r t e d about 96% o f i r o n t o Fe 15. 3+ Methods t o reduce Fe to the d i v a l e n t s t a t e i n c l u d e h e a t i n g the c r y s t a l i n argon atmosphere a t around 1000°C and h e a t i n g the c r y s t a l 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 such as I^CO-} and around 500°C f o r 40 hours ( P h i l l i p s and S t a e b l e r 1974). R e d u c t i o n o f more than 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 . L a t e l y , Holman e t a l . (1978) suggested a new c h e m i c a l treatment t h a t s i g n i f i c a n t l y a f f e c t s the s u s c e p t i b i l i t y o f LiNbOg t o l i g h t i n d u c e d i n d e x changes. The treatment i n v o l v e s h e a t i n g the c r y s t a l below i t s m e l t i n g temperature i n c l o s e p r o x i m i t y w i t h another s o l i d o f mixed l i t h i u m n i o b a t e phases, the mass o f which c o n s i d e r a b l y exceeds t h a t o f the c r y s t a l . The c o m p o s i t i o n o r s t o i c h i o m e t r y o f the LiNbO^ c r y s t a l i s a l t e r e d as a r e s u l t o f a l i t h i u m o x i d e t r a n s p o r t t h a t o c c u r s between the two s o l i d s . The s u s c e p t i b i l i t y was g r e a t l y reduced by i n c r e a s i n g the n o n - s t o i c h i o m e t r y ( l i t h i u m o x i d e o u t d i f f u s i o n ) and enhanced by d e c r e a s i n g the n o n - s t o i c h i o m e t r y ( l i t h i u m o x i d e i n d i f f u s i o n ) . II-7 F o r m a t i o n o f Holograms by D r i f t or D i f f u s i o n I I - 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 the phenomenon o f hologram 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 to a n a l y z e the f o r m a t i o n o f a p r o t o t y p e hologram formed by c a u s i n g two c o h e r e n t , monochromatic, i n f i n i t e p l a n e waves to i n t e r f e r e i n the volume o f the c r y s t a l . As shown i n F i g . 2.2, the two waves are s y m m e t r i c a l l y i n c i d e n t on the c r y s t a l and i n t e r s e c t a t an a n g l e 2 0 o . The two waves a r e i n the y - p l a n e (which c o n t a i n s a l s o the c r y s t a l c - a x i s ) . The two p l a n e waves a r e commonly c a l l e d the r e f e r e n c e wave R and the s u b j e c t or s i g n a l wave S. They may be r e p r e s e n t e d by 16. 17. R = r e x p - i (cot - p . r ) , (2.1) S = s e x p - i (cot - o . r ) , (2.2) where r and s a r e the complex a m p l i t u d e s of the two waves. p and a a r e the wavevectors o f the 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 , co i s t h e r a d i a n f r e q u e n c y of l i g h t and r = ( x , y , z ) . The i n t e n s i t y of the i n t e r f e r e n c e p a t t e r n o f the two waves i s * ,- -> 2 I = |R + S| (2.3) s u b s t i t u t i n g Eqs. 2.1 and 2.2 i n 2.3 and c o n s i d e r i n g the c o o r d i n a t e system o f F i g . 2.2, we a r r i v e a t I(.x) = 1 (1 + m cos.Kx) (2.4) o ••• i - i 2 ' i - i 2  where I . => r + s i s the average l i g h t i n t e n s i t y and m=2r.s/I i s the m o d u l a t i o n r a t i o . K=2Tr/A=4iTsin9 0/X Q i s the g r a t i n g v e c t o r w i t h A the g r a t i n g s p a c i n g and XQ the vacuum wavelength of l i g h t . 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 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 than 1600 l i n e p a i r s mm ~. T h i s l e d them to assume t h a t the d i s p l a c e m e n t o f e l e c t r o n s must be a s m a l l f r a c t i o n o f a m i c r o n to be a b l e to r e c o r d the v a r i a t i o n i n the l i g h t 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 . (1977a) e s t i m a t e d , from p h o t o c u r r e n t measurement, t h a t . o the t r a n s p o r t l e n g t h of the e l e c t r o n s was o f the o r d e r 0.8 and 2.4 A 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 out t h a t , i f the quantum e f f i c i e n c y was l e s s than u n i t y , the d i s p l a c e m e n t would i n c r e a s e . R e c e n t l y , Young e t a l . (1979) e s t i m a 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 of the e l e c t r o n s was about 24 nm ( t h i s w i l l be d i s c u s s e d i n Chapter 5 ) . i n h o l o g r a p h y i t i s the custom to d e f i n e the i n t e n s i t y as i n Eq. 2.3 and omit the f a c t o r (1/2R) on the r i g h t hand s i d e of the same e q u a t i o n where R i s the c h a r a c t e r i s t i c impedence of the medium. 18. T h i s i s s t i l l a s m a l l f r a c t i o n o f the g r a t i n g s p a c i n g A ( f o r most p r a c t i c a l eases A ranges from 0.5 to 5 i_m). As w i l l be d i s c u s s e d l a t e r i n t h i s c h a p t e r , S t a e b l e r and Amodei (1972a) o b s e r v e d energy t r a n s f e r between the r e f e r e n c e and s u b j e c t waves d u r i n g hologram 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 (1971b) 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 beam cou-p l i n g i s caused o n l y by the d i f f u s i o n mechanism. Young e t a l . (1974) devel o p e d a model w i t h o u t the 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 . account f o r the energy t r a n s f e r between the two beams even when d r i f t i s the dominant mechanism. S t u d i e s o f the dependence o f the 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 the g r a t i n g p e r i o d (Alphonse e t a l . 1975) were taken as t e n d i n g t o su p p o r t Young e t a l . ' s (1974) model s u g g e s t i n g t h a t the 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 the 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 . L a t e r , i n Cha p t e r s 3, 4 and 5, we w i l l d e v e l op a m a t h e m a t i c a l model and g i v 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 r e l a t e d to the e l e c t r o n t r a n s p o r t l e n g t h . 11-7.2 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 (1971b) assumed t h a t the r a t e o f g e n e r a t i o n o f 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 the l i g h t i n t e n s i t y , Eq. 2.4. The assumption t h a t the t r a n s p o r t l e n g t h o f the 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 than the g r a t i n g s p a c i n g , A, i m p l i e s t h a t the f r e e c a r r i e r c o n c e n t r a t i o n would remain a f a i t h f u l r e p l i c a o f the g e n e r a t i o n r a t e and would be g i v e n by n(x)= g 0 x (1 + m cos K x ) , (2.5) where x i s the c a r r i e r l i f e t i m e and i s assumed t o be independent o f the 19. c a r r i e r c o n c e n t r a t i o n . g D i s the average volume 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 to I Q . The s p a t i a l d i s t r i b u t i o n o f 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 was taken as the sum of d r i f t and d i f f u s i o n components J ( x , t ) = q u n ( x ) E , ( x , t ) ,+ qD , (2.6) where q i s the e l e c t r o n i c c h a r g e , D i s the d i f f u s i v i t y , y i s the m o b i l i t y and E ( x , t ) i s the t o t a l e l e c t r o s t a t i c f i e l d . The r a t e a t which the charge d e n s i t y P g c accumulates a t any p o i n t i s g i v e n by the c o n t i n u i t y e q u a t i o n 9 p s c _ _ 3 J ( x , t ) _ ( 2 > 7 ) oat 3x The space charge f i e l d s u p p o r t e d by the space charge d e n s i t y i s E s c Psc ^ dx . (2.8) Amodei assumed t h a t , i n the l i n e a r i n i t i a l s t a g e s of hologram f o r m a t i o n , the space charge f i e l d may be n e g l e c t e d i n the t r a n s p o r t e q u a t i o n , Eq. 2.6. He c o n s i d e r e d (a) t r a n s p o r t due to d r i f t o n l y J=-qunE a (where E a i s the e l e c t r o s t a t i c f i e l d due to an a p p l i e d v o l t a g e V a w i t h the p o l a r i t y i n d i c a t e d i n F i g . 2.2); and (b) t r a n s p o r t due to 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. 2.7 and 2.8 y i e l d s f o r d r i f t o n l y E s c = (Wig0/e) t. m E a cos Kx, (2.9) and f o r d i f f u s i o n o n l y E g c = (quxg 0/e) t m s i n Kx, (2.10) where E D = (KkT'/q) i s the 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 i s Boltzmann's 20. c o n s t a n t and T ! i s the a b s o l u t e temperature. Thus, w i t h the assumption of s h o r t t r a n s p o r t l e n g t h , a space charge f i e l d i s o b t a i n e d w i t h a phase 0 o r TT/2 a c c o r d i n g t o whether d r i f t o r d i f f u s i o n i s o p e r a t i v e . II-7.3 Young e t a l . ' s A r b i t r a r y T r a n s p o r t Length Model Young e t a l . (1974) removed the need f o r the assumption t h a t the t r a n s p o r t l e n g t h be s h o r t compared to the g r a t i n g s p a c i n g by u s i n g the c o n t i n u i t y e q u a t i o n £ - g o ( 1 + m cos Rx) • (2.1D In the i n i t i a l l i n e a r s t a g e s of hologram f o r m a t i o n , the r a t e of change of the 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 band 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 n . 1 8J q 3x 0 = g Q (1 + m cos Kx) - 2. + ± ^ . (2.12) Amodei's (1971b) assumption t h a t n = g T (1+ m cos Kx) (Eq. 2.5) o 1 9 J c o r r e s p o n d s to d r o p p i n g the term — — i n Eq. 2.12 which i s r e s p o n s i b l e f o r the change i n the space charge and hence f o r th e f o r m a t i o n o f the space charge f i e l d c o n s t i t u t i n g the hologram. 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 the space charge f i e l d i n the i n i t i a l s t a g e s o f hologram f o r m a t i o n f o r (a) t r a n s p o r t due to d r i f t o n l y ; 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 . S o l u t i o n o f Eqs. 2.7, 2.8 and 2..12 y i e l d s f o r d r i f t o n l y tm E a E = (qjixg /e) c o s (K^ -*) > (2.13) S C ° [ I + ( K L J 2 ] * 21. where <f> = t a n K L a , and f o r d i f f u s i o n o n l y tm Erj E a „ = ( q y x g 0 / e ) - — — s i n Kx, S l + ( K L d ) 2 where L a = y x E a i s the d r i f t t r a n s p o r t l e n g t h and t r a n s p o r t l e n g t h . II-7.4 D i s c u s s i o n Eq. 2.13 shows t h a t , f o r space charge f i e l d d e v e l o p e d by d r i f t , t h e r e i s a phase l a g t a n ^ K L a between the space charge f i e l d and the l i g h t i n t e n s i t y p a t t e r n t h a t produced i t . I f KL a<<l, Amodei's e x p r e s s i o n i s o b t a i n e d w i t h E « cos Kx. F o r KL >>1, the f i e l d w i l l be E = (qg t m / e K ) s i n Kx. sc a ' sc , o Here, the phase l a g w i l l be TT/2, and E g c w i l l be independent o f E a ( t h e d r i f t f i e l d ) and x. F o r i n c r e a s e d g r a t i n g s p a t i a l f r e q u e n c y (K l a r g e r ) the f i e l d d e c r e a s e s as 1/K. In the case of d i f f u s i o n - f o r m e d space charge f i e l d , i f KL^>>1, the same l i m i t i n g c ase 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 a>>l. F o r s h o r t d i f f u s i o n l e n g t h (KL CJ<<1), Eq. 2.14 r e d u c e s t o Amodei's e x p r e s s i o n (Eq. 2.11). However, whatever the magnitude of KL<j, the phase s h i f t remains the same because the 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 . The magnitude of L a ( f o r the d r i f t case) to which S t a e b l e r and Amodei's (1972a) r e s u l t s on c o u p l e d wave a n a l y s i s a p p l y a r e c r u c i a l . A phase s h i f t h a l f w a y between the two l i m i t i n g cases i s a c h i e v e d when K L a = l . In t h i s c a s e , the magnitude of the f i e l d w i l l be 3 dB below i t s v a l u e f o r s h o r t t r a n s p o r t l e n g t h . The d r i f t l e n g t h i n t h i s c ase would be e q u a l 154 nm -11 2 -1 ( f o r \ =.5u and 9 = 1 5 ° ) . T a k i n g yx=8. x 10 cm V (from r e s u l t s o f Chapter 5 ) , (2.14) i = ( x D ) 2 i s the d i f f u s i o n 22. the v a l u e o f the e l e c t r i c f i e l d r e q u i r e d to a c h i e v e K L a = l i s 200 kV cm F i e l d s below t h i s v a l u e would produce c o r r e s p o n d i n g l y s m a l l e r phase s h i f t s . I I - 8 E f f e c t s o f Beam C o u p l i n g II-8.1 I n t r o d u c t i o n C o n s i d e r the s i t u a t i o n 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 ± 0 O r e l a t i v e t o the z - a x i s ( F i g . 2.2) 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 ( S t a e b l e r and Amodei 1972a) An = n 1 cos Kx, (2.15) t h a t extends from z=0 to z=D. Both waves a r e p o l a r i z e d normal to the p l a n e of i n c i d e n c e . The g r a t i n g i s assumed to be u n i f o r m throughout the t h i c k n e s s of the 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 p e r f e c t Bragg c o n d i t i o n s p r e v a i l , the two waves R and S may be r e p r e s e n t e d by R = r ( z ) exp ( i p . r ) , (2.16) S = s ( z ) exp ( i a . r ) , (2.17) w i t h p and a, p r e v i o u s l y d e f i n e d , a r e g i v e n by p = ( 2 i m 0 A 0 ) ( - s i n 0 , p , cosO ) , (2.18) a = ( 2 7 m 0 / A 0 ) ( s i n 6 , 0 - , cos0 ) , (2.19) where n Q i s the average index of r e f r a c t i o n o f the medium and 0 i s the i n c i d e n c e a n g l e , due t o r e f r a c t i o n , i n the medium. S i n c e Bragg c o n d i t i o n s p r e v a i l , we have K=4irsin0_/A.^=2iT/A. 23. K o g e l n i k (1969) (see Appendix D) u s i n g c o u p l e d wave t h e o r y has shown t h a t i n the 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 Bragg c o n d i t i o n s d r ( z ) • / \/ Q — T ~ = - i Y s ( z ) / c o s 6 , dz (2.20) d s ( z ) • f \ I a , = -x Y r ( z ) / c o s 6 , dz where y ^ n A 0 ^ s t ' i e c o u p l i n g c o n s t a n t . S o l u t i o n of the c o u p l e d wave e q u a t i o n s , Eqs. 2.20, f o r the g e n e r a l boundary c o n d i t i o n s r(0)=A, S(0)=B exp-i<j i s • r ( z ) = A cos6z - i B exp (-i<(>) s i n 6z, (2.21) s ( z ) = B exp(-ic(>) cos 6z - i A s i n 6z, where 6= y / c o s 8 . 2 I = | r ( z ) | = A 2 c o s 2 6 z + B 2 c o s 2 6 z 4- AB s i n 2 6 z sintj), I = | s (z) | 2 = A 2 s i n 2 6 z + B 2 c o s 2 6 z + AB sin26z.'sincj). I f the two beams have e q u a l i n c i d e n t a m p l i t u d e s (A=B=1), then Eqs. 2.22 g i v e I r = 1 - s i n 26z sintj), (2.22) (2.23) 1 = 1 + s i n 26z sin<j). S t a e b l e r and Amodei (1972a) have shown t h a t t h e s e e q u a t i o n s a l s o d e s c r i b e the s i t u a t i o n where the two i n c i d e n t waves a r e i n phase but the g r a t i n g i s moveable a l o n g the x - a x i s w i t h An= n^cos(Kx-<|>) . Eqs. 2.22 and 2.23 now i n d i c a t e 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 the g r a t i n g r e s u l t s i n energy t r a n s f e r between the two beams. The amount and d i r e c t i o n o f energy t r a n s f e r depend on the a m p l i t u d e and s p a t i a l phase 24. of the g r a t i n g . The phase f a c t o r cj> r e p r e s e n t s the s p a t i a l phase s h i f t between the l i g h t i n t e n s i t y p a t t e r n and the in d e x m o d u l a t i o n i t pro d u c e s . S t a e b l e r and Amodei p o i n t e d o ut t h a t t h i s . x 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 the r e l a t i v e importance of the d i f f e r e n t mechanisms proposed f o r s t o r i n g phase holograms by the p h o t o r e f r a c t i v e e f f e c t . For example, they found t h a t , d u r i n g hologram f o r m a t i o n , t h e r e i s energy t r a n s f e r between the two beams. S i n c e the g r a t i n g produced by d r i f t o n l y has 0 or TT phase s h i f t (Eq. 2.9), f o r t h i s c a s e Eq. 2.23 p r e d i c t s no energy t r a n s -f e r , and they c o n c l u d e d t h a t the holograms were s t o r e d by d i f f u s i o n . II-8.2 C o u p l i n g D u r i n g W r i t i n g U s i n g the e x p e r i m e n t a l arrangement shown i n F i g . 2.3, S t a e b l e r and Amodei found t h a t d u r i n g hologram f o r m a t i o n t h e r e was energy t r a n s f e r between t h e two beams. From Amodei's (1971b) a n a l y s i s o f the t r a n s p o r t p r o c e s s d u r i n g r e c o r d i n g ( S e c t i o n I I - 7 . 2 ) , holograms formed by d r i f t produce a phase s h i f t 0 or TT. Eq. 2.23 i n d i c a t e s t h a t o n l y v a l u e s o f tj) o t h e r than 0 o r IT cause energy t r a n s f e r . T h i s was t a k e n as e v i d e n c e o f hologram s t o r a g e by d i f f u s i o n . I I - 8 . 3 C o u p l i n g D u r i n g Read Out When a p r e v i o u s l y r e c o r d e d hologram i s r e a d o u t , i t i s p o s s i b l e f o r a new hologram t o be w r i t t e n by the i n t e r f e r e n c e o f the r e a d i n g beam and t h e d i f f r a c t e d beam. There a r e two p o s s i b i l i t i e s f o r r e a d o u t , u s i n g the R beam or u s i n g S beam ( F i g . 2.3). Below, we t r e a t each case s e p a r a t e l y to see the i n f l u e n c e o f the r e a d out beam on the c o u p l i n g . 25. ARGON L A S E R (48 8 nm ) C R Y S T A L F i g . 2.3 E x p e r i m e n t a l a p p a r a t u s f o r r e c o r d i n g s i m p l e phase g r a t i n g s i n LiNbC- . 26. II-8.3.A R Beam Read Out In t h i s c a s e , the S beam i s b l o c k e d so t h a t t h e boundary c o n d i t i o n s a r e s(0) = 0 and r ( 0 ) = 1. From Eq. 2.21, t h e am p l i t u d e s o f the two beams i n s i d e the d i f f r a c t i o n g r a t i n g An = rijCos Kx a r e g i v e n by r ( z ) = cosSz , (2.24) s ( z ) = - i ; s i n f i z . The i n t e r f e r e n c e p a t t e r n produced by t h e s e two beams i s g i v e n by s u b s t i t u t i n g Eq. 2.24 i n t o Eqs. 2.16 and 2.17 w i t h the r e s u l t 2 I ( x , y ) = |r+s| = 1 - s i n 2 6 z . s i n Kx. (2.25) I f the mechanism o f hologram f o r m a t i o n produces an i n d e x m o d u l a t i o n due to d r i f t (Eq. 2.13) then the new g r a t i n g An£ w i l l be A n 2 = n 2 s i n (Kx-cf> a) , (2.26) where <j>a=tan ^KL a. Note h e r e t h a t , f o r t h e geometry o f Fig._,2.2 and from the e l e c t r o - o p t i c t e n s o r o f LiNbOg, t h a t An^ - E • The t o t a l phase modula-t i o n w i l l be An f c = n ^ o s Kx + n 2 s i n (Kx-(f> a) • (2.28) S t a e b l e r and Amodei argued t h a t the e f f e c t o f A n 2 was to bend the phase g r a t i n g . The e f f e c t on the t o t a l d i f f r a c t i o n e f f i c i e n c y was n o t pursued. In the case where the hologram f o r m a t i o n produces an i n d e x m o d u l a t i o n due to d i f f u s i o n (Eq. 2.10 o r 2.14), the new g r a t i n g A n 3 w i l l be A n 3 = - n 3 c o s Kx , (2.29). and the t o t a l phase m o d u l a t i o n w i l l be An = ( n 1 - n^) cos Kx . (2.30) The e f f e c t of n 3 i s to d e c r e a s e the e f f e c t i v e n e s s o f the g r a t i n g ( l e s s e r d i f f r a c t i o n ) and disenhancement o f the d i f f r a c t e d power takes p l a c e . II-8.3.B'.: S Beam Read Out In t h i s case the R beam i s b l o c k e d and t h e new boundary c o n d i t i o n s w i l l be s ( 0 ) = l and r(0)=0. From Eq. 2.21, the am p l i t u d e s o f the two beams w i l l be r ( z ) = - i sincSz, (2.31) s ( z ) = co s 6 z , and the i n t e r f e r e n c e p a t t e r n due to t h e s e two beams w i l l be I ( x , z ) = 1 + s i n 26'z s i n Kx. (2.32) The new g r a t i n g A n 2 ' produced by d r i f t i s A n 2 = -n.2 s i n (Kx -<j>a) , (2.33) and the t o t a l phase m o d u l a t i o n i s A n t = n^cos Kx - n 2 s i n ( K x - i | > a ) . (2.34) The bending o f the phase g r a t i n g w i l l be e q u a l , but i n the o p p o s i t e d i r e c t i o n , t o t h a t produced by the R beam r e a d o u t , Eq. 2.28. i The new g r a t i n g An^ produced by d i f f u s i o n w i l l be An 3 = + n 3 c o s Kx, (2.35) and the n e t phase m o d u l a t i o n i s 28. A n t = ( n 1 + n 3 ) cos Kx. (3.36) The e f f e c t o f 113 i s to i n c r e a s e the e f f e c t i v e n e s s of the g r a t i n g (more d i f f r a c t i o n ) and enhancement o f the d i f f r a c t e d power t a k e s p l a c e . II-8.3.C D i s c u s s i o n The r e s u l t s o f the l a s t two s e c t i o n s a r e b e s t r e p r e s e n t e d g r a p h i -c a l l y f o r d i s c u s s i o n p u r p o s e s . F i g . 2.4 s c h e m a t i c a l l y i l l u s t r a t e s the r e a d o u t / e r a s u r e p r o c e s s . The o r i g i n a l i n d e x m o d u l a t i o n ( o r i g i n a l hologram) i s n^ cos- Kx, r e p r e s e n t e d as a h o r i z o n t a l v e c t o r i n phasor form. The u n i f o r m p a r t of the i l l u m i n a t i o n b o t h f o r on and o f f Bragg a n g l e i l l u m i n a t i o n causes e r a s u r e o f the hologram by amount -An cos Kx which depends on the p h o t o i n d u c e d c o n d u c t i v i t y to r e l a x the space charge (Amodei and S t a e b l e r 1972a, S t a e b l e r and P h i l l i p s 1974a). 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 , which e x i s t s o n l y a t Bragg a n g l e i l l u m i n a t i o n , w r i t e s a new hologram through the d i f f e r e n t mechanisms t h a t c o n t r i b u t e i n hologram f o r m a t i o n . The r a t e a t which t h i s new hologram de v e l o p s depends on the w r i t i n g s e n s i t i v i t y o f the c r y s t a l ( P h i l l i p s e t a l . 1972, Kurz 1976, S t a e b l e r and P h i l l i p s 1974a). F i g . 2.4 i l l u s t r a t e s the combined e f f e c t o f e r a s u r e o f the hologram by the u n i f o r m p a r t o f the i l l u m i n a -t i o n and v e c t o r i a l a d d i t i o n o f the new hologram ( i n case o f Bragg a n g l e i n c i d e n c e ) f o r e r a s u r e by the R o r S waves. There a r e two i m p o r t a n t cases of o p t i c a l e r a s u r e a t the Bragg a n g l e (Moharam and Young 1978a). The f i r s t o c c u r s when the e f f e c t o f the u n i f o r m i l l u m i n a t i o n i s s t r o n g e r than the e f f e c t o f the new hologram w r i t t e n d u r i n g the e r a s u r e p r o c e s s . We would expect an e x p o n e n t i a l - l i k e decay of the space charge f i e l d f o r e r a s u r e w i t h e i t h e r the two beams but w i t h HOLOGRAM BEFORE ERASURE _^ N 1 COS (Kx) HOLOGRAM "ADDED" DURING ERASURE DC ILLUMINATION ___ - AN COS (Kx) SINUSOIDAL COMPONENTS (AT BRAGG ANGLE ONLY) a - DRIFT ± N S l N ( K x - $ ) b - DIFFUSION * N 3 cos(Kx) OFF BRAGG ANGLE ERASURE ERASURE AT BRAGG ANGLE ERASURE S - BEAM ENHANCEMENT R - BEAM DI SENHANCEMENT Fig. 2.4 A schematic representation of a qualitative model for the opt-i c a l erasure process of holograms stored by the photorefractive effect. 30. d i f f e r e n t time c o n s t a n t s , because the r e s u l t o f v e c t o r i a l a d d i t i o n o f the new hologram i s d i f f e r e n t depending on which one o f the two beams i s used ( F i g . 2.2). The second case o f importance i s when the e f f e c t o f the 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 than the e r a s u r e o f the o r i g i n a l hologram due to the u n i f o r m p a r t o f the i l l u m i n a t i o n . T h i s case i s a c h i e v e d when the w r i t i n g s e n s i t i v i t y i s much l a r g e r than the e r a s u r e s e n s i t i v i t y ( S t a e b l e r and P h i l l i p s 1974a) i n a t h i c k c r y s t a l t h a t i s h e a v i l y doped (0.1 m o l e % i r o n ) and l i g h t l y reduced (Alphonse and P h i l l i p s 1976). I t can be deduced from F i g . 2.4, f o r such a c r y s t a l , t h a t the combined e f f e c t o f d r i f t and d i f f u s i o n can be s t r o n g e r than the e r a s u r e due to the u n i f o r m p a r t of the l i g h t , the r e s u l t w i l l be c l e a r enhancement o f the o r i g i n a l hologram f o r r e a d i n g w i t h one beam ( t h e S beam) or c l e a r disenhancement f o r r e a d i n g w i t h the o t h e r beam (R beam). However, the enhancement does not c o n t i n u e i n d e f i n i t e l y . The space charge f i e l d i s e r a s e d a t the f r o n t s u r f a c e o f the hologram s i n c e the 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 p l a n e . The new hologram 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 proceeds i n t o t h e c r y s t a l away from the f r o n t s u r f a c e , and hence does not overcome the e r a s u r e caused by the u n i f o r m i l l u m i n a t i o n . I I T 9 The B u l k P h o t o v o l t a i c E f f e c t II--9 .1 I n t r o d u c t i o n As was mentioned b e f o r e , i l l u m i n a t i o n o f some p y r o - o r f e r r o -e l e c t r i c c r y s t a l s y i e l d s 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 v o l t a i c c u r r e n t s , a l t h o u g h t h e r e a r e no m a c r o s c o p i c e l e c t r i c f i e l d or c o n c e n t r a t i o n g r a d i e n t s p r e s e n t . I n c o n t r a s t to the p h o t o v o l t a i c e f f e c t o c c u r i n g i n p-n j u n c t i o n s , 31. t h i s phenomenon i s a b u l k p r o p e r t y o f the c r y s t a l , which i s d i r e c t l y r e l a t e d to the o p t i c a l e x c i t a t i o n p r o c e s s . Under o p e n - c i r c u i t c o n d i t i o n s anomalously l a r g e p h o t o v o l t a g e s up to s e v e r a l thousand v o l t s have been r e p o r t e d ( G l a s s e t a l . 1974, 1975, Kock e t a l . 1976, Kra'tzig and Kurz 1977b). R e c e n t l y , t h i s anomalous b u l k p h o t o v o l t a i c e f f e c t has r e c e i v e d g r e a t i n t e r e s t s i n c e i t p l a y s an e s s e n t i a l r o l e i n u n d e r s t a n d i n g the s t o r a g e mechanism o f volume phase holograms i n f e r r o e l e c t r i c s (von der L i n d e and G l a s s 1975b). S e v e r a l m i c r o -s c o p i c models e x i s t now e x p l a i n i n g t h i s e f f e c t . I t i s our purpose to r e v i e w t h e s e models. II-9.2 The Asymmetric P h o t o - D e l o c a l i z a t i o n Model ( G l a s s e t a l . 1974) To account f o r the e x i s t e n c e of a p h o t o r e f r a c t i v e p r o c e s s i n i r o n - d o p e d LiNbOg, G l a s s e t a l . (1974) i n t r o d u c e d a b u l k p h o t o v o l t a i c . e f f e c t based on the asymmetry of the l a t t i c e . The b a s i c i d e a s o f the model a r e as f o l l o w s . In a L i N b O ^ F e sample and below i t s C u r i e temperature the c r y s t a l has a unique p o l a r a x i s , and the d i s t a n c e between the i r o n i o n and the n i o b i u m i o n i n the +c and - c d i r e c t i o n a r e d i f f e r e n t . The r e s u l t i n g m i c r o -2+ s c o p i c asymmetry of the p o t e n t i a l a t the Fe i o n s i t e a l l o w s a p r e f e r r e d d i r e c t i o n o f t r a n s f e r o f the e l e c t r o n s f o l l o w i n g o p t i c a l e x c i t a t i o n . The e l e c t r o n i c c o n t r i b u t i o n to the p h o t o c u r r e n t i s due to a l a r g e r o v e r l a p 2+ between the o r b i t a l s o f the Fe i o n w i t h the o r b i t a l s o f t h e n i o b i u m i o n i n the +c d i r e c t i o n than i n the -c d i r e c t i o n . T h i s c u r r e n t component i s J e = (al / t i u > ) (p, I, - p i ) (2.37) e x + + where £ i s tbe quantum e f f i c i e n c y , a i s the a b s o r p t i v i t y , "hco i s the quantum of l i g h t , p + and p_ a r e the p r o b a b i l i t i e s of charge t r a n s f e r i n the +c and -c d i r e c t i o n s ; r e s p e c t i v e l y , and Z+ and %_ are the e l e c t r o n mean f r e e paths a l o n g t h e s e d i r e c t i o n s . 32. F o l l o w i n g t h i s e x c i t a t i o n ( F i g . 2.5) Franck-Condon (F.C.) r e l a x a t i o n o f the i o n s f o l l o w s . The n e t d i s p l a c e m e n t o f the-.ions a l o n g the p o l a r a x i s g i v e s an a d d i t i o n a l c u r r e n t component J = c3(aI/n .oj)(Z 1.A£,) , (2.38) where A£_^  i s the d i s p l a c e m e n t of the i t h i o n o f charge Z_^ , and the p r o d u c t Zn-A£. i s summed over a l l i o n s . A f t e r the e l e c t r o n i s s c a t t e r e d i t s motion becomes i s o t r o p i c and does not c o n t r i b u t e to the p h o t o c u r r e n t u n t i l r e c o m b i n a t i o n ( G l a s s e t . a l . 1975). I f the p r o b a b i l i t i e s o f r e c o m b i n a t i o n a t the i m p u r i t y form the ± p o l a r d i r e c t i o n s a r e p^ _ and p^ then the r e c o m b i n a t i o n c u r r e n t J r i s g i v e n by e q u a t i o n s s i m i l a r t o Eqs. 2.37 and 2.38 w i t h &+P+ r e p l a c e d by £.J_P+ and Z^ r e p l a c e d by z! . Of c o u r s e A&. i s the same a f t e r b o t h e x c i t a t i o n and recom-i i b i n a t i o n s i n c e the i m p u r i t y moves between two l o c a t i o n s . The s t e a d y - s t a t e p h o t o c u r r e n t may then be w r i t t e n ( G l a s s e t a l . 1975) J P = J E - J R = KOI , (2.39) w i t h K=(q5/n;w)(£ +p + - l_ P_ + A|p| - A V - ) + ( S A A ^ M ) (Z±-Z]_) , (2.40) i s a c o n s t a n t depending on the n a t u r e of the a b s o r b i n g c e n t e r , the l o c a l environment and the photon energy. The dependence o f K on t h e s e f a c t o r s have been i n v e s t i g a t e d by K r a t z i g and Kurz (1976, 1977a,b) and Kurz e t a l . (1975, 1977b). 33. .o5- O L i % Fe 6 ELECTRON e C u N b E o 3 O ° O o o . ° o EXCITATION P - P * *- O ° 0*~*T) ° O THERMAL RECOMBINATION F . C . SHI FT F i g . 2.5 The asymmetric p h o t o d e l o c a l i z a t i o n model i n LiNbO.^. 34. I I - 9 .3 The C o l l e c t i v e Franck-Condon R e l a x a t i o n Model (Chanussot and  G l a s s 1976) The p h o t o r e f r a c t i v e e f f e c t e x i s t s , a l s o , i n h i g h p u r i t y c r y s t a l s by v a l e n c e b a n d - t o - c o n d u c t i o n band t r a n s i t i o n s . The f r e e c a r r i e r s were generated by a two-photon p r o c e s s (von der L i n d e e t a l . 1974, 1975a, 1975b and 1976) and by x - r a y i r r a d i a t i o n (Ohmori e t a l . 1977 and von der L i n d e e t a l . 1976). I t was s uggested t h a t a b u l k p h o t o v o l t a i c e f f e c t e x i s t s i n undoped LiNbO-3 c r y s t a l s o f h i g h p u r i t y due to an i n t r i n s i c p r o p e r t y of the i d e a l c r y s t a l (von d e r L i n d e e t a l . 1975b, 1978 and Chanussot and G l a s s 1976). T h i s e f f e c t was assumed t o be caused by Franck-Condon r e l a x a t i o n o f e x c i t e d s t a t e s f o l l o w i n g o p t i c a l e x c i t a t i o n . The p h o t o v o l t a i c c u r r e n t was assumed c to be due t o a c o h e r e n t r e l a x a t i o n o f t h e l a t t i c e . A s i m p l e model demonstrat-i n g the p h y s i c a l b a s i s o f the model i s d e s c r i b e d as f o l l o w s (Chanussot and G l a s s 1976). The l i n e a r atomic model of F i g . 2.6 (a) r e p r e s e n t s the ground s t a t e (G.S.) o f the p o l a r s t r u c t u r e where each n e i g h b o u r i n g a n i o n 0 and c a t i o n B c o n s t i t u t e a u n i t c e l l . Upon o p t i c a l e x c i t a t i o n a charge i s t r a n s -f e r e d from the 0 a n i o n to the n e a r e s t n e i g h b o u r c a t i o n . F i g s . 2.6 (b) and 2.6 (c) and 2.7 show the charge d i s t r i b u t i o n i n the Franck-Condon s t a t e (F.C.S.) and r e l a x e d e x c i t e d s t a t e (R.E.S.) f o l l o w i n g e x c i t a t i o n . D u r i n g the t r a n s i t i o n G.S. - F.C.S. the i o n i c p o s i t i o n s a r e f i x e d . The i o n i c s h i f t i n the .relaxed e x c i t e d - s t a t e , A x , i s t h e • c a t i o n , s h i f t r e l a t i v e to t h e ~ "anion framework. The e x c i t a t i o n thus r e s u l t s i n a c u r r e n t J due to d i r e c -. ' - ee t i o n a l e l e c t r o n i c c h a r g e " t r a n s f e r as w e l l - a s an i o n i c c u r r e n t J ^ e due to r e l a x a t i o n . O A N I O N ( O ) O CATION (B ) ' • ELECTRON e a . G.S. 0 O o o o o b- EXCITED o ® F . C S . o o o o 0 o c. R.E.S . o o O o o o 0 o d- GROUND 0 « 0 F. C.S. o O o o o o e. G.S. o O o . o 0 o o F i g . 2.6 C o l l e c t i v e Franck-Condon Model. Q F i g . 2.7 C o o r d i n a t e c o n f i g u r a t i o n diagram f o r the Franck-Condon r e l a x a t i o n . 36. The r e c o m b i n a t i o n proceeds v i a the ground F.C.S. to the i n i t i a l ground s t a t e as i n F i g s . 2.6 (c) and 2.6 ( e ) . Recombination r e s u l t s i n e l e c t r o n i c and i o n i c c u r r e n t s J and J . . er i r The t o t a l c u r r e n t ( J + J . + J '+ J . ) i s non-zero i f the ee i e e r i r r e c o m b i n a t i o n p a t h o f the e l e c t r o n i c charge d i f f e r s from the e x c i t a t i o n p a t h . The case o f F i g . 2.6 shows the charge p r o c e e d i n g c o n t i n u o u s l y t o the r i g h t , but i n g e n e r a l the p r o b a b i l i t y o f charge t r a n s f e r t o the r i g h t and l e f t w i l l d i f f e r d u r i n g e x c i t a t i o n and r e c o m b i n a t i o n , r e s u l t i n g i n a s t e a d y - s t a t e p h o t o c u r r e n t . I I - 9.4 The P h o t o f l u c t u a t i o n Model F r i d k i n (1977) i n t e r p r e t e d the b u l k p h o t o v o l t a i c e f f e c t i n some oxygen o c t a h e d r a f e r r o e l e c t r i c s on the b a s i s o f p h o t o - i n d u c e d p o l a r i z a t i o n f l u c t u a t i o n s . A b r i e f d e s c r i p t i o n o f t h i s model i s as f o l l o w s . I n an n-type f e r r o e l e c t r i c c r y s t a l the a b s o r p t i o n o f l i g h t l e a d s to the c r e a t i o n o f f r e e and t r a p p e d e l e c t r o n s . I n a c c o r d a n c e w i t h Chanussot (1974) the i n t e r a c -t i o n o f a t r a p p e d e l e c t r o n w i t h a t r a n s v e r s a l o p t i c a l phonon l e a d s (due to pseudo Jahn - T e l l e r e f f e c t ) t o the c r e a t i o n o f the p h o t o - i n d u c e d f l u c t u a t i o n o f p o l a r i z a t i o n l o c a l i z e d near t o the t r a p . I n the volume V o f the f l u c t u a t i o n t h e r e i s a change o f the spontaneous p o l a r i z a t i o n , Ap, and e l e c t r i c f i e l d E - Ap/e. In a l l p h o t o - i n d u c e d f l u c t u a t i o n s the e l e c t r i c f i e l d s h o u l d , f o r reasons o f symmetry, have the same d i r e c t i o n . The f r e 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 c o u p l e t o t h e i r f l u c t u a t i o n s and move i n the d i r e c t i o n o f the e l e c t r i c f i e l d . 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 caused by the movement of the p h o t o - e x c i t e d e l e c t r o n s . 37. A c c o r d i n g to t h i s model i t would be r e a s o n a b l e to expect a d d i t i o n a l R a y l e i g h s c a t t e r i n g o f the l i g h t by the p h o t o f l u c t u a t i o n (Chanussot e t a l . 1977). Both o f th e s e e f f e c t s have been observed i n BaTiO^ • No photo-i n d u c e d R a y l e i g h s c a t t e r i n g has been o b s e r v e d , however, i n LiN b p ^ s u g g e s t i n g t h a t t h i s model i s i n a p p r o p r i a t e f o r LiNbOg (Chanussot e t a l . 1977). I I - 9.5 The P o l a r i z e d I m p u r i t i e s Model (von B a l t z 1978) The most r e c e n t model f o r the p h o t o v o l t a i c e f f e c t has been p r e s e n t e d by von B a l t z (1978). He c l a i m s t h a t t h i s model e x p l a i n s why t h e p h o t o v o l t a i c c u r r e n t may have d i f f e r e n t s p e c t r a l p r o p e r t i e s f o r l i g h t p o l a r i z e d p a r a l l e l and normal t o the d i r e c t i o n o f spontaneous p o l a r i z a t i o n . T h i s model i s cl a i m e d a l s o to g i v e reasons f o r the p o s s i b l e s i g n change o f the c u r r e n t , as i t i s o b s e r v e d , e.g.; i n BaTiC>3 (Koch e t a l . 1976). A c c o r d i n g t o t h i s model, no u n p h y s i c a l m a c r o s c o p i c dark c u r r e n t i s g e n e r a t e d due to the r m a l e x c i t a t i o n , u n l i k e the G l a s s model. The e s s e n t i a l s o f t h i s model a r e as f o l l o w s . I m p u r i t i e s i n t h e m a t e r i a l which s u p p l y the p h o t o c o n d u c t i o n e l e c -t r o n s a r e assumed t o be randomly d i s t r i b u t e d and t h e i r ground s t a t e \.I> to be p o l a r i z e d , i n the f e r r o e l e c t r i c phase o n l y , v i a an asymmetric s h o r t - r a n g e p o t e n t i a l . The p h o t o v o l t a i c c u r r e n t i s due to an asymmetry i n the photo-c r o s s - s e c t i o n S(fi) f S ( - f i ) , f o r i o n i z a t i o n p r o c e s s from | I > to d e l o c a l i z e d f i n a l s t a t e s |f>. fi denotes the d i r e c t i o n o f the p h o t o e l e c t r o n . The f i n a l e l e c t r o n s t a t e s |f> may be r e p r e s e n t e d as a s u p e r p o s i t i o n o f s t a t i o n a r y s t a t e s |E,£,m> w i t h energy E and a n g u l a r momenta £=0,1,2: |f> =1 a(E) |E,0,0> + b m |E,l,m> + C m | E , 2 , m > . (2.41) E,m 38. The asymmetry i n the p h o t o - c r o s s - s e c t i o n i s due to i n t e r f e r e n c e of s t a t e s |E,£,m> i n Eq. 2.41 which i s s o l e l y d etermined by the s c a t t e r i n g phases. The r e s u l t i n g p h o t o v o l t a i c c u r r e n t d e n s i t y may be w r i t t e n i n t h e form: Jp = KOII = q5IaLp/1"ito , where K i s G l a s s ' s a n i s o t r o p y 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 , a=N/S(fi)d 2Q i s the a b s o r p t i v i t y w i t h N the i m p u r i t y c o n c e n t r a t i o n , i s the e l e c t r o n t r a n s p o r t l e n g t h due to the p h o t o v o l t a i c e f f e c t " s c h u b l a n g e " and £ i s the quantum e f f i c i e n c y . In t h i s model, the p r o p e r t i e s of the p o t e n t i a l w e l l s o f the donors and t h e i r i n f l u e n c e on the s c a t t e r i n g phases e x p l a i n why t h e p h o t o v o l t a i c c u r r e n t may have d i f f e r e n t s p e c t r a l p r o p e r t i e s f o r l i g h t p o l a r i z e d p a r a l l e l and p e r p e n d i c u l a r to the d i r e c t i o n of spontaneous p o l a r i z a t i o n . 39. I l l MATHEMATICAL MODELS FOR THE BULK PHOTOVOLTAIC EFFECT WITH ARBITRARY ELECTRON TRANSPORT LENGTH I I I - l I n t r o d u c t i o n The c u r r e n t d e n s i t y due 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 may be d e s c r i b e d by the f o l l o w i n g e q u a t i o n ( G l a s s e t a l . 19 74). J (x) = K a I Ox) . . P (3.1) where K i s some a n i s o t r o p y c o n s t a n t , a i s the a b s o r p t i v i t y and I ( x ) i s t h e l i g h t i n t e n s i t y a t p o s i t i o n x i n s i d e the c r y s t a l . T h i s e x p r e s s i o n f o r t h e p h o t o v o l t a i c c u r r e n t i s a c c e p t e d by o t h e r workers i n the f i e l d as a d e s c r i p t i o n o f t h e .bulk p h o t o v o l t a i c e f f e c t i n the c r y s t a l . .. We en c o u n t e r e d two problems w i t h t h i s e x p r e s s i o n as e x p l a i n e d below. a) G l a s s ' e x p r e s s i o n f o r t h e p h o t o v o l t a i c c u r r e n t p r e d i c t s no phase s h i f t between t h e hologram and 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 a t w r i t e s i t . A c c o r d i n g l y , no beam c o u p l i n g s h o u l d be 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 (see s e c t i o n 2-9) by t h e b u l k p h o t o v o l t i a c mechanism a l o n e . We have e x p e r i m e n t a l l y o b s e r v e d s t r o n g beam c o u p l i n g d u r i n g h ologram r e c o r d i n g i n a LiNBO^ c r y s t a l under c o n d i t i o n s making 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 the dominant w r i t i n g mechanism ( s e e Chapter 5 ) . b) The e x p r e s s i o n f o r t h e p h o t o v o l t a i c c u r r e n t p r e d i c t s t h a t l i g h t p a t t e r n s o f v a r y i n g s p a t i a l f r e q u e n c i e s ( d i f f e r e n t g r a t i n g p e r i o d s ) g i v e r i s e t o holograms w i t h e q u a l i n t e n s i t y , i . e . t h e response o f the r e c o r d i n g medium v s . t h e s p a t i a l f r e q u e n c y i s f l a t . The above two problems i n d i c a t e d t h a t G l a s s ' e x p r e s s i o n f o r the p h o t o v o l t a i c c u r r e n t i m p l i c i t l y ; i m p l i e s s h o r t e l e c t r o n schublange , i n schublange i s d e f i n e d h e r e as the e l e c t r o n t r a n s p o r t l e n g t h due to the b u l k p h o t o v o l t i a c e f f e c t f o l l o w i n g p h o t o e x c i t a t i o n . 40. comparison w i t h the g r a t i n g s p a c i n g . We were' thus l e d t o develop/,a d i f f e r e n t e x p r e s s i o n f o r the p h o t o v o l t a i c c u r r e n t . T h i s n e c e s s i t a t e d t h a t we assume a p h y s i c a l model f o r t h e e l e c t r o n s c a t t e r i n g o r r e t r a p p i n g mechanism f o l l o w i n g o p t i c a l e x c i t a t i o n . S e v e r a l p h y s i c a l models were t r i e d t o see t h e i r e f f e c t on the r e s u l t a n t p h o t o c u r r e n t . A l l t h e models d i s c u s s e d below r e l y on the premise t h a t the p h o t o e l e c t r o n s a r e p r e f e r e n t i a l l y e j e c t e d from t h e i r t r a p s a l o n g the H-c-axls ( x - a x i s ) d i r e c t i o n . I t would seem r e a s o n a b l e then to d e f i n e the schuhlange as the mean d i s t a n c e t r a v e l l e d by the e l e c t r o n b e f o r e b e i n g r e t r a p p e d o r i t s motion c o m p l e t e l y randomized. A f t e r the e l e c t r o n s motion has been randomized, they can be c o n s i d e r e d as f r e e c a r r i e r s c o n t r i b u t i n g to t h e p h o t o c u r r e n t through d r i f t i n any e l e c t r o s t a t i c f i e l d s p r e s e n t a n d . d i f f u s i o n i n c o n c e n t r a t i o n g r a d i e n t s . As w i l l be shown i n the a n a l y s i s to f o l l o w , t h e two problems mentioned e a r l i e r a r e s o l v e d t h r o u g h c e r t a i n m o d i f i c a t i o n s to G l a s s ' e x p r e s s i o n . F i r s t , t h e p h o t o v o l t a i c c u r r e n t d i s t r i b u t i o n i s s p a t i a l l y s h i f t e d from 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 t h a t g enerated i t , and hence beam c o u p l i n g can take p l a c e due t o t h e b u l k p h o t o v o l t i a c mechanism. Second, the ac c u r r e n t d e c r e a s e s i n a m p l i t u d e as the s p a t i a l f r e q u e n c y o f th e i n c i d e n t l i g h t i n c r e a s e s . The response o f the m a t e r i a l ( i . e . the r e s u l t i n g hologram) to the i n c i d e n t l i g h t can thus be l i k e n e d to a low-pass f i l t e r s i n c e holograms w i t h h i g h s p a t i a l f r e q u e n c i e s a r e l e s s e f f i c i e n t l y r e c o r d e d . 41. I I I - 2 The Continuous S c a t t e r i n g Model I I I - 2 . 1 Assumptions The p h o t o e x c i t e d e l e c t r o n s a r e assumed t o t r a v e l a l o n g t h e + c - a x i s ( x - a x i s ) where they m e e t ' s c a t t e r i n g c e n t e r s . The movement of the e l e c t r o n s c o n s t i t u t e s the p h o t o v o l t a i c c u r r e n t and a f t e r an e l e c t r o n i s s c a t t e r e d i t s motion i s randomized and i t no l o n g e r p a r t i c i p a t e s i n the p h o t o v o l t a i c c u r r e n t . I n s t e a d i t becomes a f r e e e l e c t r o n a v a i l a b l e to the d i f f u s i o n and d r i f t mechanisms. I I I - 2 . 2 M a t h e m a t i c a l A n a l y s i s F o r a D i r a c d e l t a l i g h t i n t e n s i t y d i s t r i b u t i o n 6(x) at the o r i g i n , ( F i g . 3.1) t h e volume e l e c t r o n g e n e r a t i o n r a t e i m p u l s e r e s p o n s e i s h (x) = £ a 6(x)/fTco, (£ Is t h e quantum e f f i c i e n c y , a i s t h e a b s o r p t i v i t y and "ftco i s t h e quantum of l i g h t energy) . The p h o t o v o l t a i c c u r r e n t i mpulse response, h ( x ) , at p o s i t i o n x i s governed by the d i f f e r e n c e e q u a t i o n S h ^ x ) = - h j ( x ) 6 x / L p where L i s a p r o p o r t i o n a l i t y c o n s t a n t h a v i n g dimensions of l e n g t h . S o l v i n g P t h e above e q u a t i o n we o b t a i n h j ( x ) = h j ( x •= 0) exp-x/L^; x >_ 0; . ^ ^) The c o n s t a n t h j ( x = 0) i s e q u a l t o t h e charge g e n e r a t i o n r a t e and thus we o b t a i n h j ( x ) = - ( q a E, /"ft co) exp - x / L p . I ( x ) a ) F i g . 3.1 (a) L i g h t i mpulse e x c i t a t i o n . (b) Impulse r e s p o n s e o f f r e e c a r r i e r b u l k g e n e r a t i o n r a t e f o r d r i f t and d i f f u s i o n . (c) P h o t o v o l t a i c c u r r e n t d e n s i t y i mpulse r e s p o n s e . 43. The n e g a t i v e s i g n i n d i c a t e s t h a t t h e c u r r e n t i s f l o w i n g a n t i p a r a l l e l t o t h e x - a x i s . The c o n s t a n t L h e r e r e p r e s e n t s the e l e c t r o n mean f r e e p a t h . P F o r a l i g h t i n t e n s i t y d i s t r i b u t i o n I ( x ) = I Q ( 1 + m cos Kx):, t h e r e s u l t i n g p h o t o v o l t a i c c u r r e n t d e n s i t y i s the c o n v o l u t i o n o f the c u r r e n t i m pulse response, eq. 3.4, w i t h the g i v e n i n t e n s i t y d i s t r i b u t i o n . J p ( x ) = h j ( x ) * I ( x ) , ( 3 5 ) = - K a I [ 1 + m'cos(Kx -<(>)], m c\ o T p / J ' ( 3 . 6 ) i 2 - ^ where K = £ qL^/tico , m = m [ l + (KL^) ] ~ and d> = t a n - 1 K L . P P I I I - 2 . 3 D i s c u s s i o n The p h o t o v o l t a i c c u r r e n t d e n s i t y t h a t i s measurable a t t h e c r y s t a l t e r m i n a l s e q u a l s K. a I q i n Eq. 3.6 above. T h i s i s the dc p a r t of the c u r r e n t ( i . e . does not v a r y w i t h x ) . The a c p a r t i s r e s p o n s i b l e f o r t h e hologram f o r m a t i o n . T h i s l a t t e r component i s s p a t i a l l y s h i f t e d from the l i g h t i n t e n s i t y p a t t e r n t h a t g enerates i t and d e c r e a s e s m o n o t o n i c a l l y w i t h the s p a t i a l f r e q u e n c y of the i n t e n s i t y p a t t e r n . Note a l s o from Eq.3.6 t h a t b o t h the phase s h i f t , cj)^, and the b u l k 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 the a n i s o t r o p y c o n s t a n t b o t h depend on the e l e c t r o n s c h u b l a n g e , . To assume a s h o r t schublange i s to assume a d i m i n i s h i n g b u l k p h o t o v o l t a i c e f f e c t , i . e . i n c r y s t a l s e x h i b i t i n g a s t r o n g b u l k p h o t o v o l t a i c e f f e c t i t i s e s s e n t i a l t o r e a l i z e t h a t t h e r e i s a c e r t a i n non-zero phase s h i f t between the i n t e n s i t y p a t t e r n and the r e s u l t i n g hologram p r i n c i p a l l y due to t h e b u l k p h o t o v o l t a i c mechanism. 44. I I I - 3 The F i x e d T r a n s p o r t Length Model I I I - 3 . 1 Assumptions The p h o t o e x c i t e d e l e c t r o n s a r e assumed t o t r a v e l a l o n g t h e + c - a x i s a f i x e d d i s t a n c e L^ b e f o r e b e i n g s c a t t e r e d o r r e t r a p p e d . I I I - 3 . 2 M a t h e m a t i c a l A n a l y s i s F o r a D i r a c d e l t a f u n c t i o n l i g h t i n t e n s i t y d i s t r i b u t i o n 6(x) at the o r i g i n , F i g . 3.2, t h e b u l k g e n e r a t i o n r a t e i s g i v e n as b e f o r e by h (x) = 5 a 6(x)/ftco. The r e s u l t i n g c u r r e n t d e n s i t y i mpulse response i n t h i s g "-case i s g i v e n by h j ( x ) = -(q5a/Sa>)[u(x) - u(x - L ) ] ; x > 0, ( 3 ? ) where u(x) i s the H e a v i s i d e u n i t f u n c t i o n . F o r a l i g h t i n t e n s i t y d i s t r i b u t i o n I ( x ) = I (1 + mcos Kx), the o p h o t o v o l t a i c c u r r e n t d e n s i t y w i l l be J (x) = - K a I [1 + m'cos(Kx - cf> ) ] ,o o\ p o P » (J.O; t • where K = £ q L /tico , m = m s i n c ( K L /2TT) and <j> = tan . (KL /2) . Here P P P P s i n c ( x ) = s i n TTX/CTTX). I I I - 3 . 3 D i s c u s s i o n Eq. 3.8 f o r the r e s u l t i n g p h o t o v o l t a i c c u r r e n t i n t h i s model i s s i m i l a r t o Eq. 3.6 o f the p r e v i o u s a n a l y s i s . However, h e r e t h e a c p a r t of t h e c u r r e n t r e s p o n s i b l e f o r hologram f o r m a t i o n does n o t d e c r e a s e m o n o t o n i c a l l y w i t h the s p a t i a l f r e q u e n c y o f the i n t e n s i t y p a t t e r n s . I t i s , 45. K x ) 4 a ) x F i g . 3.2 (a) L i g h t i mpulse e x c i t a t i o n . (b) Impulse r e s p o n s e o f the b u l k g e n e r a t i o n r a t e o f f r e e c a r r i e r s . (c) P h o t o v o l t a i c c u r r e n t d e n s i t y i mpulse response. 46. n e v e r t h e l e s s , phase s h i f t e d from the i n t e n s i t y p a t t e r n . I1I-4 The D i s c r e t e S c a t t e r i n g C e n t e r s Model I I I - 4 . 1 Assumptions Here i n t h i s model the p h o t o l i b e r a t e d e l e c t r o n s a r e assumed, as b e f o r e , to be t r a v e l l i n g a l o n g the + c - a x i s where they meet d i s c r e t e s c a t t e r i n g c e n t e r s . These c e n t e r s are assumed to be o f the same ty p e , e q u a l l y s paced a. d i s t a n c e a p a r t . The p r o b a b i l i t y t h a t an e l e c t r o n i s s c a t t e r e d i s p < 1 and p(=l-p') i s the p r o b a b i l i t y t h a t the e l e c t r o n i s u n a f f e c t e d by t h e c e n t e r . We n e g l e c t e f f e c t s of o t h e r types o f s c a t t e r e r s as w e l l as t h e e f f e c t o f s t a t i s t i c a l v a r i a t i o n s i n the d i s t a n c e s between the c e n t e r s under c o n s i d e r a t i o n . I I I - 4 . 2 M a t h e m a t i c a l A n a l y s i s The p h o t o v o l t a i c c u r r e n t r e s p o n s e due t o a D i r a c d e l t a f u n c t i o n l i g h t i n t e n s i t y 6(x) a t the o r i g i n i s ( F i g . 3.3) oo h (x) = - ( q g a/tlco)' E p n [ u ( x - n L )-u(x-nL - L ) ] ; x >_ 0 J n=o P P P (3.9) F o r a l i g h t i n t e n s i t y d i s t r i b u t i o n I ( x ) = 1 (1 + m cos Kx) the o p h o t o v o l t a i c c u r r e n t d e n s i t y w i l l be J (x) = — K a I [1 + nr c os(Kx - <j> ) ] , P ' o L V J ' (3.10) where K , = q £ L / (p"tl-uj) ? i , 2  1y m = m p sinc(KLp/2-rr) (1 + p - 2p cos KL^) 2 and cj> = KL 11 + t a n _ 1 [ ( p s i n KL ) / (1 - p cos KL ) ] . • ,.• . P P P p • T -• F i g . 3.3 (a) L i g h t i mpulse e x c i t a t i o n . (b) Impulse r e s p o n s e o f f r e e c a r r i e r b u l k g e n e r a t i o n r a t e . (c) P h o t o v o l t a i c c u r r e n t d e n s i t y i m p u l s e r e s p o n s e . 48. III-4.3 D i s c u s s i o n E q u a t i o n 3.10 g i v e s an e x p r e s s i o n f o r the p h o t o v o l t a i c c u r r e n t t h a t i s e s s e n t i a l l y s i m i l a r t o t h o s e o b t a i n e d from the p r e v i o u s models. I I I - 5 Summary and C o n c l u s i o n s The t h r e e m a t h e m a t i c a l models 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 t r e a t e d i n t h i s c h a p t e r gave e s s e n t i a l l y s i m i l a r r e s u l t s . F i r s t , the r e s u l t a n t h ologram i s s p a t i a l l y s h i f t e d from the l i g h t i n t e n s i t y p a t t e r n t h a t produces i t . Second, the r e s u l t i n g p h o t o v o l t a i c c u r r e n t , and c o n s e q u e n t l y t h e a b i l i t y o f the c r y s t a l t o s t o r e holograms, d e c r e a s e s as the s p a t i a l f r e q u e n c y o f the l i g h t i n t e n s i t y m o d u l a t i o n i n c r e a s e s . Both e f f e c t s mentioned above depend on the e l e c t r o n schublange L^. L a r g e r L^ g i v e s l a r g e r b u l k p h o t o v o l t a i c e f f e c t , w i t h an accompanying phase s h i f t , and s m a l l e r L^ g i v e s the o p p o s i t e r e s u l t s . G l a s s ' e x p r e s s i o n f o r the b u l k p h o t o v o l t a i c e f f e c t has t o be m o d i f i e d to the form K ( K , L ) a I ( x - d> ( K , L ) / K ) . The e x p r e s s i o n s f o r i c and $ can p p P be determined i f the a c t u a l mechanisms o f e l e c t r o n s c a t t e r i n g o r r e t r a p p i n g are known. Ohmori (1976) c l a i m e d t h a t c o n d u c t i o n e l e c t r o n s i n l i t h i u m n i o b a t e might be s c a t t e r e d by LO-phonons and by i m p u r i t y s c a t t e r i n g . I t i s q u i t e p l a u s i b l e t h a t two or more s c a t t e r i n g and r e t r a p p i n g mechanisms are s i m u l t a n e o u s l y o p e r a t i v e i n the c r y s t a l . In t h i s case the r e s u l t i n g p h o t o v o l t a i c c u r r e n t impulse r e s p o n s e o f the c r y s t a l w i l l be the p r o d u c t o f the i m p u l s e r e s p o n s e s o f each i n d i v i d u a l mechanism assuming they are s t a t i s t i c a l l y i n d e p e n d e n t . The g e n e r a l form f o r t h e a n i s o t r o p y c o n s t a n t io i s q £ L^/ftco, from -1 -3 the e x p e r i m e n t a l r e s u l t s of C h a p t e r 5 < - 1 pA cm(mw) ; £ ^ 10 and "tl co = 2.5 ev ( f o r X = 0.5m), we o b t a i n L^ - 24 nm. 49. IV HOLOGRAM WRITING IN PHOTOREFRACTIVE CRYSTALS WITH ARBITRARY ELECTRON TRANSPORT LENGTHS IV-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 we model the development o f h o l o g r a m g r a t i n g s i n p h o t o r e f r a c t i v e c r y s t a l s a t the i n i t i a l and s t e a d y - s t a t e l i m i t s w i t h a r b i t r a r y e l e c t r o n t r a n s p o r t l e n g t h s . The model to be d e r i v e d makes use o f the r e s u l t s of C h a p t e r 3 ; •namely t h a t a c o r r e c t d e s c r i p t i o n of the<bulk p h o t o v o l t a i c - e f f e c t i n a " c r y s t a l an a n i s o t r o p i c p h o t o v o l t a i c c u r r e n t . w i t h a non-zero s p a t i a l phase s h i f t between i t and the l i g h t i n t e n s i t y p a t t e r n . P r e v i o u s attempts by o t h e r a u t h o r s to a n a l y z e hologram r e c o r d i n g i n p h o t o r e f r a c t i v e c r y s t a l s e x h i b i 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 assumed t h a t ^ i t can be - d e s c r i b e d by (1) an e q u i v a l e n t i n t e r n a l f i e l d o r (2) an a n i s o t r o p i c p h o t o c u r r e n t t h a t i s a r e p l i c a o f the i n t e n s i t y p a t t e r n , ( i . e . z e r o phase s h i f t ) . Both these assumptions were not c o r r e c t , i n the l i g h t o f r e s u l t s o f Chapter 3, s i n c e the n e g l e c t o f the phase s h i f t between the i n t e n s i t y p a t t e r n and p h o t o v o l t a i c c u r r e n t gives r i s e t o m i s l e a d i n g r e s u l t s c o n c e r n i n g beam c o u p l i n g and e f f e c t of f i n i t e e l e c t r o n t r a n s p o r t l e n g t h s on hologram r e c o r d i n g . The a n a l y s i s o f Amodei (1971b) f o r the s h o r t w r i t i n g time l i m i t and s h o r t e l e c t r o n m i g r a t i o n l e n g t h p r e d i c t e d phase q u a d r a t u r e between the hologram component w r i t t e n by the d i f f u s i o n mechanism and the s i n u s o i d a l l i g h t i n t e n s i t y p a t t e r n , and no phase s h i f t due to t h e d r i f t mechanism. The r e s u l t s o f Young e t a l . (1974) ( f o r the s h o r t time l i m i t a l s o ) showed t h a t f o r a r b i t r a r y e l e c t r o n m i g r a t i o n l e n g t h s , t h e d r i f t mechanism can g i v e r i s e t o a phase s h i f t between the hologram and the l i g h t i n t e n s i t y p a t t e r n . Moharam e t a l . (1979) extended the a n a l y s i s o f Young e t a l . by c o n s i d e r i n g the c o n t r i b u t i o n of the b u l k p h o t o v o l t a i c e f f e c t , (employing assumption (2) mentioned e a r l i e r ) and n u m e r i c a l l y s o l v i n g f o r the f u l l time development o f the hologram. I t i s shown h e r e t h a t the b u l k p h o t o v o l t a i c e f f e c t can cause beam c o u p l i n g and energy t r a n s f e r . T h i s i s e s p e c i a l l y so i n c r y s t a l s e x h i b i t i n g a s t r o n g b u l k p h o t o v o l t a i c e f f e c t and under s h o r t -c i r c u i t and u n i f o r m i l l u m i n a t i o n c o n d i t i o n s . IV-2 P h y s i c a l Model The c o n f i g u r a t i o n f o r hologram w r i t i n g i s as shown i n F i g . 4.1. A c o n s t a n t v o l t a g e V w i t h the i n d i c a t e d p o l a r i t y i s a p p l i e d a c r o s s the c - f a c e s o f the c r y s t a l . The c r y s t a l i s o r i e n t e d so t h a t the i n t e r f e r e n c e between the two i n c i d e n t p l a n e waves at a n gles ± 0 q r e l a t i v e t o t h e normal o f the sample forms a standing-wave p a t t e r n a l o n g t h e c - a x i s ( x - a x i s ) . U n i f o r m f u l l i l l u m i n a t i o n i s assumed s i n c e f o r the p r a c t i c a l cases o f non-u n i f o r m i l l u m i n a t i o n , " s e v e r a l ; c o m p l i c a t i o n s i n the a n a l y s i s a r i s e due t o • the i m p o r t a n t e f f e c t of the l a r g e s c a l e space charge f i e l d ( s e e Chapters 6 and 8) . The p h y s i c a l mechanisms c o n t r i b u t i n g t o t h e f o r m a t i o n of the hologram are assumed to be d i f f u s i o n , d r i f t ( i n e l e c t r o s t a t i c and s p a c e charge f i e l d s ) and the b u l k p h o t o v o l t a i c e f f e c t . E f f e c t s o f beam c o u p l i n g on the c o n t i n u o u s r e c o r d i n g o f the hologram w i l l be i g n o r e d f o r s i m p l i c i t y . The r e s u l t s o f t h i s model, however, can be i n c o r p o r a t e d i n a n u m e r i c a l model t h a t t a k e s beam c o u p l i n g e f f e c t s i n t o account, e.g., the n u m e r i c a l model of Moharam and Young (1977, 1978a) . 51. •o V o F i g . 4.1 C o n f i g u r a t i o n f o r hologram r e c o r d i n g . R and S beams a r e a l l o c a t e d i n r e l a t i o n t o the c o o r d i n a t e system which a l s o s e r v e s to d e f i n e t h e s i g n of the a p p l i e d v o l t a g e . 52. IV.3 M a t h e m a t i c a l A n a l y s i s With an i n t e n s i t y p a t t e r n I ( x ) = 1 Q ( 1 + m c o s K x ) » ^ 2.) where I i s t h e average 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 and K = 4TT s i n 8 /X i s the wavevector o f the i n t e r f e r e n c e p a t t e r n w i t h X the o O 0 vacuum wavelength o f l i g h t . The s p a t i a l d i s t r i b u t i o n o f 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 a l o n g the c - a x i s i s g i v e n by J ( x ) = q D 3 n ^ X > t ) + q u n ( x , t ) [ E ( x , t ) - V ] + J ( x ) , 3 x s c — p (4.z) c o n s i s t i n g o f d i f f u s i o n , d r i f t and the b u l k p h o t o v o l t a i c e f f e c t , q i s the e l e c t r o n i c charge, D i s t h e d i f f u s i v i t y , p i s the m o b i l i t y , V i s the e x t e r n a l l y a p p l i e d v o l t a g e , L i s the c r y s t a l l e n g t h , E s c ( x , t ) i s the p h o t o i n d u c e d space charge f i e l d , n ( x , t ) = n^ + n ^ ( x , t ) i s t h e f r e e e l e c t r o n c o n c e n t r a t i o n w i t h n the c o n c e n t r a t i o n o f e l e c t r o n s i n the dark ,~ n ( x , t ) i s the p h o t o e x c i t e d e l e c t r o n c o n c e n t r a t i o n due to i n c i d e n t i l l u m i n a t i o n and J (x) i s the p h o t o v o l t a i c c u r r e n t d e n s i t y g i v e n i n g e n e r a l by t h e e x p r e s s i o n P (see Chapter 3) J (x) = -K a I [1 + m f 1 (K,L )cos (Kx - <)>,)], O N p o 1 p <p (4.3) where K i s the a n i s o t r o p y c o n s t a n t of the b u l k p h o t o v o l t a i c e f f e c t (Chapters 2 and 3 ) , a i s the a b s o r p t i v i t y , i s the schublange .of the e l e c t r o n , 0 <_ f ^ ( K , L ^ ) <_ 1 i s a r e a l f u n c t i o n o f K and ( t a k e n e q u a l t o one i n t h i s a n a l y s i s ) a n d $ i s the phase s h i f t due 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 . P The p h o t o e x c i t e d e l e c t r o n - c o n c e n t r a t i o n i s governed by the e q u a t i o n 9 n L ( x > t ) , , V * ' 1 ^ M 1 8J(x,t) 3 t ~ g U ; " x + q 3 x ' (4.4) 53. where T i s the f r e e e l e c t r o n l i f e t i m e and gl(x) i s the volume g e n e r a t i o n r a t e and i s g i v e n i n g e n e r a l by (see Chapter 3 a l s o ) g(x) = g Q [ l + m f 2 ( K , L p ) c o s ( K x - ^ ) ] , ( 4 > 5 ) where g i s the average g e n e r a t i o n r a t e and f„(K,L ) i s a f u n c t i o n s i m i l a r o I p t o f ^ ( K , L p ) ' (and assumed e q u a l t o u n i t y a l s o i n t h i s a n a l y s i s ) . We assume t h a t the p h o t o e x c i t e d e l e c t r o n c o n c e n t r a t i o n n ( x , t ) i s c o n s t a n t w i t h time 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 such t h a t Eq. 4.4 becomes g ( ' x q 9 x (4.6) The c o n t i n u i t y e q u a t i o n f o r the t r a p p e d charge d e n s i t y p g c due to t h e c a r r i e r m i g r a t i o n i s 9 P s c _ 9 J ( x , t ) 9t 9x (4.7) P o i s s o n ' s e q u a t i o n i s i _ E T = ! s c 9 x - i e • ' (4.8) where E,^  = E ( x , t ) - j- i s the t o t a l e l e c t r o s t a t i c f i e l d (space charge and X S C J_j a p p l i e d ) and.-e'-is the p e r m i t t i v i t y . Combining E q s . 4.6, 4.7 and 4.8 and i n t e g r a t i n g w i t h r e s p e c t to space and time we o b t a i n 1 f C E ( x , t ) = - -S C ' £ J J ( x , t ) d t + A ( t ) , ° (4.9) where A ( t ) i s d e t e r m i n e d from t h e boundary c o n d i t i o n which i s the 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 (zero o r o t h e r w i s e ) f L - E dx = V, (4.10) 54. wh i ch imp l i e s •L J E ( x , t ) d x = o,' From Eqs. 4.9 and 4.11 we o b t a i n E ( x , t ) = - i f [ J ( x , t ) - J ( t ) ] d t , S £ So ° (4.12) where J (t) i s the average c o n d u c t i o n c u r r e n t d e n s i t y f l o w i n g i n the c r y s t a l . IV.4 R e s u l t s IV.4.1 I n i t i a l Stages I n the i n i t i a l s t a g e s o f hologram w r i t i n g we n e g l e c t the r e s u l t i n g space charge f i e l d i n comparison w i t h the e l e c t r o s t a t i c f i e l d ^ i n the c u r r e n t d e n s i t y e q u a t i o n , Eq. 4.2. S o l v i n g f o r n ( x , t ) and E g c ( x , t ) we o b t a i n and n ( x ) = I L + T g [1 + m'cos(Kx - <j> - cj?.... - cj))], D O p V p T v Y / J ' (4.13) q g Q t E ( x , t ) = — — [mKL c os(Kx - cj); ) + m'KL cos (Kx - <j) - cf) - <j>.) s c e K p p a p v + m ' ( K L d ) 2 s i n ( K x - < O p - ^ - c j ) ) ] , where k V L , = (xD) , L = ux - and L a r e t h e t r a n s p o r t l e n g t h s d a L p a s s o c i a t e d w i t h d i f f u s i o n , d r i f t and 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 , r e s p e c t i v e l y . The phase angles a r e <f> = t a n " 1 KL and * = t a n " 1 KL / [ l + ( K L , ) 2 ] . v p a < i $ i s t h e phase s h i f t due t o t h e f i n i t e b u l k p h o t o v o l t a i c t r a n s p o r t l e n g t h 55. w h i l e <}> i s t h e phase s h i f t due to t h e f i n i t e d r i f t and d i f f u s i o n t r a n s p o r t l e n g t h s . m' = m [ l + (KL . ) 2 ] % / { [ l + ( K L , ) 2 ] 2 + (KL ) 2 } ^ . p u a ,For .the c a s e o f hologram w r i t i n g by the d r i f t mechanism o n l y ( L , = L =0 and £ = 0) Eq. 4.14 g i v e s d p p qg tm E ( x , t ) = — — r - r r c o s ( k x - f „ ) , U + ( K L a ) 2 ] ^ ( 4 . 1 5 ) where <t> = t a n KL . a a I f d i f f u s i o n was the o n l y o p e r a t i v e mechanism (L = L =0 and J a p <t> = 0) Eq. 4.14 y i e l d s p 2 q g o t m ( K L d r E ( x , t ) = — H — - s ^ s i n K x . l + C K L d ) 2 ( 4 i l 6 ) Eqs. 4.15 and 4.16 a r e s i m i l a r t o those o b t a i n e d by Young e t a l . (1974) f o r t h e ca s e s o f d r i f t o n l y and d i f f u s i o n o n l y , r e s p e c t i v e l y . E qs. 4.13 and 4.14 are e q u i v a l e n t t o t h o s e o b t a i n e d by Moharam e t al.(1979) except f o r the e x t r a phase s h i f t w hich i s added t o a l l mechanisms c o n t r i b u t i n g t o t h e w r i t t e n hologram. O f t e n i n our e x p e r i m e n t s , t h e c r y s t a l i s s h o r t - c i r c u i t e d and u n i f o r m l y i l l u m i n a t e d such t h a t t h e o n l y o p e r a t i v e mechanisms a r e d i f f u s i o n and the b u l k p h o t o v o l t a i c e f f e c t . F o r t h i s case Eq. 4.14 y i e l d s q g o t m [ ( K L p ) 2 + ( K L d ) 4 ] 3 [ 1 + ( K L / ] * (4.17) 2 E > , t ) = — ~ ~ 2 C ° S ( K X -1 ( K V 2 -1 ED where $ = tan ' — : = tan — w i t h E = K kT' i s the d i f f u s i o n KL E 13 P v q 56. e q u i v a l e n t f i e l d , k i s Boltzmann's c o n s t a n t T" i s t h e a b s o l u t e temperature and E i s the " v i r t u a l " f i e l d (E = L /UT) a s s o c i a t e d w i t h the b u l k v v p p h o t o v o l t a i c e f f e c t ( C o r n i s h et a l . 1 9 7 6 a ) . IV.4.2 S t e a d y - S t a t e L i m i t In t h e s t e a d y - s t a t e , t h e time v a r i a t i o n o f the p h o t o e x c i t e d e l e c t r o n s v a n i s h e s and t h e 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 u n i f o r m throughout the c r y s t a l . S o l v i n g E qs. 4.2, 4.4 and 4.11 s u b j e c t to these.two assumptions we o b t a i n the space charge f i e l d a t s t e a d y - s t a t e as 2 J" m E s i n ( K x - A ) C 1 " ^ ) 2 s c W 1 + m i c o s ( K x - * ) ^ v L n 1 + m l C o s ( K x - <t> )J> 1 p 1 p (4.20) where m^ = mxg^/^n^ + x g Q ) i 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 . C l e a r l y , t h e s a t u r a t i o n v a l u e o f the f i e l d i s independent of the d r i f t o r d i f f u s i o n t r a n s p o r t l e n g t h s , o r o f the way t h e f i e l d d e v elops w i t h t i m e . The f i e l d depends, however, on the t r a n s p o r t l e n g t h " s c h u b l a n g e " connected 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 t hrough the two terms E and 4> . A l s o , i t i s phase v p s h i f t e d r e l a t i v e t o 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 by an-,amount cj> p . IV.5 D i s c u s s i o n Comparing Eqs. 4.14 and 4.20 i n d i c a t e s t h a t the'phase s h i f t o f t h e hologram i s l a r g e r d u r i n g the e a r l i e r s t a g e s o f hologram f o r m a t i o n than at s t e a d y - s t a t e . I n the i n i t i a l s t a g e s the phase s h i f t i s due 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 , d r i f t and d i f f u s i o n mechanisms. W h i l e i n the s t e a d y -s t a t e the phase s h i f t i s p r i n c i p a l l y due to the b u l k p h o t o v o l t a i c e f f e c t a l o n e . Thus e x a m i n a t i o n o f beam c o u p l i n g i n the s t e a d y - s t a t e s h o u l d y i e l d d i r e c t i n f o r m a t i o n about 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 , w h i l e beam c o u p l i n g i n the i n i t i a l s t a g e s y i e l d s i n f o r m a t i o n about t h e r e l a t i v e importance o f the o t h e r mechanisms p a r t i c i p a t i n g i n hologram w r i t i n g . The r e s u l t s o f V i n e t s k i i e t a l . (1977a) and Moharam e t a l . (1979) i n d i c a t e t h a t at s t e a d y -s t a t e t h e hologram phase s h i f t i s due t o the d i f f u s i o n mechanism, t h i s i s because t h e i r assumed models 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 do not t a k e i n t o c o n s i d e r a t i o n the phase s h i f t a s s o c i a t e d w i t h t h i s e f f e c t . The p h y s i c a l reasons why t h e hologram phase s h i f t undergoes a t r a n s i e n t change from the i n i t i a l s t a g e s t o t h e s t e a d y - s t a t e v a l u e a r e as f o l l o w s . When two beams i n t e r s e c t t o form a s t a n d i n g wave p a t t e r n the p h o t o -c u r r e n t w i l l be modulated by t h i s i n t e r f e r e n c e p a t t e r n and a t r a p p e d charge w i l l r e s u l t t h a t forms the hologram. The d i f f e r e n t mechanisms c o n t r i b u t i n g t o t h i s p h o t o c u r r e n t p r o d u c e a c e r t a i n phase s h i f t a s s o c i a t e d w i t h each mechanism. The s t e a d y - s t a t e i s approached when the feedback e f f e c t o f the r e s u l t i n g s p a c e charge f i e l d c a n c e l s the e f f e c t o f a l l mechanisms.:and t h e i r accompanying phase s h i f t s . One phase component i s n o t c a n c e l l e d , however, which i s the phase s h i f t i n h e r e n t w i t h the b u l k p h o t o v o l t a i c e f f e c t . As a r e s u l t , a t r a n s i e n t phase w i l l t a k e p l a c e , t h a t decays as s a t u r a t i o n i s approached. The assumption o f = f = 1 i n Eqs. 4.3 and 4.5 o f the a n a l y s i s i s j u s t i f i e d by t h e f o l l o w i n g argument. I f we use the r e s u l t s o f Chapter 3 and i n p a r t i c u l a r the model o f s e c t i o n 3.2 we f i n d t h a t <j> = t a n "ScL and P P 2 - J " f . = f„ = [1 + (KL ) ] 2 . Thus f o r - 4 = 10° the energy t r a n s f e r between x z p p the two w r i t i n g beams can be up t o 18% (from Eq. 2.23 i n c h a p t e r 2) w h i l e f ^ (or fy) i s s m a l l e r than u n i t y by l e s s than 2%. The r e d u c t i o n i n the m o d u l a t i o n r a t i o i s thus l e s s than 2%. 58. V HOLOGRAPHIC MEASUREMENTS V - l I n t r o d u c t i o n In t h i s c h a p t e r we d i s c u s s e x p e r i m e n t a l l y o b t a i n e d r e s u l t s on the d i f f r a c t i o n e f f i c i e n c y and beam c o u p l i n g d u r i n g hologram w r i t i n g i n a LiNbOgiFe c r y s t a l . T h i s c r y s t a l e x h i b i t e d s t r o n g beam c o u p l i n g d u r i n g the w r i t i n g p r o c -ess under c o n d i t i o n s which .precluded d r i f t o r d i f f u s i o n from b e i n g the cause. The dominant mechanism f o r hologram s t o r a g e was the b u l k p h o t o v o l t a i c e f f e c t and i t i s b e l i e v e d t h a t t h i s mechanism i s a l s o r e s p o n s i b l e f o r the ob s e r v e d l a r g e beam c o u p l i n g . The o b t a i n e d e x p e r i m e n t a l r e s u l t s were compared w i t h the computer model o f Moharam and Young (1977, 1978a), which attempts to d e s c r i b e the hologram w r i t i n g p r o c e s s t a k i n g i n t o account the i n t e r a c t i o n t h a t t a k e s p l a c e between the hologram and the r e c o r d i n g beams. The 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 was c o n s i s t e n t w i t h the computed d a t a but t h e amount of beam c o u p l i n g was much l a r g e r than t h a t p r e d i c t e d by t h e computer model. The r e a s o n f o r t h i s disagreement i s t h a t the computer model was based on t h e assumption t h a t the t r a n s p o r t l e n g t h i n the b u l k p h o t o v o l t a i c e f f e c t was s h o r t on the s c a l e o f the hologram g r a t i n g . We have m o d i f i e d t h i s computer model by r e l a x i n g the r e s t r i c t i o n o f s h o r t e l e c t r o n t r a n s p o r t l e n g t h u s i n g the r e s u l t s o f Chapter 4. The amount o f beam c o u p l i n g computed from t h i s m o d i f i e d computer model was of the same o r d e r as the e x p e r i m e n t a l l y o b t a i n e d r e s u l t s . V-2 E x p e r i m e n t a l P r o c e d u r e s The l i g h t s o u r c e was a S p e c t r a P h y s i c s Model 166 argon i o n l a s e r 59. used a t 514.5 nm. As shown i n F i g . 5.1, the beam was s p l i t u s i n g a commercial beam s p l i t t e r and then expanded u s i n g beam expanders w i t h s p a t i a l f i l t e r s t o 25 mm d i a m e t e r , the c r y s t a l used was No. 6 (see Appendix E ) . The aim was to o b t a i n as near as p o s s i b l e u n i f o r m i n c i d e n t i l l u m i n a t i o n w h i l s t r e t a i n i n g s u f f i c i e n t i n t e n s i t y i n o r d e r to m i n i m i z e the e f f e c t o f the 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 the envelope o f l i g h t i n t e n s i t y ( Chapters 6 and 8 ) . The i n c i d e n t i n t e n s i t i e s o f the two beams were made e q u a l . Both beams were measured and r e c o r d e d s i m u l t a n e o u s l y a t the e x i t f a c e , u s i n g A l p h a m e t r i c s i l i c o n p-n j u n c t i o n d e t e c t o r s . The d i f f r a c t i o n e f f i c i e n c y was measured by i n t e r r u p t i n g one beam. 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 h e r e as the r a t i o o f the i n t e n s i t y o f the r e c o n s t r u c t e d S beam to the sum of the i n t e n s i t i e s o f the R and S beams l e a v i n g the c r y s t a l . The R and S beams d e f i n e d a p l a n e o f i n c i d e n c e which i n c l u d e d the c - a x i s , F i g . 5.1, l y i n g i n the p l a n e o f the major f a c e s o f the c r y s t a l . The R beam was d e f i n e d as the beam a p p r o a c h i n g from t h e p o s i t i v e end o f the c - a x i s . The r e s u l t s g i v e n a r e f o r e l e c t r i c v e c t o r p e r p e n d i c u l a r t o the p l a n e o f i n c i d e n c e ( i . e . o r d i n a r y p o l a r i z a t i o n w i t h i n t h e c r y s t a l ) . The ( e x t e r n a l ) a n g l e o f i n c i d e n c e was 14.9° f o r b o t h R and S beams. T h i s g i v e s a g r a t i n g s p a c i n g o f about lum. The c r y s t a l was s h o r t c i r c u i t e d i n a l l o f the experiments r e p o r t e d h e r e . I t was found n e c e s s a r y t o s h i e l d the l i g h t beams from v a r i a b l e temperature a i r c u r r e n t s which o t h e r w i s e produced s e r i o u s n o i s e i n the e x i t beams. T h i s was done u s i n g tubes t o e n c l o s e the l i g h t beams and by c o v e r i n g the whole t a b l e w i t h a p e r s p e x / m e t a l "greenhouse". M e c h a n i c a l v i b r a t i o n d i d not g i v e o b s e r v a b l e n o i s e on a CRO ( u n l e s s the t a b l e was tapped) and was, t h e r e f o r e c o n c l u d e d n o t to be a problem. I t would g i v e beam c o u p l i n g f l u c -t u a t i n g w i t h the m e c h a n i c a l d i s p l a c e m e n t . 5.1 O p t i c a l arrangement for the-holographic storage. • 61. From the observed crystal absorptivity of 21.6 cm * at 450 nm (measured on a spectrometer) one can use the data of Phi l l i p s and Staebler (1974) (their Fig. 6) to obtain an estimate of 0.01 mole% F e 2 + i . e . > a ratio of F e 2 + / F e 3 + of about 1:9. The crystal was antireflection coated to minimize interference between beams multiply reflected at the crystal faces which can otherwise seriously change the observed diffraction efficiency, an effect exacerbated by thermal expansion due to heating by the laser beam (Cornish and Young 1975a, Moharam 1976b). The absorptivity " - at the laser wavelength 514.5 nm was measured using the laser beam as 15 cm 1 and this value-was used in the computer program. V-3 Results V-3.1 Photocurrent Measurements (Moharam 1978b) The magnitude of the bulk photovoltaic effect was investigated for the crystal under study by measuring the photocurrent i along the c-axis of the crystal as a function of voltage V for a uniform incident light intensity I. The results are shown in Fig. 5.2. They were least mean squares fitted by i = a l + bVI + cV, (5.1) giving a=(H.2±0.4)xl0~ 1 0 A cm2 W~\ b= (25± 2 ) x l 0 _ 1 5 A cm2 W-1 and c=(0.1 _15 ±0.01)xl0 mho. The third term in Eq. 5.1 is the dark current due to the 100 F i g . 5.2 P h o t o c u r r e n t between e l e c t r o d e s on c f a c e s o f i r o n - d o p e d LiNbO c r y s t a l ' as a f u n c t i o n o f l i g h t i n t e n s i t y f o r t h r e e a p p l i e d v o l t a g e s . f i n i t e conductivity of the c r y s t a l plus contributions from the leakage conductance of the p l a s t i c c r y s t a l holder and the surface conductance of the c r y s t a l . The apparent conductivity of the c r y s t a l (neglecting the l a s t - 1 5 . - 1 two factors above) was 10 mho cm . However, t h i s f i g u r e would give a time constant for hologram decay i n the dark of about 45 min. which was much shorter than observed (days). The dark conductivity i s used i n the computer model i n the r a t i o of l i g h t to dark conductivity o^/a^ . The value used was a la = 1000. The above data shows that (with the l i g h t i n t e n s i t i e s used) the r a t i o was l a r g e r than t h i s but further increase i n the value put i n t o the computer model would not a f f e c t the r e s u l t (although a decrease i n o^/a^ to,say,10 would). With uniform i l l u m i n a t i o n throughout the c r y s t a l the photocurrent density would be given by J •= ical + a TV/L , (5.2) where K i s the anisotropy constant c h a r a c t e r i s t i c of the impurity, a i s the a b s o r p t i v i t y , and L i s the length of the c r y s t a l . Allowing f o r the nonuniform l i g h t i n t e n s i t y due to absorption, and for the dimensions of the c r y s t a l used, the photocurrent becomes i = 0.75 KI + 0.05 a LV, (5.3) - 9 - 1 which on comparison with Eq. 5.1 gives K =(1.5 ± 0.06)x 10 A cm W which i s about one-half the value reported by Glass et a l . (1974) but i s very close to the value reported by Kra'tzig and Kurz (1976, 1977b). The photo-- 1 3 conductivity as a function of l i g h t i n t e n s i t y i s found as(5.0 ± 0.7)x 10 mho cm-* per W cm 2 . i Experimentally, the d i f f r a c t i o n e f f i c i e n c y and amount of beam coupling were obtained versus exposure ( i n t e n s i t y x time), while the computer 64'.; model gave those q u a n t i t i e s v e r s u s n o r m a l i z e d time t/T, Q, T q b e i n g the d i -e l e c t r i c r e l a x a t i o n time under i l l u m i n a t i o n . Here T = e/a T and from the o i -above experiments a =(5.0 ± 0.7)x 1 0 - 1 3 (I.(W cm 2 ) 1 )(mho cm 1 ) . The i n t e n s i t y I c a n c e l s i n t h i s c o n v e r s i o n when the r e s u l t s a r e p l o t t e d a g a i n s t —2 exposure. The i n c i d e n t beam i n t e n s i t i e s were b o t h 33 m W cm f o r the r u n -2 o f F i g . 5.3 and 25 m W cm i n a l l subsequent r u n s . From the above d a t a the v a l u e o f the v i r t u a l f i e l d (Chapter 4) was E V = 45 ± 5 kV cm" 1. 5.3.2 H o l o g r a p h i c Measurements F i g . 5.3 i s f o r the f i r s t exposure o f the c r y s t a l t o l a s e r l i g h t a f t e r a p e r i o d o f s e v e r a l months d u r i n g which the c r y s t a l was r e p o l i s h e d and i t s a n t i r e f l e c t i o n c o a t i n g s r e p l a c e d . F i g . 5.3 (a) shows the e v o l u t i o n o f the 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 exposure p l u s c u r v e s c a l c u l a t e d from t h e computer model f o r v a l u e s o f the v i r t u a l f i e l d b r a c k e t i n g the v a l u e e s t i m a t e d from p h o t o c u r r e n t measurements. There i s c l e a r l y no problem i n o b t a i n i n g a r e a s o n a b l y good f i t between measured d i f f r a c t i o n e f f i c i e n c y and t h a t from v. the model ( t o which the i n p u t parameters are E , o^/op, a, a p p l i e d v o l t a g e and the g e o m e t r i c a l f a c t o r s i . e . no parameters not e s t i m a t e d from s e p a r a t e e x p e r i m e n t s ) . When the beam c o u p l i n g d u r i n g the w r i t i n g p r o c e s s i s examined, however, a s e r i o u s apparent anomaly was i n v a r i a b l y o b s e r v e d . T h i s i s i l l u s t r a t e d i n F i g . 5.3 (b) which shows the i n t e n s i t i e s o f the R and S beams a f t e r l e a v i n g the c r y s t a l n o r m a l i z e d by d i v i d i n g each by h a l f t h e i r sum. The t r a n s f e r o f energy i s much l a r g e r than computed w i t h E^ = 55 kV cm 1 , the magnitude o f E i t g i v i n g the b e s t f i t to the d i f f r a c t i o n e f f i c i e n c y >-o z UJ 100 0 E y = 6 5 k V / c m E v = 45 k V / e m 10 2 E X P O S U R E ( J / c m ) 15 20 F i g . 5.3 (a) C i r c l e s i n d i c a t e the 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 w i t h exposure d u r i n g t h e f i r s t run where t h e c r y s t a l was f r e e o f p h o t o i n d u c e d r e f r a c t i v e - i n d e x changes. The l i n e s a r e c a l c u l a t e d from the computer model w i t h E as shown. S h o r t - c i r c u i t e d c r y s t a l w i t h as ne a r as p o s s i b l e uniform, i l l u m i n a t i o n o f t h e whole c r y s t a l and e q u a l i n t e n s i t y beams o f 33mW/cm . in ui \— cn z LU h-Z < LU m LU UJ rr 5 - B E A M 11 ' " • i n : R - B E A M 0 5 10 2 E X P O S U R E (J / c m ) 15 F i g . 5.3 (b) S o l i d l i n e s r e p r e s e n t the o b s e r v e d beam i n t e n s i t i e s d u r i n g the same r u n as (a) n o r m a l i z e d by d i v i d i n g each beam by o n e - h a l f t h e i r sum a t each t i m e . D o t t e d l i n e s r e p r e s e n t the v a l u e c a l c u l a t e d from the computer model w i t h E =55 kVcm . -v • / • 66. F i g . 5.3 (c) Same experimental data as (b) normalized by d i v i d i n g the i n t e n s i t y of each beam by i t s i n i t i a l v a l u e . 67. ( v a l u e s o f 45 or 65 kV cm 1 would n o t be much d i f f e r e n t ) . The s i g n i f i c a n c e o f t h i s anomaly i s d e a l t w i t h i n t h e D i s c u s s i o n s e c t i o n below. One o f the drawbacks o f l i t h i u m n i o b a t e as a hologram s t o r a g e m a t e r i a l i s the development o f a l i g h t s c a t t e r i n g p r o c e s s on exposure t o ... l a s e r l i g h t (see Chapter 10). The e f f e c t o f t h i s s c a t t e r i n g may be seen by comparing the n o r m a l i z e d beam i n t e n s i t i e s o f F i g . 5.3 (b) w i t h the a c t u a l beam i n t e n s i t i e s o f F i g . 5.3 ( c ) . The t o t a l beam i n t e n s i t y f a l l s below t w i c e the o r i g i n a l e x i t i n t e n s i t y o f e i t h e r beam as s c a t t e r i n g d e v e l o p s . With v e r y l o n g exposures the s c a t t e r i n g can remove most o f the i n c i d e n t beam. F i g . 5.4 shows the r e s u l t s o b t a i n e d i n a s e r i e s o f f i v e hologram w r i t i n g e x p e r i m e n t s . B e f o r e each new hologram was w r i t t e n , the o l d hologram was e r a s e d by exposure to the R beam a l o n e u n t i l momentary exposure t o R and S beams t o g e t h e r produced no change i n the R beam i n t e n s i t y i n d i c a t i n g no c o u p l i n g between the two beams and hence p r a c t i c a l l y complete e r a s u r e o f the hologram. F i g . 5.4 (a) shows the 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 i n the f i v e s u c c e s s i v e runs ( t h e f i r s t r u n i s t h a t shown i n F i g . 5.3). F i g . 5.4 (b) shows the i n t e n s i t i e s o f the R and S beams d u r i n g w r i t i n g n o r m a l i z e d by d i v i d i n g by t h e i r i n i t i a l v a l u e s . In some cases t h e i r sum i n c r e a s e s above 2 as the exposure i n c r e a s e s . T h i s i s due to t h e e r a s u r e , when b o t h R and S beams a r e p r e s e n t , o f s c a t t e r i n g p a r a s i t i c g r a t i n g s (Chapter 10) w r i t t e n d u r i n g the e r a s u r e s t e p w i t h o n l y R i n c i d e n t . T h i s problem can be a m e l i o r a t e d by e r a s i n g the p a r a s i t i c g r a t i n g s by e x p o s i n g t h e c r y s t a l to a s i n g l e beam a f t e r r o t a t i o n about a v e r t i c a l a x i s . A new s e t o f p a r a s i t i c g r a t i n g s i s w r i t t e n d u r i n g t h i s p r o c e s s b u t they a r e i n e f f e c t i v e i n s c a t t e r -i n g when one r e t u r n s to the o r i g i n a l c r y s t a l o r i e n t a t i o n because the Bragg c o n d i t i o n which c o u p l e s them t o the main beam i s no l o n g e r s a t i s f i e d . T h i s 1 0 0 >-CJ 2: LU o t—« u_ UJ 2: o < a ) 5 •6 0 10 E X P O S U R E 1 5 ( J / c r r i ) F i P 5 4 (a) D i f f r a c t i o n e f f i c i e n c y v e r s u s exposure d u r i n g f i v e c o n s e c u t i v e r u n ; w i t h o p t i c a l e r a s u r e between r u n s . The f i r s t r un i s t h a t o f F i g . 5 . 3 ( a ) , equal i n c i d e n t beam i n t e n s i t i e s : 33 mW/cm* i n the f i r s t and 25 mW cm i n l a t e r r u n s . (b) Beam i n t e n s i t i e s n o r m a l i z e d by d i v i d i n g - by i n i t i a l v a l u e s . ID LU IT) LD 2 < LU m LU b, 2 < _i LU CC 1-5 0 - 5 S - B E A M R - B E A M 0 5 1 0 2 E X P O S U R E ( J / c m z ) 15 69, F i g . 5.4 (c) Beam i n t e n s i t i e s n o r m a l i z e d by d i v i d i n g each by o n e - h a l f t h e i r sum. 70. t e c h n i q u e can p r e v e n t the sum o f R and S beams n o r m a l i z e d by t h e i r i n i t i a l v a l u e s from r i s i n g above 2. F i g . 5.4 (c) shows the R and S beams n o r m a l i z e d by d i v i d i n g each by h a l f t h e i r sum a t each time. The i n i t i a l v a l u e s o f R and S a t the e x i t s i d e now d i f f e r from each o t h e r , even though the i n c i d e n t R and S beams a r e e q u a l , by an amount which depends on how much o f the p r e v i o u s s c a t t e r i n g was l e f t unerased. F i n a l l y , F i g . 5.5 (a) shows the e v o l u t i o n o f the d i f f r a c t i o n e f f i c i e n c y and F i g . 5.5 (b) shows the R and S beams from the f i r s t run ( F i g . 5.3) p l u s two f u r t h e r runs each c a r r i e d out a f t e r the c r y s t a l had been l e f t f o r s e v e r a l days f o r holograms t o decay. In summary, the e v o l u t i o n o f the d i f f r a c t i o n e f f i c i e n c y c o u l d be f i t t e d by the model b u t the beam c o u p l i n g was always l a r g e r than p r e d i c t e d . The e v o l u t i o n o f the d i f f r a c t i o n e f f i c i e n c y was r e a s o n a b l y r e p r o d u c i b l e b u t the e v o l u t i o n o f the R and S beams was l e s s r e p r o d u c i b l e and was i n f l u e n c e d by s c a t t e r i n g by p a r a s i t i c g r a t i n g s . V T4 D i s c u s s i o n In t h i s c h a p t e r we have measured the amount o f beam c o u p l i n g t h a t takes p l a c e d u r i n g hologram r e c o r d i n g . The e x p e r i m e n t a l l y o b t a i n e d r e s u l t s were compared w i t h a computer model a p p l i c a b l e t o the whole time e v o l u t i o n . S i n c e the c r y s t a l was s h o r t c i r c u i t e d and almost u n i f o r m l y i l l u m i n a t e d , d r i f t o f p h o t o e l e c t r o n s i n a p p l i e d or p h o t o g e n e r a t e d l a r g e s c a l e f i e l d s was mi n i m i z e d . D i f f u s i o n was r e l a t i v e l y u n i m p o r t a n t , as can be seen by comparing the v i r t u a l f i e l d , e s t i m a t e d as 45 ± 5kV cm 1 from p h o t o c u r r e n t measurements, w i t h the d i f f u s i o n e q u i v a l e n t f i e l d f o r the e x p e r i e m n t a l c o n d i t i o n s used o f 100 0 5 10 15 20 EXPOSURE ( J / c m Z ) F i g . 5.5 (a) D i f f r a c t i o n e f f i c i e n c y f o r t h r e e runs i n c l u d i n g t h a t o f F i g . 5.3(a) (cr o s s e d ) p l u s two f u r t h e r runs w i t h t h e c r y s t a l l e f t f o r a p e r i o d o f s e v e r a l days b e f o r e each run to a l l o w decay o f r e f r a c t i v e - i n d e x changes. E q u a l beam i n t e n s i t i e s : 33 mW/cm2 i n the f i r s t r un and 25 mW/cm2 i n l a t e r r u n s , (b) Beam i n t e n s i t i e s n o r m a l i z e d by d i v i d i n g b y . t h e i r i n i t i a l v a l u e s . 72. 1.6 kV cm 1 . D i f f u s i o n c o u l d n o t have produced more than a s m a l l f r a c t i o n o f the observed d i f f r a c t i o n e f f i c i e n c y . T h i s l e a v e s the b u l k p h o t o v o l t a i c e f f e c t as the dominant p r o c e s s . The hologram t h a t i s s t o r e d i n t h i s c ase has been d e r i v e d i n Chapter 4 (Eq\ 4.17) . T h i s hologram i s phase s h i f t e d from the l i g h t i n t e n s i t y p a t t e r n t h a t g e n e r a t e d i t by an amount + <j>^ ) , where <j)p i s the phase s h i f t a s s o c i a t e d w i t h the b u l k p h o t o v o l t a i c e f f e c t (Chapter 3) and d>^  = t a n " 1 E^/E (E, , i s the v i r t u a l f i e l d and E ^ i s the d i f f u s i < TD D v 'v D iion e q u i v a l e n t f i e l d ) . The anomalous s t r o n g beam c o u p l i n g e x p e r i m e n t a l l y o b served may be e x p l a i n e d now i f we use the p u b l i s h e d formulae f o r beam c o u p l i n g d u r i n g hologram w r i t i n g ( S t a e b l e r and Amodei 1972a). We use t h e c o u p l e d wave formulae f o r the i n t e n s i t i e s I and I„ i n terms o f the i n i t i a l i n t e n s i t y I , n e g l e c t i n g a b s o r p t i o n (Eqs. 2.23 i n Chapter 2 r e p e a t e d h e r e f o r conven-i e n c e ) . I _ / I = 1 - s i n 26D sin<f> R ° (2.23) I g / I Q = 1 + s i n 26D sin<J> where 6 = 7 m , / A cos8. Here n, i s the a m p l i t u d e o f the s i n u s o i d a l r e f r a c t i v e i o J-index m o d u l a t i o n , 6 i s a n g l e of r e f r a c t i o n i n s i d e the c r y s t a l , D i s c r y s t a l t h i c k n e s s and cj> i s the amount o f phase l a g between the hologram and th e l i g h t i n t e n s i t y p a t t e r n ( i n our case <j> = d>p + (f)^ ) . With the 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 ( s i n 2 6 D , see Appendix D) o f 3.9% a t an exposure o f 1.2 J/cm 2 and u s i n g t h e measured r a t i o o f the i n t e n s i t i e s o f t h e beams e x i t -i n g from the c r y s t a l , we o b t a i n <j)=10.4° compared to the computer model v a l u e o f 1.7° (which i s e q u a l to <j>^  p r e v i o u s l y d e f i n e d ) . The phase s h i f t due to the b u l k p h o t o v o l t a i c e f f e c t a l o n e i s <j>p = t a n ^KL p (see Chapter 3) where K i s the g r a t i n g v e c t o r and L p i s the e l e c t r o n s chublange. From the above r e s u l t s we o b t a i n a v a l u e f o r Lp = 24 nm. T h i s seems a s u p r i s i n g l y l a r g e v a l u e i n view of the low m o b i l i t y (about 1 cm 2V ^sec . r e p o r t e d f o r r e d u c e d • 73.; c r y s t a l s by Ohmori e t a l . (1976). We have m o d i f i e d the computer model o f Moharam and Young (1977) m a i n l y by i n t r o d u c i n g a phase l a g , c|)p, to the hologram component produced by the b u l k p h o t o v o l t a i c e f f e c t . t o a l l o w f o r the f i n i t e s chublange of the e l e c t r o n i n accordance w i t h the r e s u l t s o f Chapter 4. F i g . 5.6 shows the e f f e c t o f the phase s h i f t on the computed beam c o u p l i n g p l o t t e d a g a i n s t exposure f o r ^=55 kV cm 1 . The e s t i m a t e of the schublange i n the b u l k p h o t o v o l t a i c p r o c e s s a l l o w s us to e s t i m a t e the quantum e f f i c i e n c y E,, d e f i n e d as the p r o b a b i l i t y t h a t an e l e c t r o n i s e j e c t e d when a photon i s absorbed by an i r o n c e n t e r , f o r the case t h a t a l l e l e c t r o n s a r e e j e c t e d i n the same d i r e c t i o n . F o r t h i s — 3 c a s e , J p = Kal = ^(al/lTw) L p q . We o b t a i n £ about 10 . F o r the case t h a t Lp r e p r e s e n t s a s t a t i s t i c a l d i f f e r e n c e , w i t h e l e c t r o n s e j e c t e d i n b o t h +c and -c d i r e c t i o n s , L = p,£, - p £ , where p, and p a r e t h e r e l a t i v e p r o -p r + + - - + -b a b i l i t i e s and £ + and £ a r e the mean t r a n s p o r t l e n g t h s f o r the two d i r e c t i o n s (see s u b s e c t i o n 2.10.2 f o r a d i s c u s s i o n o f t h i s model). For example, the l e s s e q u a l p + and p , the l o n g e r £ + and Jl , i f t h e s e two a r e assumed about e q u a l . In p r e v i o u s work, Alphonse e t a l . (1975) s t u d i e d the e f f e c t o f s p a t i a l f r e q u e n c y on w r i t i n g e f f i c i e n c y f o r a c r y s t a l i n which holograms were w r i t t e n p r e d o m i n a n t l y by d i f f u s i o n . They found a r o l l - o f f i n w r i t i n g e f f i c i e n c y a t about 1600 l i n e p a i r s (Ap) mm 1 from which they e s t i m a t e d an o r d e r o f magnitude of d i f f u s i o n l e n g t h as 40 nm. T h i s , however, does n o t s u p p o r t our v a l u e f o r L p s i n c e one must c l e a r l y d i s t i n g u i s h between t r a n s p o r t l e n g t h s i n t r u e d r i f t and d i f f u s i o n (where the t r a n s p o r t i s t e r m i n a t e d by c a p t u r e of the e l e c t r o n by some t r a p p i n g c e n t e r ) and the t r a n s p o r t l e n g t h i n the b u l k p h o t o v o l t a i c e f f e c t (schublange) which i s presumed (and o n l y presumed) to i n v o l v e momentum s c a t t e r i n g p r o c e s s . F i g 5.6 Beam i n t e n s i t i e s c a l c u l a t e d from t h e m o d i f i e d computer mo d i f f e r e n t assumed v a l u e s of. the phase s h i f t <(> a s s o c i a t e d w i t h t h p h o t o v o l t a i c e f f e c t . 75. Vahey (1975) o b t a i n e d an e s t i m a t e o f t r a n s p o r t l e n g t h u s i n g e x p e r i m e n t a l d a t a p u b l i s h e d by Amodei e t a l . f o r a c r y s t a l i n which we would now b e l i e v e the b u l k p h o t o v o l t a i c p r o c e s s to be dominant. ( T h i s p r o c e s s was not then r e c o g n i z e d and the p r o c e s s was assumed to be d r i f t i n a b u i l t - i n f i e l d ) . Vahey used a dynamic model ( i . e . one a l l o w i n g f o r i n t e r a c t i o n o f the w r i t i n g l i g h t w i t h the hologram b e i n g w r i t t e n ) but merely assumed t h a t the r e f r a c t i v e i n d e x p a t t e r n d e v e l o p e d as n 1 = n Q ( l - e x p - t / T g ) . From the time 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 he o b t a i n e d , c o i n c i d e n t a l l y , 24 nm. Because of the approximate n a t u r e o f the t h e o r y and the f a c t t h a t d a t a on beam c o u p l i n g was not g i v e n , i t i s d i f f i c u l t t o a s s e s s t h e degree o f s u p p o r t which t h i s g i v e s to our v a l u e . 76. VI HOLOGRAM WRITING WITH NONUNIFORM ILLUMINATION VI-1 Introduction In t h i s chapter, hologram w r i t i n g with an a r b i t r a r y one-dimensional nonuniform ..light i l l u m i n a t i o n geometry i s studied. In Chapter 4, we developed a model for hologram w r i t i n g with a r b i t r a r y electron transport lengths under uniform l i g h t i l l u m i n a t i o n . Most of the models given i n the l i t e r a t u r e are for t h i s type of i l l u m i n a t i o n . Hologram w r i t i n g with nonuniform i l l u m i n a t i o n i s , however, the most l i k e l y geometry to be employed i n pr a c t i c e i n holographic memory systems, say, where a number of small holograms i n the form of a g r i d would probably be stored i n the c r y s t a l . Nonuniform i l l u m i n a t i o n produces, i n addition to the sin u s o i d a l space charge f i e l d c o n s t i t u t i n g the Hologram, a large scale space charge f i e l d associated with the envelope of the l i g h t pattern. Such a f i e l d was observed by Chen (1969) i n h i s o r i g i n a l compensator experiments. The feed-back e f f e c t of t h i s large scale f i e l d on the r e d i s t r i b u t i o n of photoliberated electrons was neglected i n most previous models since uniform i l l u m i n a t i o n was assumed. Experimental investigations of the e f f e c t of the f r a c t i o n a l i l l u m i n a t i o n of the c r y s t a l on hologram w r i t i n g were made by Cornish et a l . (1976a) and i t was reported that t h i s large scale f i e l d associated with the l i g h t envelope has a s i g n i f i c a n t e f f e c t on hologram w r i t i n g . A numerical model for hologram w r i t i n g i n the case of a Gaussian i l l u m i n a t i o n geometry was obtained by Moharam and Young (1976a). We propose i n t h i s chapter an a n a l y t i c model for hologram w r i t i n g under constant applied v o l t a g e . The model Is f o r a c r y s t a l h a v i n g a f i n i t e dark c o n d u c t i v i t y under an a r b i t r a r y o n e - d i m e n s i o n a l l i g h t i n t e n s i t y d i s t r i b u t i o n . The feedback e f f e c t o f the space charge f i e l d i s i n c l u d e d , beam c o u p l i n g e f f e c t s a r e , however, not c o n s i d e r e d s i n c e we a r e m a i n l y i n t e r e s t e d i n s t u d y i n g the l a r g e s c a l e f i e l d and i t s i n f l u e n c e on hologram w r i t i n g . VI-2 Model The i n t e r f e r e n c e p a t t e r n o f two c o h e r e n t beams s y m m e t r i c a l l y i n c i d e n t on the c r y s t a l w i t h a n g l e s ±9 and t h e i r p l a n e of i n c i d e n c e i n c l u d e s o the c - a x i s ( x - a x i s ) as i n F i g . 6.1 i s I ( x ) = I Q ( x ) ( l + m cos K x ) , (6.1) where I Q ( x ) r e p r e s e n t s the d i s t r i b u t i o n o f the e n v e l o p e of the l i g h t i n t e n s i t y and m i s t h e m o d u l a t i o n r a t i o (assumed independent o f x ) . K=4TT s i n G / A where o o A q i s the vacuum wavelength o f l i g h t . Assuming the f r e e c a r r i e r c o n c e n t r a t i o n not to v a r y w i t h time d u r i n g the hologram f o r m a t i o n and u s i n g the r e s u l t s o f Chapter 4 we can w r i t e n(x) = n D + n L ( x ) , (6.2) where n^ i s the 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 the dark and nj j(x)=g(x)x i s the c o n c e n t r a t i o n of the p h o t o l i b e r a t ' e d . c a r r i e r s , T i s the c a r r i e r l i f e t i m e and g(x) i s the volume g e n e r a t i o n r a t e g i v e n by 5«I (x) g(x) = ^—f [1 + m cos (Kx - cj>p)] , (6.3) •fyiLo where E, i s the quantum e f f i c i e n c y , a ,is the a b s o r p t i v i t y , "fico i s the quantum of l i g h t energy and <j) i s the s p a t i a l phase s h i f t i n t r o d u c e d due to the 78. X F i g . 6.1 C o n f i g u r a t i o n f o r hologram r e c o r d i n g w i t h a g e n e r a l i z e d l i g h t i n t e n s i t y d i s t r i b u t i o n . The c o o r d i n a t e system used i n r e l a t i o n t o the c - a x i s s e r v e s t o d e f i n e the s i g n o f t h e a p p l i e d v o l t a g e . 79. f i n i t e t r a n s p o r t l e n g t h o f the p h o t o l i b e r a t e d e l e c t r o n s [see Chapter 4 ] . The e f f e c t o f changes i n the t r a p occupancy on the 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 these s o u r c e s of 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 rtant than the feedback e f f e c t o f the space charge 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 by J ( x , t ) = qD q u n [ E s c ( x , t ) - j- ] + J p ( x ) , (6.4) which i n c l u d e s the d i f f u s i o n , d r i f t and the p h o t o v o l t a i c c u r r e n t J p ( x ) which i s g i v e n by [see Chapter 4] J p ( x ) = - K a I o ( x ) [1 + m cos (Kx - cf>p) ] . (6.5) E q u a t i o n (6.5) can be put i n a more a p p r o p r i a t e form f o r t h i s a n a l y s i s , w i t h the h e l p o f Eqn. (6.3) as J p ( x ) = - q y n L ( x ) E v , (6.6) where E v = KKw/qyx^ i s a c o n s t a n t h a v i n g the dimensions v o l t s ( u n i t l e n g t h ) which i s termed the v i r t u a l f i e l d ( C o r n i s h e t a l . 1976a). Eq. (6.6) . i m p l i e s t h a t s h o r t e l e c t r o n t r a n s p o r t l e n g t h i s assumed i n our a n a l y s i s . 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 charge d e n s i t y p due to c a r r i e r m i g r a t i o n i s • P s c = _ 3 J ( x , t ) . (6.7) 3x P o i s s o n ' s e q u a t i o n i s 3 E T P S C , (6-8) 3x 8 0 . where E = E ( x , t ) - — i s the t o t a l e l e c t r o s t a t i c f i e l d and e i s the p e r m i t -T sc L . . . . t i v i t y o f the medium. Combining Eqs. 6.7 and 6.8 and i n t e g r a t i n g w i t h r e s p e c t to space we o b t a i n e9E ( x , t ) ^ = - J ( x , t ) + J ( t ) , ( 6 . 9 ) 3 t where J ^ ( t ) i s the t o t a l c u r r e n t d e n s i t y f l o w i n g i n the c r y s t a l which c o n s i s t s o f the c o n d u c t i o n c u r r e n t and the d i s p l a c e m e n t c u r r e n t d e n s i t i e s . P e r f o r m i n g L a p l a c e t r a n s f o r m a t i o n on Eq. 6.9 we o b t a i n 1 T T J - J SE (x,s) - E ( x , 0 ) = - rrf-r- [E ( x , s ) - -^r- ] - „ T D • s c v ' s c v ' ' T(x) L s c v ' ' s L J sKT(x) d£n n(x) J y _ ^ l £ i + J ( s ) / e dx ' sT(x) ' n(,x) 1 - T \ D / , ° ' ( 6 . 1 0 ) where/ s i s the complex f r e q u e n c y , T(x) = e/qun(x) i s the d i e l e c t r i c r e l a x a t i o n time, E ( x , 0 ) i s the i n i t i a l v a l u e o f the space charge f i e l d due to p r e v i o u s sc , KkT exposure and E ^ = — - — i s the d i f f u s i o n e q u i v a l e n t f i e l d w i t h k Boltzmann's c o n s t a n t , T 1 i s the a b s o l u t e temperature. Eq. 6.10 can be r e w r i t t e n as E s c ( x , s ) = [ E ^ x . O ) + ( J + ^ - E , ) / ^ ) - ^ 2 _ _ • d £ n n ( x ) + j T - ( s ) / e ] i ( S + 1 / T ( X ) ) . ( 6 . 1 1 ) The boundary c o n d i t i o n which i 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 ( z e r o o r o t h e r w i s e ) g i v e n by E T dx = V , ( 6 . 1 2 ) 81. t h i s means t h a t and E ( x , t ) dx = 0. " (6.13) L S C A p p l y i n g Eq. 6.13 to Eq. 6.11 we o b t a i n J T (s)= - e Q 1 ( s ) Q 2 ( s ) - ~ ( E y + J ) [ Q ^ s J - s ] , . (6.14) E s c ( s ) " ifiTrTxT " Q 1 ( s ) Q 2 ( s ) / ( ^ i / T W ) + ( E v + I ) [17a— Q 1 ( s ) / s ( s + l / T ( x ) ) ] . -E ° d £ n r i ( - x ) / s ( s + l / T ( x ) ) 5 (6.15) KT(x) ' dx Q = L [ 1 d x / ( s + l / T ( x ) ) ] 1 and Q 2 = — L L E s c ( x , 0 ) d x L s + l / T ( x ) In Eqs. 6.14 and 6.15 we n e g l e c t e d a s m a l l e x t r a d i f f u s i o n term caused by the p o s s i b l e i n e q u a l i t y o f the i l l u m i n a t i o n c o n d i t i o n s a t the c-edges o f the c r y s t a l . Eqs. 6.14 and 6.15 a r e g e n e r a l s o l u t i o n s , w i t h i n the l i m i t s o f the model, to the r e s u l t a n t p h o t o c u r r e n t and space charge f i e l d a t any l o c a t i o n i n the c r y s t a l a t a l l time s t a g e s d u r i n g hologram w r i t i n g o r e r a s i n g . To o b t a i n a complete s o l u t i o n we have to know the d i s t r i b u t i o n o f the l i g h t i n t e n s i t y i n v o l v e d so as to be a b l e t o c a l c u l a t e the i n t e g r a l s i n Q ^ s ) and Q 2 ( s ) . 82.. VI-2.1 The S m a l l Time A p p r o x i m a t i o n In t h e i n i t i a l s t a g e s o f exposure the t o t a l c u r r e n t d e n s i t y and the r e s u l t i n g space charge f i e l d can be deduced from Eqs. 6.14 and 6.15 r e s p e c t i v e l y by assuming s > > l / T ( x ) , (note t h a t T(x) i s never z e r o due to the f i n i t e dark c o n d u c t i v i t y ) . J T ( t ) * J A V ( 0 ) [1 - t ( i ) A V ] - a A V ( E v + I ) , (6.16) E s c ( S , t ) = E s c(x.,0) [ l - t ( i ) A V ]+ ( E v + 1) fcj^j- - ) A V ] t . qD_ d£n n ( x ) t ( 6 > l y ) e n (x) where J A y ( 0 ) = £ qyn(x)E (x,0)dx i s the average c o n d u c t i o n c u r r e n t T S c L 1 produced by the i n i t i a l space charge f i e l d . a ^ v = ^qyn(x)dx i s the average 1 0AV c o n d u c t i v i t y o f the c r y s t a l and ( — ) = i s the average r a t e o f charge J- A V £ r e l a x a t i o n due to the c o n d u c t i v i t y o f the medium. VI-2.2 I n t e r m e d i a t e and S a t u r a t i o n Stages.. E q u a t i o n s 6.14 and 6.15 d e s c r i b e t h e development o f the p h o t o -c u r r e n t and the space charge f i e l d i n the.complex f r e q u e n c y domain ( s -domain). The time domain s o l u t i o n can be found by p r e f o r m i n g the i n v e r s e L a p l a c e t r a n s f o r m on the two e q u a t i o n s . The e x p r e s s i o n s a r e complex and n u m e r i c a l t e c h n i q u e s , such as FFT, a r e p o s s i b l e , no new i n s i g h t i n t o the problem i s , however, p r o v i d e d t h a t cannot be g a i n e d from s t u d y i n g the s a t u r a t i o n or s t e a d y - s t a t e . The s t e a d y - s t a t e i s found by a p p l y i n g the f i n a l v a l u e theorem of the L a p l a c e t r a n s f o r m , which y i e l d JT • - ( E v + r ) / ( 1 / a ) .83. (6.18) E (x) = .(E. + J ) (1 s c v ' v v L T(x) . _ qD d t o a ( x ) •, (6.19) T AV i s VI-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 From Eqs. 6.14 and 6.15 we can determine the parameters t h a t govern the development o f the p h o t o c u r r e n t and space charge f i e l d a t a l l exposure t i m e s . These a r e the i l l u m i n a t i o n d i s t r i b u t i o n , the c r y s t a l con-d u c t i v i t y (dark and p h o t o i n d u c e d ) , a p p l i e d v o l t a g e , the b u l k p h o t o v o l t a i c e f f e c t and t h e v a l u e and d i s t r i b u t i o n o f any i n i t i a l space charge f i e l d s . d e n s i t y t h a t flows i n the c r y s t a l i s n o t c o n s t a n t w i t h time, i t has an i n i t i a l and a f i n a l v a l u e . T h i s i s t r u e even i f t h e r e were no i n i t i a l f i e l d s to be r e l a x e d o r any a p p l i e d v o l t a g e s (V=0), t h i s i s because o f the feedback e f f e c t o f the i n d u c e d space charge f i e l d . The p h o t o c u r r e n t w i l l n ot change w i t h ..... time and w i l l be e q u a l to the p h o t o v o l t a i c c u r r e n t o n l y f o r the case o f a s h o r t - c i r c u i t e d c r y s t a l t h a t i s 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 w i t h no space charge f i e l d s p r e s e n t from p r e v i o u s exposures. These c o n d i t i o n s := s h o u l d be f o l l o w e d when a t t e m p t i n g t o measure the 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 s e c r y s t a l s . In Eqs. 6.17 and 6.19 the term r e p r e s e n t i n g the hologram component produced by d i f f u s i o n mechanism i s a " l o c a l " term. That i s t h i s term depends o n l y on the l o c a l i l l u m i n a t i o n s t a t e o f the sample, and i s independent o f the Comparing Eqs. 6.16 and 6.18 r e v e a l s t h a t the t o t a l p h o t o c u r r e n t 84. l i g h t i n t e n s i t y d i s t r i b u t i o n i n o t h e r r e g i o n s o f t h e sample. In a d d i t i o n , t h i s d i f f u s i o n c o n t r i b u t i o n term i s independent o f the e l e c t r i c a l c o n d i t i o n s at the c r y s t a l t e r m i n a l s , i . e . whether t h e r e i s a v o l t a g e a p p l i e d o r the c r y s t a l i s s h o r t - o r - o p e n - c i r c u i t e d . In f a c t , t h i s term i s i d e n t i c a l t o i t s c o u n t e r p a r t i n t h e a n a l y s i s o f B l ^ t e k j a e r (1977) ( h i s Eq. 13) f o r the case of an o p e n - c i r c u i t e d c r y s t a l . The terms d e s c r i b i n g the t o t a l c u r r e n t d e n s i t y i n Eqs. 6.16 and 6.18 a r e " g l o b a l " terms, i . e . they depend on the s t a t e o f i l l u m i n a t i o n o f the whole sample as e v i d e n c e d by the a v e r a g i n g i n t e g r a l s i n the e x p r e s s i o n s . Of c o u r s e the c u r r e n t s t r o n g l y , depends on the e l e c t r i c a l c o n d i t i o n s of t h e c r y s t a l t e r m i n a l s . In Eqs. 6.17 and 6.19 the terms d e s c r i b i n g the hologram component due to d r i f t o r the b u l k p h o t o v o l t a i c e f f e c t a r e "mixed" terms. They not o n l y depend on the l o c a l l i g h t i n t e n s i t y but a l s o on the i l l u m i n a t i o n s t a t e throughout the c r y s t a l . These terms depend a l s o on the e l e c t r i c a l c o n d i t i o n s a t the c r y s t a l t e r m i n a l s . D u r i n g the development of the hologram we have v a r i o u s e f f e c t s due to the feedback e f f e c t o f the space charge f i e l d . These i n c l u d e b o t h the 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 the i n t e r f e r e n c e o f the beams and the 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 the envelope of the l i g h t i n t e n s i t y p a t t e r n . To a n a l y z e the e f f e c t on the hologram, we assume t h a t the e n v e l o p e of the l i g h t i n t e n s i t y may be t aken 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 to the s c a l e o f the v a r i a t i o n s of the l i g h t i n t e n s i t y due to i n t e r f e r e n c e . Thus the envelope may be c o n s i d e r e d as c o n s t a n t over a d i s t a n c e g i v i n g a few p e r i o d s o f the s i n u s o i d a l components. T h i s a l l o w s the p h o t o i n d u c e d space charge f i e l d to 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 fundamental and h i g h e r harmonics s i n u s o i d a l components (Moharam 1978b). 85. From Eqs. 6.17 and 6.19 i t i s deduced t h a t t h i s e nvelope f i e l d i s produced o n l y by the terms r e s u l t i n g from the d r i f t mechanism due to a p p l i e d v o l t a g e s and the b u l k p h o t o v o l t a i c e f f e c t . The d i f f u s i o n mechanism produces o n l y a v e r y s m a l l c o n t r i b u t i o n n e g l i g i b l e i n comparison w i t h the o t h e r two mechanisms. The r e s u l t s to be p r e s e n t e d i n the s u b s e c t i o n s below a r e l i m i t e d to the s t e a d y - s t a t e space charge f i e l d . D i f f e r e n t i l l u m i n a t i o n geometries are c o n s i d e r e d f o r comparison purposes. VI-3...1 E f f e c t o f E x t e n t of F r a c t i o n a l I l l u m i n a t i o n E x a m i n a t i o n o f Eq. 6.19 shows t h a t the c r y s t a l l e n g t h a f f e c t s the v a l u e o f the space charge f i e l d through the f i e l d component due to d r i f t , m a i n l y through the parameter T the average d i e l e c t r i c r e l a x a t i o n time o f the i l l u m i n a t e d c r y s t a l . The d i f f u s i o n c o n t r i b u t i o n to the sapce charge f i e l d i s independent of the c r y s t a l l e n g t h . We s h a l l i n v e s t i g a t e the e f f e c t o f the r a t i o o f the i l l u m i n a t e d p a r t o f the c r y s t a l t o the c r y s t a l l e n g t h f o r two types o f i l l u m i n a t i o n g e o m e t r i e s . G a u s s i a n i l l u m i n a t i o n d i s t r i b u t i o n o f the form I ( x ) = I (1+ m cos Kx) exp ( - 2 x 2 / p 2 ) , (6.20) where I Q i s the average l i g h t i n t e n s i t y a t the c e n t r e o f the beam (x=0), m i s the m o d u l a t i o n r a t i o a l s o atJthe c e n t e r of the beam and p i s the beam r a d i u s a t e - 2 p o i n t s . The o t h e r type of i l l u m i n a t i o n geometry c o n s i d e r e d i s the r e c t a n g u l a r i l l u m i n a t i o n 86. =. I0(l+mcosKx); 0<|x|<£ I(x) . (6.21) = 0 ; £<|x|<L For this case closed form expressions for the large scale f i e l d ) and the fundamental component of the hologram can be obtained Edc = ( E v + I > U-[2£/L + (l-2£/L)(l-m'2)*(n/nD)]~]-}; (6.22) for 0<|x|<l and E d c=(E v+ I ){l-[(l-2£/L) + (2£/L)(l-m'2) (n^n)]" 1} , (6.23) for £<|x|<_L , where m'=m(nL/n). The fundamental f i e l d component amplitude due to d r i f t i s E, = 2(E + J )(l/m ,)[l-(l-m , 2) i][2£/L+(l-2£/L)(n/n n)(l-m , 2) i]" 1 . (6.24) This component exists only in the illuminated area. In Fig. 6.2 the spatial distribution of the amplitude of the local Fourier dc component of the photoinduced space charge f i e l d at satura-tion is plotted for a Gaussian illumination. Plotted also i s the contribution of d r i f t and diffusion to the fundamental sinusoidal components. This i s for beams having m=l, c r L / a D = 5 and 2p/L=.4. Here is the dark conductivity of the crystal and o =qunT is the maximum value of the photoinduced conduct-i v i t y . The parameters were chosen identical to those used by Moharam and Young (1976a). Fig. 6.2 shows that the maximum of the fundamental sinusoidal component due to d r i f t is not at the center of the beam (maximum intensity). This i s due to the negative feedback effect of the local dc f i e l d component 87, - -5 - -3 - -1 0 -1 -3 -5 DISTANCE ALONG CRYSTAL LENGTH Fig. 6.2 Spatial distribution 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 diffusion (in units of E n) for o /o = 5, 2p/L = 0.4, and m=lYo. which i s maximum a t the center o f the beam. The d i f f u s i o n component i s not a f f e c t e d by the l o c a l dc space charge f i e l d and i t s maximum c o r r e s p o n d s t o the i n t e n s i t y maximum. F i g . 6.3 shows the manner i n which the am p l i t u d e o f the envelope f i e l d ( l o c a l dc f i e l d ) a t s a t u r a t i o n v a r i e s w i t h t h e r a t i o o f beam w i d t h t o c r y s t a l l e n g t h . T h i s r a t i o i s d e f i n e d as 2&/L f o r r e c t a n g u l a r i l l u m i n a t i o n and 2p/L f o r G a u s s i a n i l l u m i n a t i o n . Note t h a t f o r narrow beams the two types o f i l l u m i n a t i o n produce n e a r l y e q u a l envelope f i e l d a m p l i t u d e s a t the c e n t e r o f the beam. F o r b o t h g e o m e t r i e s the env e l o p e f i e l d s d e c r e a s e i n amp l i t u d e as the f u l l i l l u m i n a t i o n s t a t e i s approached, t h i s e f f e c t has been e x p e r i m e n t a l l y o b s e r v e d by Krumins e t a l . 1978. The f i e l d s a r e e x a c t l y z e r o when t h e c r y s t a l i s f l o o d e d w i t h u n i f o r m i n t e n s i t y l i g h t , t h i s i s s a t i s f i e d f o r the case o f r e c t a n g u l a r i l l u m i n a t i o n and the r e s u l t i n g e n v e l o p e f i e l d d e c r e a s e s f a s t e r than i t s c o u n t e r p a r t i n the G a u s s i a n i l l u m i n a t i o n c a s e . F i g . 6.4 shows the v a r i a t i o n o f the fundamental component amplitude due to d r i f t a t s a t u r a t i o n w i t h the r a t i o o f beam w i d t h t o c r y s t a l l e n g t h f o r b o t h types o f i l l u m i n a t i o n a t the c e n t e r o f the beam a l s o . F o r bo t h types of: i l l u m i n a t i o n the fundamental component i n c r e a s e s as t h e c r y s t a l approaches the f u l l i l l u m i n a t i o n s t a t e . The r e a s o n f o r t h i s t r e n d i s the d e c r e a s i n g feedback e f f e c t o f the envelope f i e l d s ( F i g . 6.3) and more e f f i c i e n t r e c o r d i n g r e s u l t s . R e c t a n g u l a r i l l u m i n a t i o n produces more e f f i c i e n t r e c o r d i n g near the f u l l i l l u m i n a t i o n s t a t e s i n c e f o r t h i s geometry the envelope f i e l d approaches the z e r o v a l u e f a s t e r than i n the G a u s s i a n i l l u m i n a t i o n c a s e . 1 89. GAUSS IAN NORMALIZED BEAM WIDTH F i g . 6.3 The l o c a l dc component 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 o f E + V/L) at the c e n t e r o f t h e beam v e r s u s the n o r m a l i z e d beam w i d t h f o r the G a u s s i a n and r e c t a n g u l a r i l l u m i n a t i o n g e o m e t r i e s , a fa = 10, m=1.0. LU £ O u o _ l UJ u_ Ul < o zz ZD 1 \-rr -5 h NORMALIZED Wl DTH F i g 6 . 4 The fundamental F o u r i e r component o f the space charge f i e l d due t o d r i f t ( i n u n i t s o f E +V/L) a t t h e c e n t e r o f t h e beam v s . the n o r m a l i z e d beam w i d t h . Two d i f f e r e n t : i l l u m i n a t i o n g e o m e t r i e s a r e assumed the G a u s s i a n and t h e r e c t a n g u l a r . 0 T/a_=lO, m=1.0. 90. 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 . (1976a) c o n f i r m t h i s dependence o f the fundamental component o f the f i e l d on the f r a c t i o n a l i l l u m i n a t i o n o f the c r y s t a l . T h e i r r e s u l t s show t h a t the l a r g e r the r a t i o of beam diameter to c r y s t a l s i z e , the l a r g e r the p h o t o i n d u c e d space charge f i e l d and the l a r g e 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 the g r a t i n g . VI-3.2 The E f f e c t o f the Dark C o n d u c t i v i t y E x a m i n a t i o n o f Eq. 6.24 shows t h a t , i n the i l l u m i n a t e d a r e a , an i n c r e a s e i n the 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 the dark 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 the m o d u l a t i o n r a t i o m. An accompanying d e c r e a s e i n the a b i l i t y o f the c r y s t a l t o s t o r e holograms s h o u l d thus r e s u l t . F i g . 6.5 d e p i c t s such a t r e n d f o r the fundamentmental f i e l d component due to d r i f t f o r the case o f a f u l l y i l l u m i n a t e d c r y s t a l ( r a t i o 2£/L=l i n Eq. 6.24). When the r a t i o o f the p h o t o c o n d u c t i v i t y to the dark c o n d u c t i v i t y (a /a-) exceeds a p p r o x i m a t e l y 10, the e l e c t r i c f i e l d depends l e s s on t h i s Li D r a t i o . F i g . 6.6 shows the e f f e c t o f a /a on the s a t u r a t i o n v a l u e o f the e nvelope f i e l d i n the i l l u m i n a t e d a r e a f o r beam w i d t h e q u a l one t e n t h the c r y s t a l l e n g t h f o r the case o f r e c t a n g u l a r i l l u m i n a t i o n . G a u s s i a n i l l u m i n a t i o n g i v e s an envelope f i e l d dependence v e r y much s i m i l a r t o t h a t i n F i g . 6 . 6 . The fundamental f i e l d component of the space charge f i e l d due to d r i f t f o r the case of nonuniform i l l u m i n a t i o n shows a d i f f e r e n t depend-ence on o ^ / O p f ° r r e c t a n g u l a r i l l u m i n a t i o n and G a u s s i a n i l l u m i n a t i o n as shown i n F i g s . 6.7 and 6.8 r e s p e c t i v e l y . T h i s can be a s c r i b e d t o the s c r e e n i n g or feedback e f f e c t o f the e n v e l o p e f i e l d . 91. 0-1 1 0 1 0 1 0 F i g . 6.5 The fundamental F o u r i e r component o f the space charge f i e l d due t o d r i f t [ i n u n i t s o f m(E +V/L)] under f u l l i l l u m i n a t i o n c o n d i t i o n s v s . o^/a^. The c u r v e s a r e g i v e n i n terms o f t h r e e d i f f e r e n t v a l u e s o f m. UJ z o Q. o o LU C J Q •5 U 0 0-1 cr* 1 0 1 0 10 F i g . 6.6 The l o c a l dc component o f the space charge f i e l d ( i n u n i t s o f E^+V/L) v s . o /a . 2£/L=0.1, t h r e e d i f f e r e n t v a l u e s o f m a r e assumed. Li U 1 0 1 1 0 1 0 1 0 F i g . 6.7 The fundamental component o f the space charge f i e l d ( i n u n i t s o f m(E +V/L) v s . c L / a D f o r r e c t a n g u l a r i l l u m i n a t i o n . 2£/L=0.1, t h r e e d i f f e r e n t v a l u e s o f m a r e assumed. F i g . 6.8 The fundamental component o f the space charge f i e l d due to d r i f t [ i n u n i t s o f m(E +V/L) ] and due to d i f f u s i o n ( i n u n i t s o f mE];)) v s . o^/a f o r G a u s s i a n i l l u m i n a t i o n geometry. 2p/L=0.1, t h r e e d i f f e r e n t v a l u e s o f m are assumed. As was s t a t e d p r e v i o u s l y , the d i f f u s i o n c o n t r i b u t i o n to the fundamental component of the space charge f i e l d i s independent o f the i l l u m i n a t i o n geometry. In F i g . 6.8 t h i s component i s p l o t t e d to show i t s dependence on the r a t i o ala. From the p r e v i o u s diagrams we draw the f o l l o w i n g c o n c l u s i o n s . F o r hologram r e c o r d i n g i n a f u l l y i l l u m i n a t e d c r y s t a l i t i s advantageous to have a la > 10 ( F i g . 6.5) to o b t a i n good s t o r a g e e f f i c i e n c y . U s u a l l y the Lt JJ dark c o n d u c t i v i t y o f the c r y s t a l i s so low t h a t the r a t i o a /a > 1 0 3 . F o r the more p r a c t i c a l s i t u a t i o n of hologram r e c o r d i n g i n a p a r t i a l l y i l l u m i n a t e d c r y s t a l , F i g s . 6.6, 6.7 and 6.8 i n d i c a t e t h a t when a /a ^ 1 good r e c o r d i n g J_i JJ r e s u l t s . T h i s i s so s i n c e i n t h i s neighbourhood o f the r a t i o a /a the s a t u r a t i o n v a l u e o f the envelope f i e l d i s o n l y a s m a l l f r a c t i o n o f the v i r t u a l or a p p l i e d f i e l d and the fundamental f i e l d component c o n s i t i t u t i n g the hologram i s maximum. P r a c t i c a l l y , we can v a r y the r a t i o a /a by u n i f o r m l y f l o o d i n g Lt JJ the c r y s t a l w i t h a n o t h e r n o n - i n t e r f e r i n g l i g h t o f i n t e n s i t y e q u a l to t h a t o f the r e c o r d i n g l i g h t . T h i s b i a s i n g l i g h t i n t r o d u c e s an a d d i t i o n a l p h o t o -v o l t a i c c u r r e n t and p h o t o c o n d u c t i v i t y components which we can f o r m a l l y r e -p r e s e n t as a " d a r k c o n d u c t i v i t y " d u r i n g r e c o r d i n g . The b i a s i n g l i g h t can be o b t a i n e d from a non-coherent s o u r c e , another l a s e r or from the same l a s e r s o u r c e but w i t h a p o l a r i z a t i o n normal t o t h a t used f o r th e r e c o r d i n g beams. F i g . 6.9 shows the s p a t i a l d i s t r i b u t i o n o f the a m p l i t u d e o f the l o c a l F o u r i e r dc component of the p h o t o i n d u c e d space charge f i e l d a t s a t u r a -t i o n t o g e t h e r w i t h the c o n t r i b u t i o n of d r i f t and d i f f u s i o n to the fundamental s i n u s o i d a l components p l o t t e d f o r G a u s s i a n i l l u m i n a t i o n . T h i s i s f o r beams h a v i n g m=l, a la = 1000 and beam di a m e t e r to c r y s t a l l e n g t h 2p/L=.4. Com-Li JJ p a r i n g t h i s f i g u r e w i t h the p r e v i o u s F i g . 6.2 shows the e f f e c t of the dark - • 5 - 3 -.1 0 -1 -3 -5 DISTANCE ALONG CRYSTAL L E N G T H F i g . 6.9 S p a t i a l d i s t r i b u t i o n o f some F o u r i e r components o f the p h o t o -induced f i e l d due t o d r i f t ( i n u n i t s o f E +V/L) and due t o d i f f u s i o n ( i n u n i t s o f E ) f o r a /o =1000> 2p/L=0.4Y and m=1.0. c o n d u c t i v i t y on 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 f i e l d components. I t s e f f e c t i s most pronounced on the fundamental f i e l d component due to d r i f t which i s g r e a t l y i n f l u e n c e d by the n e g a t i v e feedback e f f e c t o f the envelope f i e l d . V I I AN INTERFEROMETRIC METHOD FOR OBSERVING THE SPACE CHARGE FIELD V I I - 1 I n t r o d u c t i o n R e f l e c t i o n s a t the s u r f a c e s o f the c r y s t a l produce l i g h t and dark f r i n g e s t h a t depend on the t h i c k n e s s o f the c r y s t a l and the i l l u m i n a t i o n geometry. The f r i n g e s s h i f t u n i f o r m l y on h e a t i n g the c r y s t a l o r on a p p l y i n g an e l e c t r i c f i e l d . The geometry o f t h e f r i n g e s i s an i n d i c a t i o n o f the l o c a l v a r i a t i o n o f the o p t i c t h i c k n e s s o f the c r y s t a l . The arrangement to be d e s c r i b e d i s s i m p l e and does not cause any i n t e r f e r e n c e w i t h the hologram r e c o r d i n g p r o c e s s . Furthermore the l o c a l v a r i a t i o n s i n the r e f r a c t i v e i n d e x can be v i s u a l l y examined. Changes i n the 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 can be viewed i n d e p e n d e n t l y and the s i g n and a m p l i t u d e o f the i n d e x change e s t i m a t e d . The i n t e r f e r o m e t r i c method to be d e s c r i b e d i n t h i s c h a p t e r i s adapted from the method used by C o r n i s h e t a l . (1976b). VII-2 F r i n g e F o r m a t i o n Arrangement A monochromatic l i g h t s o u r c e S i s assumed to be p l a c e d i n f r o n t o f the c r y s t a l as shown i n F i g . 7.1. The i n c i d e n t s p h e r i c a l wave i s p a r t i a l l y r e f l e c t e d a t the f i r s t s u r f a c e o f the c r y s t a l and appears t o be p r o p a g a t i n g from the secondary s o u r c e S^. The r e m a i n i n g p o r t i o n o f the wave i s t r a n s m i t t e d through the c r y s t a l and i s p a r t i a l l y r e f l e c t e d a t the second s u r f a c e and the r e f l e c t e d s p h e r i c a l wave appears t o be p r o p a g a t i n g from the secondary s o u r c e S 2 . The 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 s i s n e g l e c t e d i n t h i s a n a l y s i s . F i g . 7.1 I l l u s t r a t i n g r e f l e c t i o n o f l i g h t from a p o i n t s o u r c e from the f r o n t and back s u r f a c e s o f a LiNbO c r y s t a l . . The i n t e r f e r e n c e p a t t e r n can be seen on a s c r e e n p l a c e d i n the i n t e r f e r e n c e r e g i o n (the shaded r e g i o n ) . 98. I n t e r f e r e n c e f r i n g e s a r e v i s i b l e on any p l a n e i n the shaded r e g i o n common to the two waves from the s o u r c e s and Such f r i n g e s a r e s a i d t o be n o n - l o c a l i z e d (Born and Wolf 1965). In the p l a n e o f o b s e r v a t i o n , F i g . 7.2, the l o c u s o f p o i n t s P f o r which the phase d i f f e r e n c e between the waves from S^ and S^ i s c o n s t a n t is, the s u r f a c e d e f i n e d by S^P - S^P = c o n s t a n t . Hence, the maxima and minima o f the r e s u l t a n t i n t e n s i t y form a f a m i l y o f h y p e r b o l o i d s o f r e v o l u t i o n about S^ and S^ as common f o c i . F r i n g e s i n a p l a n e normal t o the p e r p e n d i c u l a r b i s e c t o r o f S^S^ a r e s e c t i o n s o f t h e s e h y p e r b o l o i d s , and a r e themselves h y p e r -b o l a e as shown i n F i g . 7.4(a) f o r the case of a f r e s h c r y s t a l . V I I -3 E x p e r i m e n t a l P r o c e d u r e s The e x p e r i m e n t a l arrangement f o r the photographs taken i n t h i s c h a p t e r i s shown i n F i g . 7.3. An argon i o n l a s e r beam (514.5 nm, 2.8 mm diameter) i s used to i n d u c e the i n d e x inhomogeniety a t normal i n c i d e n c e . The beam from a He-Ne l a s e r (632.8 nm) i s s p a t i a l l y f i l t e r e d and expanded t o i l l u m i n a t e the whole c r y s t a l . VII-4 E x p e r i m e n t a l R e s u l t s Photographs o f the f r i n g e s a r e shown i n F i g . 7.4, The photos were o b t a i n e d .with a 35 mm-SLR camera w i t h a macro l e n s a t t a c h e d . The photos on page 100 ( F i g - 7.4 ( a ) , 7.4 ( c ) , 7.4 ( e ) ) c o r r e s p o n d to the e x t r a o r d i n a r y i n d e x and those on page 101 ( F i g . 7.4 ( b ) , 7.4 ( d ) , 7 . 4 / ( f ) ) to the o r d i n a r y i n d e x . F i g . 7.2 Geometry o f t h e i n t e r f e r e n c e f r i n g e s t h a t can be seen on the s c r e e n o f F i g . 7.1. SPATIAL F I L T E R HE-NE L A S E R A/2 PLATE CRYSTAL ARGON L A S E R F i g . 7.3 The e x p e r i m e n t a l arrangement employed t o i n d u c e an i n d e x change by t h e . a r g o n l a s e r beam and to be viewed by l i g h t from a He-Ne l a s e r . F i g . 7.4 I n t e r f e r e n c e f r i n g e s i n a Fe-doped c r y s t a l showing t h e o p t i c a l l y induced changes i n the r e f r a c t i v e i n d i c e s . ( a ) , (c) and (e) a r e photos taken w i t h e x t r a o r d i n a r y p o l a r i z e d l i g h t and show the v a r i a t i o n s i n n F i g . 7.4 ( b ) , (d) and ( f ) a r e photos t a k e n w i t h o r d i n a r y p o l a r i z e d l i g h t and show the v a r i a t i o n s i n n n . 102. Figs.7.4 (a) and 7.4 (b) show the c r y s t a l a f t e r any o p t i c a l l y -i n d u c e d inhomogeneity has been annealed out o f the c r y s t a l by h e a t i n g i t f o r t h r e e hours a t 275°C. The v e r t i c a l band i s due to f a u l t y p o l i s h i n g and o p t i c a l i n h o m o g e n e i t i e s i n the b u l k produced d u r i n g growth. The photos i n F i g . 7.4 (c) and 7.4 (d) have been o b t a i n e d by s l i g h t l y r o t a t i n g the c r y s t a l around an a x i s i n the p l a n e o f the page and normal to the i n c i d e n t beam d i r e c t i o n i n F i g . 7.3. F i g . 7.4 (c) e s s e n t i a l l y shows the 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 change a l o n g the c - a x i s d i r e c t i o n which i s p a r a l l e l to the u n d i s t o r t e d f r i n g e s . W h i l e F i g . 7.4 (d) shows the v a r i a t i o n s i n the o r d i n a r y i n d e x a l o n g the same d i r e c t i o n . F i g . 7.4 (e) and 7.4 ( f ) have been o b t a i n e d by s l i g h t l y r o t a t i n g the c r y s t a l around an a x i s normal to the p l a n e o f t h e page i n F i g . 7.3. The two f i g u r e s show the same changes i n the r e f r a c t i v e i n d i c e s as viewed a l o n g a d i r e c t i o n normal to the c - a x i s d i r e c t i o n . I n t h e l a s t two f i g u r e s the r e v e r s a l o f t h e f r i n g e c u r v a t u r e i n d i c a t e s , as w i l l be d i s c u s s e d i n the next s e c t i o n , a r e v e r s a l o f s i g n i n the r e f r a c t i v e i n d e x change which i s analogous t o the e x p e r i m e n t a l r e s u l t s o f Chen (1969) except t h a t h e r e the v a r i a t i o n i n n and n a r e seen s e p a r a t e l y r a t h e r than t h e v a r i a t i o n i n e o the b i r e f r i n g e n c e . The change i n the e x t r a o r d i n a r y i n d e x i s more than t h e change i n the o r d i n a r y i ndex because the e l e c t r o - o p t i c c o e f f i c i e n t o f the f i r s t ( r ^ ) . i s about t h r e e times as l a r g e as the e l e c t r o - o p t i c c o e f f i c i e n t o f the l a t t e r ( r ) . 103. VII-5 D i s c u s s i o n The f r i n g e changes produced by the p h o t o r e f r a c t i v e e f f e c t were ob s e r v e d i n a Fe-doped c r y s t a l (#3, see appendix E ) . S i m i l a r e f f e c t s were observed i n o t h e r c r y s t a l s (#1 and #2) ., No i n d e x change c o u l d be seen, however, by t h i s method i n the o t h e r a v a i l a b l e c r y s t a l s because they e i t h e r p o s s e s s e d a h i g h a b s o r p t i v i t y (low f r i n g e v i s i b i l i t y ) o r they p o s s e s s e d a weak b u l k p h o t o v o l t a i c e f f e c t ( s m a l l f r i n g e s h i f t ) . VII-5.1 E s t i m a t i o n of the magnitude o f the Space Charge F i e l d An e s t i m a t e o f t h e magnitude of the space charge f i e l d a t a c e r t a i n l o c a t i o n i n t h e c r y s t a l can be o b t a i n e d from the s h i f t i n the f r i n g e s . To move a dark f r i n g e t o the p o s i t i o n of t h e next dark f r i n g e , the o p t i c p a t h must change by A/2, where A i s the vacuum wavelength o f l i g h t . I f t h e t h i c k -ness d remains c o n s t a n t then And = A/2 and An = A/2d. F o r A = 632.8 nm and -k _ d = 2.5 mm, An = 1.3x10 . The space charge f i e l d would be E = 2An / r 0 0 n d = e • r ° . e 33 e 7.1 kV c m - 1 , f o r n = 2.24 and r,, = 3 0 . 8 x l 0 _ 1 ° cm V " 1 . e 4 In our c r y s t a l s the s i t u a t i o n i s n o t q u i t e t h a t s i m p l e . As was e x p l a i n e d i n Chapter 6, when a c r y s t a l i s p a r t i a l l y i l l u m i n a t e d an e l e c t r i c f i e l d w i l l d e v e l o p i n the i l l u m i n a t e d and dark a r e a s of the c r y s t a l . Thus the o b s e r v e d s h i f t i n the f r i n g e s c o r r e s p o n d s to the d i f f e r e n c e between these two e l e c t r i c f i e l d s . To o b t a i n the v a l u e o f the f i e l d i n each r e g i o n i t i s n e c e s s a r y t o know e i t h e r the f i e l d d i s t r i b u t i o n b e f o r e h a n d o r to compare the f r i n g e p a t t e r n s b e f o r e and a f t e r exposure to l i g h t . VII-5.2 D e t e r m i n i n g the S i g n of the Index Change The s i g n o f the i n d e x change An can be deduced w i t h r e f e r e n c e to F i g . 7.2. Assume t h a t i n some p a r t o f t h e c r y s t a l the r e f r a c t i v e i n d e x has i n c r e a s e d 104. from i t s o r i g i n a l v a l u e . At t h i s p o i n t the o p t i c p a t h d i f f e r e n c e S^P - S^P i n c r e a s e s a l s o from i t s o r i g i n a l v a l u e . But the p a t h d i f f e r e n c e i n c r e a s e s when p o i n t P moves p a r a l l e l t o S^S^ d i r e c t i o n and as i t moves i t meets h y p e r -b o l a e w i t h i n c r e a s i n g c u r v a t u r e . Thus, a t the l o c a t i o n o f the i n d e x i n c r e a s e the f r i n g e c u r v a t u r e w i l l a l s o i n c r e a s e . I f the i n d e x change was n e g a t i v e t h e o p p o s i t e takes p l a c e and t h e f r i n g e c u r v a t u r e d e c r e a s e s or even r e v e r s e s i t s d i r e c t i o n . T h i s i s e x a c t l y what happened i n the photos o f F i g . 7.4 (e) and F i g . 7.4 ( f ) . 105. V I I I INFLUENCE OF THE ENVELOPE FIELD ON HOLOGRAM STORAGE IN LiNb03 V I I I - 1 I n t r o d u c t i o n The l o c a l dc f i e l d t h a t d e v e l o p s d u r i n g p a r t i a l i l l u m i n a t i o n o f LiNb03 c r y s t a l s (Chapter 6) tends to c a n c e l the combined e f f e c t o f 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 and any e l e c t r o s t a t i c f i e l d s i n i t i a l l y p r e s e n t due to a p p l i e d v o l t a g e s . As a r e s u l t f u r t h e r r e c o r d i n g produces holograms w i t h lower d i f f r a c t i o n e f f i c i e n c i e s . F a t i g u e e f f e c t s due t o t h i s f i e l d have been r e p o r t e d w i t h L i T a O ^ ( S p i n h i r n e e t a l . 1974) and L i N b 0 3 (Nogami e t a l . 1976). In t h i s c h a p t e r we i n v e s t i g a t e the f a t i g u e e f f e c t f o r two types o f i l l u m i n a t i o n g e o m e t r i e s : o n e - d i m e n s i o n a l r e c t a n g u l a r and c i r c u l a r q u a s i G a u s s i a n . In the former, the c r y s t a l i s u n i f o r m l y i l l u m i n a t e d o v e r p a r t o f i t w i t h dark r e g i o n s on e i t h e r s i d e s a l o n g the c - a x i s . T h i s type of i l l u m -i n a t i o n geometry i s easy to a n a l y z e (Chapter 6 ) . The G a u s s i a n type of i l l u m i n a t i o n i s more l i k e l y t o be used i n p r a c t i c a l a p p l i c a t i o n s , a l t h o u g h i t i s much h a r d e r to model. The envelope f i e l d was measured u s i n g the i n t e r f e r o m e t r i c t e c h -n i q u e of Chapter 7. V I I I - 2 E x p e r i m e n t a l P r o c e d u r e s and R e s u l t s Two samples of LiNbO^rFe were used, Nos. 3 and 5 (see Appendix E ) . Both samples were baked i n a i r w h i l e packed i n L i 2 C 0 g a t 520°C f o r 20 h o u r s , a treatment due t o S t a e b l e r and P h i l l i p s 1974a, t o enhance hologram w r i t i n g and o p t i c a l e r a s u r e . 106. VTII-2.1 P h o t o c u r r e n t Measurements The magnitude 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 was i n v e s t i g a t e d f o r sample No.3 by measuring the c u r r e n t i a l o n g the c - a x i s o f the c r y s t a l as a f u n c t i o n o f the u n i f o r m l i g h t i n t e n s i t y I(A.=5l4.5 nm) w i t h the a p p l i e d v o l t a g e V as a parameter. The r e s u l t s a r e shown i n F i g . 8.1 which were l e a s t mean square f i t t e d by i = a l + bVI + cV, (8.1) g i v i n g a=0.33 pAcm 2mW - 1, b=8xl0 - 3pAcm 2V - 1mW _ 1 and c=0.34x10 1 5mho cm - 1. F o l l o w i n g the same a n a l y s i s t h a t was employed i n s e c t i o n V-3.1 o f Chapter 5 we o b t a i n a v a l u e f o r the a n i s o t r o p y c o n s t a n t K = 0.6 pA cm mW 1 which i s l e s s than h a l f the v a l u e r e a d from the curves o f F i g . 1 o f K r a t z i g and Kurz (1976, 1977b) f o r t h i s d oping and wavelength. The v i r t u a l f i e l d E^=20 ± 1 kV cm from the measurements^above • VIII-2.2 R e c t a n g u l a r I l l u m i n a t i o n Sample No.3 was p l a c e d i n the apparatus shown i n F i g . 8.2. The argon l a s e r (A=514.5 nm) was p o l a r i z e d normal to the p l a n e o f i n c i d e n c e ( o r d i n a r y p o l a r i z a t i o n ) . The beams from the v a r i a b l e beam s p l i t t e r were s p a t i a l l y f i l t e r e d and expanded to 25 mm d i a m e t e r . The a n g l e o f i n c i d e n c e i n a i r was 12 . 5 ° . Two r e c t a n g u l a r a p e r t u r e s 2 mm wide were p l a c e d i n the path o f each beam w i t h the l o n g e r d i m e n s i o n normal to the p l a n e o f i n c i d e n c e such t h a t the two bands o f l i g h t i n t e r s e c t w i t h i n the c r y s t a l . The r e f e r e n c e beam was d e f i n e d as the beam a p p r o a c h i n g the c r y s t a l from the + c - a x i s end. The r a t i o o f the r e f e r e n c e to s u b j e c t beam i n t e n s i t i e s was a d j u s t e d to 10:1 g i v i n g a m o d u l a t i o n r a t i o 0.6. The s u b j e c t beam i n t e n s i t y was chosen much lower than t h a t o f the r e f e r e n c e so t h a t the l i g h t i n t e n s i t y d i s t r i b u t i o n i s .107. F i g . 8.-i P h o t o c u r r e n t between e l e c t r o d e s on c f a c e s o f i r o n - d o p e d l i t h i u m n i o b a t e c r y s t a l as a f u n c t i o n o f l i g h t i n t e n s i t y f o r t h r e e a p p l i e d v o l t a g e s . 108. tr o Fig. 8.2 E x p e r i m e n t a l arrangement f o r s t o r i n g holograms by r e c t a n g u l a r l i g h t i n t e n s i t y d i s t r i b u t i o n t h a t p a r t i a l l y i l l u m i n a t e s t h e 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 i s c o n t i n u o u s l y m o n i t o r e d by the a u x i l i a r y He-Ne beam. 109. not g r e a t e l y d i s t u r b e d when the s u b j e c t beam i s i n t e r r u p t e d d u r i n g r e a d o u t . Hence we can assume t h a t the envelope f i e l d tends to d e v e l o p towards the same f i n a l v a l u e b o t h d u r i n g w r i t i n g and e r a s u r e . The d i f f r a c t i o n e f f i c i e n c y was measured by a He-Ne l a s e r beam (A=632.8 nm) a t i t s own Bragg a n g l e w i t h e x t r a o r d i n a r y p o l a r i z a t i o n i n s i d e the c r y s t a l . The He-Ne beam was o f c i r c u l a r c r o s s - s e c t i o n w i t h diameter s l i g h t l y l e s s than the w i d t h o f the i l l u m i n a t e d a r e a . n was d e f i n e d as t h e r a t i o o f the d i f f r a c t e d He-Ne l i g h t t o t h e i n c i d e n t beam i n t e n s i t y t a k i n g r e f l e c t i o n i n t o a c c o u n t . F i g . 8.3 d e p i c t s the 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 v e r s u s exposure d u r i n g hologram r e c o r d i n g and subsequent e r a s u r e . P l o t t e d a l s o i s the r e s u l t i n g e n v e l o p e f i e l d . The envelope f i e l d b u i l d s up w i t h exposure d u r i n g r e c o r d i n g and e r a s u r e . The s a t u r a t i o n v a l u e from t h e i n t e r -f e r e n c e f r i n g e s was about 21 kV cm - 1. L e a s t mean square f i t t i n g to the f u n c t i o n A [ l - e x p ( - B U ) ] , where U i s the exposure, gave A=23 kV c m - 1 and B= 0.05 J - 1 cm 2 w i t h s t a n d a r d d e v i a t i o n f o r A 1.5 kV cm - 1. From the above f o r m u l a , the p r e d i c t e d s a t u r a t i o n v a l u e of the envelope f i e l d i s h i g h e r than observed but w i t h i n the bounds o f e x p e r i m e n t a l e r r o r s . To i n v e s t i g a t e the i n f l u e n c e o f the envelope f i e l d on hologram s t o r a g e , the f o l l o w i n g s e t o f experiments was performed ( F i g . 8.4). The c r y s t a l was f i r s t a n n e a l e d a t 275°C f o r two hours t o e r a s e any r e s i d u a l space charge f i e l d s t h a t might be p r e s e n t due to p r e v i o u s e x p o s u r e s . A hologram was r e c o r d e d w i t h an exposure 2.3 J cm 2 and a d i f f r a c t i o n e f f i c i e n c y n n — 2% r e s u l t e d . : no. 0 10 20 E X P O S U R E ( J / c m 2 ) F i g . 8.3 Developement 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 and the env e l o p e f i e l d v s . exposure f o r r e c t a n g u l a r i l l u m i n a t i o n geometry. T o t a l power d e n s i t y 18.5 mW/cm2 and m o d u l a t i o n r a t i o 0.6. 5 10 15 EXPOSURE ( J / c m 2 ) F i g . 8.4 E f f e c t o f the p r e s e n c e o f an i n i t i a l e n v e l o p e f i e l d i n the i l l u m i n a t e d a r e a on the 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 w i t h exposure. 111. The c r y s t a l was annealed a g a i n and w i t h the r e f e r e n c e beam s h i n i n g -2 -1 f o r an exposure 3.8 Jem an envelope f i e l d E 2-7 kVcm d e v e l o p e d . A _2 hologram was then r e c o r d e d w i t h the same exposure (2.3 Jem ) as above. The r e s u l t i n g d i f f r a c t i o n e f f i c i e n c y was n 2 - 1% . The c r y s t a l was annealed once more and w i t h the r e f e r e n c e beam s h i n i n g f o r an exposure 14.4 Jcm -^ an envelope f i e l d E 3 ~ 16 k V c m - 1 d e v e l o p e d . A hologram was then r e c o r d e d w i t h exposure 2.3 Jcm -^ as b e f o r e . The r e s u l t i n g d i f f r a c t i o n e f f i c i e n c y was n 3 = 0.1% .about 24 times lower than the e f f i c i e n c y t h a t can be s t o r e d i n a f r e s h c r y s t a l . V I I I - 2 . 3 Quasi G a u s s i a n I l l u m i n a t i o n The setup o f F i g . 8.2 was m o d i f i e d by removing the beam expanders and r e p l a c i n g the r e c t a n g u l a r a p e r t u r e s w i t h c i r c u l a r ones h a v i n g a 2.8 mm di a m e t e r . The beam r a t i o was a d j u s t e d t o u n i t y . 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 h e r e as the r a t i o o f the d i f f r a c t e d argon l i g h t i n t e n s i t y t o the 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. The f o l l o w i n g s e t o f experiments was conducted on sample No. 3 ( t o t a l power i n each beam was 15 mW). In the f r e s h c r y s t a l a hologram was r e c o r d e d w i t h an exposure l e v e l 0.2 Jcm~2 and a d i f f r a c t i o n e f f i c i e n c y rij'jj 0.19% r e s u l t e d . A f t e r t h e c r y s t a l has been a n n e a l e d , the r e f e r e n c e beam i l l u m i n a t e d the c r y s t a l f o r an exposure 3.8 Jem -?- s i m i l a r t o the p r e v i o u s experiments —7 ( V I I I - 2 . 2 ) . R e c o r d i n g a hologram w i t h 0.2 Jem exposure y i e l d e d n ~J0.15%. In the f r e s h c r y s t a l the r e f e r e n c e beam i l l u m i n a t e d the c r y s t a l -2 f o r an exposure 14.4 Jem . The r e s u l t i n g d i f f r a c t i o n e f f i c i e n c y was n 3= 0.08% . 112. C r y s t a l No. 5 was p l a c e d i n the m o d i f i e d apparatus o f F i g . 8.2 and the power i n each beam was a d j u s t e d t o 17 mW. The v i r t u a l f i e l d i n t h i s sample was ^ 5 kVcm ^ as e s t i m a t e d from the s h i f t i n the f r i n g e s (Chapter 7) t h a t r e s u l t e d due t o exposure t o the q u a s i G a u s s i a n beams. F a t i g u e e f f e c t s i n t h i s sample were lower i n conroarison to sample _2 No. 3. Pre-exposure to the r e f e r e n c e beam a l o n e f o r an exposure o f 15 Jem r e s u l t e d i n a d i f f r a c t i o n e f f i c i e n c y 80% o f the . d i f f r a c t i o n e f f i c i e n c y t h a t can be w r i t t e n i n a f r e s h c r y s t a l . ( E x p o s u r e s d u r i n g r e c o r d i n g i n b o t h -2 cases were 0.3 Jem ) . A s e r i e s o f hologram r e c o r d - e r a s e c y c l e s were conducted w i t h a minimum bf 17 runs a t each v a l u e of the a p p l i e d v o l t a g e . P r i o r t o t a k i n g the measurements, however, at l e a s t 5 r e c o r d - e r a s e c y c l e s were performed to a l l o w the response o f the c r y s t a l t o s e t t l e into s t e a d y - s t a t e c o n d i t i o n s . The exposure d u r i n g h o l o g r a m . r e c o r d i n g i n .these runs was 0.3 J e m - 2 a n c j 9 i J c m - ^ d u r i n g e r a s u r e . With a s h o r t - c i r c u i t a p p l i e d the average d i f f r a c t i o n e f f i c i e n c y was 0.44%. With +3 kV a p p l i e d an. e l e c t r i c f i e l d 3 kVcm-'*' was e s t a b l i s h e d a n t i p a r a l l e l t o the c - a x i s , and the average d i f f r a c t i o n e f f i c i e n c y was 0.62% . When the d i r e c t i o n o f the f i e l d was r e v e r s e d the average d i f f r a c t i o n e f f i c i e n c y was 0.3% . F i n a l l y , when the- a p p l i e d v o l t a g e was +3 kV d u r i n g r e c o r d i n g and -3 kV d u r i n g e r a s u r e , t h e average d i f f r a c t i o n e f f i c i e n c y was 0.64% . I n . . a l l o f the above r u n s , the s t a n d a r d d e v i a t i o n i n the d i f f r a c t i o n e f f i c i e n c y d i d not exceed 0.02% . 113. V I I I - 3 D i s c u s s i o n I n the experiments o f s e c t i o n ( V I I I - 2 . 2 ) the l i g h t i n t e n s i t y en-v e l o p e i s a p p r o x i m a t e l y r e c t a n g u l a r and so i s the r e s u l t i n g e n v e l o p e f i e l d . The s c r e e n i n g o f 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 s thus u n i f o r m t h r o u g h o u t 2 the i l l u m i n a t e d a r e a . We can w r i t e n a ( E v - E ) , where E i s the env e l o p e f i e l d i n the i l l u m i n a t e d a r e a p r e s e n t a t the s t a r t o f a new hologram r e c o r d -i n g c y c l e . T a k i n g an average v a l u e f o r E ^ - 21.4 kVcm ^ from the p r e v i o u s r e s u l t s we can now q u a n t i t a t i v e l y study the r e s u l t s o f the e x p e r i m e n t s . 2 2 We had n /n - 0.5 w h i l e (E -E.) ^ 0.4. A l s o , n / n = 0.04 and (E -E,) -2. i "V o 1 . V ^ 0.06. C l e a r l y , the observed r e d u c t i o n i n the 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 c o n s e c u t i v e w r i t i n g c y c l e s can be c o r r e l a t e d w i t h the envelope f i e l d p r e s e n t at the s t a r t o f r e c o r d i n g a hologram. S c a t t e r i n g e f f e c t s ( C h a p t e r 10) a r e not a problem h e r e s i n c e the He-Ne beam does n o t s a t i s f y the Bragg c o n d i -t i o n s f o r the p a r a s i t i c holograms g e n e r a t e d by the argon l i g h t . For the case o f hologram r e c o r d i n g w i t h q u a s i G a u s s i a n beams reduced f a t i q u e response i s o b s e r v e d . n 2 / n 1 - 0.8 and M g / n - ^ - 0.4, under s i m i l a r c o n d i t i o n s o f p r e - e x p o s u r e to the r e f e r e n c e beam a l o n e b e f o r e hologram r e c o r d i n g . We can q u a l i t a t i v e l y e x p l a i n t h i s as f o l l o w s . F o r q u a s i G a u s s i a n i l l u m i n a t i o n the envelope f i e l d i s maximum a t the beam c e n t e r and d e c r e a s e s towards the edges o f the beam where i t changes s i g n on b o t h s i d e s a l o n g the c - a x i s . The s i n u s o i d a l f i d l d r e s p o n s i b l e f o r the hologram has a d i f f e r e n t d i s t r i b u t i o n ( F i g . 6.9 i n Chapter 6) -as compared^ .to the f i e l d " t h a t r e s u l t s .due"'.to r e c t a n g u l a r i l l u m i n a t i o n . At the beam c e n t e r the f i e l d has a l o c a l minimum w h i l e i t s maximum v a l u e s a r e s h i f t e d towards 1 1 4 . t h e edges o f the beam a l o n g the c-axis;where t h e r e i s l e s s s c r e e n i n g caused by the envelope f i e l d . The s c r e e n i n g e f f e c t o f the envelope f i e l d f o r q u a s i G a u s s i a n i l l u m i n a t i o n i s l e s s than i n the case o f r e c t a n g u l a r i l l u m i n a t i o n thus e x p l a i n i n g t h e observed reduced f a t i g u e . As was r e p o r t e d by S t e a b l e r and P h i l l i p s 1974a, the enhanced s t o r a g e 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 brought about by a p p l y i n g a v o l t a g e a c r o s s the c r y s t a l t e r m i n a l s was g r e a t l y reduced a f t e r o p t i c a l exposure e q u i v a l e n t t o a few w r i t e - e r a s e c y c l e s . They r e p o r t e d a l s o t h a t r e v e r s a l o f the f i e l d brought about the f i e l d enhanced s t o r a g e . In our own experiments on sample No. 5, the e f f e c t o f the a p p l i e d v o l t a g e on enhanced s t o r a g e was s t i l l s i g n i f i c a n t even w i t h f a t i g u e e f f e c t s a f t e r a few r e c o r d i n g c y c l e s (±40% 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 w i t h no v o l t a g e a p p l i e d ) . R e v e r s a l o f the v o l t a g e r e s u l t e d i n a mere 3% i n c r e a s e i n the d i f f r a c t i o n e f f i c i e n c y as compared t o i t s v a l u e when t h e a p p l i e d v o l t a g e d i d not change d u r i n g r e c o r d i n g o r e r a s i n g . VTII-4 C o n c l u s i o n s F a t i g u e e f f e c t s a r e r e l a t e d t o the envelope f i e l d t h a t r e s u l t s from nonuniform i l l u m i n a t i o n o f the c r y s t a l and depend on the type o f i l l u m i n a t i o n employed. The e x p e r i m e n t a l r e s u l t s a r e c o n s i s t e n t w i t h the assumption t h a t f a t i g u e i s caused by the s c r e e n i n g e f f e c t o f the envelope f i e l d . R e v e r s i n g the a p p l i e d v o l t a g e d u r i n g r e c o r d i n g and e r a s u r e can cause f a t i g u e e f f e c t s t o be reduced. More stu d y i s r e q u i r e d o f hologram s t o r a g e by G a u s s i a n beams i n p h o t o r e f r a c t i v e m a t e r i a l s b e f o r e t h e i r p e r -formance i s o p t i m i z e d f o r the v a r i o u s a p p l i c a t i o n s . 115. IX HOLOGRAM FIXING IN LITHIUM NIOBATE IX-1 Introduction A volume hologram must be reconstructed with the same wavelength that was used during recording to ensure that a l l frequency components of the hologram satisfy the Bragg conditions simultaneously. Optical erasure takes place, however, since the crystal i s necessarily sensitive t o l i g h t o f t h i s wavelength-• Continuous., read out thus causes loss of stored information. Techniques for fixing the holograms (i.e.,rendering them insensitive to light) have been developed. We review these techniques here and develop a mathematical model for hologram read out after the fixing process is performed. The results of this model are used to explain observations reported in the literature concerning the fixing results in LiNbO^. IX-2 Two-Photon Recording 1 The models for hologram recording mentioned so far were based on the assumption that electrons are promoted from the donor ground state to the conduction band in a single direct step, induced by absorption of one photon as indicated schematically in F i g . 9.1(a). Two alternative methods (von der Linde et a l . 1974, 1976a,..1976b) of photoionization are depicted in Fig. 9.1(b) and 9.1(c) . On the one hand, the donor can be ionized by simultaneous absorption of two photons having frequencies co^ and co^ such that the combined energy 'ft-(a)^ + a^) is sufficient to promote the electron to the conduction band, Fig. 9.1(b). On the other hand, i f there is a suitable excited state between the ground state and the conduction band, then the transition can be accomplished in a stepwise fashion. The intermediate state is populated f i r s t by absorption of a photon of frequency co^, and subsequently 116. F i g . 9.1 D i f f e r e n t p h o t o i o n i z a t i o n p r o c e s s e s 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 : (a) d i r e c t s i n g l e - p h o t o n a b s o r p t i o n , (b) two-photon a b s o r p t i o n v i a a v i r t u a l i n t e r m e d i a t e l e v e l , (c) two-photon two s t e p a b s o r p t i o n v i a a r e a l i n t e r m e d i a t e s t a t e . WRITE READ E R A S E a) b) c) F i g . 9.2 Two-photon hologram r e c o r d i n g : (a) w r i t i n g w i t h two i n t e r f e r i n g beams a t ui and an a u x i l i a r y beam a t d i f f e r e n t f r e q u e n c y a>2: (b) n o n - d e s t r u t i v e r e a d i n g a t o i l ; (c) e r a s u r e by u n i f o r m two-photon e x c i t a t i o n w i t h and u)2,. the e x c i t e d donor i s p h o t o i o n i z e d by another photon a t u^, F i g . 9 . 1 ( c ) . Once m o b i l e c a r r i e r s a r e produced, t h e r e s p e c t i v e t r a n s p o r t mechanisms d i s c u s s e d i n Chapter 2 can r e d i s t r i b u t e the e l e c t r i c a l c h a r g e . Holograms w r i t t e n by t h i s mechanism have i m p o r t a n t advantages over holograms r e c o r d e d i n t h e o r d i n a r y s i n g l e photon f a s h i o n . C o n s i d e r the w r i t i n g of an elementary g r a t i n g as i n d i c a t e d i n F i g . 9 . 2 ( a ) . Two c o h e r e n t l i g h t beams o f f r e q u e n c y u>^  i n t e r s e c t ' i n the medium f o r m i n g an i n t e r f e r e n c e ; p a t t e r n . But the photon energy "nu^ i s s m a l l e r than t h e i o n i z a t i o n energy, and no p h o t o e x c i t a t i o n can o c c u r w i t h the beams a l o n e . Photons o f fre q u e n c y form a t h i r d , s p a t i a l l y u n i f o r m l i g h t beam t h a t p r o v i d e s the e x t r a energy f o r two-photon p h o t o i o n i z a t i o n . The r a t e at which p h o t o e l e c t r o n s a r e g e n e r a t e d v a r i e s s p a t i a l l y a c c o r d i n g t o the l i g h t d i s t r i b u t i o n a t w ^ . T h e r e f o r e the r e c o r d e d hologram r e p r o d u c e s t h e i n t e r f e r e n c e p a t t e r n at j u s t as i n t h e case o f s i n g l e photon e x c i t a t i o n , and the s p a t i a l f r e q u e n c y o f the hologram g r a t i n g i s g i v e n by K = 4Trsin0/ A ^ where 6 i s the i n c i d e n c e a n g l e o f the beams at f r e q u e n c y co^ and i s t h e i r w a v e l e n g t h . As n o t e d e a r l i e r , r e c o n s t r u c t i o n of the holograms under the Bragg c o n d i t i o n s c a l l s f o r t h e same wavelength and angle o f the r e a d out beam. I n t h i s s i t u a t i o n t h e hologram can be r e c o n s t r u c t e d u s i n g the same wavelength w i t h o u t the accompanying e r a s u r e . O p t i c a l e r a s u r e can be performed w i t h u n i f o r m l i g h t beams at and , F i g . 9 . 2 ( c ) . The pho t o -c u r r e n t s g e n e r a t e d by t h e two-photon p h o t o e x c i t a t i o n w i l l n e u t r a l i z e the space charge s i m i l a r t o the s i n g l e photon c a s e . Two-photon r e c o r d i n g e n a b l e s us t o n o n d e s t r u c t i v e l y read, w i t h o u t c h a n g i n g the w a v e l e n g t h f o r r e c o r d i n g and r e a d o u t . 118. Two-photon hologram r e c o r d i n g i n n o m i n a l l y undoped LiNbO^ r e q u i r e s s t r o n g m u l t i p h o t o n e x c i t a t i o n through p i c o second o p t i c a l p u l s e s (von der L i n d e e t a l . 1974) i n o r d e r to a c h i e v e h i g h i n t e n s i t i e s ..This case corresponds to F i g . 9 . 1 ( b ) . However.with an i n t e r m e d i a t e l e v e l such as i n 3+ LiNbO^: Cr (von d e r L i n d e e t a l . 1976a, 1976b) t h e r e q u irement of h i g h pumping i n t e n s i t y i s r e l a x e d . IX-3 Thermal F i x i n g IX-3.1 I n t r o d u c t i o n A t h e r m a l f i x i n g t e c h n i q u e f o r LiNbO^ was s u c c e s s f u l l y implemented by Amodei and S t a e b l e r (1971d) and S t a e b l e r and Amodei (1972b) t o o b t a i n permanent holograms t h a t are i n s e n s i t i v e to o p t i c o r ^ t h e r m a l r e l a x a t i o n . In t h i s t e c h n i q u e , holograms are f i x e d by h e a t i n g the c r y s t a l d u r i n g o r a f t e r h o logram r e c o r d i n g t o 100 - 200°C f o r about 30 m i n u t e s . The model which Amodei and S t a e b l e r p r o p o s e d f o r t h i s p r o c e s s i s t h a t a t a temperature above. 100°C, LiNbO^ has an i o n i c c o n d u c t i v i t y ( a c t i v a t i o n energy ^ 1 . 1 ev; S t a e b l e r and Amodei 1972b) t h a t q u i c k l y n e u t r a l i z e s the space charge p a t t e r n s o f the hologram. S i n c e the e l e c t r o n s a r e i n deep t r a p s ( a c t i v a t i o n energy M..4 ev; S t a e b l e r and Amodei 1972b), t h e r m a l decay o f the e l e c t r o n i c space charge c o n s t i t u t i n g the h ologram t o be f i x e d p roceeds at a r a t e s l o w e r than t h a t o f the i o n i c compensation p r o c e s s . The r e s u l t i s an i o n i c charge p a t t e r n t h a t p e r f e c t l y m i r r o r s the p a t t e r n of the r e c o r d e d hologram. Assuming the space charge f i e l d due to the t r a p p e d e l e c t r o n s i s E o c o s K x , where K i s the g r a t i n g v e c t o r , then the e q u i l i b r i u m v a l u e of the amplitude of the 2 2 -1 r e l a x e d f i e l d i s E (1 + Z. N /,'eR' kT') . Here Z . i s t h e i o n i c c h a r g e , N i s o 1 o' - . ' i o the c o n c e n t r a t i o n of i o n i c d e f e c t s , e i s the p e r m i t t i v i t y , k i s Boltzmann's c o n s t a n t and T' i s the a b s o l u t e temperature. Assuming 119. 6 -1 K ^ 10 m T' = 400°K and Z. = q (the e l e c t r o n i c charge) t h e n N must be l ' o 15 -3 » 10 cm f o r complete r e l a x a t i o n of the f i e l d . Upon c o o l i n g to room temperature, the i o n i c p a t t e r n i s f r o z e n i n and a permanent r e p l i c a o f the hologram i s s t o r e d . Exposure to u n i f o r m c o h e r e n t or i n c o h e r e n t l i g h t w i l l r e d i s t r i b u t e the e l e c t r o n s so as t o r e v e a l a c e r t a i n f r a c t i o n o f the i o n i c f i e l d . The i o n i c d e f e c t s i n v o l v e d i n the f i x i n g p r o c e s s have not been p o s i t i v e l y i d e n t i f i e d . Some e v i d e n c e was o b t a i n e d , however, by the use o f the s c a n n i n g e l e c t r o n m i c r o s c o p e ( W i l l i a m s e t a l . 1976) t h a t S i i o n s a r e at l e a s t one i o n i c s p e c i e s c o n t r i b u t i n g t o the f i x i n g p r o c e s s . L i , Nb o r oxygen v a c a n c i e s c o u l d a l s o c o n t r i b u t e t o f i x i n g b u t cannot be i n v e s t i g a t e d by the e l e c t r o n m i c r o s c o p e . IX-3.2 F i x e d Hologram Read Out S h i n i n g u n i f o r m l i g h t on a LiNbO^ c r y s t a l i n which a hologram has been f i x e d i n d u c e s p h o t o c u r r e n t s and p h o t o c o n d u c t i v i t y which p a r t i a l l y n e u t r a l i z e s the i o n i c charge p a t t e r n . The amount of the r e t r i e v e d hologram depends on the c o n c e n t r a t i o n of e l e c t r o n donors as w e l l as on the mechanisms i n v o l v e d i n t h e t r a n s p o r t o f f r e e e l e c t r o n s . The assumptions t h a t were employed i n the p r e v i o u s c h a p t e r s of m a t h e m a t i c a l models f o r h ologram w r i t i n g i n e l e c t r o - o p t i c c r y s t a l s p r o v e d i n a d e q u a t e t o g i v e a t r u e p i c t u r e of hologram r e t r i e v a l a f t e r f i x i n g . I t i s e s s e n t i a l t o take i n t o account the e f f e c t s o f v a r i a t i o n s i n the c o n c e n t r a t i o n of e l e c t r o n donors and t r a p s on the b u l k g e n e r a t i o n r a t e and c a r r i e r l i f e t i m e . , r e s p e c t i v e l y . . 120. CO L e t n = n + E n ' v exp i p , L o P p=—oo r . - oo , • f = f Q +-' E, f exp i p Kx , ( 9 i l ) p=_oo " where n i s the c o n c e n t r a t i o n of p h o t o l i b e r a t e d e l e c t r o n s and f i s the r a t i o of t r i v a l e n t i r o n c o n c e n t r a t i o n to t h e t o t a l i r o n d o p i n g . F o r n^ and f to be r e a l we must have n = n* and f = f* where * denotes P -P P -P complex' c o n j u g a t e . The r a t e e q u a t i o n s a r e 3n T n T , 1 3J L - - a — -L, + — 3t & x q 3x ' N — ^ = g - -L , ... 3E '" £ V i i r - - J +A(O , ( 9 > 2 ) where N i s the t o t a l i r o n d o p i n g c o n c e n t r a t i o n , J i s 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 , A ( t ) i s the t o t a l c u r r e n t d e n s i t y ( c o n d u c t i o n and d i s p l a c e m e n t ) and E g c i s the s p a c e charge f i e l d due to f r e e and t r a p p e d e l e c t r o n s . g = 5 S' IN(1 - f)/"fto) i s the volume g e n e r a t i o n r a t e w i t h £ the quantum e f f i c i e n c y , S' the c a p t u r e c r o s s - s e c t i o n of photons and I i s the l i g h t i n t e n s i t y . The s p a t i a l phase s h i f t between the g e n e r a t i o n r a t e and the l i g h t i n t e n s i t y p a t t e r n , as d i s c u s s e d i n Chapter 3 » due"to 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 e g l e c t e d h e r e , x = 1/SvNf i s the c a r r i e r l i f e t i m e w i t h S the c a p t u r e c r o s s - s e c t i o n of f r e e e l e c t r o n s and v the f r e e e l e c t r o n mean square v e l o c i t y . A s l i g h t l y s i m i l a r a n a l y s i s to the one we w i l l p e r f o r m was o b t a i n e d by S t a e b l e r (1977) . 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 J i s g i v e n by J = q y n ( E s c - \ + E.cosKx) - q y n ^ + qD | | , ( 9 < 3 ) 121. where p i s the e l e c t r o n m o b i l i t y , D i s the d i f f u s i v i t y , V i s the a p p l i e d v o l t a g e (+c- end p o s i t i v e ) , L i s the c r y s t a l l e n g t h , i s the amplitude of the e l e c t r o s t a t i c f i e l d due to t h e f i x i n g i o n s » E i s the " v i r t u a l " v f i e l d r e p r e s e n t a t i v e 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 ( x - a x i s i s p a r a l l e l t o c - a x i s ) . ( C o r n i s h e t a l . 1976a) and n i s the d e n s i t y o f f r e e e l e c t r o n s . The e l e c t r o n i c space charge f i e l d s a t i s f i e s the e l e c t r o s t a t i c e q u a t i o n 3 E o „ s c e-x = q Z (Nf - n ) exp i p Kx. oX p r> p= _ oo f f (9.4) S o l v i n g Eqs. 9.1 to 9.4, i n c l u s i v e , - s i m u l t a n e o u s l y we can o b t a i n the time development o f the f i x e d h ologram d u r i n g r e a d o u t . We w i l l s t u d y o n l y the s t e a d y - s t a t e s o l u t i o n . I n the s t e a d y - s t a t e the time v a r i a t i o n s i n Eq. 9.2 v a n i s h as w e l l as the d i v e r g e n c e o f 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 . We thus o b t a i n the e q u i l i b r i u m e l e c t r i c f i e l d ( f o r p = ± 1 terms o n l y c o n s i d e r e d ) where E t ( x ) E cos(Kx + cf>) [.(1 + a ) 2 + b V qE (1 + n /n ) [ N f (1 - f ) - n ] JJ D O O Oj (3 - * ( E v + L ) / ED ' = t a n - 1 b / ( l + a) , (9.5) KkT' i s the d i f f u s i o n e q u i v a l e n t f i e l d , and q i s the 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 the dark. 122 U s u a l l y , 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 s the dominant mechanism i n V LiNbCL ; i . e . E » - , E . Thus Eq. 9.5 becomes 3 v L D E. cos(Kx + <t>') E = _ i . -t ? V ' (1 + CT) 2 , (9.6) -^g- ( i + n / n ) [Nf (1 - f ) - n ] , and £ is-JCj D o o o o cf>' = t a n 1C, IX-3.3 D i s c u s s i o n From Eq. 9.5 we can enumerate the parameters t h a t govern the a m p l i t u d e o f the r e t r i e v e d hologram. These parameters a r e : the i r o n d oping c o n c e n t r a t i o n N; o x i d a t i o n r a t i 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 E^; the V a p p l i e d e l e c t r o s t a t i c f i e l d - , i f any; t h e e f f e c t o f d i f f u s i o n E ; and the Li JJ c o n c e n t r a t i o n of p h o t o l i b e r a t e d c a r r i e r s n . o An a p p r e c i a b l e f r a c t i o n o f the f i x e d h ologram i s r e t r i e v e d o n l y when the terms a,b << 1 i n Eq. 9.7. We can e x p l a i n 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 f i x i n g r e s u l t s r e p o r t e d i n the l i t e r a t u r e i n t h e l i g h t o f Eqs. 9.5 and 9.6. S t a e b l e r and Amodei (1972b) found e x p e r i m e n t a l l y t h a t n o m i n a l l y undoped c r y s t a l s gave good f i x i n g r e s u l t s when c r y s t a l s were h e a t e d t o 100°C f o r 30 min. T h i s t e c h n i q u e was u n s u c c e s s f u l , however, ( P h i l l i p s e t a l . 1972) f o r i r o n - d o p e d c r y s t a l s (0.01 mode% and u p ) . The r e s u l t a n t d i f f r a c t i o n e f f i c i e n c y a f t e r f i x i n g was t y p i c a l l y one to two o r d e r s o f magnitude lower than the o r i g i n a l 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 t o be f i x e d . B e t t e r r e s u l t s were a c h i e v e d when holograms were f i x e d w h i l e the f i x i n g p r o c e s s was o p e r a t i v e 123. above 100°C. In n o m i n a l l y undoped c r y s t a l s most of the i r o n i o n s are i n the d i v a l e n t s t a t e (through c h e m i c a l t r e a t m e n t ) to o b t a i n adequate s e n s i t i v i t y 17 -3 f o r r e c o r d i n g . I r o n d o p i n g would be ^ 0.001 mole "A, i . e . N ^ 10 cm and f < 0.01. A l s o t y p i c a l g r a t i n g v e c t o r s a r e K y 5 ym 1 f o r most r e c o r d i n g s i t u a t i o n s . Thus, i n Eq. 9.6, the term C i s C * 2 / E v ' ' (9.7) where E i s i n kVcm ^ u n i t s . In t h i s case i f the v i r t u a l f i e l d E > 2 kVcm ^ v v — good f i x i n g r e s u l t s a r e achievable.- U s u a l l y E^ v a l u e s a r e h i g h e r than t h i s l i m i t . F o r i r o n - d o p e d c r y s t a l s , however, N i n c r e a s e s f a s t e r than the i n c r e a s e i n E ^ and the term C i n Eq. 9.6 i s no l o n g e r s m a l l compared to u n i t y . As a r e s u l t , the amount of the r e t r i e v e d h o l o g r a m w i l l be a f r a c t i o n o f the o r i g i n a l hologram as was e x p e r i m e n t a l l y o b s e r v e d . A p r a c t i c a l s o l u t i o n t o o b t a i n good f i x i n g r e s u l t s i s by i n c r e a s i n g the f a c t o r E^ i n t h e numerator of Eq. 9.6 T h i s i s a c h i e v e d by i n c r e a s i n g the a mplitude of the hologram to be f i x e d by i n c r e a s e d exposure at e l e v a t e d temperatures -so t h a t t h e i o n i c c o n d u c t i v i t y c o n t i n u a l l y n e u t r a l i z e s the r e s u l t a n t e l e c t r o s t a t i c f i e l d . S t a e b l e r e t a l . (1975) r e p o r t t h a t holograms s t o r e d by t h i s t e c h n i q u e p o s s e s s e d an i n d e x m o d u l a t i o n s i x times l a r g e r t h a n holograms s t o r e d a t room temperature. We were a b l e t o f i x a hologram i n c r y s t a l No.2 (see Appendix E) by h e a t i n g the c r y s t a l a t 100°C f o r 30 minutes a f t e r a hologram has been s t o r e d w i t h a d i f f r a c t i o n e f f i c i e n c y 52%. The r e s u l t i n g d i f f r a c t i o n e f f i c i e n c y a f t e r f i x i n g r e a c h e d a maximum 33%. "HoweverV c o n t i n u o u s read out 124. gave r i s e 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 (see Chapter 10) and the d i f f r a c t i o n e f f i c i e n c y dropped to below 4%. T h i s c r y s t a l i s 0.015 mole % 1 8 — 3 i r o n - d o p e d , which means N - 3 x 10 cm and f ^ 0.1 as de t e r m i n e d from i t s o a b s o r p t i v i t y (2.5 cm and the d a t a of P h i l l i p s and S t a e b l e r (1974) . F o r t h i s c r y s t a l the term C i n Eq. 9.6 i s C - 30/E (E i s i n kv c m _ 1 ) . v v From f r i n g e : measurements- . (see Chapter .7 ) the e s t i m a t e d v a l u e o f E ^ ^ 15 kv cm *" which means t h a t f o r t h i s c r y s t a l C - 2 . A c c o r d i n g to Eq. 9.6, t h e d i f f r a c t i o n e f f i c i e n c y a f t e r f i x i n g s h o u l d have been o n l y about 10%. However, beam enhancement (Chapter 2 ) i s r e s p o n s i b l e f o r the i n c r e a s e d v a l u e of the f i x e d hologram d i f f r a c t i o n e f f i c i e n c y . 125. X OPTICALLY INDUCED LIGHT SCATTERING AND BEAM DISTORTION IN IRON-DOPED LITHIUM NIOBATE CRYSTALS X - l I n t r o d u c t i o n A l t h o u g h i r o n - d o p e d l i t h i u m n i o b a t e as a r e u s a b l e phase hologram s t o r a g e m a t e r i a l has been d e s c r i b e d as h a v i n g some o f the " g r o s s p h y s i c a l a t t r i b u t e s o f the i d e a l m a t e r i a l " ( C o l l i e r e t a l . 1971), one of i t s a t t r i b u t e s which i s l e s s than i d e a l i s t h a t i t shows a p r o g r e s s i v e b u i l d - u p o f d i f f u s 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 and beam d i s t o r t i o n d u r i n g exposure t o l i g h t . These e f f e c t s i n c r e a s e t h e more the c r y s t a l i s made more e f f i c i e n t t o s t o r e holograms (Amodei e t a l . 1972b, S t a e b l e r and P h i l l i p s 1974a, Alphonse and P h i l l i p s 1976). S c a t t e r i n g can be a d i f f i c u l t p r o b l e m which we have found to reduce the s i g n a l - t o - n o i s e r a t i o t o unusable l e v e l s ; a r e c o n s t r u c t e d image can t o t a l l y d i s a p p e a r i n the s c a t t e r i n g n o i s e . I n p r e v i o u s work, C a r l s e n (1974) and d ' A u r i a e t a l . (1974) r e p o r t e d the s e r i o u s n e s s of t h i s e f f e c t on the s i g n a l - t o - n o i s e r a t i o i n r e a d / w r i t e h o l o g r a p h i c memories employing LiNbO^: Fe c r y s t a l s . C a r l s e n r e p o r t e d t h a t d u r i n g r e a d out of holograms s t o r e d i n a Fe-doped LiNbO^ c r y s t a l , t h e image was l o s t p r i m a r i l y due t o the i n d u c e d s c a t t e r i n g .and o n l y s e c o n d a r i l y to o p t i c a l e r a s u r e . The s c a t t e r i n g n o i s e has a c u m u l a t i v e e f f e c t d u r i n g r e p e a t e d w r i t e -r e a d / e r a s e c y c l e s ( H u i g n a r d e t a l . 1975a) or d u r i n g m u l t i p l e hologram s t o r a g e ( H u i g n a r d e t a l . 1975b, M i c h e r o n e t a l . 1976). P h i l l i p s e t a l . (1972) c o n c l u d e d from "the f a c t t h a t the s c a t t e r i n g appears o n l y d u r i n g exposure 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 n s i 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 the 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 i n t e r f e r e n c e f r i n g e s 126. p r e s e n t i n the beam." I n t h e i r 19 76 r e v i e w , A l p h o n s e and P h i l l i p s (1976) su g g e s t e d t h a t the e f f e c t i s due t o p a r a s i t i c holograms produced by i n t e r f e r e n c e between p o r t i o n s of the beam which i n t e r s e c t because o f d e f l e c t i o n by " o p t i c a l l y i n d u c e d l o c a l i n d e x i n h o m o g e n e i t i e s due t o the p o s s i b l y n o n u n i f o r m c r o s s - s e c t i o n o f the beam." The i d e a of p a r a s i t i c holograms was f i r s t i n t r o d u c e d i n c o n n e c t i o n w i t h LiNbO^ by Magnusson and G a y l o r d ( 1 9 7 4 ) . They p o s t u l a t e d some form of "bulk s c a t t e r i n g from i n h o m o g e n e i t i e s " as the t r i g g e r f o r s t a r t o f s c a t t e r i n g phenomenon. They used an a n a l y s i s g i v e n by Forshaw (1973) i n c o n n e c t i o n w i t h PMMA ( p o l y m e t h y l m e t h a c r y l a t e ) . T h i s o b s e r v e d s c a t t e r i n g phenomenon i s n o t unique o n l y to LiNbO^, o t h e r workers r e p o r t e d i t s o c c u r r e n c e i n PLZT (Kumada et a l . 1976), SBN(Micheron and Bismuth 1973) and B S O ( H e r r i a u e t a l . 1978). Ano t h e r s c a t t e r i n g phenomenon was observed by Chanussot e t a l . (1977). They r e p o r t e d R a y l e i g h s c a t t e r i n g i n B a T i O ^ due t o f l u c t u a t i o n s i n t h e r e f r a c t i v e i n d e x p r o d u c e d by o p t i c a l l y i n d u c e d f l u c t u a t i o n s i n the spontaneous p o l a r i z a t i o n . T h i s s c a t t e r i n g mechanism, however, as r e p o r t e d by the a u t h o r s , does n o t take p l a c e i n LiNbO ^ j and i s thus d i s t i n c t from t h e s c a t t e r i n g phenomenon d i s c u s s e d h e r e . I n t h i s c h a p t e r we propose a model f o r the s c a t t e r i n g phenomenon. T h i s model i s b a s e d on the premise t h a t s c a t t e r i n g i s t r i g g e r e d o r i n i t i a t e d by l i g h t i n t e r f e r e n c e w h i c h r e s u l t s from t h e d e f o c u s i n g l e n s a c t i o n o f the r e f r a c t i v e i n d e x i n h o m o g e n e i t i e s r e l a t e d t o the l a r g e s c a l e f i e l d ( C hapter •-6.). R a y - t r a c i n g c a l c u l a t i o n s c o n f i r m e d t h a t c r o s s o v e r s between p o r t i o n s of t h e l i g h t beam take p l a c e because o f t h i s l a r g e s c a l e i n d e x inhomogeneity. E x p e r i m e n t a l o b s e r v a t i o n s , however, i n d i c a t e d t h a t t h e f i n a l s c a t t e r i n g t o 127. be l i t t l e dependent on the i n t e n s i t y p a t t e r n o f the i n c i d e n t i l l u m i n a t i o n . X-2 Some B a s i c E x p e r i m e n t a l O b s e r v a t i o n s The p r o g r e s s i v e development o f beam d i s t o r t i o n and 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 i s i l l u s t r a t e d i n F i g . 1 0 . 1 ( a ) t o ( e ) . A s i n g l e beam p o l a r i z e d p a r a l l e l t o t h e c r y s t a l c - a x i s ( e x t r a o r d i n a r y p o l a r i z a t i o n ) and at normal i n c i d e n c e was used. I n F i g . 1 0 . 1 ( a ) a photograph o f t h e t r a n s m i t t e d beam p r i o r t o any d i s t o r t i o n i s g i v e n . With i n c r e a s i n g exposure, the edges o f the t r a n s m i t t e d beam s p l i t o f f two " h a l f moons" as i l l u s t r a t e d by the photograph i n F i g . 1 0 . 1 ( b ) and the scan i n F i g . 1 0 . 1 ( c ) . O b s e r v a t i o n s of t h i s t y p e o f d i s t o r t i o n were r e p o r t e d by A s k k i n e t a l . (1966), Chen (1969), T h a x t e r (1969). In a d d i t i o n , and c o n c u r r e n t l y w i t h the development of the beam d i s t o r t i o n , two d i f f u s e l o b e s o f s c a t t e r e d l i g h t appear as shown i n Fig.10.1 ( d ) . [ T h i s photograph was t a k e n a t t h e same s t a g e o f l a s e r exposure as t h a t of F i g . 1 0 . 1 ( b ) but w i t h o u t beam a t t e n u a t i o n . F i g . 1 0 . 1 ( e ) i s an e n l a r g e d p i c t u r e o f the beam a f t e r s c a t t e r i n g d e v e l o p e d . The d i s t o r t e d beam a f t e r l e a v i n g the c r y s t a l f e l l n o r m a l l y on a h i g h r e s o l u t i o n p h o t o g r a p h i c p l a t e (Kodak type 649-F). We observe a s p e c k l e p a t t e r n which i s n o t an a r t i f a c t s i n c e no imaging system was employed. I f s c a t t e r i n g i s a l l o w e d t o develop f u l l y by p r o l o n g e d exposure up to 90% o f the t r a n s m i t t e d l a s e r power may be d i s s i p a t e d by s c a t t e r i n g i n s t e a d o f b e i n g u t i l i z e d t o r e c o n s t r u c t t h e s t o r e d hologram. Fig.10.9 d e p i c t s such a s i t u a t i o n . T r a n s m i t t e d l i g h t was measured u s i n g an A l p h a m e t r i c s model dc 1010 r a d i o m e t e r w i t h an a p e r t u r e to s e l e c t the t r a n s m i t t e d beam and r e j e c t the d i f f u s e l y s c a t t e r e d l i g h t . The s c a t t e r i n g i s g r e a t e r when the i n i t i a l exposure and r e a d out l i g h t a r e e x t r a o r d i n a r i l y p o l a r i z e d . T h i s i s due to t h e n a t u r e o f the e l e c t r o - o p t i c t e n s o r o f LiNbO, (see Appendix B ) . F i g . 10.1 Progression of beam d i s t o r t i o n during exposure of c r y s t a l #3 (at X= 514 5 nm). (a) Transmitted beam i n the i n i t i a l stages of exposure before any d i s t o r t i o n took place. (b) S p l i t t i n g o f f of the beam edges a f t e r 11 Jem"2 exposure ( c r y s t a l c-axis i s h o r i z o n t a l ) . ( c ) Beam scan along the c-axis. ( i ) Scan f o r case (1-a), ( i i ) scan f o r case (1-b). Fig. 10.1 (e) An e n l a r g e d photograph o f th e d i s t o r t e d beam f o r a c a s e s i m i l a r t o (1-b), n o t e the s t r o n g s p e c k l e p a t t e r n . 130. X-2.1 I d e n t i f i c a t i o n of t h e S c a t t e r i n g as Due to P a r a s i t i c G r a t i n g s The f a c t t h a t s c a t t e r i n g e x h i b i t s a n g u l a r s e n s i t i v i t y s i m i l a r t o volume holograms ( P h i l l i p s et a l . 1972) s u g g e s t s t h a t i n t e r f e r e n c e f r i n g e s s t o r e d i n the c r y s t a l are r e s p o n s i b l e f o r the phenomenon. The s i g n i f i c a n c e of cones o f d i f f r a c t e d l i g h t , r e p o r t e d by Magnusson and G a y l o r d (1974), i s t h a t they show t h a t when t h e c r y s t a l i s i r r a d i a t e d , even w i t h a s i n g l e beam, p a r a s i t i c g r a t i n g s are p r e s e n t w i t h g r a t i n g v e c t o r s c a p a b l e o f p r o d u c i n g s c a t t e r i n g over a l a r g e range of. f o r w a r d d i r e c t i o n s . These g r a t i n g s must be w r i t t e n by i n t e r f e r e n c e w i t h l i g h t w h i c h has been s c a t t e r e d by some mechanism to be determined. The p roblem o f s c a t t e r i n g from t h i c k phase holograms was t r e a t e d by Forshaw (1973a) u s i n g t h e Ewald sphere c o n s t r u c t i o n . In b r i e f , i f the r e f r a c t i v e i n d e x v a r i a t i o n s are r e s o l v e d by a F o u r i e r a n a l y s i s i n t o p l a n e wave components e x p - i K-r, t h e g r a t i n g v e c t o r s K may be r e p r e s e n t e d by v e c t o r s i n r e c i p r o c a l space s t a r t i n g from the o r i g i n . The Bragg c o n d i t i o n 2TT(S - S ) A = K, where s i s a u n i t v e c t o r i n t h e d i r e c t i o n of the i n c i d e n t o o wave, s i s a p o s s i b l e d i r e c t i o n of s c a t t e r e d r a d i a t i o n and A i s t h e wavelength, i s r e a l i z e d by drawing b ack from the o r i g i n a v e c t o r -ZTTS^/X and c o n s t r u c t i n g a Ewald sphere w i t h r a d i u s 2TTA about the end of t h i s v e c t o r . P o s s i b l e d i r e c t i o n s o f 2TTS/A are g i v e n by t h e i n t e r s e c t i o n of the s p h ere w i t h the end of a g r a t i n g v e c t o r K. The ends of g r a t i n g v e c t o r s p r o d u c e d by i n t e r f e r e n c e between a s i n g l e s t r o n g beam S q and s m a l l amounts of r a d i a t i o n w i t h a l l p o s s i b l e d i r e c t i o n s l i e on two spheres t o u c h i n g a t the o r i g i n and whose c e n t r e s j o i n t o produce t h e d i r e c t i o n S q . R e c o n s t r u c t i o n w i t h the o r i g i n a l r a d i a t i o n g i v e s a Ewald s p h e r e which c o i n c i d e s w i t h one o f t h e s e spheres thus p r o d u c i n g d i f f r a c t i o n i n a l l d i r e c t i o n s . 131. Fig.10.2 shows the s c a t t e r e d l i g h t i n the r e c o n s t r u c t i o n o f a hologram formed by t h e i n t e r s e c t i o n of two beams of s i m i l a r d i m e n s i o n s . In p r o t o t y p e holograms, formed by two p l a n e waves, d i f f u s e s c a t t e r i n g i s ma i n l y c o n c e n t r a t e d between the emerging beams and r i n g s o f d i f f r a c t e d l i g h t appear even w i t h o u t m i s a l i g n m e n t . ( I n F i g . 10.2 the p h o t o g r a p h i c exposure was such as to show the d i f f u s e s c a t t e r i n g b u t not the r i n g s ) . With a c r y s t a l w h i c h has been damaged by a s i n g l e beam, a s l i g h t r o t a t i o n causes the b r o a d l o b e s t o d i s a p p e a r and f i r s t one th e n more r i n g s t o appear. An example i s shown i n F i g . 1 0 . 3 . I f the damage pro d u c e d by an argon i o n l a s e r (A = 514.5nm) i s viewed w i t h a He-Ne l a s e r (A. = 632.8nm) the l o b e s a r e reduced i n a n g u l a r e x t e n t and l e s s w e l l d e v e l o p e d r i n g s appear o u t s i d e the l o b e s as shown i n F i g . 1 0 . 4 . The Ewald s p h e r e i n t h i s case has ( a d i f f e r e n t r a d i u s and hence would i n t e r c e p t o n l y the " b l u r " which the f i n i t e g r a t i n g imposes on the spheres o f g r a t i n g v e c t o r s . An a d d i t i o n a l f a c t o r comes i n t o p l a y , s i n c e , as w i l l be e x p l a i n e d i n the n e x t s e c t i o n , t h e l a r g e s c a l e i n d e x inhomogeneity d i s t r i b u t i o n a c t s as a d i v e r g i n g l e n s on the i n c i d e n t beam and the r e s u l t i n g d e c o l l i m a t i o n m o d i f i e s the p i c t u r e o f the Ewald s p h e r e i n the r e c i p r o c a l s p a c e i n the manner s u g g e s t e d i n F i g . 1 0 . 5 (Cowley 1975). Drawing the i n c i d e n t beam d i r e c t i o n s as v e c t o r s (2T\/\)S^ t o t h e r e c i p r o c a l l a t t i c e o r i g i n , 0, t h e end p o i n t s are d i s t r i b u t e d over a d i s c h a v i n g the shape o f the i n c i d e n t beam and a r e l a t i v e w e i g h t i n g o f the p o i n t s g i v e n by t h e i n t e n s i t y d i s t r i b u t i o n o f the beam. C o r r e s p o n d i n g t o each p o i n t of t h i s d i s c t h e r e w i l l be a d i f f e r e n t l y o r i e n t e d Ewald s p h e r e , so t h a t we may t h i n k o f an Ewald sphere t h i c k e n e d int o a s p h e r i c a l s h e l l which v a r i e s w i t h d i s t a n c e from;the ' o r i g i n 0. Now as we read w i t h a - d i f f e r e n t wavelength, and n e g l e c t i n g f o r s i m p l i c i t y any d i v e r g e n c e t h a t might be i n c u r r e d on the beam, a Ewald sphere w i t h , 10.2 S c a t t e r i n g d u r i n g read out o f a p r o t o t y p e hologram. The beam on the r i g h t i s the read out beam and the d i f f r a c t e d beam i s on the l e f t . 10.3 S c a t t e r i n g a f t e r r o t a t i n g the c r y s t a l i n which the hologram o f F i g . 10.2 i s s t o r e d . D i f f r a c t e d beam v a n i s h e s and a r i n g appears t o u c h i n g the t r a n s m i t t e d r e f e r e n c e beam. 10.4 He-Ne l a s e r beam s c a t t e r i n g p a t t e r n (X=632.8 nm) a f t e r the damage was i n d u c e d by an argon l a s e r (A=514.5 nm). 133. P F i g 10 5 The e f f e c t o f d e c o l l i m a t i o n o f a beam i n s p r e a d i n g the Ewald sphere i n t o a s p h e r i c a l s h e l l o f v a r y i n g t h i c k n e s s . 134. d i f f e r e n t r a d i u s w i l l i n t e r s e c t t h i s s p h e r i c a l s h e l l . T h i s e x p l a i n s why, when the He-Ne beam i s s u b s t i t u t e d f o r the argon beam, the l o b e s are reduced i n a n g l e . The r i n g s seen i n t h i s case around t h e two l o b e s , even w i t h the absence of m i s a l i g n m e n t , might be e x p l a i n e d by the f i n i t e n e s s o f the hologram 2 volume g i v i n g a ( s i n x / x ) i n t e n s i t y d i s t r i b u t i o n o r i t may be due to t h e p r e v a l e n c e o f a Raman-Nath s c a t t e r i n g regime (Moharam and Young 1978c). X-2.2 Lens A c t i o n of the Large S c a l e P a t t e r n of O p t i c a l l y Induced R e f r a c t i v e  Index Change I n t h i s s e c t i o n we c o n f i r m t h a t the beam d i s t o r t i o n can be e x p l a i n e d i n terms of l e n s a c t i o n by t h e l a r g e s c a l e i n d e x changes p r o d u c e d by the l i g h t beam. The p a t t e r n o f the r e f r a c t i v e i n d e x change which i s p r o d u c e d by a s i n g l e beam of c i r c u l a r c r o s s - s e c t i o n has, i n a l l r e p o r t e d cases, shown the form o f a c e n t r a l r e g i o n i n which the i n d e x i s reduced, more f o r e x t r a o r d i n a r y than f o r o r d i n a r y p o l a r i z a t i o n , w i t h two s m a l l r e g i o n s on e i t h e r s i d e a l o n g the c - a x i s i n which the i n d e x i s i n c r e a s e d ( s e e F i g . 2 . 1 i n C hapter 2 ) . T h i s p a t t e r n (as f i r s t found and e x p l a i n e d by Chen 1969) i s of the form e x p e c t e d f o r t r a n s p o r t of p h o t o r e l e a s e d e l e c t r o n s by d r i f t i n an e l e c t r i c f i e l d o r by t h e b u l k p h o t o v o l t a i c p r o p e r t y o f the c r y s t a l . D i f f u s i o n would be e x p e c t e d t o be r e l a t i v e l y i n e f f e c t i v e because of the low s p a t i a l f r e q u e n c i e s of the e n v e l o p e o f the l i g h t p a t t e r n . F i g . 1 0 . 6 shows the m o d e l l e d form of the r e f r a c t i v e i n d e x a l o n g the c - a x i s and n o r m a l to i t i s t h e o p t i c f a c e p l a n e . The r e f r a c t i v e i n d e x i s assumed t o be of the form n = n° + & n ( r ) , where n° i s the 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 b e f o r e any, damage has o c c u r e d and 6 n ( r ) << n° i s t h e r e s u l t i n g change. 135. 0 F i g . 10.6 The assumed r e f r a c t i v e i n d e x v a r i a t i o n a l o n g t h e c - a x i s ( x - a x i s ) and a l o n g t h e o t h e r major a x i s l y i n g i n the o p t i c f a c e (the shown d o t t e d l i n e ) . The o r i e n t a t i o n o f the c r y s t a l major axes r e l a t i v e to the c o o r d i n a t e axes used i s shown i n the n e x t figure'. 136. The o r i e n t a t i o n of the c o - o r d i n a t e axes w i t h r e s p e c t t o the p r i n c i p a l axes of the c r y s t a l i s as shown i n F i g . 1 0 . 7 . A s i n g l e beam o f _2 e - r a d i u s r ^ i s assumed to be p r o p a g a t i n g a l o n g t h e z - a x i s d i r e c t i o n . As a r e s u l t o f the exposure t o l i g h t the f o l l o w i n g d i s t r i b u t i o n o f 6 n ( r ) i s 2 2 assumed. 6n = 6n n exp(-y /8r ) a l o n g the y - a x i s . A l o n g the x - a x i s v r:. . 4 4 ( c - a x i s ) we assume 6n = 6n_exp(-r2x / r ) f o r 0 < x < .75r and 1 . / . o _ l l o 6 = - ( 6 n 1 / 4 ) e x p ( l - |x|/r ) f o r r < Ixl and a l i n e a r v a r i a t i o n i n t h e n 1 1 1 o o ' 1 r e g i o n .75r < x < r o — — o T h i s p a t t e r n of the r e f r a c t i v e i n d e x change would a c t e s s e n t i a l l y as a n e g a t i v e s p h e r i c a l l e n s , as a r e s u l t o f w h i c h t h e i n c i d e n t beam s h o u l d d i v e r g e . The ray paths i n the c r y s t a l a r e governed by the ray e q u a t i o n (Born and Wolf 1965) h [ n ( £ ) ] = g r a d n ( ? ) ' (10.1) where s i s the d i s t a n c e a l o n g the r a y , r i s the p o s i t i o n v e c t o r o f a p o i n t on the r a y and n ( r ) i s t h e r e f r a c t i v e i n d e x a t p o i n t r . To t r a c e the r a y s a l o n g t h e z - a x i s i n the y - p l a n e , we c o n s i d e r the x-component of Eq.10.1, w i t h the p a r a x i a l r a y a p p r o x i m a t i o n assumed, i . e . - . The ray e q u a t i o n thus reads d r / J \ i dn — [ n ( d x ) ] - — • d Z dz d X (10.2) The ray paths a l o n g the z - a x i s i n the x - p l a n e a r e found from the y-component o f Eq.10.1 which i s T T tn(dy_)] * £ . d z d l d y (10.3) 137. F i g . 10.7 The c o o r d i n a t e axes chosen and t h e i r r e l a t i o n to the c r y s t a l major axes. 138. E q u a t i o n s 10.2 and 10.3 were n u m e r i c a l l y s o l v e d , the r e s u l t i n g r a y paths a r e p l o t t e d i n Fig.10.8 f o r the assumed r e f r a c t i v e i n d e x inhomogeneity and beam d i a m e t e r as a parameter. Note from F i g . 10.8 t h a t as the beam becomes more narrow, i . e . i t s s p a t i a l f r e q u e n c y i n c r e a s e s , the d i v e r g e n c e o f the beam i n c r e a s e s . We conclude t h a t the l e n s a c t i o n accounts f o r the d i s t o r t i o n o f t h e beam. The s p l i t t i n g o f f o f the h a l f moons i s due t o t h e l a r g e i n d e x g r a d i e n t at the o u t e r p o r t i o n s o f the damage p a t t e r n as compared t o the i n n e r p o r t i o n s . The h a l f moons mainly o c c u r a l o n g t h e c - a x i s because the i n d e x g r a d i e n t i s l a r g e r a l o n g t h i s d i r e c t i o n than a l o n g the d i r e c t i o n normal t o i t . X-2.3 Source o f the I n i t i a l S c a t t e r i n g I n t h i s s e c t i o n we c o n s i d e r the source o f the i n i t i a l s c a t t e r e d l i g h t which, by i n t e r a c t i o n w i t h u n s e a t t e r e d l i g h t , causes p a r a s i t i c holograms t o be w r i t t e n . The a n a l y s i s i n the p r e v i o u s s e c t i o n o f t h e l e n s a c t i o n o f the l a r g e s c a l e i n d e x p a t t e r n produced by a s i n g l e beam shows t h a t c r o s s o v e r s o f p o r t i o n s o f t h e beam w i l l o c c u r once the exposure has re a c h e d a c e r t a i n l e v e l and below t h i s exposure v a l u e no s c a t t e r i n g s h o u l d take p l a c e . Such c r o s s o v e r s were p o s t u l a t e d by A l p h o n s e and P h i l l i p s (1976). They would cause p a r a s i t i c g r a t i n g s to be w r i t t e n . F urthermore, t h e c r o s s o v e r s w i l l o c c u r p r e f e r e n t i a l l y a l o n g the c - a x i s s i n c e the beam b e n d i n g i s g r e a t e s t i n t h i s d i r e c t i o n . T h i s , t h e r e f o r e , i s one f a c t o r which would h e l p e x p l a i n the p r e f e r e n t i a l s c a t t e r i n g a l o n g the c - a x i s . The a n g u l a r d e v i a t i o n s i n v o l v e d i n the c r o s s o v e r s are ±4° (depending on t h e beam r a d i u s ) as opposed t o about ±17° i n t h e o b s e r v e d s c a t t e r i n g . However, i t s h o u l d be noted t h a t as the l a r g e s c a l e o p t i c a l l y 0-1 O o - J —-< O X z < < t-D R , = 0 - 0 5 c m 0-0 - 0-1 o O E o 0-1 R 0 = 0- 05 c m ui in < I— co Q < I X 0-0 _ 0.1 0-03 0 0 0-2 0-4 0- 6 0-8 10 ' DISTANCE ALONG Z-AXI5 (cm) Fig. 10.8 Computed ray paths i n the c r y s t a l as they t r a v e l i n a 1cm th i c k c r y s t a l along the z-axis. Two d i f f e r e n t beam r a d i i were used f o r comparison. 140. i n d u c e d r e f r a c t i v e i n d e x p a t t e r n d e v e l o p s , the r a y p a t t e r n w i l l change and the p a r a s i t i c g r a t i n g s w i l l be c o n t i n u a l l y m o d i f i e d by o v e r - w r i t i n g o f g r a t i n g s of d i f f e r e n t d i r e c t i o n s and p e r i o d . T h i s p r o c e s s s h o u l d , t h e r e f o r e , p roduce a v e r y complex p a t t e r n o f i n d e x changes, c o r r e s p o n d i n g t o a wide range o f K - v e c t o r s p r e f e r e n t i a l l y d i r e c t e d a l o n g the c - a x i s . An i m p o r t a n t t e s t t o a s c e r t a i n the r e l e v a n c e of the l e n s a c t i o n on 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 i s to r e l a t e the e f f e c t o f v a r i a t i o n of the i n c i d e n t l i g h t geometry on the s c a t t e r i n g . Fig.10.9 shows the e f f e c t o f the s i z e o f the beam u s i n g c r y s t a l No.3(see Appendix E ) , t h i s c r y s t a l was i l l u m i n a t e d by 1 and 7mm d i ameter beams t o compare the dependence of s c a t t e r i n g on the u n i f o r m i t y o r s p a t i a l v a r i a t i o n of the l i g h t i n t e n s i t y . In t h i s c r y s t a l the exposure r e q u i r e d to w r i t e a hologram w i t h 1% d i f f r a c t i o n e f f i c i e n c y was .14Jcm f o r s y m m e t r i c a l beams s e p a r a t e d by 30° e x t e r n a l t o t h e c r y s t a l . Somewhat u n e x p e c t e d l y we o b t a i n e d a comparable degree o f s c a t t e r i n g f o r t h e two beam d i a m e t e r s . Fig.10.10 r e f e r s to an experiment i n which another c r y s t a l was used (No.6) where the beam was s p a t i a l l y f i l t e r e d and expanded t o i l l u m i n a t e the whole c r y s t a l . F i g 10.11(a) -shows the s c a t t e r i n g p a t t e r n w i t h the o r i g i n a l damaging beam. As f o r s m a l l beams p a r t i a l l y i l l u m i n a t i n g the c r y s t a l , wide l o b e s of s c a t t e r e d l i g h t appear a l o n g the c - a x i s . F i g s . 10 .11(b) and 10.11(c) were taken w i t h t h e c r y s t a l r o t a t e d as d e s c r i b e d i n the c a p t i o n s . The a t t e n u a t i o n o f t h e d i f f u s e s c a t t e r i n g and the appearance of r i n g s are c l e a r l y s i m i l a r t o the b e h a v i o u r found w i t h a beam t h a t i l l u m i n a t e s o n l y p a r t o f t h e c r y s t a l . However, i t s h o u l d be n o t e d t h a t Amodei e t a l . (1972b) have o b s e r v e d the s c a t t e r i n g to be more pronounced when the l i g h t t r a v e r s i n g the c r y s t a l has sharp i n t e n s i t y v a r i a t i o n s . P h i l l i p s e t a l . (1972) r e p o r t e d t h a t i n t e r f e r e n c e e f f e c t s due to m u l t i p l e r e f l e c t i o n s d i d not cause the development of d i f f u s e s c a t t e r i n g . 141. Q LU N CC O Z to 2: LU X o LU in < cc -J 0-5 0 10 20 E X P O S U R E ( J / c m ) F i g . 10.9 R a t i o o f t r a n s m i t t e d beam i n t e n s i t y t o the i n i t i a l t r a n s m i t t e d i n t e n s i t y I ( E ) / I ( G ) v s . exposure i n c r y s t a l #3 w i t h e i t h e r a 1 o r 7 mm beam d i a m e t e r s . The numbers between b r a c k e t s denote which beam was shone f i r s t . Each beam was shone i n a f r e s h s p o t i n the c r y s t a l . £ to z: UJ 0-5 0-4 0-3 0-2 0.1 L 0 F U L L I L L U M I N A T I O N / P A R T I A L I L L U M I N A T I O N / 10 20 30 AO E X P O S U R E ( J / c m * ) F i g . 10.10 R a t i o o f t r a n s m i t t e d i n t e n s i t y , to i n i t i a l t r a n s m i t t e d i n t e n s i t y I ( t ) / I ( 0 ) v s . exposure i n c r y s t a l #6. The d i s c o n t i n u i t y i n the c u r v e a t t h e end o f exposure r e s u l t e d when the f u l l i l l u m i n a t i o n was c u t down to p a r t i a l i l l u m i n a t i o n (spot d i a m e t e r =5mm). a) b) c) 143. F i g . 10.11 The r e s u l t i n g s c a t t e r i n g i n c r y s t a l #6 under f u l l u n i f o r m i l l u m i n a t i o n (A=514.5 nm, e x t r a o r d i n a r y p o l a r i z a t i o n ) , a) At 12° i n a p l a n e c o n t a i n i n g the c-a x i s and p o l a r i z a t i o n d i r e c t i o n , b) S i m i l a r to (a) but the c r y s t a l i s r o t a t e d 0.15° about an a x i s normal to the i n c i d e n c e p l a n e , c) S i m i l a r to (a) a l s o but the c r y s t a l i s r o t a t e d 8.5° about an a x i s i n the i n c i d e n c e p l a n e and normal to the p r o p a g a t i o n d i r e c t i o n of l i g h t . 144. We have c o n f i r m e d t h i s . Much the same r e s u l t s were shown by a c r y s t a l 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 s and f a c e s s l i g h t l y o f f - p a r a l l e l as by p a r a l l e l f a c e d c r y s t a l s w i t h o u t c o a t i n g s . I n f a c t , t h i s i s t o be e x p e c t e d s i n c e the g r a t i n g s which w i l l be formed by the s t a n d i n g p a t t e r n due to m u l t i p l e r e f l e c t i o n s would be p a r a l l e l t o the major s u r f a c e s and would i n t e r a c t o n l y c o l l i n e a r l y w i t h the i n c i d e n t l i g h t . X- 2.4 D i s c u s s i o n We have seen from t h e model and experiments o f the p r e v i o u s s e c t i o n s t h a t the l e n s a c t i o n o f t h e l a r g e s c a l e i n d e x v a r i a t i o n s accounted f o r the observed beam d i s t o r t i o n upon e x i t i n g from the c r y s t a l . However, t h i s model was l e s s than s a t i s f a c t o r y i n a c c o u n t i n g f o r two o b s e r v a t i o n s : (1) The l a r g e a n g u l a r s p r e a d o f the o b s e r v e d d i f f u s e s c a t t e r i n g . (2) The apparent independence o f the s c a t t e r i n g on the non-u n i f o r m i t y o r 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 . L e t us c o n s i d e r another p o s s i b l e mechanism t h a t was s u g g e s t e d t o t r i g g e r s c a t t e r i n g . Magnusson and G a y l o r d (1974) proposed t h a t l i g h t i s b e i n g i n i t i a l l y s c a t t e r e d due t o m a t e r i a l i n h o m o g e n e i t i e s , t h e n a t u r e o f these s c a t t e r i n g c e n t e r s was not d e t e r m i n e d . I f t h e s e c e n t e r s were r e s p o n s i b l e f o r t r i g g e r i n g the s c a t t e r i n g then we s h o u l d expect s c a t t e r i n g t o be independent of the l i g h t e n v e l o p e v a r i a t i o n s . The wide s c a t t e r i n g a n g l e s c o u l d be a t t r i b u t e d t o t h e n a t u r e of the c e n t e r s , f u r t h e r m o r e , t h e f a c t t h a t s c a t t e r i n g i s d i r e c t e d p r e f e r e n t i a l l y a l o n g the c r y s t a l c - a x i s c o u l d be e x p l a i n e d by the a n i s o t r o p i c n a t u r e of hologram r e c o r d i n g i n LiNbO^ (see Appendix C ) . We might add a l s o i n t h i s c o n n e c t i o n our own o b s e r v a t i o n on s c a t t e r i n g a f t e r f u l 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 : we n o t e d t h a t every exposed p o i n t i n the c r y s t a l gave e s s e n t i a l l y the same s c a t t e r i n g 145. p a t t e r n produced by t h e f u l l y i l l u m i n a t e d c r y s t a l . T h i s seems to i n d i c a t e t h a t s c a t t e r i n g i s p r o d u c e d from e v e r y exposed p o i n t i n the c r y s t a l . There i s o n l y , however, one c r u c i a l d i s c r e p a n c y w i t h t h i s model which the l e n s a c t i o n model e x p l a i n s . Alphonse and P h i l l i p s (1976) r e p o r t e d t h a t the time development of the s c a t t e r i n g n o i s e i s d i s t i n c t l y d i f f e r e n t from the time development o f an o r d i n a r y hologram. Below a c e r t a i n exposure l e v e l t h e r e i s no o p t i c a l l y i n d u c e d n o i s e , o n l y " s t a t i c " n o i s e t h a t does not v a r y w i t h exposure i s p r e s e n t which i s p r o b a b l y caused by m a t e r i a l b u l k and s u r f a c e inhomogeneity. Above t h i s exposure l e v e l 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 n o i s e s t a r t s b u i l d i n g up w i t h exposure. An o r d i n a r y hologram b u i l d s up w i t h exposure the i n s t a n t t h e exposure i s a p p l i e d . Now i f s c a t t e r i n g i s caused by th e s c a t t e r i n g c e n t e r s of Magnusson and G a y l o r d t h e n t h e p a r a s i t i c s c a t t e r i n g holograms would develop w i t h e x p o s u r e i n a manner s i m i l a r t o t h a t of an o r d i n a r y hologram. On t h e o t h e r hand, t h e l e n s a c t i o n t h e o r y i n d i c a t e s t h a t p a r a s i t i c holograms a r e due t o beam b e n d i n g which takes p l a c e o n l y some time a f t e r the exposure s t a r t e d . We can r e c o n c i l e the v a r i o u s o b s e r v a t i o n s b r i e f l y as f o l l o w s . Due t o m a t e r i a l b u l k and s u r f a c e i n h o m o g e n e i t i e s and p o s s i b l y a l s o due t o dust p a r t i c l e s at t h e c r y s t a l s u r f a c e some form of s p e c k l e p a t t e r n d e v e l o p s i n s i d e t h e c r y s t a l (Forshaw 1975) as can be seen i n F i g . 1 0 . 1 ( e ) . The h i g h i n t e n s i t y r e g i o n s of t h i s s p e c k l e p a t t e r n produce an i n d e x i nhomgeneity, s i m i l a r t o t h a t o f F i g . 1 0 . 6 , which d e c o l l i m a t e s the l i g h t and t r i g g e r s the f o r m a t i o n of p a r a s i t i c g r a t i n g s . I f t h i s model i s t r u e , t h e n the two d i s c r e p a n c i e s o f the l e n s a c t i o n model can now be e x p l a i n e d . From Fig.10.8 we see t h a t as the s p a t i a l e x t e n t of t h e i n d e x inhomogeneity becomes narrower the d i v e r g e n c e o f the l i g h t i n c r e a s e s , t h e e x t e n t o f the h i g h 146. i n t e n s i t y r e g i o n s o f the s p e c k l e p a t t e r n i s u s u a l l y s m a l l to cause the wide a n g u l a r s p r e a d o f the s c a t t e r i n g p a t t e r n . A l s o , i n t h i s s c a t t e r i n g regime, 1. the t r i g g e r i n g o f s c a t t e r i n g i s independent o f the l i g h t i n t e n s i t y e n v e l o p e as we have e x p e r i m e n t a l l y o b s e r v e d . X-3 O p e r a t i o n o f P a r a s i t i c G r a t i n g s The p a r a s i t i c g r a t i n g s c l e a r l y dominate t h e s c a t t e r i n g as obse r v e d , s i n c e the t r i g g e r i n g p r o c e s s which caused t h e p a r a s i t i c g r a t i n g s t o be w r i t t e n i s s t i l l o p e r a t i v e , we have t o e x p l a i n how t h e g r a t i n g s can, i n f a c t , a c h i e v e t h i s dominance, the f i n a l r e s u l t a c c o r d i n g t o t h e above experiments, b e i n g a p p a r e n t l y almost independent o f the i n i t i a l t r i g g e r i n g . The r e a s o n why p a r a s i t i c g r a t i n g s dominate the s c a t t e r i n g p r o c e s s i s the beam enhancement, d i s c u s s e d i n Chapter 2"., which causes p a r a s i t i c g r a t i n g s o f 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 i e s t o be w r i t t e n accompanied by l a r g e energy t r a n s f e r from the main beam t o the s c a t t e r e d n o i s e . As we have d i s c u s s e d i n Chapter 2 , t h e enhancement e f f e c t d u r i n g hologram w r i t i n g takes p l a c e o n l y i f t h e r e i s a s p a t i a l phase s h i f t between the w r i t t e n hologram and 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 t h a t causes i t . From t h e models o f Chapters ; 3 ' and 4 and 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 Chapter 5- we c o n c l u d e d t h a t 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 s the main mechanism t h a t causes beam c o u p l i n g i n l i t h i u m n i o b a t e . T h i s agrees w i t h the o b s e r v a t i o n t h a t s c a t t e r i n g i n LiNbO^ was prominent i n c r y s t a l s o p t i m i z e d f o r h o l o g r a p h i c s t o r a g e (Amodei e t a l . 1972b, S t a e b l e r and P h i l l i p s 1974a, A l p h o n s e and P h i l l i p s 1976). Beam enhancement e f f e c t s depend on the d i r e c t i o n o f t h e c - a x i s b o t h d u r i n g r e c o r d i n g and r e a d i n g o f ho l o g r a m s . S p e c i f i c a l l y , t h e beam a p p r o a c h i n g the c r y s t a l from t h e -ve c - a x i s end i s a m p l i f i e d on the expense 147. o f the o t h e r beam. T h i s s h o u l d produce s c a t t e r i n g l o b e s o f unequal i n t e n s i t i e s on b o t h s i d e s o f the normal t o t h e c - a x i s c o n t r a r y to what i s a c t u a l l y o b s e r v e d . I t may be i m p o r t a n t i n t h i s c o n n e c t i o n t h a t r e c e n t work (K u k h t a r e v 1977) seems t o i n d i c a t e t r a n s f e r o f l i g h t to whichever, beam i s weaker i r r e s p e c t i v e o f the o r i e n t a t i o n o f the beams t o the c - a x i s . 148-XI SPATIAL FILTERING PROPERTIES OF VOLUME HOLOGRAMS Xl-1 Introduction Materials capable of s t o r i n g volume (thick) holograms,such as the class of f e r r o e l e c t r i c photorefractive c r y s t a l s , might be used for applications formerly employing t h i n holograms (e.g., matched f i l t e r i n g and d i s p l a y s ) , i n addition to applications for which th i n holograms are unsuited (e.g., multiple storage). As f a r as I know no analysis f or the behaviour of a volume hologram employed i n the Vander Lugt f i l t e r or s i m i l a r devices i s a v a i l a b l e i n the l i t e r a t u r e . XI-2 The Analysis XI-2.1 The Recording Process Figure 11.1 i s a schematic of the system to be studied. It i s the arrangement employed for synthesizing a matched f i l t e r as was suggested by Vander Lugt (1964) with the exception that the storage medium i s now a material capable of storing volume holograms. Let a plane object with amplitude transmittance h(x^,y^) (which i s the required impulse response of the f i l t e r ) be placed at the front f o c a l plane of lens L. When t h i s transparency i s illuminated from the l e f t by a plane wave of monochromatic coherent r a d i a t i o n of amplitude S, the r a d i a t i o n e l e c t r i c f i e l d at the back f o c a l plane of the lens represents the two-dimensional Fourier transform of h(x^,y^), i . e . , v g(x2,y2,0)= C O C O 2E.0 '-Lni T.TV, Q*- Q J1D\J H(T¥2,T?2) where H ( — 2 , ^ 2 ) = 1XF- XF 'XF ' " V \ F 'XF dx x dy x h ( x 1 , y 1 ) e x p - i 2 i r ( x ^ "1 " 7 ! ^ » A — O Q ' _ C O LU cr 2 o Fig. 11.1 Optical system to be studied for investigating the spatial f i l t e r i n g properties of volume holograms. 150. F i s the f o c a l l e n g t h o f l e n s L, A i s the r a d i a t i o n wavelength and i = / ^ l . The time-dependent p e r i o d i c f a c t o r o f the f i e l d exp-iwt i s o m i t t e d t h r o u g h -out the a n a l y s i s and p o l a r i z a t i o n e f f e c t s w i l l be n e g l e c t e d ( i . e . , the e l e c t r i c f i e l d i s t r e a t e d as a s c a l a r ) . Lower-case l e t t e r s a r e used h e r e t o denote t w o - d i m e n s i o n a l f u n c t i o n s o r d i s t r i b u t i o n s and upper-case l e t t e r s t o denote the two-dimensional F o u r i e r t r a n s f o r m o f the f u n c t i o n o r t h e . d i s t r i b u t i o n . At the back f o c a l p l a n e o f l e n s L t h e r a d i a t i o n e l e c t r i c f i e l d a 6 has the a n g u l a r spectrum A(——) g i v e n by (Goodman 1968, P.48) A A Af--^> = Ah'X} _ c o _oo d x 2 d y 2 v g ( x 2 , y 2 , 0 ) e x p - i 2 T T ( q x y f g y 2 ) , (11.1) A where a and 3 a r e t h e d i r e c t i o n c o s i n e s o f the p l a n e wave w i t h a m p l i t u d e , .a da dg ., , •, •, A ( - r - , — ) — r - r e l a t i v e t o the x„ and y 0 - a x e s , r e s p e c t i v e l y . A A A A L 2-S x y S u b s t i t u t i n g -r^ H ^ X F 2 'AF 2^ ' I ? S T E A D O F v s ^ x 2 ' y 2 ' ° ^ i n E ( 1 * 1 1 ' J -we o b t a i n A' A i A F __co CO dx 2dy 2 H ^ , ^ ) exp-i2T L(ax 2+By 2) , A =-iAFS h ( a F , g F ) , (11.2) where we have used the s c a l e p r o p e r t y o f the F o u r i e r t r a n s f o r m . At any p o i n t ( x 2 , y 2 , z 2 ) i n the s t o r a g e medium the r a d i a t i o n e l e c t r i c f i e l d can be d e s c r i b e d i n terms o f the a n g u l a r spectrum o f Eq.11.2 (Goodman 1968, P.51) v s ( x 2 , y 2 , z 2 ) =-iAFS ^ £ | h(aF,gF) expi2iKax 2 + e y 2+Yz 2) , \ —OO _oo (11.3) 151. where Y = ( l - a 2 - B 2 ) 1 / 2 f o r a 2+B 2 <_1 and 1/2 Y = i ( a 2 + $ 2 - l ) f o r a 2 + 3 2 > 1 (evanescent waves). The r e f e r e n c e wave i s assumed to be o f the form r ( x 2 , y 2 , z 2 ) = r e x p i ^ t a ^ + S ^ + Y ^ ) . (11.4) X R e c o r d i n g media a r e s e n s i t i v e t o the l i g h t i n t e n s i t y which i s g i v e n by I ( x 0 , y 0 , z 0 ) = | r ( x 9 , y 9 , z 9 ) + v o ( x 9 , y 9 , z 9 ) | 2 , ^2*y2' 2' I J " V A 2 " 2 ' 2' s v 2 , J ,2» 2 = | r ( x 2 , y 2 , z 2 ) | 2 + | v g ( x 2 , y 2 . z ^ ) | 2 -- i A F S r * 00 OO ^ ^ h(aF,3F) e x p i M ( a - a 1 ) x 2 + ( 6 - B 1 ) y 2 + ( Y - Y 1 ) z 2 ] + _co _co o o o o +iAFS*r ^ ~ ^ h*(aF,BF) e x p - i 2 T L [ ( a - a 1 ) x 2 + ( B - 3 1 ) y 2 + ( Y - Y 1 . ) z 2 ] • X _ o o — OO (11.5) As a r e s u l t " o f the i n t e n s i t y d i s t r i b u t i o n o f Eq.11.5 a change i n the p r o p e r t i e s o f the medium takes p l a c e . As i n t h e t h i n hologram case (Goodman 1968, P.175) the l a s t two terms i n Eq.11.5 a r e the ones o f i n t e r e s t . When a t h i n hologram i s used i n th e f i l t e r one term g i v e s the c r o s s -c o r r e l a t i o n and the o t h e r g i v e s the c o n v o l u t i o n o f h w i t h the t e s t s i g n a l g. XI-2.2 Hologram Read Out D u r i n g a p p l i c a t i o n as a matched f i l t e r we p l a c e a t r a n s p a r e n c y w i t h amplitude t r a n s m i t t a n c e g(x^,y^) a t the f r o n t f o c a l p l a n e of l e n s L (Fig.11.2) i n p l a c e of h(x^,y^) and b l o c k the r e f e r e n c e wave. We s h i n e a c o l l i m a t e d , monochromatic and c o h e r e n t wave of a m p l i t u d e S' on g(x^,y^) from the l e f t and measure the i n t e n s i t y o f the d i f f r a c t e d l i g h t i n the p l a n e ( x ^ , y ^ ) . Our o b j e c t i v e i s t o f i n d how t h i s d i f f r a c t e d l i g h t i n t e n s i t y 152. F i g . 11.2 O p t i c a l arrangement f o r the matched f i l t e r i n g o p e r a t i o n . 153. i s r e l a t e d t o Mx^y-^) and gCx^jYi). The unexposed medium i s c h a r a c t e r i z e d by i t s complex p e r m i t t i v i t y eQ = e' + i e " , e" « e' , e" > 0. (11.6) To e x c l u d e m u l t i p l e r e f l e c t i o n s a t the medium b o u n d a r i e s we assume the r e a l p a r t o f t o be e q u a l t o the p e r m i t t i v i t y o f the s u r r o u n d i n g space ( a i r ) . A f t e r exposure, the hologram r e s u l t s due t o m o d u l a t i o n o f £ q by the i n t e r f e r e n c e p a t t e r n o f Eq.11.5. e - l Q + T I ( x 2 , y 2 , z 2 ) , | TI <* 2 ,ZJ | ^ « ^ , (11.7) where T=T'+iT" i s the m a t e r i a l c o e f f i c i e n t . L i n e a r r e c o r d i n g c o n d i t i o n s a r e assumed. F o l l o w i n g the same s t e p s o f the a n a l y s i s o f the p r e c e d i n g s e c t i o n l e a d i n g t o Eq.11.3, t h e e l e c t r i c f i e l d o f the t e s t s i g n a l g(x^,y^) i n the hologram medium i s g i v e n by v s ( x 2 , y 2 , z 2 ) =-iAFS' ^ 2 ^ 2 g ( a 2 F , B 2 F ) e x p i 2 £ ( a 2 x 2 + B 2 y 2 + Y 2 z 2 ) , \ _ c o _ o o (11.8) where Y 2 = ( l - a 2 - B 2 ) 1 / 2 f o r a 2+B 2 < 1 and Y 2 = i ( a 2 + B 2 - l ) 1 / 2 f o r a 2 + 3 2 > 1. We d i s c u s s the o p e r a t i o n o f the hologram f o l l o w i n g the methods g i v e n by Gabor and S t r o k e (1968), Wolf (1969), Forshaw (1973b), Shono (1976) and L a n g b e i n and Lederer. (1979) . 154. In Eq.11.8 we have assumed t h a t the i n c i d e n t f i e l d i s not m o d i f i e d by the hologram ( i . e . , s m a l l p e r t u r b a t i o n and weak s c a t t e r i n g a r e assumed: k i n e m a t i c o r Born a p p r o x i m a t i o n d i f f r a c t i o n ) . The t o t a l r a d i a t i o n e l e c t r i c f i e l d E ( x 2 , y 2 , z 2 ) due t o the unperturbed i n c i d e n t wave v ^ ( x 2 , y 2 , z 2 ) and the d i f f r a c t e d wave v ( x 2 , y 2 , z 2 ) i s g i v e n by E ( x 2 , y 2 , z 2 ) = v s ( x 2 » y 2 ' Z 2 ) + v ( x 2 ' y 2 ' Z 2 ) ' 0-1-9) The t o t a l f i e l d E ( x 2 , y 2 , z 2 ) s a t i s f i e s the time-independent wave e q u a t i o n ( V 2 + o ) 2 v e o ) E ( x 2 , y 2 , z 2 ) = 0, (11.10) o u t s i d e the hologram medium ( E q i s t h e p e r m i t t i v i t y o f a i r ) and (V 2+ a)V;e ) E ( x 2 , y 2 , z 2 ) = 0, (11.11) i n s i d e the medium. While the i n c i d e n t f i e l d v ^ ( x 2 , y 2 , z 2 ) s a t i s f i e s the f o l l o w i n g wave e q u a t i o n i n s i d e the medium (V 2+ u) 2y e o ) v ^ ( x 2 , y 2 , z 2 ) = 0, (11.12) which d e s c r i b e s the p r o b a g a t i o n o f the i n c i d e n t f i e l d i n the u n p e r t u r b e d medium. In the above e q u a t i o n s V 2 i s t h e L a p l a c i a n o p e r a t o r , V i s the p e r m e a b i l i t y and o t h e r symbols have been p r e v i o y s l y d e f i n e d . From Eqs.11.9, 11.11 and 11.12, the d i f f r a c t e d f i e l d v ( x 2 , y 2 , z 2 ) obeys the e q u a t i o n ( V 2 + w 2 u e o ) v ( x 2 , y 2 , z 2 ) = - o o 2 u T I ( x 2 ,y2 , z 2 ) E ( x 2 ,y2 ,7.^) , (11.13) i n s i d e the hologram medium. 155, S o l u t i o n s o f V g ( x 2 » y 2 ' Z 2 ^ a n d V ^ X 2 , y 2 ' Z 2 ^ m u s t s a t i s f y the c o n t i n u i t y c o n d i t i o n s a t the boundaries" o f the s t o r a g e medium. The hologram i s assumed to have dimensions X, Y and Z a l o n g the , and z 2 ~ a x e s , r e s p e c t i v e l y . From Eq.11.13, the d i f f r a c t e d l i g h t a t the p o i n t ( x o , y o , z Q ) o u t s i d e the hologram medium i s g i v e n by.(Wolf 1969) v ( x .y .z ) = tu2yT otJo* o' 4TT X Y Z d x 2 d y 2 d z 2 I ( x 2 , y 2 , z 2 ) E ( x 2 , y 2 5 z 2 ) e x p i K | r - r ? | (11.14) | r - r J _ _ 1 o 2 1 e x p i K | r o - r 2 | _ _ where — — — — i s the a p p r o p r i a t e form o f Green's f u n c t i o n , G ( | r - r „ | ) , r e p r e s e n t i n g a s p h e r i c a l wave emanating from the elementary volume 6x 26y 2<5z 2 c e n t e r e d a t the p o i n t ( x 2 , y 2 , z 2 ) whose p o s i t i o n v e c t o r i s , r 2 , r Q i s the 2TT p o s i t i o n v e c t o r o f the o b s e r v a t i o n p o i n t ( x ^ y ^ j Z ^ ) and K=—— i s the p r o b a g a t i o n v e c t o r o f the s p h e r i c a l waves. The l i m i t s o f i n t e g r a t i o n i n Eq.11.14 a r e o v e r the hologram volume o n l y s i n c e the f u n c t i o n T I ( x 2 , y 2 , z 2 ) i s non-zero o n l y i n t h i s r e g i o n . We assume the f o l l o w i n g a p p r o x i m a t i o n s to be v a l i d f o r Eq.11.14. 1) v ( x 2 , y 2 , z 2 ) « v ^ ( x 2 , y 2 , z 2 ) , i . e . , E ( x 2 , y 2 , z 2 ) = v g ( x 2 , y 2 , z 2 ) 2) K | r o - r 2 | - . K . r , - K ^ , - • = c o n s t a n t r- (a^K^+Q^y^+j^z^), where K i s the wavevector i n the d i r e c t i o n of r . o o 3) r Q » r 2 , i . e . , 1 — = l / r Q , l . r o " r2> 156. where b o t h v e c t o r s a r e measured from the o r i g i n o f the c o o r d i n a t e axes x 2 , y 2 and . Eq.11.14 can now be w r i t t e n as v ( x .y , z J = o/uT b o ' o -•• 4irr d x 2 d y 2 d z 2 I . ( x 2 ,y 2 ,2 2> v 8 ( x 2 » y2 X Y Z e x p - i 2 T T ( a o x 2 + g o y 2 + Y o z 2 ) . (11.15) A c o n s t a n t phase f a c t o r , e x p i K . r Q , has been o m i t t e d s i n c e i t w i l l d i s a p p e a r when the i n t e n s i t y o f the d i f f r a c t e d l i g h t i s measured. As was shown by Gabor and S t r o k e (1968) , o n l y the f o u r t h term i n Eq.11.5 i s r e s p o n s i b l e f o r the d i f f r a c t e d l i g h t from the volume hologram S u b s t i t u t i n g t h i s term i n s t e a d o f I.(x y z-p i n Eq.11.8 we o b t a i n v ( v W = co 2pTX 2F 2S*S ' r 47rr CO CO CO CO da dj3 X X da dp_ X X d x 2 d y 2 d z 2 _oo _CO —CO —CO X Y Z where h ( a F , B F ) g ( a 2 F , g 2 F ) e x p - i 2 T L [ ( a - a 1 + a o - a 2 ) x 2 + . . .] (11.16) P e r f o r m i n g the innermost i n t e g r a l we o b t a i n v ( v 3 o > V = i o2 y T X 2 F 2 S * S 'rXYZ 4irr "da d3 X X « da dg X X —CO —OO —CO —CO h ( a F , B F ) g ( a 2 F , B £ F ) s i n e ( a X / X ) s i n e ( b Y / X ) s i n e ( c Z / X ) , (11.17) a = a-a^+a Q-a 2, b = B-B 1 +B o-B 2, c = Y-Y 1+Y 0-Y 2, and s i n e x = s i n TTX/TTX. 157. We a r e i n t e r e s t e d i n volume holograms f o r which X, Y and Z >> X such t h a t t h e s i n e f u n c t i o n s i n Eq.11.17 become D i r a c d e l t a f u n c t i o n s . A p p l y i n g the s i f t i n g p r o p e r t y o f the D i r a c d e l t a f u n c t i o n s on Eq.11.17 we o b t a i n v ( o t , 3 ,y ) = ui2uTA 2T 2S*S'rXYZ o ' o ' ' o 4iTr o _oo _ oo * | h*(aF,BF) where we s u b s t i t u t e d g [ ( a Q - a 1 + a ) F , ( B o - B 1 + 6 ) F ] 6 ( Y - Y 1 + Y 0 - Y 2 ) , (11.18) a =a -a..+a and z o 1 s =g - p +g Z O X i n Eq.11.18 i s g i v e n from t h e e q u a t i o n a 2 + 3 2 + Y 2 = l . The p r e v i o u s a n a l y s i s a p p l i e s a l s o t o the case o f a t h i n hologram i f we take the hologram t h i c k n e s s Z=0. In t h i s case the i n t e g r a l over t h e hologram volume i n Eq.11.16 reduces t o an i n t e g r a l over the hologram a r e a and the d i f f r a c t e d f i e l d can be d e s c r i b e d as , . . c j 2yTA 2 F 2S*S'rXY v(a , 3 ,Y )= — o ' o ' ' o 47rr o _ 0 0 _ 0 0 ^ * f h*(aF,BF) g[ ( a Q - a 1 + a ) F , ( B ^ B ^ F ] . (11.19) T h i s e q u a t i o n shows t h a t the c r o s s - c o r r e l a t i o n o f the two f u n c t i o n s appearson the ( x ^ y ^ ) p l a n e c e n t e r e d a t x^=-aL^fr. Y ^ " ^ ^ Fo r the volume hologram case (Eq.11.18), the d i f f r a c t i o n c o n d i -t i o n s a r e more r e s t r i c t i v e because o f the p r e s e n c e o f the D i r a c d e l t a 158. f u n c t i o n . D i f f r a c t i o n i s o n l y f e a s i b l e when t h e argument o f the D i r a c d e l t a f u n c t i o n i s i d e n t i c a l l y e q u a l t o z e r o . The above shows t h a t r e p l a -cement o f t h e t h i n h ologram by a volume ho l o g r a m i n t h e Vander L u g t f i l t e r changes t h e r e s p o n s e o f t h e f i l t e r i n a manner t h a t needs f u r t h e r s t u d y . Below we d i s c u s s o t h e r p o s s i b l e a p p l i c a t i o n s o f volume holograms. XI-3 A p p l i c a t i o n s i n H o l o g r a p h i c I n t e r f e r o m e t r y H o l o g r a p h i c i n t e r f e r o m e t r y i s a n o n - d e s t r u c t i v e t e s t i n g t e c h n i q u e w i t h a growing number o f i n d u s t r i a l a p p l i c a t i o n s . The advantages o f u s i n g p h o t o r e f r a c t i v e m a t e r i a l s f o r t h i s a p p l i c a t i o n o v e r p h o t o s e n s i t i v e p l a t e s a r e m a i n l 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 y f o r b e t t e r v i s i b i l i t y o f t h e f r i n g e s and no p r o c e s s i n g b e i n g r e q u i r e d f o r r e a l - t i m e a p p l i c a t i o n s . H u i g n a r d e t a l . (1977a, 1977b) were s u c c e s s f u l i n u s i n g B i ^ S i O ^ • c r y s t a l s (BSO) f o r r e a l - t i m e double exposure and time-average i n t e r f e r o m e t r y . We have d e m o n s t r a t e d the f e a s i b i l i t y o f employing LiNbO^ c r y s t a l s f o r i n t e r f e r o m e t r y a p p l i c a t i o n s u s i n g the se t - u p o f F i g . U . 3 . The c r y s t a l used was Fe-doped (0.1 mole %) L i N b 0 3 No.6 (see Appendix E) under s h o r t - c i r c u i t c o n d i t i o n s . A S p e c t r a P h y s i c s model 166 argon i o n l a s e r ( A = 514.5nm) i s used f o r t h e holo g r a m s t o r a g e . The beam p o l a r i z a t i o n was normal t o t h e c - a x i s i n s i d e the c r y s t a l ( o r d i n a r y p o l a r i z a -t i o n ) t o reduce e f f e c t s o f s c a t t e r i n g (Chapter 10). Beam s p l i t t e r B s p l i t the l a s e r beam i n t o t h e r e f e r e n c e and s i g n a l beams and each 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 m a t e d by s p a t i a l f i l t e r s SF t o 25mm d i a m e t e r . The r e f e r e n c e t r a n s p a r e n c y o f t h e i n t e r f e r o m e t e r , F i g . 1 1 . 4 ( a ) , was p l a c e d i n the path o f t h e s i g n a l beam at the f r o n t f o c a l p l a n e P^ o f l e n s L^ ( f o c a l l e n g t h = 195mm, dia m e t e r = 40mm). A t the back f o c a l p l a n e P^ i s the F o u r i e r t r a n s f o r m o f t h e t r a n s p a r e n c y , n o t e t h a t the c r y s t a l does not c o n t a i n :1S9. 160. F i g . 12.6 (a) The o r i g i n a l transparency to be recorded as viewed a f t e r being t r a n s m i t t e d through the c r y s t a l . (b) The r e c o n s t r u c t e d image of the recorded hologram. Note the added noise due to o p t i c a l s c a t t e r i n g , (c) The i n t e r f e r o g r a m obtained by coherently adding the d i f f r a c t e d l i g h t from the s t o r e d hologram a n d the t r a n s m i t t e d l i g h t from the o r i g i n a l transparency. p l a n e P^, i . e . d e f o c u s e d F o u r i e r t r a n s f o r m hologram r e c o r d i n g i s employed, i n o r d e r t o a v o i d the s e v e r e d i s t o r t i o n i n t r o d u c e d on the output i n f o r m a t i o n t h a t r e s u l t s b e c a u s e o f the peaked dc i n t e n s i t y d i s t r i b u t i o n at p l a n e P^ . F i n a l l y l e n s ( f o c a l l e n g t h 270mm, diameter 55mm) p r o j e c t s the i n v e r t e d image o f p l a n e P^ onto p l a n e P^. A d i f f u s i n g s c r e e n i s p l a c e d at P^ and a camera i s used t o r e c o r d the ou t p u t r e s u l t s . An i n t e r f e r o g r a m was o b t a i n e d by the f o l l o w i n g p r o c e d u r e . The t r a n s p a r e n c y o f F i g . 1 1 . 4 ( a ) was used t o s t o r e the hol o g r a m o f F i g . 1 1 . 4 ( b ) . The s i g n a l t o n o i s e r a t i o o f t h e r e c o n s t r u c t e d image ( i . e . image c o n t r a s t ) i s lower than t h a t o f the o r i g i n a l p i c t u r e m a i n l y b e c a u s e o f s c a t t e r i n g e f f e c t s (see Cha p t e r 10) . A s l i g h t d i s t o r t i o n of the o r i g i n a l image was i n t e n t i o n a l l y i n t r o d u c e d f o r comparison purposes w i t h the o r i g i n a l image, t h i s was done by p l a c i n g a s l i g h t l y d i s t o r t e d t r a n s p a r e n t p l a s t i c s h e e t i n the pa t h o f t h e s i g n a l beam between p l a n e P^ and l e n s T h i s phase v a r i a t i o n was c o n v e r t e d t o am p l i t u d e v a r i a t i o n as a r e s u l t o f the c o h e r e n t summation o f t h e phase d i s t o r t e d image and the o r i g i n a l s t o r e d u n d i s t o r t e d image t h a t r e s u l t s due to d i f f r a c t i o n o f the r e f e r e n c e beam. The v i s i b i l i t y of t h e f r i n g e s was maximized by a d j u s t i n g the i n t e n s i t i e s o f th e r e f e r e n c e and s i g n a l beams by t h e v a r i a b l e a t t e n u a t o r s A^ and A^, F i g . 1 1 . 4 ( c ) . We s h o u l d note t h a t p h o t o r e f r a c t i v e m a t e r i a l s c a p a b l e o f hologram s t o r a g e s t o r e t w o - d i m e n s i o n a l o b j e c t s t h a t a r e t r a n s p a r e n t to l i g h t o r can s p e c u l a r l y r e f l e c t i t . No attempt t o r e c o r d d i f f u s e l y s c a t t e r e d l i g h t o f f 3-D o b j e c t s i s r e p o r t e d up t i l l now; t h i s i s m a i n l y due t o the low s e n s i t i v i t y o f t h e a v a i l a b l e s t o r a g e media. Thus, p o s s i b l e i n t e r f e r o m e t r y 162. a p p l i c a t i o n s employing t h e s e p h o t o r e f r a c t i v e media seem t o be c o n f i n e d t o f l u i d mechanics, a c o u s t i c resonances and t h e d e s i g n and p r o d u c t i o n o f o p t i c a l components. XI-4 Volume Holograms as Mo d u l a t o r s f o r O p t i c Communications Recent advances i n l a s e r s , f i b e r o p t i c s and s o l i d - s t a t e p h o t o d e c t o r s have made the c o n s t r u c t i o n o f wideband o p t i c a l l i n k s a p r a c t i c a l r e a l i t y . The o p t i c a l l i n k c o u l d be through f i b e r o p t i c s , f r e e space o r s h o r t - h a u l a t m o s p h e r i c t r a n s m i s s i o n . O p t i c communications p o s s e s s s e v e r a l advantages over o t h e r c o n v e n t i o n a l communications t e c h n o l o g y s uch as O p t i c a l communications p o s s e s s enormous p o t e n t i a l bandwidth and d a t a r a t e s . I t i s the o n l y u s e a b l e p o r t i o n o f t h e e l e c t r o m a g n e t i c spectrum s t i l l uncrowded. and u n a f f e c t e d by l i c e n s e r e q u i r e m e n t s . A t mospheric o p t i c a l l i n k s c o u l d be s e t up imm e d i a t e l y w i t h o u t w a i t i n g f o r fr e q u e n c y a l l o c a t i o n from the government, which might t a k e o v e r a y e a r . - Immunity a g a i n s t e l e c t r o m a g n e t i c i n t e r f e r e n c e (EMI) and jamming. - S e c u r i t y , f o r m i l i t a r y and c o r p o r a t e a p p l i c a t i o n s . - L a s e r s o u r c e s are narrow band and p o s s e s s v i r t u a l l y no s i d e l o b e s . The p o s s i b i l i t y o f s m a l l p o r t a b l e t e r m i n a l s ; the same microwave antenna g a i n and i n f o r m a t i o n r a t e s can be a t t a i n e d w i t h much s m a l l e r and l i g h t e r o p t i c a l r a d i a t o r s and d e t e c t o r s . One o f the key l i n k s i n the r e a l i z a t i o n o f t r a n s m i t t i n g l a r g e . amounts o f i n f o r m a t i o n i s a means o f m o d u l a t i n g the l i g h t beam. In t h i s ,163. s e c t i o n we d i s c u s s the a b i l i t y o f a volume hologram to f u n c t i o n as an o p t i c l i g h t m odulator. T h e . p r i n c i p l e i s b a s e d on the energy i n t e r c h a n g e between two i n t e r s e c t i n g l i g h t beams i n a volume hologram i f b o t h beams s a t i s f y the Bragg c o n d i t i o n s o f the hologram b u t t h e r e i s a s p a t i a l phase s h i f t between 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 and the hologram. T h i s was f i r s t r e c o g n i z e d by S t a e b l e r and Amodei (1972k) as was d i s c u s s e d i n Chapter 2 • A s c h e m a t i c f o r such an o p t i c modulator i s shown i n F i g . 1 1 . 5 . The output o f t h e l a s e r i s s p l i t i n t o two beams, 1^ and 1^, and the i n f o r m a t i o n to be.impressed on t h e l i g h t i s f e d t o an e l e c t r o - o p t i c l i g h t modulator so as t o change the phase, <j>, o f one o f the l a s e r beams, I 2 , say. The time v a r i a t i o n i n the phase i s assumed t o be <f>(t) = a s i n o)t, where a i s some p r o p o r t i o n a l i t y c o n s t a n t t h a t depends on the n a t u r e and geometry of the phase modulator and to i s the f r e q u e n c y o f the i n p u t i n f o r m a t i o n . I f we assume the i n t e n s i t y o f t h e l a s e r beams to be e q u a l ( n o r m a l i z e d t o u n i t y ) , t h e n t h e l i g h t i n t e n s i t y i n s i d e the hologram volume i s g i v e n by I ( z , t ) = 1 + b sin<}>(t) , I 2 ( z , t ) = 1 - b sin<J,(t) , ( 1 1 > 2 0 ) z i s the d i s t a n c e a l o n g the hologram t h i c k n e s s , b = sin(2Trn^z/A ocos6), n^ i s the amplitude o f the r e f r a c t i v e i n d e x s i n u s o i d a l change, 8 i s the h a l f a n gle between the two i n t e r s e c t i n g beams i n s i d e - t h e hologram medium and X^ i s the vacuum w a v e l e n t h of l i g h t . We have assumed phase hologram t y p e . S u b s t i t u t i n g t h e f u n c t i o n a l form of the phase v a r i a t i o n i n Eq. 11.20 we o b t a i n 1^ • - 1 + 2b J^(a) s i n w t , I 2 = 1 - 2b J x ( a ) s i n w t , 1 } r -164. VOLUME HOLOGRAM P H A S E MODULATOR F i g . 11.5 An o p t i c a l m o d u l a t o r scheme t h a t a f f e c t s the l i g h t i n t e n s i t y o f e i t h e r one of the o u t g o i n g l a s e r beams from the volume hologram. where use have been made o f t h e i d e n t i t y s i n ( x s i n y ) = 2 E J 2 ^ + - ^ ( x ) s i n ( 2 h + l ) y h=o where J i s the B e s s e l f u n c t i o n of the f i r s t k i n d o f o r d e r n. I n o r d e r t o f i n d n the f r e q u e n c y response o f volume hologram, i . e . , how the amount of energy i n t e r c h a n g e v a r i e s w i t h the f r e q u e n c y CJ o f i n p u t i n f o r m a t i o n , we make the f o l l o w i n g assumptions. The amount of energy t r a n s f e r due to beam c o u p l i n g i s s m a l l i n comparison to the average beam i n t e n s i t y such t h a t the beam i n t e n s i t y i s assumed c o n s t a n t throughout t h e hologram. The f a c t o r s 'b' i s E q . l l . 2 1 a r e t aken e q u a l to t h e i r v a l u e s at the hologram e x i t , i . e . , z i s taken e q u a l to t h e hologram t h i c k n e s s . The two beams have s i m i l a r l a t e r a l d i m ensions. With t h e s e assumptions we f i n a l l y a r r i v e at e x p r e s s i o n s f o r 1^ and 1^ i n t h e form: 11,2 e 1 1 e S l n w t' CH.22) where e ^ 2b J ^ ( a ) s i n 6 . I t was assumed t h a t ndu/c << 1, where d i s the l a s e r beam w i d t h , n i s the average v a l u e of the r e f r a c t i v e i n d e x and c i s the v e l o c i t y of l i g h t i n vacuum. T h i s l a s t assumption i m p l i e s t h a t the l i g h t must pass the i n t e r f e r e n c e r e g i o n i n a time much s h o r t e r than the p e r i o d i c time o f the s i g n a l s . F o r n - 2, d ^ 1mm and c = 3 x lO^mm sec \ we o b t a i n maximum o p e r a t i n g f r e q u e n c y ^ 10 GHz. I f r e q u i r e d , t h i s f r e q u e n c y can be i n c r e a s e d by m a i n l y d e c r e a s i n g the beam w i d t h as i s p o s s i b l e through i n t e g r a t e d o p t i c s t e c h n i q u e s . F o r h i g h e r f r e q u e n c i e s the response o f the f i l t e r s t a r t s to f o l l o w a | s i n x/x| type o f d e c r e a s e . •"16.6. T h i s m o d u l a t i n g scheme does away w i t h . l i g h t p o l a r i z e r s and a n a l y z e r s employed i n o t h e r l i g h t m o d u l a t o r s . I t i s amenable t o be implemented through i n t e g r a t e d o p t i c s t e c h n i q u e s . The l a s e r s o u r c e can be a l a s e r d i o d e o r a compact Nd: YAG l a s e r . The o t h e r components needed, t h e beam s p l i t t e r , l i g h t r e f l e c t o r s and the hologram can a l l be c o n s t r u c t e d from i n t e g r a t e d o p t i c s components (Wood et a l . 1975, V e r b e r e t a l . 1977). The l i g h t beams can be gu i d e d a l o n g o p t i c a l waveguides ( C a r r u t h e r s e t a l . 1974). And the phase modulator, F i g . 1 2 . 8 , can be c o n s t r u c t e d by t h i n f i l m t e c h n i q u e s (Kaminow et a l . 1973). One d i s a d v a n t a g e o f t h i s t y p e o f modulator i s the low m o d u l a t i o n depth o f the i n t e n s i t y o f t h e l a s e r beam c a r r y i n g the i n f o r m a t i o n . X I I . CONCLUSIONS The o b j e c t i v e s o f t h i s r e s e a r c h were t o g a i n f u r t h e r u n d e r s t a n d i n g o f the mechanisms of o p t i c a l s t o r a g e i n l i t h i u m n i o b a t e and t o i n v e s t i g a t e 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 . They i n v o l v e , i n each c a s e , both m o d e l l i n g and e x p e r i m e n t a t i o n . The f o l l o w i n g s p e c i f i c t o p i c s were judged l i k e l y to reward i n v e s t i g a t i o n . 1. The e f f e c t o f n o n u n i f o r m i l l u m i n a t i o n on h ologram s t o r a g e ( w r i t e - r e a d / e r a s e ) . Nonuniform i l l u m i n a t i o n i s the most p r o b a b l e arrangement to be employed f o r h o l o g r a p h i c s t o r a g e i n p r a c t i c e , but i s more d i f f i c u l t to model than the case of f u l l u n i f o r m i l l u m i n a t i o n of t h e c r y s t a l . 2. The mechanisms of hologram f i x i n g and subsequent r e t r i e v a l d u r i n g read o u t . 3. The s o u r c e s of o p t i c a l l y g e n e r a t e d n o i s e ( s c a t t e r i n g ) and accompanying beam d i s t o r t i o n . 4. The p o t e n t i a l use o f e r a s a b l e o r f i x e d holograms i n d a t a p r o c e s s i n g a p p l i c a t i o n s . L a t e r i n the r e s e a r c h i t was r e a l i z e d t h a t the b u l k p h o t o v o l t a i c e f f e c t p r e s e n t i n LiNbO^ was accompanied by a f i n i t e e l e c t r o n t r a n s p o r t l e n g t h ,and" f u r t h e r experiments were c a r r i e d out to i n v e s t i g a t e the e f f e c t of t h e f i n i t e t r a n s p o r t l e n g t h on hologram s t o r a g e . T h e o r e t i c a l models of t h i s e f f e c t were a l s o d e v e l o p e d . , 168. The c o n t r i b u t i o n s w h i c h were made to the s u b j e c t may be summarized as f o l l o w s : a) An a n a l y s i s of t h e p h o t o i n d u c e d space charge f i e l d d u r i n g hologram w r i t i n g was d e v e l o p e d f o r the s h o r t w r i t i n g time a p p r o x i m a t i o n and a t s a t u r a t i o n c o n d i t i o n s . T h i s a n a l y s i s i n c o r p o r a t e d f i n i t e e l e c t r o n t r a n s p o r t l e n g t h s due to 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 , d r i f t and the d i f f u s i o n mechanisms. I t was shown t h a t the r e s u l t a n t hologram i s s p a t i a l l y s h i f t e d from 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 a t produces i t . T h i s s h i f t c o n s i s t s o f a t r a n s i e n t component gen e r a t e d by the above mentioned mechanisms and a s t a t i o n a r y or f i x e d component due to the b u l k p h o t o v o l t a i c e f f e c t a l o n e . E x p e r i m e n t a l r e s u l t s s u p p o r t e d t h e f i n i t e t r a n s p o r t l e n g t h model due t o the b u l k p h o t o v o l t a i c e f f e c t and gave an approximate e s t i m a t e o f i t s v a l u e . b) An a n a l y t i c a l treatment f o r t h e f u l l time development of the p h o t o i n d u c e d space charge f i e l d and p h o t o c u r r e n t d u r i n g exposure t o l i g h t was d e v e l o p e d . The t r e a t m e n t i s a p p l i c a b l e t o any o n e - d i m e n s i o n a l l i g h t i n t e n s i t y d i s t r i b u t i o n i n c i d e n t on a c r y s t a l w i t h f i n i t e dark c o n d u c t i v i t y and 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 e d f o r the feedback e f f e c t o f the r e s u l t i n g space charge 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 at a l l exposure t i m e s . Short e l e c t r o n t r a n s p o r t l e n g t h was assumed. I t was shown t h a t t h e l o c a l dc component of the space charge f i e l d ( t h e envelope f i e l d ) has a s t r o n g i n f l u e n c e on the hologram b e i n g w r i t t e n . I t was a l s o shown t h a t f o r p a r t i a l l y i l l u m i n a t e d c r y s t a l s i t i s advantageous t o make the dark and l i g h t c o n d u c t i v i t i e s have, e q u a l v a l u e s . Experiments were c a r r i e d o ut to i n v e s t i g a t e the development o f the en v e l o p e f i e l d d u r i n g hologram w r i t i n g and i t s e f f e c t on subsequent hologram r e c o r d i n g a t t e m p t s . A m a t h e m a t i c a l model f o r hologram r e t r i e v a l a f t e r the f i x i n g p r o c e s s was d e v e l o p e d . T h i s model took i n t o account the e f f e c t o f v a r i a t i o n s of e l e c t r o n donor and t r a p c o n c e n t r a t i o n on the r a t e o f e l e c t r o n g e n e r a t i o n and r e t r a p p i n g d u r i n g h o logram r e a d o u t . I t was shown t h a t the amount of t h e r e t r i e v e d hologram depends on m a t e r i a l parameters (e.g.; i r o n d o p i n g , o x i d a t i o n / r e d u c t i o n r a t i o and 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 ) and on e x t e r n a l parameters (e.g.; t h e a p p l i e d v o l t a g e and the i n t e n s i t y o f l i g h t ) . T h e o r e t i c a l m o d e l l i n g and e x p e r i m e n t a l o b s e r v a t i o n s were g i v e n f o r the phenomena o f beam d i s t o r t i o n and 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 . I t was shown t h a t beam d i s t o r t i o n i s due to the d e f o c u s i n g a c t i o n o f t h e l a r g e s c a l e r e f r a c t i v e i n d e x change due to t h e l i g h t envelope. L i g h t s c a t t e r i n g was s u g g e s t e d t o be due to t h e l e n s a c t i o n o f i n d e x v a r i a t i o n s due to l a s e r s p e c k l e s i n s i d e the c r y s t a l . 170. e) An a n a l y s i s o f t h e b e h a v i o u r o f a volume h o l o g r a m empl-oyed i n t h e Vander L u g t f i l t e r was d e v e l o p e d . I t was shown t h a t r e p l a c e m e n t o f the t h i n h o l o g r a m i n the f i l t e r changes i t s r e s p o n s e . A p p l i c a t i o n s f o r volume holograms i n h o l o g r a p h i c i n t e r f e r o m e t r y and o p t i c a l communications were s u g g e s t e d . XII -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 the mechanisms o f the 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 the p h y s i c s 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 a r e not a d e q u a t e l y u n d e r s t o o d . F u r t h e r i n v e s t i g a t i o n of holograms s t o r e d by c i r c u l a r G a u s s i a n beam geometry i s r e q u i r e d t o o p t i m i z e the p e r f o r m a n c e o f LiNbO^ f o r a p p l i c a t i o n s . The models f o r the b u l k p h o t o v o l t a i c e f f e c t 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 a p p l i e d t o ho 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 b u t beam c o u p l i n g e f f e c t s have t o be i n c o r p o r a t e d . 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They c i t e the f o l l o w i n g p a r t i c u l a r s e t o f c o n d i t i o n s which they c o n c l u d e s a t i s f a c t o r y , but not always s u f f i c i e n t , f o r the growth o f s i n g l e domain LiNbO^: 1) 0.5 a t % Mo0 3 added to the m e l t , 2) growth d i r e c t i o n 20-40° from the c - a x i s , 3) n e g a t i v e d i p o l e end o f c r y s t a l f a c i n g the m e l t , 4) f l a t m e l t - c r y s t a l i n t e r f a c e , 5) seed f r e e from t w i n b o u n d a r i e s , and v e r y s t a b l e temperature so t h a t twins w i l l n o t form, 6) s t e a d y slow p u l l i n g r a t e , e.g. 1/3-3/4 i n h r \ The a u t h o r s r e p o r t e d t h a t s i n g l e domain c r y s t a l s can be p r e p a r e d by two a d d i t i o n a l methods: p o l i n g a t e l e v a t e d temperatures, and growth by the C z o c h r a l s k i t e c h n i q u e i n an e l e c t r i c f i e l d . In the l a t t e r method, the one commonly used, i f no f i e l d i s a p p l i e d d u r i n g p u l l i n g m u l t i - d o m a i n c r y s -t a l s .form. 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 growth, a 180° domain w a l l i s produced. A c r y s t a l grown from 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 % I ^ 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 and Burke 1974) than t h a t grown from s t o i c h i m e t r i c m e l t (50 mole % L i 9 0 ) . 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 L i t h i u m n i o b a t e i s a f e r r o e l e c t r i c c r y s t a l , C u r i e temperature ~1210°C and 1260°C m e l t i n g p o i n t (Nassau e t a l , 1966b), o f the oxygen o c t a h e d r a group ABO3 ( C l a r k e t a l . 1973) where A i s an a l k a l i or a l k a l i n e - e a r t h m e t a l i o n and B i s a t r a n s i t i o n m e t a l i o n . At room temperature, the c r y s t a l l i n e s t r u c t u r e c o n 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 hexa-g o n a l c l o s e p a c k 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, one t h i r d by L i and the remainder a r e v a c a n t (Abrahams e t a l . 1966a, 1966b). A l l 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 . LiNbG-3 i s an o p t i c a l l y u n i a x i a l c r y s t a l n e g a t i v e , w i t h l i t t l e a b s o r p t i o n , f o r the pure c r y s t a l , from 350 nm to 5 ym (Nassau e t a l . 1966b). The r e f r a c t i v e i n d i c e s a t 500 nm a r e r e p o r t e d to be (Boyd e t a l . 1967) n Q=2.3444 and n e=2.2446 ( i . e . i t has n e g a t i v e b i r e f r i n g e n c e ) . B a r k e r and Loudon (1967) have shown t h a t the r e f r a c t i v e i n d e x i n the t r a n s p a r e n t r e g i o n o f l i t h i u m n i o b a t e can be e x p l a i n e d i n terms of a s i n g l e uv o s c i l l a t i o n term and s e v e r a l i n f r a r e d terms. The d i e l e c t r i c 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 the c - a x i s as 78 and a l o n g the c - a x i s as 32 (Nassau e t a l . 1966b). H i g h temperature 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 b o t h i o n i c and e l e c t r o n i c c o n 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 c m 2 V _ 1 s e c - 1 a t 1000°k i n a -2/3 5 0 % C O f 5 0% C O 2 a t m and e x h i b i t s a T dependence. At room temperature e l e c t r o n m o b i l i t y 0.8 c m 2 V - 1 s e c - 1 was r e p o r t e d by Ohmori e t a l . (1976) , where i t s temperature dependence i n d i c a t e s s c a t t e r i n g o f p h o t o e l e c t r o n s by o p t i c a l phonons and p o s s i b l y i m p u r i t y s c a t t e r i n g . APPENDIX B ELECTRO-OPTIC EFFECT IN LITHIUM NIOBATE The change i n the r e f r a c t i v e i n d e x o f a c r y s t a l produced by an e l e c t r i c f i e l d i s known as the e l e c t r o - o p t i c e f f e c t . The d i e l e c t r i c proper 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 s . .E. , ( B . l ) 1 o I J j where D i s the d i s p l a c e m e n t , E i s the e l e c t r i c f i e l d , -e i s the p e r m i t t i v i t y of f r e e space and e „ the t e n s o r p e r m i t t i v i t y o f the medium. The p r o p a g a t i o n o f an e l e c t r o m a g n e t i c wave 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 the p o l a r i z a t i o n and d i r e c t i o n o f p r o p a g a t i o n o f the wave w i t h r e s p e c t t o the c r y s t a l axes. 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 , propagate f o r a g i v e n wave normal. The r e f r a c t i v e i n d i c e s o f the two waves may be o b t a i n e d by drawing an e l l i p s o i d known as the i n d i c a t r i x . I f x^ , x 2 and x^ a r e the p r i n c i p a l d i r e c t i o n s o f the p e r m i t t i v i t y t e n s o r , the i n d i c a t r i x i s d e f i n e d by the e q u a t i o n 2 2 2 ' X l X2 X 3 + ~ + - f " = 1 , (B.2) nr n 2 n 3 where n^=/e^T , \\<i=fz^ and n3=v £ -I f a s t r a i g h t l i n e i s drawn from the c e n t e r o f the e l l i p s o i d p a r a l l e l t o the wave normal o f the p r o p a g a t i n g wave, then an e l l i p s e may be formed by c l e a v i n g the e l l i p s o i d through i t s c e n t e r , p e r p e n d i c u l a r to t h i s l i n e . The semi-axes o f t h i s e l l i p s e d e f i n e the 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 which 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 the two p r o p a -g a t i n g waves a r e then g i v e n by the l e n g t h s o f the semi-axes. - 183. 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 , the r e f r a c t i v e i n d e x o f the c r y s t a l i s a l t e r e d , and the g e n e r a l e q u a t i o n o f the i n d i c a t r i x then becomes . I g [ .nr + z i j k \ + R i j k £ E k E £ + • • • • i v r 1 ' x> J j k j J i , • . . n^j (B.3) where the i n d i c e s i , j , k, £, .... r u n from 1 t o 3. The c o e f f i c i e n t s z . M and R a r e the l i n e a r and 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 . 1 j iC 1JKJ6 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 , r , •«-*• z... N, and mk ( i j ) k R R/ . . w i n\ where m . and n r u n from 1 t o 6, m i s r e l a t e d t o ( i i ) and mn ( I J ) ( k £ ) ' J / n t o (k£) as f o l l o w s . 1 11, 2 22, 3 -M- 33, 4 ^ 23, 5 - ^ - 1 3 , 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 the l i n e a r e l e c t r o - o p t i c e f f e c t (such as those w i t h a c e n t e r 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 the q u a d r a t i c e l e c t r o - o p 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 the 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 s r e q u i r e t h a t some o f the 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 zero as shown by the f o l l o w i n g m a t r i x ( C l a s s 3m). 0 0 0 0 •42 - r 22 - r 22 "22 "42 0 0 13 L13 33 0 0 0 where r 1 3 = 8 . 6 x 10 1 0 cmV - 1, r 2 2 = 3 . 4 x 10 1 0cmV l, ^^=28 x l O 1 0cmV 1 and r 3 3 = 3 0 . 8 x l O " 1 0 c m V _ 1 ( T u r n e r 1966). 184. 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 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 - ^ , ^ and n e=n-j. The i n d i c a t r i x i s thus g i v e n by 1 2 1 2 ( — ~ r 2 2 E 2 + r 1 3 E 3 ) x 1 + ( — + r 2 2 E 2 + r i 3 E 3 ) x 2 + n n o o 1 2 ( — 2 + r 3 3 E 3 ) x 3 + 2 ( - r 2 2 E 1 ) x 1 x 2 + 2 ( r i t 2 E 2 ) x 2 x 3 + n e 2 ( r i + 2 E 1 ) x 3 x 1 = 1 . (B.4) From t h i s e q u a t i o n i t can be seen t h a t , i f E^ 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 ma j o r 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 would 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 , however, E-^  o r E^ a r e p r e s e n t , t h e n a r o t a t i o n t a k e s p l a c e . F o r h o l o g r a m s t o r a g e i n LiNbO^ , 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 . j ( 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 d u c e s t o ( - ~ + r i 3 E 3 ) x 2 + ( + r 1 3 E 3 ) x 2 + ( + r 3 3 E 3 ) x 2 = 1. (B.5) V " n 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, A n Q and A n £ i n 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 r e f r a c t i v e i n d i c e s , r e s p e c t i v e l y . 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 1 3 n e r 3 3 An = E 3 and An = E 3 • (B.6) o 2 2 The change i n i n d e x i s 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 hol 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 6 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 the wave normal c o i n c i d e s w i t h the new x^ , the new e q u a t i o n o f the i n d i c a t r i x i s C J Y + r 1 3 ) x ' 2 + ( - ^ 2 + r f 3 ) x ' ^ = 1 , (B . 7 ) n Q n f where - — = ~ + c o s ^ and r^3 = r i 3 s i n 2 0 + r33Cos 26 , and the change n - f n p n e i n the i n d i c e s o f r e f r a c t i o n A n Q and A n e i s 3 3 n e r 1 3 n f r f 3 A n o - = 2 — E s a n d A n e = 2 — E s * S i n c e f o r l i t h i u m n i o b a t e n Q > n e and r33 > r ^ 3 then r ^ ^ < r 3 3 and n f < n r e Thus, n e g l e c t i n g the e f f e c t o f non-normal i n c i d e n c e o f the l i g h t waves, r e s u l t s i n an over e s t i m a t i o n o f t h e change i n the r e f r a c t i v e i n d e x f o r e x t r a o r d i n a r y wave p o l a r i z a t i o n . However, i t does not a f f e c t the change i n the r e f r a c t i v e i n d e x f o r o r d i n a r i l y p o l a r i z e d waves. The u s u a l hologram r e c o r d i n g geometries i n v o l v e s a n g l e s o f i n c i d e n c e 6=15° o u t s i d e the c r y s t a l , which c o r r e s p o n d s to 6.6° i n s i d e the c r y s t a l and the e r r o r i n t r o d u c e d i n A n e i s .6% o n l y . 186. APPENDIX C HOLOGRAM RECORDING IN L i N b 0 3 WITH GENERAL GRATING VECTOR ORIENTATION C l I n t r o d u c t i o n The a b i l i t y of l i t h i u m n i o b a t e to s t o r e holograms i s g r e a t e s t when the g r a t i n g v e c t o r s o f 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 a r e p a r a l l e l o r near p a r a l l e l t o the c r y s t a l c - a x i s . When the g r a t i n g v e c t o r s a r e normal to the c - a x i s holograms a r e r e c o r d e d w i t h v e r y weak d i f f r a c t i o n e f f i c i e n c y o r not r e c o r d e d a t . a l l . T h i s i s due to the s t r o n g a n i s o t r o p y 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 s i n c e the p h o t o v o l t a i c c u r r e n t flows o n l y a n t i p a r a l l e l to the c - a x i s and t h e s t r o n g e s t m o d u l a t i o n o f t h i s c u r r e n t by 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 o c c u r s o n l y when p l a n e s o f maximum l i g h t i n t e n s i t y a r e normal to the c - a x i s d i r e c t i o n . F o r the case o f holograms s t o r e d by d i f f u s i o n , the s i n u s o i d a l e l e c t r i c f i e l d produced by a g i v e n l i g h t i n t e n s i t y p a t t e r n would have an am p l i t u d e which i s independent o f the d i r e c t i o n o f the g r a t i n g v e c t o r except i n s o f a r as i t was a f f e c t e d by the p r o b a b l y s l i g h t a n i s o t r o p y of the d i f f u -KkT' s i v i t y . The d i f f u s i o n e q u i v a l e n t f i e l d E^= — - — , where K=2TT/A i s the g r a t -i n g v e c t o r and A the g r a t i n g s p a c i n g , k i s Boltzmann's c o n s t a n t , T* i s the a b s o l u t e temperature and q i s the e l e c t r o n i c charge. F o r a 1pm g r a t i n g -1 s p a c i n g 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 would be 1.6 kVcm which i s more than enough to produce 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 10 mm t h i c k c r y s t a l (Amodei 1971c). However, some c o n f u s i o n over the 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 because holograms a r e not r e a d i l y s t o r e d when the c - a x i s o f the c r y s t a l i s p a r a l l e l t o p l a n e s o f maximum l i g h t i n t e n s i t y i n t e r f e r e n c e p a t t e r n . T h i s a n i s o t r o p y , as w i l l be shown, would a r i s e from the e l e c t r o -o p t i c p r o p e r t i e s . C.2 The A n a l y s i s L e t us c o n s i d e r t h e r e c o r d i n g o f a hologram h a v i n g the g e n e r a l g r a t i n g v e c t o r o r i e n t a t i o n shown i n F i g . C . l . The s i n u s o i d a l space charge f i e l d , E , o f the hologram t h a t i s g e n e r a t e d due to the b u l k p h o t o v o l t a i c s c e f f e c t and t h e d i f f u s i o n mechanisms i s d i r e c t e d a l o n g the d i r e c t i o n o f t h e g r a t i n g v e c t o r . As was mentioned above, the magnitude o f the component o f E s c g e n e r a t e d by d i f f u s i o n i s independent of the g r a t i n g v e c t o r o r i e n t a t i o n . The p h o t o v o l t a i c c u r r e n t w i l l be modulated by the x^-component o f the g r a t i n g v e c t o r o n l y , t h a t i s to say the " v i r t u a l f i e l d " seen by the g r a t i n g w i l l be e q u a l E y cos0 c o s c f i o n l y , where E v t h e v i r t u a l f i e l d 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 ( C o r n i s h e t a l . 1976). The space charge f i e l d component due to t h i s mechanism, which i s d i r e c t e d a l o n g the g r a t i n g v e c t o r , can be r e s o l v e d a l o n g the major axes o f the c r y s t a l as E s c a E v ^ \ cos8sin2(f>, y sin20co,s 2<f>,cos 20cos 2 < j ) ]. ( C . l ) F o r a g r a t i n g v e c t o r normal to the c - a x i s d i r e c t i o n , e i t h e r 0 o r if w i l l be e q u a l ir/2 and from Eq. C . l the c o n t r i b u t i o n 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 v a n i s h e s . E x a m i n a t i o n o f the e l e c t r o - o p t i c t e n s o r of 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 . The e q u a t i o n o f t h e i n d i c a t r i x (see Appendix B) i s ( - \ ~ r 2 2 E 2 + r 1 3 E 3 ) x J + ( - \ + r 2 2 E 2 + r ^ E ^ + ( - \ + r 3 3 E 3 ) x ^ + 2 ( - r 2 2 E 1 ) x 1 x 2 + 2 ( r 4 2 E 2 ) x 2 x 3 + n e 2 ( r 4 2 E i ) x x x 3 = 1 , (C.2) C . l G r a t i n g v e c t o r o r i e n t a t i o n r e l a t i v e t o the c r y s t a l major axes. where E^, and E^ a r e the e l e c t r i c f i e l d components i n the x-^, x 2 and x^ d i r e c t i o n s r e s p e c t i v e l y ; n Q and n g a r e the 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 . In the 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 holograms, the c - a x i s (x^) i s i n the p l a n e o f i n c i d e n c e and normal to the b i s e c t o r o f the two beams as shown i n F i g . C.2 ( 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 E3. From Eq. C..',2 the major e f f e c t s o f t h i s f i e l d a r e changes i n n £ p r o p o r t i o n a l to r ^ ^ (30.8 x 1 0 - 1 0 c m V - 1 ) and a change i n n Q p r o p o r t i o n a l to r-^3(8.6. x 10 ^cmV" 1) (see Appendix B ) . The changes i n n Q and n g o c c u r whether the l i g h t i s p r o p a g a t i n g i n t h e x^ or x 2 d i r e c t i o n s , and from Eq. C . l we deduce t h a t the c o n t r i b u t i o n 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 s t r o n g e s t i n t h i s c o n f i g u r a -t i o n . I f t h e c r y s t a l i s t u r n e d through 90° so t h a t the c - a x i s i s normal to the p l a n e o f i n c i d e n c e , 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 no l o n g e r o p e r a t i v e , as can be deduced from Eq. C . l , and hologram s t o r a g e i s by d i f f u s i o n o n l y . The f i e l d component t h a t r e s u l t s depends upon the d i r e c t i o n i n which l i g h t i s p r o p a g a t i n g . F i g . C.2(b) shows the case where the g r a t i n g v e c t o r i s i n the x^ d i r e c t i o n . T h i s g e n e r a t e s a f i e l d E-^. In t h i s case 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 g or n Q , t h e o n l y change i n the r e f r a c t i v e i n d e x b e i n g a s m a l l r o t a t i o n due to t h e c r o s s terms. 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 . C.2(c) shows the case where the g r a t i n g v e c t o r i s i n the x 2 d i r e c t i o n . T h i s g e n e r 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 to r r >„(3.4 x 1 0 - 1 ° c m V _ 1 ) w i l l r e s u l t . The 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 to s m a l l r o t a t i o n o f the p r i n c i p a l axes o f the i n d i c a t r i x . To r e a d a hologram i n t h i s c o n f i g u r a t i o n , the e l e c t r i c v e c t o r o f the l i g h t would have to be p o l a r i z e d i n the x„ d i r e c t i o n ( o r d i n a r y p o l a r i z a t i o n ) . .'1-9.0. F i g . C.2 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 s o f f o r m i n g holograms. The e f f i c i e n c y would be a t most 100 times l e s s than f o r holograms s t o r e d i n the c o n f i g u r a t i o n o f F i g . C.2(a) (Moharam 1978b). I t i s e v i d e n t t h a t , from the 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 , the E^ component o f the space charge f i e l d has the g r e a t e s t e f f e c t on the i n d i c e s o f r e f r a c t i o n . Very weak holograms c o u l d be s t o r e d w i t h the g r a t i n g v e c t o r normal to the c - a x i s under c e r t a i n c o n d i t i o n s , namely, the l i g h t must pr o p a g a t e i n the d i r e c t i o n o f the a - a x i s (x-^) and be p o l a r i z e d p a r a l l e l to the b - a x i s ( x 2 ) • I n t h i s case holograms a r e s t o r e d by the d i f f u s i o n mechanism o n l y . In most p r a c t i c a l s i t u a t i o n s , 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 the dominant mechanism f o r hologram s t o r a g e . The E^ component ge n e r a t e d due t o t h i s p r o c e s s , from Eq. C . l , i s p r o p o r t i o n a l t o E v c o s 2 0 c o s 2 ( p . U s u a l l y 8=cp=0 and maximum c o n t r i b u t i o n r e s u l t s . 192. APPENDIX D COUPLED WAVE THEORY FOR THICK HOLOGRAMS K o g e l n i k (1969) has t r e a t e d the Bragg d i f f r a c t i o n o f p l a n e waves by volume phase holograms u s i n g c o u p l e d wave t h e o r y . A condensed v e r s i o n o f h i s treatment w i l l be o u t l i n e d f o r t h e case o f p e r f e c t Bragg c o n d i t i o n s and f o r the case where the g r a t i n g p l a n e s a r e p a r a l l e l t o the b i s e c t o r o f the two i n c i d e n t beams R and S, as i n d i c a t e d i n F i g . D . l . The a n a l y s i s assumes monochromatic l i g h t t o be i n c i d e n t on a phase hologram g r a t i n g o f t h i c k n e s s d a t an a n g l e 8 q p o l a r i z e d p e r p e n d i c u l a r to the p l a n e o f i n c i d e n c e . Only two waves a r e assumed t o be p r e s e n t : the r e f e r e n c e wave R and the s i g n a l wave S. The assumption l i m i t s the a n a l y s i s to holograms w i t h the c o n d i t i o n 2 2 X" » A n Q n ^ (Moharam and Young 1978c), where XQ i s the vacuum wavelength of l i g h t , A i s the g r a t i n g s p a c i n g , n Q i s the mean r e f r a c t i v e i n d e x o f the hologram medium and n^ i s the m o d u l a t i o n a m p l i t u d e o f the r e f r a c t i v e i n d e x . The a n a l y s i s i s v a l i d (under the above c o n d i t i o n ) t o t h i n o r t h i c k holograms as was d i s c u s s e d by S t o r k and W o l f f (1975). Wave p r o p a g a t i o n i n the g r a t i n g i s d e s c r i b e d by the s c a l a r wave e q u a t i o n V 2 E + T 2 E = 0 , ( D . l ) where T 2 = g 2 - 2iaf3. ; a « B , (D.2) and E ( x , z ) i s the complex a m p l i t u d e o f the y component o f the e l e c t r i c f i e l d , a i s the 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 and g= 2im/A 0, w i t h n the r e f r a c t i v e i n d e x . The f r i n g e s o f the phase g r a t i n g r e s u l t from a s p a t i a l m o d u l a t i o n o f n n = n Q + n^ cos K .. r , (D.3) F i g . D . l The r e l a t i o n between the p r o p a g a t i o n v e c t o r s p and a and the g r a t i n g v e c t o r K a t e x a c t Bragg i n c i d e n c e . - 194. where |KI=2IT/A i s the g r a t i n g v e c t o r . S u b s t i t u t i n g Eq. D.3 i n D.2, and assuming t h a t n Q >> n-^  and terms 2 1 i n v o l v i n g n, a r e n e g l i g i b l e , we a r r i v e a t r 2 = 6 2 - 2 i a g + 43 (frn /A ) cos K.r , (D.4) o o o v ± where g = 2irn /A i s the average p r o p a g a t i o n c o n s t a n t , o o o The e x a c t Bragg r e l a t i o n i m p l i e s a = p + K , (D.5) where p and a are the p r o p a g a t i o n v e c t o r s o f the r e f e r e n c e beam R and s u b j e c t beam S. r e s p e c t i v e l y . | p | = |o| = g Q and from t h e Bragg r e l a t i o n , Eq. D.5, p = a = 6 co s 0 : and , K=47rsin8./A n , Z Z O ° U where 6 i s the. i n c i d e n c e angle- i n the medium due to r e f r a c t i o n . The s p a t i a l m o d u l a t i o n i n the r e f r a c t i v e i n d e x forms a g r a t i n g which c o u p l e s the two waves R and S and causes an exchange o f energy between them. The complex a m p l i t u d e s o f these two waves, R(z) and S(z) v a r y a l o n g z as a r e s u l t o f the energy i n t e r c h a n g e and a b s o r p t i o n . The t o t a l e l e c t r i c f i e l d i n the medium i s E ( x , z ) = R(z) exp ( - i p . r ) + S ( z ) exp ( ^ i o . r ) . (D.6) To s o l v e the c o u p l e d wave e q u a t i o n s , Eqs. D . l , D.4 and D.6 a r e combined and by comparing terms i n e x p ( - i p . r ) and e x p ( - i p . r ) we o b t a i n the c o u p l e d wave e q u a t i o n s : R* cos0 + aR = - i y S , (D.7) S* cos6 + aS = - i y R ? (D.8) ;195. where the primes i n d i c a t e d i f f e r e n t i a t i o n w i t h r e s p e c t to z and Y = 1 m i / A 0 l s the c o u p l i n g c o n s t a n t which d e s c r i b e s the c o u p l i n g between the two beams. The i n t e r a c t i o n between the beams i s assumed slow t o w a r r a n t n e g l e c t i n g R" and S" i n the above two e q u a t i o n s . P h y s i c a l l y , the R and S waves change t h e i r a m p l i t u d e a l o n g z due to c o u p l i n g t o the o t h e r wave (yR,yS) and due to a b s o r p t i o n (aR,aS). The c o u p l e d wave e q u a t i o n s (Eq. D.7 and Eq. D.8) can be extended t o a p p l y t o the case 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 the p l a n e o f i n c i d e n c e when y i s r e p l a c e d by y' = Y c o s 2 6 . The g e n e r a l s o l u t i o n t o the c o u p l e d wave e q u a t i o n s i s R(z) = r ^ expd^z + r 2 expo^z, (D.9) S(z) = s^ expo^z + s 2 e x p a 2 z , (D.10) the c o n s t a n t s r_^ and s_^ depend on the boundary c o n d i t i o n s . The c o n s t a n t s o\ may be o b t a i n e d by s u b s t i t u t i n g Eqs. D.9 and D.10 i n t o the c o u p l e d wave e q u a t i o n s . The r e s u l t i s 1 ,(.- a ± iy). ( D . l l ) 1,2 cos To f i n d r . and s. the boundary c o n d i t i o n s f o r . t r a n s m i s s i o n h o l o -x i grams a r e i n t r o d u c e d . The r e f e r e n c e wave R i s assumed to s t a r t w i t h u n i t a m p l i t u d e a t z=0. At i t pro p a g a t e s through the phase g r a t i n g i t decays as i t c o u p l e s energy i n t o S which i s assumed t o be z e r o a t z=0. The boundary c o n d i t i o n s a r e R(0) = 1 , S(0) = 0 . (D.12) S o l v i n g f o r r ^ and s^ and s u b s t i t u t i n g i n Eq. D.10 the am p l i t u d e o f the s i g n a l wave as i t l e a v e s the g r a t i n g i s g i v e n by S ( d ) = (a -Ocos ^ x P ( ° 2 d ) " e xP(°i d)]- (D.13) The d i f f r a c t i o n e f f i c i e n c y o f the g r a t i n g i s d e f i n e d as n=SS In the p r e s e n t case t h i s r e duces to n = exp ( C Q S Q ) s i n 2 yd/cos0 197. APPENDIX E SOURCES OF THE LITHIUM NIOBATE CRYSTALS The crystals used in this study were obtained from Crystal Technology Inc., Mountain View, California and from Harshaw Chemical Company, Solon, Ohio. Table E...1 l i s t s the nominal dimensions, orientation and impurity doping of the crystals. The f i r s t crystal was purchased from Harshaw and a l l other crystals were purchased from Crystal Technology. A l l the crystals were grown by the Czochralski technique. The composition of the melt from which the crystals were grown is given in Table E . l . ,A stoichiometric melt contains more L i than does a congruent melt (Redfield and Burke 1974). In our crystals, the stoichio-metric melt was 49.0 mole % H 2 O while the congruent melt was 48.6 mole % L i 0. 198. TABLE E.1 L i t h i u m N i o b a t e C r y s t a l s A v a i l a b l e to T h i s Study C r y s t a l Dimensions(mm) P o l i s h e d f a c e s ( s ) I r o n - d o p i n g (mole- %) Co m p o s i t i o n o f , the melt-a b c 1 15 3 20 b undoped congruent 2 10 5 10 a,b 0.015 congruent 3 10 2.5 20 b 0.015 congruent 4 10 1.5 10 b 0.015 congruent 5 10 1 10 b undoped s t o i c h i m e t r i c 6 1 10 10 a 0.1 s t o i c h i m e t r i c APPENDIX F DISTORTION PROPERTIES OF LITHIUM NIOBATE F - l I n t r o d u c t i o n In t h i s appendix we d i s c u s s the f a c t o r s t h a t might cause d i s t -o r t i o n o r a b e r r a t i o n o f t h e i n f o r m a t i o n o r images as a r e s u l t o f u s i n g l i t h i u m , n i o b a t e as a s t o r a g e element i n the o p t i c a l system. F-2 Sources o f . D i s t o r t i o n i n LiNbO^ The s o u r c e s o f d i s t o r t i o n a r e due t o t h e f o l l o w i n g f a c t o r s . The i n t e n s i t y o f t h e s i g n a l beam c a r r y i n g t h e i n f o r m a t i o n d u r i n g r e c o r d i n g i s n o t u n i f o r m , i . e . i t has a f r e q u e n c y spectrum and t h a t i t s p l a n e wave s p e c t r a l components a r e o f u n e q u a l a m p l i t u d e s , T h i s l e a d s to u n e q u a l m o d u l a t i o n depths o f the v a r i o u s s i m p l e s i n u s o i d a l holograms r e p r e s e n t i n g the i n f o r m a t i o n t o be s t o r e d . D u r i n g r e a d o u t , t h e r e c o n s t r u c t i o n e f f i c i e n c y o f t h e s e v a r i o u s components w i l l n o t be e q u a l s i n c e the e f f i c i e n c y i s a q u a d r a t i c f u n c t i o n o f the m o d u l a t i o n d e p t h . T h i s f a c t , however, can be u s e d . t o our advantage t o i n c r e a s e t h e s i g n a l - t p - n o i s e r a t i o (SNR) o f i n p u t i n f o r m a t i o n i f we a d j u s t the r e f e r e n c e beam i n t e n s i t y so as t o p r o d u c e maximum m o d u l a t i o n depths f o r the f r e q u e n c y components o f i n t e r e s t , no f u r t h e r p r o c e s s i n g would be r e q u i r e d (Vander Lugt and R o t z 1970). The r e c o r d i n g e f f i c i e n c y depends on t h e magnitude and o r i e n t a t i o n o f t h e g r a t i n g . v e c t o r s o f 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 d u r i n g r e c o r d i n g . The dependence on the magnitude o f the g r a t i n g v e c t o r i s a r e s u l t of the f i n i t e c a r r i e r t r a n s p o r t l e n g t h (see Chapters 2 ,4 ) . The dependence on the o r i e n t a t i o n of the g r a t i n g v e c t o r i s due t o t h e a n i s o t r o p i c n a t u r e o f t h e hologram . r e c o r d i n g mechanisms as w e l l as the n a t u r e o f the e l e c t r o - o p t i c t e n s o r o f LiNbO^ (see Appendices B and C ) . Both of t h e s e two f a c t o r s a r e not v e r y i m p o r t a n t , however, i n t h e c o u r s e o f r e c o r d i n g a s i n g l e hologram s i n c e the g r a t i n g v e c t o r c o r r e s p o n d s to a " h i g h f r e q u e n c y c a r r i e r " 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 v e r t h i s g r a t i n g v e c t o r i s imposed s m a l l e r g r a t i n g v e c t o r s , by comparison, c o r r e s p o n d i n g to the s p a t i a l f r e q u e n c y o f the i n f o r m a t i o n to be r e c o r d e d . Thus the above two f a c t o r s would a f f e c t s t o r a g e o f m u l t i p l e holograms o n l y . The e n v e l o p e of the l i g h t i n t e n s i t y d u r i n g r e c o r d i n g i n f l u e n c e s t h e r e c o r d e d hologram i n two ways. F i r s t , the d i f f r a c t i o n e f f i c i e n c y shows a dependence on t h e i n t e n s i t y o f l i g h t because o f the non-zero dark c o n d u c t i v i t y o f the c r y s t a l as we have seen i n Chapter 6 . Second, the n o n u n i f o r m l i g h t i n t e n s i t y g e n e r a t e s a l a r g e s c a l e space charge f i e l d (Chapter'6.) which causes unequal r e c o r d i n g of the d i f f r a c t i o n e f f i c i e n c y i n the v a r i o u s p a r t s of the c r y s t a l as i s c l e a r l y apparent i n the p r e v i o u s F i g . 6 . 9 . I n a d d i t i o n , t h i s l a r g e s c a l e f i e l d causes a l a r g e s c a l e v a r i a t i o n i n the r e f r a c t i v e i n d e x which causes image d i s t o r t i o n by t h e l e n s a c t i o n (see Chapter 10 ) . F o u r i e r t r a n s f o r m h o l o g r a p h y i s u s u a l l y used i n the r e c o r d i n g o f d i g i t a l o r a n a l o g i n f o r m a t i o n . G e n e r a l l y , i n such a case, an i n t e n s e l i g h t s p o t % c o r r e s p o n d i n g to the dc component o f t h e i n p u t i n f o r m a t i o n i s c o n c e n t r a t e d a t the c e n t e r o f the hologram. To smooth o u t t h e v a r i a t i o n s i n the l i g h t i n t e n s i t y , a phase s h i f t e r i s used o f t h e random t y p e ( B u r k h a r d t 19 70, Takeda 1972a, Takeda e t a l . 1972b) o r d e t e r m i n i s t i c t y p e ( D a l l a s 1973, Yonezawa 1976, T o r i i 1978) i n the p a t h o f the beam c a r r y i n g the i n p u t i n f o r m a t i o n . P o l a r i z a t i o n e f f e c t s o f t h e l i g h t might l e a d t o u n e q u a l r e c o r d i n g e f f i c i e n c y o f the i n f o r m a t i o n . B a s i c a l l y , f o r an extended o b j e c t , the a n g l e between the s i g n a l and r e f e r e n c e , beam i s n o t c o n s t a n t t hroughout the i n t e r f e r e n c e r e g i o n . As a r e s u l t , p o l a r i z a t i o n a f f e c t s the m o d u l a t i o n depth through m u l t i p l y i n g f a c t o r which v a r i e s between z e r o and one. R e f r a c t i o n e f f e c t s , however, r e d u c e the v a r i a t i o n s i n a n g l e s between the r e f e r e n c e and s i g n a l beams s u c h t h a t t h i s p o l a r i z a t i o n f a c t o r w i l l change between one and 0.6 o n l y f o r two beams a p p r o a c h i n g t h e c r y s t a l at g r a z i n g i n c i d e n c e and o f o p p o s i t e d i r e c t i o n s w i t h t h e i r p o l a r i z a t i o n v e c t o r s i n t h p l a n e o f i n c i d e n c e . 

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