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The photorefractive effect in Lithium Niobate Cornish, William D 1976

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THE PHOTOREFRACTIVE EFFECT IN LITHIUM NIOBATE by WILLIAM D. CORNISH B.Sc- (Hon), Queen's U n i v e r s i t y at Kitgston, 1969 M.A.Sc., U n i v e r s i t y of B r i t i s h Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering We accept t h i s t hesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA NOVEMBER, 1975 In 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 an 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 that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s 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 Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a 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 be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f £ ^ c ^ r ! ^ j ^<& v>'^j 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 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date f<> CXc/** , ABSTRACT Exposure of the insulating ferroelectr ic c rys ta l , l i thium niobate, to l ight of the appropriate wavelength causes small changes in the refractive indices. This phenomenon which has recently been named the photorefractive effect allows phase holograms to be stored in the crys ta l . The work described in this thesis was undertaken to obtain an understanding of the mechanisms of the photorefractive effect in connec-tion with possible engineering applications. The process i s thought to involve the spat ia l redistr ibut ion of photo-excited electrons among traps. Space charge f ie lds develop which modulate the refract ive indices through the electro-optic ef fect . I n i t i a l l y , the mechanisms proposed for charge transport were dif fusion and dr i f t in an internal f i e l d of pyroelectric o r ig in . Using these mechanisms, Amodei had treated the i n i t i a l development of phase holograms on the assumption that the electron transport length was short. A theoretical treatment without the rest r ic t ion of short transport length is presented which shows that the eff iciency of hologram writing increases for increased transport length up to a certain l im i t . In addit ion, i t i s shown that the resolution of the recording medium is not l imited by increased transport length. More recently, Glass, von der Linde and Negran have proposed a new phenomenon, the bulk photovoltaic ef fect , as being responsible for charge transport. Photocurrent measurements are presented which provide further evidence for the existence of this ef fect . The relat ive contributions of d r i f t , d i f fusion and the bulk photovoltaic effect to the photorefractive process are investigated by applying a f i e ld during hologram formation. It i s found that the effects i of p o s i t i v e and negative applied f i e l d s are not symmetric. The degree of asymmetry depends on what f r a c t i o n of the c r y s t a l i s i l l u m i n a t e d . I t i s also found that both the voltages applied during previous ex-posures and voltages applied during the current exposure i n f l u e n c e the d i f f r a c t i o n e f f i c i e n c y . I t i s thought that these e f f e c t s are caused by large scale space charge f i e l d s which are produced by exposure to l i g h t . The development of these space charge f i e l d s i s discussed. I t Is concluded that the holograms were w r i t t e n by a combination of d i f f u -sion, d r i f t i n applied and space charge f i e l d s , and the bulk photo-v o l t a i c e f f e c t . The importance of m u l t i p l e i n t e r n a l r e f l e c t i o n s between the faces of the c r y s t a l had not previously been considered. I t i s shown that i n measuring the photorefractive s e n s i t i v i t y by holographic,, e l l i p s metric and adjustable-compensator techniques, neglecting m u l t i p l e r e f l e -ctions may cause serious e r r o r s . Two methods of probing large scale changes i n the r e f r a c t i v e ind i c e s are o u t l i n e d . In the f i r s t method an automated ellipsometer was modified and programmed to measure the b i r e f r i n g e n c e of the l i t h i u m niobate c r y s t a l s and the change i n b i r e f r i n g e n c e due to i l l u m i n a t i o n . This method har- y i e l d e d information on the extent of o p t i c a l l y - i n d u c e d space charge f i e l d s , the uniformity of the c r y s t a l and the e f f e c t s of heating c r y s t a l s under d i f f e r e n t conditions. A second and more ra p i d method of inspecting large scale changes i n the r e f r a c t i v e i n d i c e s i s based on making the c r y s t a l act as a Fabry-Perot interferometer. The s e n s i t i v i t y of photorefractive c r y s t a l s to l i g h t has been associated with impurities and defects i n the c r y s t a l l a t t i c e . Methods of modifying the valence state of i r o n impurities are important since i i 2+ the photorefractive s e n s i t i v i t y i s dependent on the amount of Fe 2+ 3+ present i n the c r y s t a l and on the r a t i o of Fe to Fe . One of the 3+ 2+ methods used to convert Fe to Fe i s heating the c r y s t a l i n l i t h i u m carbonate. I t i s shown that t h i s treatment, i n a d d i t i o n to reducing i r o n i mpurities, changes the b i r e f r i n g e n c e of the c r y s t a l and reduces the rate at which space charge f i e l d s decay. I t i s suggested that the l a s t e f f e c t i s caused by the destruction of shallow traps. A further charge transport mechanism has. been suggested'in which 2+ 3+ intervalence t r a n s f e r of electrons between Fe and Fe impurity states occurs. I t i s argued that i f , instead, electrons enter the conduction band, luminescence should be observable. I t i s shown that a photo-i luminescence band which i s associated t-;ith i r o n impurities i n the c r y s t a l may be observed i n the region of 770 nm. i i i TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENTS i v LIST OF ILLUSTRATIONS ; v i i i LIST OF TABLES x i i ACKNOWLEDGEMENT x i i i 1. INRODUCTION 1 2. PHYSICAL MODELS FOR THE PHOTOREFRACTIVE EFFECT 7 2.1 Introduction 7 2.2 The E l e c t r o - o p t i c Nature of the Photorefractive E f f e c t 7 2.3 The Internal F i e l d Theory 8 2.4 Johnston's P o l a r i z a t i o n Model 10 2.5 The Formation of Holograms By D r i f t or D i f f u s i o n . . 12 2.5.1 Introduction 12 2.5.2 Analysis f o r Short D r i f t or D i f f u s i o n Length . 14 2.5.3 The E f f e c t s of Beam Coupling 16 2.5.3.1 Coupling during Writing 16 I 2.5.3.2 Interactions during Readout 19 2.5.4 Analysis with A r b i t r a r y D r i f t or D i f f u s i o n Length 22 l 2.5.5 Discussion 24 2.6 The Bulk Photovoltaic E f f e c t 26 2.7 Transient Photorefractive E f f e c t s 28 2.8 Defect Sites 29 2.9 Discussion 31 2.9.1 Bulk Photovoltaic E f f e c t 31 2.9.2 B u i l t - i n F i e l d s of P y r o e l e c t r i c O r i g i n . . . . 33 2.9.3 D i f f u s i o n 33 3. ELLIPSOMETRIC PROBE OF THE PHOTOREFRACTIVE EFFECT IN LiNbOg. 37 3.1 Introduction . . . . . . 37 >. 3.2 Theory and Operation of the Ellipsometer 39 i v Page 3.3 S e n s i t i v i t y of the Ellipsometer 44 3.4 The Automated Ellipsometer 44 3.5 Sample Alignment 47 3.6 Temperature Control f o r Ellipsometer Measurements . . . 4 7 3.7 The E f f e c t s of Mult i p l e Internal R e f l e c t i o n s 49 3.7.1 The E f f e c t on the Measurement of the B i r e -fringence 49 3.7.2 The E f f e c t of Mu l t i p l e Internal Reflections on the Photorefractive Process 53 3.8 Birefringence Measurements Along the c-axis of the C r y s t a l 54 3.9 Optically-induced Birefringence Change Due to a One-Dimensional Gaussian Beam 59 3.9.1 Introduction 59 3.9.2 Theoretical Considerations 62 3.9.3 Experimental Results 65 3.9.4 Discussion 68 3.10 Measurements on Crystals Heated i n IJ^COg 68 3.10.1 Introduction 68 3.10.2 Experimental Procedures and Results . . . . . . 6 9 3.10.3 Discussion 71 4. THE USE OF FABRY PEROT FRINGES TO OBSERVE THE PHOTOREFRACTIVE EFFECT 77 4.1 Introduction 77 4.2 Experimental Procedures . . 77 4.3 Experimental Results 79 4.4 Discussion . . . . . 81 5. EXPERIMENTAL CONSIDERATIONS FOR HOLOGRAM FORMATION . . . . . . 83 5.1 Elementary Equations 83 5.2 The O p t i c a l System 85 5.3 Hologram Storage 90 v Page 6 . INFLUENCE OF MULTIPLE INTERNAL REFLECTIONS AND THERMAL EXPANSION ON THE EFFECTIVE DIFFRACTION EFFICIENCY OF HOLOGRAMS IN L i N b 0 3 93 6 .1 Introduction 93 6 .2 Theory 93 6 . 3 Experimental Results 100 7. PHOTOCURRENTS IN LITHIUM NIOBATE . . . . . . 103 7 .1 Introduction 103 7.2 Experimental Procedure . . . . . . . . . 103 7 . 3 Results 104 7 .4 Discussion . . 108 8 . THE EFFECTS OF INTERNAL AND APPLIED FIELDS ON HOLOGRAMS STORED IN L i N b 0 3 112 8 .1 Introduction . 112 8 .2 Experimental Procedures 114 8 . 2 . 1 Sample Preparation and Hologram Measurements . 114 8 . 2 . 2 M u l t i p l e I n t e r n a l R e f l e c t i o n s . 115 8 . 3 Results 121 8 .4 Discussion 122 9 . LUMINESCENCE DUE TO IRON CENTRES . . . . . . . . . . . . . . 129 9 . 1 Introduction 129 9 .2 Experimental Procedures 129 9 . 3 Results and Discussion . 131 1 0 . CONCLUSIONS 137 1 0 . 1 Suggestions f o r Further Research . 139 REFERENCES . . . . . . 141 vi Page APPENDIX A FURTHER PROPERTIES OF LiNb0 3 147 A . l Miscellaneous Physical Properties 147 A.2 C r y s t a l Growth 148 A.3 Thermal Bleaching and Fi x i n g of Holograms i n LiNb0 3 148 APPENDIX B THE ELECTRO-OPTIC BEHAVIOUR OF LiNb0 3 151 APPENDIX C COUPLED WAVE THEORY FOR THICK HOLOGRAM GRATINGS . . 154 APPENDIX D SOURCES OF LiNb0 3 CRYSTALS 159 APPENDIX E THE APPLICATION OF LITHIUM NIOBATE IN A HOLOGRAPHIC MEMORY SYSTEM . . . . . 160 APPENDIX F ELLIPSOMETER ALIGNMENT 166 APPENDIX G ELLIPSOMETRIC INVESTIGATION OF THE ELECTRO-OPTIC AND ELECTROSTRICTIVE EFFECTS IN T a ^ 172 G.l Introduction 172 G.2 Experimental Procedures 174 G. 3 Results 175 APPENDIX H PROGRAMS USED TO CONTROL THE ELLIPSOMETER SYSTEM . . 180 H. l Intoduction 180 H.2 Main Programs 184 H.3 Subroutines Called by the Main Programs . . . . 190 v i i LIST OF ILLUSTRATIONS Page F i g . 2.1 O p t i c a l l y induced 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 F i g . 2.2 C o n f i g u r a t i o n f o r rec o r d i n g holograms 13 F i g . 2.3 Experimental arrangement f o r rec o r d i n g simple phase g r a t i n g s 18 F i g . 2.4 S p a t i a l r e l a t i o n s of the i n t e n s i t y , space charge, space charge f i e l d , and index modulation i n a s i n u s o i d a l g r a t i n g 21 F i g . 2.5 R e l a t i o n s of 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 of forming holograms 35 F i g . 3.1 Schematic of computer-controlled e l l i p s o m e t e r system 38 F i g . 3.2 Vector r e l a t i o n s of the e l l i p s o m e t e r elements and the p r i n c i p a l axes of the c r y s t a l 41 F i g . 3.3 Schematic of the apparatus used to thermostat the l i t h i u m niobate c r y s t a l s during e l l i p -someter measurement 48 F i g . 3.4 T y p i c a l measurement of the temperature s t a b i l i t y i n s i d e the i n s u l a t e d box 48 F i g . 3.5 E f f e c t of temperature on the p o l a r i z e r reading when measuring the b i r e f r i n g e n c e i n a 3 mm t h i c k c r y s t a l of L i N b 0 3 50 F i g . 3.6 The d i f f e r e n c e i n the p o l a r i z e r readings of two scans along the c-axi s showing the s c a t t e r i n the readings 50 F i g . 3.7 C a l c u l a t e d change i n A as a f u n c t i o n of a change i n the b i r e f r i n g e n c e and the thic k n e s s 52 F i g . 3.8 M u l t i p l e r e f l e c t i o n s i n a d i e l e c t r i c s l a b . . . 54 F i g . 3.9 V a r i a t i o n w i t h thickness i n the mean i n t e n s i t y due to m u l t i p l e r e f l e c t i o n s 55 F i g . 3.10 The v a r i a t i o n i n the p o l a r i z e r reading along the c - a x i s of an undoped c r y s t a l 56 F i g . 3.11 V a r i a t i o n i n the p o l a r i z e r and analyser readings along the c-axi s of an undoped c r y s t a l 57 v i i i Page F i g . 3.12 V a r i a t i o n i n the p o l a r i z e r reading along the c-axis of a Fe-doped c r y s t a l 58 F i g . 3.13 T h e o r e t i c a l f i t of the p o l a r i z e r readings along the c-axis of an . undoped c r y s t a l 60 F i g . 3.14 Method used to i l l u m i n a t e c r y s t a l s with a narrow beam of l i g h t 61 F i g . 3.15 P r o f i l e of the l i g h t i n t e n s i t y along the c-axis of the c r y s t a l for the method of i l l u m i n a t i o n shown i n F i g . 3.14 . . . 61 F i g . 3.16 The space charge f i e l d s (E ) developed for the cases of d r i f t and d i f f u s i o n caused by the i n t e n s i t y d i s t r i b u t i o n g(x) . 64 F i g . 3.17 E l l i p s o m e t r i c scan of the o p t i c a l l y - i n d u c e d birefringence change caused by a s i n g l e l a s e r beam focussed by a c y l i n d r i c a l lens 66 F i g . 3.18 E l l i p s o m e t r i c scan of the o p t i c a l l y - i n d u c e d birefringence change i n a Fe-doped c r y s t a l . . . . 67 F i g . 3.19 The change i n the p o l a r i z e r reading caused by heating i n Li 2CC> 3 . . • 70 F i g . 3.20 T h e o r e t i c a l f i t of the p o l a r i z e r readings along the c-axis of an undoped c r y s t a l a f t e r treatment i n I^CO^ -70 F i g . 3.21 Change i n the p o l a r i z e r reading due to i r r a d i a t i o n of three places along the c-axis with a one-dimensional Gaussian beam 72 F i g . 3.22 Thermal decay of o p t i c a l damage i n an undoped ' l i t h i u m niobate c r y s t a l before heating i n LiCO^ . . 73 F i g . 3.23 Logarithm of the change i n AP as a function of time 74 F i g . 4.1 O p t i c a l arrangement for taking photographs of the Fabry-Perot interference fringes 78 F i g . 4.2 Fabry-Perot fringes i n an Fe-doped c r y s t a l showing optically-induced changes i n the r e f r a c t i v e indices 80 F i g . 5.1 Interference pattern of two plane waves 84 i x Page F i g . 5 . 2 D i f f r a c t i o n o f t h e r e f e r e n c e wave b y t h e h o l o g r a m g r a t i n g 85 F i g . 5 . 3 E x p e r i m e n t a l a r r a n g e m e n t f o r m e a s u r i n g t h e d i f f r a c t i o n e f f i c i e n c y o f p l a n e wave h o l o g r a m s . 37 F i g . 5 . 4 A l t e r n a t i v e a r r a n g e m e n t f o r m e a s u r i n g t h e d i f f r a c t i o n e f f i c i e n c y 87 F i g . 5 . 5 The a p p a r a t u s u s e d and a measurement o f t h e s t a b i l i t y o f t h e o p t i c a l b e n c h . 89 F i g . 5 . 6 B u i l d - u p o f t h e e f f e c t i v e d i f f r a c t i o n e f f i c i e n c y w i t h t i m e i n a F e - d o p e d c r y s t a l 90 F i g . 6 . 1 M u l t i p l e r e f l e c t i o n s i n a h o l o g r a m g r a t i n g . . . 95 F i g . 6 . 2 E f f e c t i v e d i f f r a c t i o n e f f i c i e n c y o f a h o l o g r a m i n a l i t h i u m n i o b a t e c r y s t a l p l o t t e d a g a i n s t t e m p e r a t u r e change . . 96 F i g . 6 . 3 E x p e r i m e n t a l l y o b t a i n e d r e f l e c t e d i n t e n s i t y o f an a r g o n i o n l a s e r beam i n c i d e n t on a 3 mm t h i c k c r y s t a l o f undoped L i N b O ^ p l o t t e d a g a i n s t t i m e o f e x p o s u r e 97 F i g . 6 . 4 E f f e c t i v e d i f f r a c t i o n e f f i c i e n c y o f a h o l o g r a m i n a F e - d o p e d L i N b O ^ c r y s t a l p l o t t e d a g a i n s t t e m p e r a t u r e 98 F i g . 7 . 1 T ime d e v e l o p m e n t o f t h e p y r o e l e c t r i c and p h o t o c u r r e n t s d u r i n g i l l u m i n a t i o n and o f t h e p y r o -e l e c t r i c c u r r e n t a f t e r t h e l i g h t i s t u r n e d o f f . .105 F i g . 7 . 2 P h o t o c u r r e n t i n an undoped L i N b O ^ c r y s t a l f o r d i f f e r e n t i n t e n s i t i e s 105 F i g . 7 . 3 I n i t i a l s t a g e o f h o l o g r a m f o r m a t i o n i n a n undoped c r y s t a l a t two w a v e l e n g t h s 106 F i g . 7 . 4 I n i t i a l s t a g e o f h o l o g r a m f o r m a t i o n i n a n F e -c doped c r y s t a l a t two w a v e l e n g t h s . . . . • . . . . .106 F i g . 8 . 1 Measu remen t o f one w r i t e , r e a d - e r a s e c y c l e . . . H 6 F i g . 8 . 2 E f f e c t o f t h e a p p l i e d f i e l d on t h e t r a n s m i t -t a n c e o f LiNb03 117 F i g . 8 . 3 E f f e c t o f p r i o r e x p o s u r e a t d i f f e r e n t v o l t a g e s on h o l o g r a m w r i t i n g f o r p a r t i a l i l l u m i n a t i o n o f t h e s a m p l e 119 x F i g . 8 . 4 N o r m a l i z e d v a l u e s o f a r c s i n n v s . a p p l i e d - . a ? . e f i e l d f o r a h o l o g r a m w r i t t e n w i t h t h e c r y s t a l o n l y p a r t l y i l l u m i n a t e d 120 1 /2 F i g . 8 . 5 T ime d e v e l o p m e n t o f a r c s i n n d u r i n g h o l o g r a m w r i t i n g f o r p a r t i a l i l l u m i n a t i o n 123 F i g . 8 . 6 I d e a l i z e d i l l u s t r a t i o n o f t h e d e v e l o p m e n t o f t h e " d c " s p a c e c h a r g e f i e l d 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 126 F i g . . 9 . 1 S c h e m a t i c o f t h a a p p a r a t u s u s e d t o m e a s u r e t h e p h o t o l u m i n e s c e n c e i n L i N b O ^ 130 F i g . 9 . 2 P h o t o l u m i n e s c e n c e s p e c t r a o f L i N b 0 3 a t 300°K . . 132 F i g . 9 . 3 A b s o r p t i o n s p e c t r u m f o r undoped and f o r a F e -doped L iNbOo c r y s t a l b e f o r e and a f t e r h e a t i n g i n L i 2 C 0 3 1 3 3 -F i g . 9 . 4 P h o t o l u m i n e s c e n c e s p e c t r a o f F e - d o p e d L i N b 0 3 a t 300 K f o r two d i f f e r e n t w a v e l e n g t h s o f e x c i t a t i o n - . . . . 135 F i g . A . l A p p a r a t u s f o r t h e C z o c h r a l s k i g r o w t h o f L i N b 0 3 i n an e l e c t r i c f i e l d 149 i F i g . C . l M o d e l o f a t h i c k h o l o g r a m g r a t i n g . . . . . . . 154 F i g . E . l S c h e m a t i c o f a h o l o g r a p h i c memory s y s t e m . . . . 161 F i g . F . l Z e r o c o r r e c t i o n f o r t h e a n g l e o f i n c i d e n c e s c a l e . 167 F i g . F . 2 C o r r e c t i o n s t o t h e a n a l y s e r and p o l a r i z e r s c a l e s . 167 F i g . F . 3 Z e r o c o r r e c t i o n f o r t h e q u a r t e r wave p l a t e s c a l e . 167 F i g . G . l Lower p a r t o f if),A doma in f o r i n c r e a s i n g t h i c k -n e s s o f t a n t a l u m up t o t h r e e c y c l e s 177 F i g . G . 2 " U p p e r p a r t o f i{i,A d o m a i n 177 F i g . G . 3 C o n t o u r s o f c o n s t a n t i n d e x and c o n s t a n t f i l m t h i c k n e s s on a t a n t a l u m s u b s t r a t e 179 x i LIST OF TABLES Page Table 5.1 S e n s i t i v i t y of LiNbO^ to hologram storage 91 Table 7.1 E f f e c t s of s h o r t - c i r c u i t and open-circuit cooling on the photocurrent 107 Table 7.2 Photocurrents measured during hologram formation i n an Fe-doped and undoped c r y s t a l at two d i f f e r e n t wavelengths 107 Table 7.3 A comparison of g L calculated from the photo-current to g L calculated from the d i f f r a c t i o n e f f i c i e n c y 110 Table D.l Lithium niobate c r y s t a l s used i n t h i s study . . . . 159 Table E . l A comparison of some computer memory systems . . . 165 Table H.l Main programs used to c o l l e c t data and co n t r o l the automated ellipsometer , 181 Table H.2 Subroutines used by the main programs l i s t e d i n Table H.l 182 x i i ACKNOWLEDGEMENT I would like to thank my supervisor, Dr. L. Young for his encouragement and guidance during the course of this research. I am grateful for the assistance received from Dr. R. Parsons in the luminescence studies. I wish to express my appreciation to Mr. D. Daines and Mr. J. Stuber for their assistance in the machine shop, to Mr. A. MacKenzie for drawing many of the graphs and to Mrs. A. Semmens for typing a portion of the thesis. The National Research Council of Canada (Grant No. A3392), the Defence Research Board of Canada(Grant No. 5501-67) and the H.R. MacMillan family (fellowship) are gratefully acknowledged for their financial support. xiii x i v 1 CHAPTER 1 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 , exposure to l i g h t of the appropriate wavelength induces small changes i n the r e f r a c t i v e indices . This phenomenon has recently been named the photorefractive e f f e c t . Phase holograms may be stored i n photorefractive c r y s t a l s by exposure to two i n t e r f e r i n g coherent l i g h t beams. The photorefractive e f f e c t has been found i n f e r r o e l e c t r i c c r y s t a l s such as l i t h i u m niobate (Ashkin et a l . 1966), strontium barium niobate (SBN) (Thaxter 1969), barium t i t a n a t e , l i t h i u m tantalate (Ashkin et a l . 1966) and potassium niobate (Ostrowsky et a l . 1970). In addition, the p a r a e l e c t r i c c r y s t a l , potassium tantalate niobate (KTN) above i t s Curie point, exhibits the e f f e c t i f an e l e c t r i c f i e l d i s applied to the c r y s t a l (Chen 1967). The work described here was undertaken to obtain an under-standing of the mechanisms involved i n the photorefractive e f f e c t i n connection with possible engineering a p p l i c a t i o n s . Lithium niobate was chosen as the material for the study because i t seemed the most promising material f o r applications and because high q u a l i t y c r y s t a l s are r e a d i l y a v a i l a b l e . Although more was known about the photorefractive e f f e c t i n l i t h i u m niobate than i n other c r y s t a l s , the basic mechanisms were not understood, the s p e c i f i c a t i o n of the material for engineering applications was incomplete and i t was not c l e a r how to optimize the properties of the c r y s t a l . * More p r e c i s e l y , the l i g h t induces a change i n the o p t i c a l i n d i c a t r i x . 2 When l i t h i u m niobate f i r s t became a v a i l a b l e i n s i n g l e cry- . s t a l s of o p t i c a l q u a l i t y , i t s high e l e c t r o - o p t i c a l c o e f f i c i e n t s and other properties made i t a t t r a c t i v e for use i n e l e c t r o - o p t i c modulators. It was found, however, that i l l u m i n a t i o n caused o p t i c a l inhomogeneities to form i n the c r y s t a l and these degraded the performance of the modulator. The c r y s t a l s were most s e n s i t i v e to the blue and green wave-lengths of l i g h t . The objective of the f i r s t i n v e s t i g a t i o n s was to f i n d ways of eliminating the photorefractive e f f e c t . Control of impur-i t i e s i n the c r y s t a l , e s p e c i a l l y i r o n , was found to be important i n t h i s regard. , Chen, LaMacchia and Fraser (1968) were the f i r s t to point out that the photorefractive e f f e c t can be used to store thick phase holograms i n l i t h i u m niobate. In contrast to photographic methods, no development or bleaching processes are required. The holograms can be o p t i c a l l y or thermally erased and new holograms written. D i f f r a c t i o n e f f i c i e n c i e s approaching 100% are t h e o r e t i c a l l y possible. Experimentally, d i f f r a c t i o n e f f i c i e n c i e s of up to 60% have been reported (Amodei et a l . 1972). Chen et a l . ' s discovery created i n t e r e s t i n the p o t e n t i a l a p p l i c a t i o n of l i t h i u m niobate as a holographic storage medium.in .' • o p t i c a l computer memories. This a p p l i c a t i o n i s being investigated by several companies. I t was shown by van Heerden (1963) that theoret- . 3 i c a l l y the ultimate storage capacity of a volume hologram i s V / A ;Vbits where V i s the volume and A i s the wavelength of l i g h t . This means that 12 3 t h e o r e t i c a l l y more than 10 b i t s can be stored i n a 1 cm c r y s t a l . One method of o p t i c a l l y s t o r i n g data i s on a page by page basis. Each page containing N b i t s would be written or read as a 3 hologram. A large number of holograms would be stored either i n an x-y array or i n superposition. For a w r i t i n g time of x sec, the w r i t i n g rate would beN/x: b i t s / s e c . For instance, i f hologram storage 4 of a page containing 10 b i t s required 1 ms, the w r i t i n g rate would be 10^ b i t s / s e c . Another method was suggested by Carlsen (1974) which would not require the page of data to be assembled before storage. Presently, only prototype page composers have been b u i l t . The data could be written one b i t at a time i n an x-y array with random access to any b i t on a page. Data would be read out i n p a r a l l e l , one page at a time. Experimentally, Carlsen has stored pages of 16,000 b i t s i n i r o n -doped l i t h i u m niobate. A number of review a r t i c l e s are a v a i l a b l e which discuss the advantages and l i m i t a t i o n s of o p t i c a l memories (Rajchman 1970, King 1972, Anderson 1972, H i l l 1972, Kiemle 1974, Chen and Zook 1975). (See Appendix E ). It i s believed that the mechanism of the photorefractive e f f e c t may be broadly described as follows. Exposure of a photorefrac-t i v e c r y s t a l to l i g h t of the appropriate wavelength causes photo-exci-t a t i o n of electrons from traps. A s p a t i a l r e d i s t r i b u t i o n of the electrons among traps sets up space charge f i e l d s which a f f e c t the r e f r a c t i v e indices v i a the e l e c t r o - o p t i c e f f e c t . The space charge f i e l d s may be a l t e r e d by further exposure to l i g h t . The o p t i c a l inhomo-geneities may be removed by heating the c r y s t a l to approximately o 200 C. Electrons are thermally excited and uniformly d i s t r i b u t e d so that on cooling, the f l u c t u a t i o n s i n the r e f r a c t i v e indices are removed. The mechanisms which r e d i s t r i b u t e the electrons are of primary i n t e r e s t i n current research. Rapid development of the subject occurred during the course of t h i s study. At i t s beginning, i t was recognized that both d r i f t and d i f f u s i o n were pos s i b l e transport mechanisms (Amodei 1971a, 1971b) but t h e i r r e l a t i v e importance was not established. A t h e o r e t i c a l treatment of the development of r e f r a c t i v e index gratings through d r i f t or d i f f u s i o n was made without the r e s t r i c t i o n of a short migration length as assumed by Amodei (1971a), This i s discussed i n Chapter 2. I t was shown that the e f f i c i e n c y of hologram w r i t i n g increases f o r increased migration length up to a c e r t a i n l i m i t . Also, increased migration length would not l i m i t the r e s o l u t i o n of the recording medium. Glass, von der Linde and Negran (1974b, 1975a) have proposed that the photorefractive e f f e c t involves an e n t i r e l y new transport mech-anism which they have l a b e l l e d the "bulk photovoltaic e f f e c t " , This mechanism i s thought to be responsible for photocurrents which were pre-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 of p y r o e l e c t r i c o r i g i n . The bulk photovoltaic e f f e c t i s outlined i n Chapter 2 and i n l a t e r chapters (7,8,9) experimental r e s u l t s which bear upon the e f f e c t are discussed. In measuring the d i f f r a c t i o n e f f i c i e n c y of phase holograms stored i n i n s u l a t i n g c r y s t a l s , the e f f e c t of m u l t i p l e i n t e r n a l r e f l e c t i o n s between the c r y s t a l faces had been neglected. Measurements of the build-up of the d i f f r a c t i o n e f f i c i e n c y with time have been used to test various p h y s i c a l models f o r the photorefractive e f f e c t . U s ually a prototype hologram has been wr i t t e n by the i n t e r f e r e n c e of two plane waves. I t i s then necessary to r e l a t e the observable d i f f r a c t i o n e f f i c i e n c y to the predicted r e f r a c t i v e index modulation. The e f f e c t s of multiple r e f l e c t i o n s are discussed i n Chapter 5. I t i s shown that neglecting multiple r e f l e c t i o n s can cause serious e r r o r s . Small o changes i n temperature (approximately 1 C) such as those that are 5 e a s i l y produced by exposure to medium i n t e n s i t y l a s e r beams produce s u f f i c i e n t change i n c r y s t a l thickness to produce s i g n i f i c a n t e f f e c t s . In Chapter 7, the e f f e c t s of applying an external f i e l d to the c r y s t a l are investigated. This i s a useful method of probing the transport mechanism and may also be a u s e f u l means of c o n t r o l l i n g hologram formation. The published data obtained from using t h i s tech-nique were i n apparent contradiction. I t i s shown that the r e s u l t s of such an experiment depend on the f i e l d applied during previous exposure to l i g h t as well as on the f i e l d applied during the current experiment. It i s also shown that the portion of the c r y s t a l illuminated w i l l determine the magnitude of the space charge f i e l d that may develop i n the c r y s t a l . D i f f e r e n t i l l u m i n a t i o n geometries give d i f f e r e n t r e s u l t s . Since the applied f i e l d changes the o p t i c a l thickness of the c r y s t a l , the e f f e c t of multiple r e f l e c t i o n s may also a f f e c t the r e s u l t s . Observations of o p t i c a l l y induced changes i n the b i r e f r i n -gence due to s i n g l e beams can produce u s e f u l information. The compen-sator method of Chen (1969) i s slow and does not allow for absorption. It i s shown (Chapter 3) that ellipsometry has c e r t a i n advantages and i t s usefulness has been considerably increased through automatic computer c o n t r o l . I t i s shown that , as i n hologram studies, the e f f e c t s of i n t e r n a l multiple r e f l e c t i o n s must be taken i n t o account i n the e l l i p s o m e t r i c and adjustable compensator methods. In addition to using the ellipsometer, i t i s shown that large scale changes i n the r e f r a c t i v e index can be r a p i d l y inspected i f the c r y s t a l i s made to act as a Fabry-Perot interferometer (Chapter 4) The s e n s i t i v i t y of photorefractive c r y s t a l s to l i g h t has been associated with impurities i n the c r y s t a l s and defects r e l a t e d to the 6 non-stoichiometry of the c r y s t a l (Peterson et a l . 1972). I t Is of primary importance to determine the nature and properties of the defect s i t e s from which electrons may be photo-excited and trapped. The most e f f i c i e n t method found to increase the photorefractive s e n s i t i v i t y has been the addition of i r o n impurities ( P h i l l i p s et a l . 1972, Peterson et a l . 1973). P h i l l i p s and Staebler (1974) have shown that heating l i t h i u m niobate c r y s t a l s while packed i n L^CO^ reduced i r o n impurities from 3 + 2 + Fe to Fe . They have used t h i s method to c o n t r o l photorefractive s e n s i t i v i t y (Staebler and P h i l l i p s 1974a). Crystals treated by t h i s method have been studied with the ellipsometer (Chapter 3) and by holo-graphic methods (Chapter 7). I t i s shown that i n a d d i t i o n to reducing the i r o n impurities, the treatment changes the birefringence of the c r y s t a l and decreases the rate at which the optically-induced space charge f i e l d s decay. To explain these r e s u l t s i t i s proposed that the treatment destroys shallow traps. To determine whether electrons enter the conduction band or whether charge transport i s the r e s u l t of intervalence t r a n s f e r , a search for luminescence was made. A luminescence band at 770 nm was observed (Chapter 8) which was associated with the i r o n impurities i n the c r y s t a l . The luminescence was stronger i n c r y s t a l s a f t e r heat t r e a t -ment i n Li oC0„. 7 CHAPTER 2 PHYSICAL MODELS FOR THE PHOTOREFRACTIVE EFFECT 2.1 Introduction As was mentioned i n the previous chapter, a number of models have been proposed to explain the photorefractive e f f e c t . The develop-ment of these models i s now outlined to show how they a f f e c t the present understanding of the process. 2.2 The E l e c t r o - o p t i c Nature of the Photorefractive E f f e c t Chen, LaMacchia and Fraser (1968) found that the recon-s t r u c t i o n of holograms stored i n l i t h i u m niobate was only 1/10 as e f f i c i e n t f o r ordinary ray i l l u m i n a t i o n as for extraordinary ray i l l u m i n -a t i o n . To explain t h i s observation, they suggested that the photo-r e f r a c t i v e process responsible for hologram storage involved 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 . The d i f f r a c t i o n e f f i c i e n c y n i s propor-2 t i o n a l to s i n (aAn), where a i s a constant and An i s the amplitude of the r e f r a c t i v e index modulation (Kogelnik 1969). This implies that for Chen et a l . ' s observation (An )/(An ) - 0.3. This ic consistent with e o the e l e c t r o - o p t i c c o e f f i c i e n t r ^ which a f f e c t s the ordinary index being 1/3 that which a f f e c t s the extraordinary index ^^2' In contrast with t h i s , Gaylord et a l . (1972) observed no dependence of the d i f f r a c t e d power on the p o l a r i z a t i o n of the readout beam. They have since found (private communication) that t h i s obser-vat i o n was p e c u l i a r to one p a r t i c u l a r c r y s t a l and a l l other measurements Appendix 2 contains an o u t l i n e of the 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 niobate. 8 they have made are consistent with Chen et a l . ' s observations. They speculated that the holograms they saw were not caused by the mechan-isms usually associated with the photorefractive e f f e c t . To j u s t i f y using the r e l a t i v e magnitudes of r ^ ^ and r ^ to account for the p o l a r i z a t i o n dependence of reading holograms, Chen et a l . proposed that the r e f r a c t i v e Indices were modulated by a space charge f i e l d directed along the x^ or c-axis of the c r y s t a l . The next section d e t a i l s further experiments conducted by Chen to v e r i f y t h i s . 2.3 The Internal F i e l d Theory Using an adjustable compensator method (Sec. 3.1), Chen (1969) observed changes i n birefringence induced with a s i n g l e l a s e r beam. F i g . 2.1(a) shows the o p t i c a l l y induced change i n birefringence along l i n e s p a r a l l e l and perpendicular to the c-axis. The birefringence reverses sign along the c-axis but not along the b-axis. To explain t h i s observation, Chen proposed a model i n which an electron could be excited from one trap and then captured i n another. He assumed that there was an i n t e r n a l e l e c t r i c f i e l d E d i r e c t e d from the p o s i t i v e end o of spontaneous p o l a r i z a t i o n of the c r y s t a l to the negative end, that i s , a n t i p a r a l l e l to the spontaneous p o l a r i z a t i o n vector P g. The d i r e c t i o n of the f i e l d was determined by applying an external f i e l d to the c r y s t a l . One d i r e c t i o n of the f i e l d retarded the change i n the b i r e f r i n -gence while the other d i r e c t i o n enhanced the change. This f i e l d would cause photo-excited electrons to d r i f t along the c-axis toward the beam periphery. Chen appears to have envisioned that a f t e r being retrapped and reexcited many times, these electrons would d r i f t out of the illuminated area and would remain trapped at l e v e l s too deep to be beam - diameter DISTANCE Fig. 2.1 (a) The solid line ( ) shows the change in birefringence along the c-axis and the dashed line ( ) the change along the b-axis due to a beam of circular symmetry (X= 488 nm). (b) Chen's postulated space charge f i e l d distribution which causes the observed change in A(n - n ). e o 10 reexcited by thermal processes. The space ,charge f i e l d , E , created SO by the trapped electrons and p o s i t i v e l y ionized centers from which the electrons originated caused the observed r e f r a c t i v e index v a r i a 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 (Fig. 2.1(b)). Since l i t h i u m niobate ex 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 -n ) i s e o 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 of the e l e c t r i c f i e l d . The space charge f i e l d required for the magnitude of the observed e f f e c t 4 was 6.7: x 10 V/cm. To v e r i f y that there 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 looked for and found a short c i r c u i t photocurrent. The d i r e c t i o n of the photocurrent was consistent with a f i e l d opposite to P . It was suggested that the f i e l d might be of p y r o e l e c t r i c o r i g i n since a portion of the l i g h t used to form the hologram would cause non-uniform heating of the c r y s t a l . However, Chen pointed out that (dP/dT< 0) and therefore the f i e l d would be i n the wrong d i r e c t i o n for h i s observations. Although i t seemed evident that an i n t e r n a l f i e l d existed, Chen did not account for i t s o r i g i n s . 2.4 Johnston'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 of unknown o r i g i n , Johnston (1970) proposed an a l t e r n a t i v e model i n which photo-induced v a r i a t i o n s i n the macroscopic p o l a r i z a t i o n caused the photo-r e f r a c t i v e e f f e c t . Illumination of the c r y s t a l would excite electrons i n the conduction band r e s u l t i n g i n a change i n the density of f i l l e d traps i n the region of i l l u m i n a t i o n . This i n turn 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 divergence of the p o l a r i z a t i o n produces a f i e l d which skews the d i f f u s i o n a l process of electrons i n 11 the conduction band. A f t e r the l i g h t i s turned o f f , there remains a change i n the macroscopic p o l a r i z a t i o n which induces 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 . Using t h i s model, Johnston was able to account q u a l i t a t i v e l y f o r the s p a t i a l l y dependent features of Chen's observations (Fig.2.1). However, Amodei(1971 a), and Amodei and Staebler (1972b) have shown that there are a number of d i f f i c u l t i e s with t h i s mechanism. They concluded that a very large number of electrons would be required to enter the conduction band to account f o r the generation of the f i e l d s necessary to skew the e l e c t r o n i c motion. The same magnitude of induced index change could r e s u l t from space charge f i e l d s created through simple d i f f u s i o n and retrapping processes. The number of electrons involved would be l e s s 3 by a f a c t o r of 10 than would be required i n Johnston's model. In 13 a d d i t i o n , they observed that Johnston's r e s i s t i v i t y measurement (10 ohms) which l e d him to conclude that any i n t e r n a l f i e l d s would r e l a x i n a short time, was abnormally low. T y p i c a l values would allow f i e l d s to remain f o r many weeks. Amodei and Staebler(1972b) suggested that the b u i l t - i n f i e l d which Chen used to explain h i s r e s u l t s was of pyroelectri?. o r i g i n and developed when the c r y s t a l was cooled from a high temperature. The development of such a f i e l d may be explained i n the following way. A rectangular 3m f e r r o e l e c t r i c c r y s t a l with no free charges and no net space charges would have a f i e l d corresponding to a p o l a r -i z a t i o n charge P per u n i t area on faces normal to the c-axis. This rem ' would, i n f a c t , be above normal d i e l e c t r i c breakdown values. In p r a c t i c e , the c r y s t a l would have been cooled from some high temperature at which appreciable con d u c t i v i t y existed, s u f f i c i e n t to cancel the f i e l d . Excess 12 charges would accumulate close to each c-face. As cooling progresses, the conductivity w i l l freeze out while the remanent p o l a r i z a t i o n P rem continues to change. F i n a l l y , an uncompensated component of P r e m w i l l e x i s t giving a b u i l t - i n f i e l d of magnitude 1 r T l 3P • r ST ( - ^ ) d T O a 1 where and T Q are the temperature - at which the conductivity disappears and the 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 . 2.5 The Formation of Holograms by D r i f t or D i f f u s i o n 2.5.1 Introduction To probe the mechanism of hologram storage i n l i t h i u m niobate, i t i s convenient to analyze the formation of a prototype hologram formed by causing two coherent plane waves to i n t e r f e r e i n the volume of the c r y s t a l . The r e s u l t i n g interference pattern i s s i n u s o i d a l and the \ . i n t e n s i t y i s of the form I = I Q ( 1 + m cos kx) 2^ ]_) where k i s the s p a t i a l frequency of the pattern, arid m"is :the _modulation r a t i o and x i s i n the plane of the two beams and perpendicular to t h e i r b i s e c t o r . For two beams i n t e r s e c t i n g at an angle 20 as shown i n Fig.2.2 k = 2n/% where % = A/(2 s i n 6 ) and A i s the wavelength of the l i g h t . Amodei (1971a) has used t h i s configuration to analyze the formation of holograms. He assumed that the d r i f t or d i f f u s i o n length of photo-excited electrons was very much smaller than the grating period i . * A d e f i n i t i o n of the modulation r a t i o i s given i n Chapter 5. 13 c+ -X F i g . 2.2 Configuration f o r recording holograms. The two plane waves R and S i n t e r f e r e to produce a s i n u s i o d a l l i g h t i n t e n s i t y pattern with a period I. The c+ end of the c r y s t a l (of thickness d) i s shown i n the p o s i t i v e x d i r e c t i o n . 14 Staebler and Amodei(1972b) have shown that depending on the mechanism of charge transport, the p e r i o d i c index modulation may be s h i f t e d i n phase with respect to the i n t e n s i t y modulation that created i t . Observations on the transfer of energy between the two w r i t i n g beams, and the assumption that free electrons moved a very short d i s t -ance before being retrapped (Amodei 1971b)led them to conclude that both d i f f u s i o n and d r i f t were responsible f o r the grating formation. In Sec.2.5.4 i t i s shown that the i n t e r p r e t a t i o n of t h e i r r e s u l t s depends on how f a r electrons move. 2.5.2 Analysis for Short D r i f t or D i f f u s i o n Length Chen et a l . (1968) found that they could store holograms with a r e s o l u t i o n of greater than 1600 lines/mm. This led them to assume that the displacement of electrons due to the i l l u m i n a t i o n must be a f r a c t i o n of a micron to be able to record the v a r i a t i o n i n i n t e n -s i t y . Because of t h i s , Amodei assumed that i n developing a theory, i t would be reasonable to r e s t r i c t the d r i f t or d i f f u s i o n length to a f r a c t i o n of the grating wavelength. Amodei assumed that the rate of promotion of electrons into the conduction band g(x) to be proportional to the i n t e n s i t y of l i g h t , at l e a s t i n the i n i t i a l stages of hologram formation when the traps may be taken as uniformly f i l l e d . The assumption of short d r i f t or d i f f u s i o n length implies that the f r e e - c a r r i e r concentration i s given by n(x) = g T(1 + m cos kx) (2.2) O i where x i s the l i f e t i m e of c a r r i e r s and g Q i s proportional to I q ( i n Eq. 2.1). The s p a t i a l d i s t r i b u t i o n of the current was taken as the sum of the d r i f t and d i f f u s i o n components, 15 J(x) =uneE(x) + eD dn(x) , 2 3* dx where e i s the e l e c t r o n i c charge, u i s the mobility for electrons, E(x) i s the t o t a l e l e c t r o n i c f i e l d and D i s the d i f f u s i o n constant for electrons. The rate at which space charge density, p accumulates at any point i s given by the continuity equation ^ = " V - J - (2.4) Combining t h i s with Eq. 2.3 the build-up of the space charge density can be expressed as •t p(t) =- d(uneE +eD dn/dx) dt (2.5) ' o dx The space charge f i e l d supported by the space charge density i s E (x) = sc _P dx (2.6) e where e i s the d i e l e c t r i c constant of the material. In Eq. 2.'4, the space charge density p which Amodei has used to c a l c u l a t e the space charge f i e l d , includes not only the trapped charge density, but also the free charge density i n the conduction band. These equations, then, describe the s i t u a t i o n during i l l u m i n a t i o n . A f t e r i l l u m i n a t i o n ceases the free electron density decays. Amodei considered ( i ) transport due to d r i f t only, J = nepE, where E = - E q (the space charge f i e l d E g c i s neglected i n the transport equation, and the b u i l t - i n f i e l d i s taken negative to cause electrons to d r i f t i n the p o s i t i v e x d i r e c t i o n ) ; and ( i i ) transport due to d i f f u s i o n only J = eD dn. Solution of <Eqs. 2.4 to 2.6 y i e l d dx for d r i f t only (eyTE Qtg om) cos kx (2.7) E = — sc e 16 and f o r d i f f u s i o n o n l y , E = (eDt/rg mk) s i n kx (2.8) sc & o e Thus, w i t h the above assumption of short d r i f t or d i f f u s i o n l e n g t h , a response i s obtained w i t h a d i f f e r e n c e of IT/2 i n phase s h i f t according to whether d r i f t or d i f f u s i o n i s o p e r a t i v e . I t i s i n t e r e s t i n g to note (Young et a l . (1974)) that the s h o r t e r the d i f f u s i o n or d r i f t lengths are assumed to be, the slower the production of the index g r a t i n g s i n c e the e l e c t r o n s are assumed to r e t u r n more n e a r l y to t h e i r o r i g i n a l p o s i t i o n s . 2.5.3 The E f f e c t s of Beam Coupling Staebler and Amodei (1972b) have considered the i m p l i c a t i o n s of beam coupling during reading and w r i t i n g holograms. They have shown that coupled wave a n a l y s i s allows the determination of whether the p e r i o d i c index g r a t i n g s are s h i f t e d w i t h respect to the p e r i o d i c i n t e n -s i t y p a t t e r n s which produce them. In the case they considered, two coherent beams R and S are symmetrically i n c i d e n t at an angle 6 r e l a t i v e to the z a x i s as shown i n F i g . 2.2. Both waves are p o l a r i z e d perpen-d i c u l a r to the plane of incidence and are i n c i d e n t 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 s of r e f r a c t i v e index, i n = cos kx (2.9) that extends from z = 0 to z = d. The two waves R and S can be w r i t t e n i n the form R = r ( z ) e x p ( - i ( 2 Tr cos 6 z + kx)) A 17 (2.12(a)) S = s(z) exp(-i(2 TT cos 6 z - kx)) (2.10) X Kogelnik (1969) (see Appendix C) using coupled wave theory has shown that for a nonabsorbing phase grating and perfect Bragg conditions, dr(z) = -i'ks(z) dz (2.11) ds(z) = -iKr(z) dz where K = frn^/(X cos 9 ). The coupled wave equations have the general solution r(z) = a exp (iio'z) + b exp ( - I K Z ) s(z) = -a exp ( I K Z ) + b exp (-i^z) If we let the boundary conditions be r(0) = 1 and s(0) = A exp (-i<f>) then the coefficients a and b can be determined and the beam amplitudes become r(z) = cos ( K Z ) - iA exp (-i<}>) sin (<z) (2.12(b)) s(z) = - i sin ( K Z ) + A exp (-i<f>) cos ( K Z ) The intensities of the two beams are given by . 12 2 2 2 1^ = |'r | = cos ( K Z ) + A sin ( K Z ) - A sin ( 2K Z ) sin <j> 2 2 2 2 (2.13) I = |s | = sin ( K Z ) + A cos ( K Z ) + A sin (2kz) sin <f> If the beams have equal incident amplitudes, then A = 1 and Eq. 2.13 gives I J J = 1 - sin ( 2 K Z ) sincf) Ig = 1 + sin (2K Z) sin<j> (2.14) Staebler and Amodei(l972b) have shown that these equations also describe the situation where the two waves have a fixed phase relationship, but the phase grating i s moveable along the x axis with An = n^ cos (kx +<j>). Eq. 2.14 now indicates that the energy transfer 18 b e t w e e n beams depends on t h e p o s i t i o n o f t h e g r a t i n g w i t h r e s p e c t t o t h e i n t e r f e r e n c e p a t t e r n . The p h a s e f a c t o r <f> I n E q . 2 . 1 4 r e p r e s e n t s t h e p h a s e s h i f t be tween t h e l i g h t i n t e n s i t y p a t t e r n and t h e i n d e x m o d u l a t i o n i t p r o d u c e s . 2 . 5 . 3 . 1 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 t h e e x p e r i m e n t a l a r r a n g e m e n t shown i n F i g . 2 . 3 , S t a e b l e r and Amode i (1972b) f o u n d t h a t d u r i n g h o l o g r a m f o r m a t i o n t h e r e was e n e r g y t r a n s f e r r e d be tween t h e beams . F rom A m o d e i ' s (1971) a n a l y s i s o f t h e 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 . 2 . 5 . 2 ) , h o l o g r a m s fo rmed by d i f f u s i o n p r o d u c e a p h a s e s h i f t <f> = TT / 2 and h o l o g r a m s f o r m e d b y d r i f t p r o d u c e a p h a s e s h i f t 4> = 0 , . E q . 2 . 1 4 i n d i c a t e s t h a t o n l y t h e v a l u e s o f <j> o t h e r t h a n z e r o o r IT c a u s e e n e r g y t r a n s f e r b e t w e e n t h e beams . T h i s was t a k e n a s e v i d e n c e f o r h o l o g r a m s s t o r e d b y d i f f u s i o n . F i g . 2 . 3 E x p e r i m e n t a l a r r a n g e m e n t f o r r e c o r d i n g s i m p l e p h a s e g r a t i n g s . 19 2.5.3.2 Interactions During Readout When a previously recorded hologram i s read out, i t i s possible f o r a new hologram to be written by the interference of the reading beam and the d i f f r a c t e d beam. During readout, the S beam i s blocked so that the boundary conditions a r e s (0) = 0, and r(0) = 1. From Eq. 2.12, the amplitude of the two beams within the d i f f r a c t i o n grating n = n^ cos(kx) are given by r (z) = cos (kz) (2.15) s(z) = - i s i n ( K Z ) The interference pattern produced by these two beams i s found by sub-s t i t u t i n g Eq. 2.15 into Eq. 2.10 with the r e s u l t X t o t a l = l r + s I = 1 + s i n ( 2 K z ) s i n (kx) (2.16) If the mechanism of hologram formation produces an index modulation proportional to the i n t e n s i t y then the new grating An 2 w i l l be An 2 = n^ s i n (kx) (2.17) The t o t a l phase modulation w i l l be An t = n^ cos (kx) + n 2 s i n (kx) (2.18) Staebler and Amodei.(1972b) argued that the e f f e c t of was to bend the phase grating. This 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 not pursued except to point out that the e f f e c t should be the same for readout with the R beam or the S beam. In the case where hologram formation produces an index modu-l a t i o n s h i f t e d by TT/2, the new grating An^ w i l l be An 3 = n 3 cos (kx) ^ ^ The t o t a l phase modulation w i l l be 20 A n t = + n3^ C O S (2.20) The e f f e c t of i s to increase or decrease the effectiveness of the grating. Although the value of n^ w i l l be larger toward the back of the grating, the d i f f r a c t i o n e f f i c i e n c y depends only on the amplitude of the grating (Kermish 1969). An example of t h i s i s shown i n F i g . 2.4. Curve 3 i l l u s t r a t e s a phase grating that resulted from d i f f u s i o n of electrons. The space charge and the space charge f i e l d producing t h i s grating are shown i n curves 1 and 2 re s p e c t i v e l y . I t i s assumed that a p o s i t i v e f i e l d decreases the index when - the c-axis i s oriented as shown. When the R beam i s used f o r readout, i t i n t e r f e r e s with the d i f f r a c t e d beam to form a l i g h t - i n t e n s i t y pattern shown by the fourth curve. The i n t e n s i t y maximum i s on the +x side of the peaks i n curve 3 because the R beam i s propagating i n the +x d i r e c t i o n . The f i f t h and s i x t h curves show the space charge and the space charge f i e l d r e -s u l t i n g from d i f f u s i o n of electrons due to the i n t e n s i t y pattern of curve 4. The induced index change An^ w i l l either enhance or degrade the o r i g i n a l index pattern fta^ depending on the d i r e c t i o n of the +c-axis. If the c r y s t a l i s oriented i n the same way as i t was for the production of A n ^ , then enhancement occurs. I f the c r y s t a l i s reversed so that the +c end of the c r y s t a l i s facing i n the -x d i r e c t i o n , the total d i f f r a c t i o n e f f i c i e n c y w i l l be degraded. The same r e s u l t i s ob-tained i f the c r y s t a l i s not reversed but the S beam i s used f o r readout. This reverses the phase of the l i g h t modulation, and the space charge that accumulates opposes the space charge already present. i Staebler and Amodei's coupled-wave analysis has shown that i n t e r a c t i o n between the two beams used i n holography depends on the sine of the phase angle $ between the i n t e n s i t y modulation and the index 21 F i g . 2.4 (1): sinusoidal space charge created by a sinusoidal l ight pattern. (2): the space charge f i e l d created by (1). (3): variat ion in the refractive index caused by (2) through the electro-opt ic ef fect . (4): a second l ight intensity pattern which creates (5), a space charge and (6), a space charge f i e l d . The space charge f i e l d either enhances of degrades the grating An^ depending oh the direct ion of the +c end of the crys ta l . 22 modulation. Using Amodei's treatment of the formation of phase holo-grams, they r e s t r i c t the value of <|> to TT/2 when d i f f u s i o n i s responsible for charge transport and to zero or TT when d r i f t i s responsible for charge transport. I t i s not obvious why <j> should be such a d i s c o n t i n -uous function. P h y s i c a l l y i t would be more reasonable for <$> to vary smoothly. In the next section i t i s shown that i f the r e s t r i c t i o n of short d r i f t or d i f f u s i o n length i s removed, then d r i f t w i l l allow any value of cj> depending on the d r i f t length where as the value of <j> for d i f f u s i o n w i l l be ±T/2 independent of the d i f f u s i o n length. 2.5.4 Analysis with A r b i t r a r y D r i f t or D i f f u s i o n Length The need for the assumption that the d r i f t or d i f f u s i o n length be short may be removed by using the continuity equation (Young, Wong, Thewalt and Cornish 1974) 3n = g - n + _! 3t T e 3x (2.21) In the i n i t i a l stages of hologram formation, the rate of change of the concentration of free electrons i n the conduction band i s zero at con-stant l i g h t i n t e n s i t y so that 0 = g - n _ + l | J x e 9x Amodei's (1971a) assumption that n = gt corresponds to dropping the term (1/e) 9J/8x i n Eq. 2.22. This term i s the negative of g - n/x, which gives the rate of trapped space-charge build-up causing the e f f e c t s of i n t e r e s t . For the d r i f t only case, J(x) = neyE where E=••-E . Writing g = g p ( l + m cos (kx)), Eq. 2.22 becomes — = —=—(1 + m cos kx) dx L y E Q 23 where L = uE^ and g Q i s proportional to I (Eq. 2.1). This may be solved to y i e l d n(x) = xg u(x) + 'n(O) - xg - -: go m exp(-x/L) •'•2 2 1 + L V xg om H (cos kx - kLsin kx) (2.24) 2 2 1 + L k where u(x) i s a unit step function and n(0) i s the i n i t i a l value of the free electron concentration. Then with 3p /t.3vt = ->9j/c3x and c3E /$X = p/e , sc the space charge f i e l d may be written E g c = _ t _ f(g T - n ) d x (2.25) e^ex . Substituting i n Eq. 2.25, and excluding terms due to the termination of the p e r i o d i c l i g h t pattern, the form of the space charge f i e l d i s E = t e g o m I k 2 L 2 s i n kx + kL cos kx V l (2.26) S C F k I 2 2 2 2 ' J e a + k L 1 + k L This equation can also be derived by taking the convolution of the l i g h t pattern with the impulse response which, f o r t h i s case, i s proportional to exp(-x/L). (In the d i f f u s i o n case, we have two exponentials back-to-back.) P h y s i c a l l y we have a p o s i t i v e space charge wave due to the removal of electrons from traps plus a negative space charge wave due to th e i r retrapping. For the d i f f u s i o n - o n l y case, J = eDdn/dx. Writing L' = (Dx) , the s o l u t i o n for the space charge f i e l d due to d i f f u s i o n i s 2 eg mtkL' E s p = ° 0 9 s i n kx (2.27) S C ( l + k 2 L , 2 ) e 24 2.5.5 Discussion For the case of space charge f i e l d s developed by d r i f t , the phase s h i f t between the per i o d i c l i g h t i n t e n s i t y pattern and the per i o -d i c space charge f i e l d depends on the d r i f t length. In Eq. 2.26, for Lk<<l, Amodei's expression i s obtained (with E c"« cos kx) . For Lk>> 1 a much larger space charge f i e l d i s obtained given by . E = (teg m/ek) s i n kx. The response f o r t h i s case i s due to the sc o p o s i t i v e space charge wave only, the negative charge due to retrapped electrons being uniform. I t i s also independent of E q and x. The phase s h i f t between the l i g h t i n t e n s i t y pattern and the index grating produced depends on the magnitude of the d r i f t length L and v a r i e s from zero (for L k « l ) to T/2 (for L k > : > l ) . In the case of diffusion-formed space charge f i e l d s , f o r large enough L'k, the same l i m i t i n g case i s obtained as for d r i f t only with Lk^>l. However, whatever the magnitude of L'k the phase s h i f t remains the same because the free electrons have equal p r o b a b i l i t y of moving i n e i t h e r d i r e c t i o n . The magnitudes of L (for the d r i f t case) to which Staebler and Amodei's r e s u l t s on coupled wave analysis apply are c r u c i a l . . A phase s h i f t halfway between the two l i m i t i n g cases i s achieved when Lk = 1. The d r i f t length i s then given by L = E Q U X= 1/2 T = X / 4 T T S r n 9 . For 9= 15° and X = 500nm a d r i f t length of 153.7 nm i s cal c u l a t e d . (For 9= 45°, L = 56.3 nm). The actual value of L w i l l depend on the distance 4 between empty traps, and t h e i r cross section. Taking E q = 10 V/cm (which i s smaller than Chen's (1969) estimate of the p y r o e l e c t r i c f i e l d i n h i s c r y s t a l s ) and u = 15 cm 2/Vsec (extrapolated from 1000°K (Jorgensen et a l . 1969)), a value of x = 10 ^ sec i s obtained. The N 25 p y r o e l e c t r l c f i e l d would be expected to be rather an uncontrolled quantity which depends on the h i s t o r y , i n p a r t i c u l a r the thermal h i s t o r y , of the c r y s t a l so that the value of L would depend on the p a r t i c u l a r specimen used. It has been suggested by Staebler and Amodei(1972b) that the p h y s i c a l processes involved i n the photorefractive e f f e c t due to s i n g l e l i g h t beam may not n e c e s s a r i l y be the same as operate i n hologram formation. More s p e c i f i c a l l y , the higher the s p a t i a l frequencies i n -volved i n the l i g h t i n t e n s i t y pattern, the more d i f f u s i o n w i l l tend to become important. The question may be tested by considering a hologram formed by two plane waves. Taking, for s i m p l i c i t y , both d i f f u s i o n length L 1 and d r i f t length L to be short compared to the r e c i p r o c a l of the s p a t i a l frequency, the space charge f i e l d due to d r i f t would be small compared to that due to d i f f u s i o n i f EQ<<Dk. Using the E i n s t e i n r e l a t i o n between u and D (D = K'Tu/e) for a wavelength X = 500nm, temperature T = 300°K and 6 = 6.75° (calculated from an angle of incidence of 15°), E o « K ' T k / e = 765 V/cm, where K' i s the Boltzmann constant. For d i f f u s i o n to dominate, the t o t a l f i e l d i n the c r y s t a l would have to be only a few 10 V/cm. I f the d r i f t length were»not n e g l i g i b l e compared to 1/k, the f i e l d would have to be even smaller. Thus a rather compl.->.te suppression of the t o t a l f i e l d would be required. In conclusion, an increase i n d r i f t or d i f f u s i o n would cause an increase i n w r i t i n g e f f i c i e n c y up to a c e r t a i n l i m i t . A lso, i n deciding whether d r i f t or d i f f u s i o n i s operative i n a p a r t i c u l a r e x p e r i -ment, evidence based on detecting a s p a t i a l s h i f t between the r e f r a c t i v e index grating 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 which produced i t must be interpreted bearing i n mind the following considerations. A grating produced by d i f f u s i o n w i l l be s h i f t e d by ±TT/2 i n the reference frame 26 defined by the l i g h t beams, the sign depending on the d i r e c t i o n of the c(+) axis with respect to the l i g h t beams. For a grating produced by d r i f t , the s h i f t depends on the d r i f t length L. For L>>l/k, the s h i f t i s the same as for d i f f u s i o n . For L<<l/k, there w i l l be either 0 or it s h i f t depending on the d i r e c t i o n of the f i e l d causing d r i f t , with respect to the c(+) d i r e c t i o n and the sign of the appropriate e l e c t r o -optic c o e f f i c i e n t . I f the f i e l d i s a 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 , then changing the d i r e c t i o n of the c(+) axis also reverses the f i e l d and so has no e f f e c t on the d i r e c t i o n of electron motion with respect to the c r y s t a l axes. Intermediate values of L give i n t e r -mediate s h i f t s . In general, other evidence w i l l be required to decide the point, for example,tests of the e f f e c t of an applied f i e l d such as that reported by Staebler and Amodei (1972b). Recently, Vahey (1975) has presented a nonlinear coupled-wave theory of hologram storage i n LiNbO^. Closed - form solutions are found to describe the energy exchange between beams during recording.:.: The"time development of the d i f f r a c t i o n e f f i c i e n c y i s also described. Using the concept of Sec. 2.5.4 (Young et a l . 1974) which involves the dependency of the phase s h i f t between the phase grating and the l i g h t i nterference pattern on the d r i f t length of electrons, Vahey's equations provide a good d e s c r i p t i o n of Staebler et al.'(1972b) coupled-wave experiments. 2.6 The Bulk Photovoltaic E f f e c t Glass, von der Linde and Negran(l974b, 1975a) have proposed that the photorefractive e f f e c t i s caused by an e n t i r e l y new transport mechanism which they have l a b e l l e d the "bulk photovoltaic e f f e c t " . 27 Chynoweth (1956) and Chen (1969) observed that photocurrents would flow i n BaTiO^ and LiNbO^ respectively,:.in the absence of applied f i e l d s . In both cases the e f f e c t was accounted for by i n t e r n a l f i e l d s due to space charge e f f e c t s . Glass et al.(1974b) stated that a f t e r 20 hours of continuous i l l u m i n a t i o n , the ;photocutrents which they measured remained constant. The photoconductivity would relax any i n t e r n a l f i e l d s and a decay of the photocurrent would be noticeable. Experimental _ observations of the photocurrent are presented i n Chapter 7. Glass et a l . have used the bulk photovoltaic e f f e c t rather than i n t e r n a l f i e l d s to account for the photocurrent. Although the physics of t h i s new phenomenon perhaps' have not been f u l l y established, i t appears that Glass et a l . are j u s t i f i e d i n proposing that i t e x i s t s . Glass, von der Linde and Negran have postulated that electrons contributing to the photocurrent and the photorefractive process reside 2+ i n asymmetric p o t e n t i a l wells (Fe ions) Upon e x c i t a t i o n , there i s a greater p r o b a b i l i t y that they w i l l move i n one d i r e c t i o n than another. 2+ They account for t h i s phenomenon by assuming that the Nb-Fe distances i n the ±c-direction are d i f f e r e n t . The asymmetry at a l l the defects has the same e f f e c t and there i s a net e l e c t r o n i c current Following the e x c i t a t i o n of the electron, the ionized impurity i s displaced along the polar axis of the c r y s t a l due to Franck-Condon r e l a x a t i o n . This gives r i s e to a displacement current A f t e r a c e r t a i n time the e l e c t r o n i c momentum w i l l become random and no longer contribute to the current u n t i l recombination. Glass et a l . also suggest that the recom-bin a t i o n process may be asymmetric. The p r o b a b i l i t i e s of recombination from electrons approaching a trap from the ±c-direction are d i f f e r e n t and a net recombination current J a r i s e s . A f t e r recombination the impurity 28 moves back to i t s o r i g i n a l p o s i t i o n but t h i s process contributes no current since the impurity and captured electron move together. The t o t a l steady-state current J i s given by J = J 1 + J 0 - J = K a l (2.28) e l e2 r where a i s the absorption and I the i n t e n s i t y of l i g h t . The term K i s dependent on the nature of the absorption centre (the d i r e c t i o n a l pro-b a b i l i t i e s and mean free paths of e l e c t r o n i c motion on e x c i t a t i o n and recombination) and the photon energy. Even i f the net e l e c t r o n i c current created by e x c i t a t i o n and recombination i s zero, there w i l l s t i l l be a net current due to the i o n i c displacement ^ n ° P e n c i r c u i t , the current J w i l l charge the c r y s t a l J = Kal + eyE (2.29) so that a f i e l d E w i l l be created which w i l l saturate at E = KcxI/ue. sat 2.7 Transient Photorefractive E f f e c t s In addition to the previous models which attempt to explain o p t i c a l inhomogeneities which p e r s i s t long a f t e r the i l l u m i n a t i o n i s removed, Glass et a l . (1975a, 1975b) have studied a photorefractive process that p e r s i s t s f o r only a very short time ('vlOysec). They suggest that the process i s caused by a v a r i a t i o n i n the p o l a r i z a t i o n . Thermal e x c i t a t i o n of the c r y s t a l r e s u l t s i n a p y r o e l e c t r i c p o l a r i z a t i o n due to thermal expansion. O p t i c a l e x c i t a t i o n produces a p o l a r i z a t i o n change due to a change of the dipole moment of the excited defect. These processes produce a displacement current J = dD e/dt, where D £ i s the e l e c t r o n i c displacement. The l i f e t i m e of the e f f e c t i s determined by the r e l a x -a t i o n time of the excited state. These transient e f f e c t s have been obser-3+ ved i n LiNbO, doped with Cr ions (Glass et a l . 1975a, 1975b) but not 29 i n "undoped" or iron-doped l i t h i u m niobate. 2.8 Defect S i t e s The photorefractive e f f e c t i s most e f f i c i e n t when l i g h t of the wavelengths 400 to 5 00nm i s used (Serreze and Goldner 1973). E x c i t a t i o n i s thought to occur from traps within the 3.72 ev band gap (Clark et a l . 1973). As was mentioned i n Chapter 1, both impurities and defects re l a t e d to the non-stoichiometry of the c r y s t a l act as defect s i t e s . P h i l l i p s et a l . (1972) have shown that the gamma i r r a d i a t i o n (of un-stated energy) of undoped LiNbO^ increases the photorefractive sensi-r.. t i v i t y by increasing the concentration of l a t t i c e defects which act as electron traps. In addition, impurity doping with elements such as i r o n , manganese, copper, rhodium and chromium ( P h i l l i p s et a l . 1972, Peterson et a l . 1971, 1973; Mikami et a l . 1973, Glass et al.(l'9;74a) improve the s e n s i t i v i t y of the c r y s t a l . As was pointed out i n Chapter 1, i r o n has so f a r proven to be the best dopant. Studies of nominally pure l i t h i u m niobate have revealed the presence of i r o n contamination of 10 ppm to 100 ppm i n a l l samples tested (Peterson et a l . 1971, 1973; Nash 1973). These impurities are thought to be c h i e f l y responsible for the photorefractive e f f e c t i n "undoped" l i t h i u m niobate. Peterson et a l . (1971) and Clark et a l . (1973) have suggested , though not conclusively shown, that the i r o n replaces a l i t h i u m ion i n the c r y s t a l l a t t i c e . Recently, Keune et a l . (1975) have 3+ suggested that the most l i k e l y s i t e of the Fe ion i s the Nb s i t e , based on the Mossbauer-effect study of i r o n impurities i n LiNbO^. 2+ They were not able to i d e n t i f y the most l i k e l y s i t e for the Fe ions. Conversion of i r o n impurities to the divalent state increases 30 the o p t i c a l absorption i n the region of 470 nm with a subsequent increase i n the p h t o t r e f r a c t i v e s e n s i t i v i t y . I t has been suggested (Clark et a l . 1973) that o p t i c a l absorption causes the intervalence transfer of an 2+ 5+ 2+ 3+ electron from a Fe ion to a Nb ion. The Fe i s converted to Fe 2+ 5+ when i t loses an electron. A f t e r the Fe -»• Nb tr a n s f e r , the electron i s free to move i n the conduction band which i s made up of Nb d o r b i t a l s . 3+ Retrapping occurs at Fe ions. As might be expected, the photorefractive s e n s i t i v i t y may be cont r o l l e d by the oxidation state of the i r o n impurities. Smith et a l . (1968) found that f i e l d annealing l i t h i u m niobate at 600°C with a current 2 density of 5 ma/cm decreased the photorefractive s e n s i t i v i t y of the c r y s t a l . Later Peterson et a l . (1971) showed that simply annealing the c r y s t a l i n a i r or oxygen at 600°C for about 75 hours decreased the induced change about 22 times. F i e l d annealing made any induced change; undetectable. The.applied f i e l d caused the ir o n ions to miagrate towards the negative electrode. A yellow-brown deposit eventually appeared on the negative electrode as ir o n came r i g h t out of the c r y s t a l . Using EPR techniques, i t was shown that annealing the c r y s t a l s i n a i r or oxygen 3+ converted about 96% of the i r o n to Fe (Clark et a l . 1973). « 3+ 2+ Methods which convert Fe to Fe include heating the c r y s t a l s i n an argon atmosphere and heating the c r y s t a l s i n a i r while packed i n a l i t h i u m s a l t such as Li2C0 3 ( P h i l l i p s and Staebler 1974a). The Li2C0 3 treatment i s discussed i n Chapter 3. * A d e f i n i t i o n of intervalence transfer i s given^iin Sec. 9.3 (Hush 1967). 31 2.9 Discussion At the present time, i t i s thought that the charge transport involved i n the photorefractive e f f e c t may be described by J = ne yE + e D ^ + Kctl . (2.30) The r e l a t i v e contributions of the d r i f t , d i f f u s i o n and bulk photovoltaic terms are unsettled. These w i l l be discussed i n the following three sections. The space charge f i e l d (E ). may be calculated using Eqs. 2.4, sc j 2.6,2.21 and 2.30 and the change i n the indices of r e f r a c t i o n found from 3 n. An. = - -jj- r . .(E ) . (2.31) where r „ i s the appropriate e l e c t r o - o p t i c c o e f f i c i e n t and n^ i s the appropriate index of r e f r a c t i o n . 2.9.1 Bulk Photovoltaic E f f e c t The new term i n Eq. 2.30 dcoO i s formally equivalent to the current density which would be produced by d r i f t i n a constant f i e l d (either b u i l t - i n or applied) provided that the l i g h t i n t e n s i t y did not change appreciably over distances comparable to that t r a v e l l e d by an electron before trapping. In t h i s case, the extra electron concentra-t i o n produced by l i g h t i s gx where g i s the l o c a l rate of generation of free electrons (proportional to ol) and x i s t h e i r l i f e t i m e . With a f i e l d E , the d r i f t current i s e g x u E v . If we assume that the electrons do not i n f a c t t r a v e l f a r i n the above sense, we may allow for the new process by considering a " virtuai'* f i e l d to be added on to whatever f i e l d s are present, due to space charge or external a p p l i c a t i o n . The magnitude of E i s uncertain and i s probably dependent on various 32 properties of the c r y s t a l . In one case Glass et a l . (1974b) claim that the photocurrent could be reduced to zero with an applied f i e l d 4 2 of 6 x 10 V/cm for a radiant i n t e n s i t y of 0.32 W/cm decreasing to / o 3.8 x 10 V/cm for 0.08 W/cm . In another case (Glass et a l . 1975a) they 3 show that 1.8 x 10 V/cm was required to cancel the photocurrent f o r 0.32 W/cm2 and 1.2 x 10 3 V/cm for 0.08 W/cm2. The experimental data i n 3 Chapter 8 suggests that the " v i r t u a l " f i e l d , E , i s approximately 10 V/cm i n the c r y s t a l investigated. The analyses of hologram development using only d r i f t and d i f f u s i o n presented e a r l i e r i n t h i s chapter are not in v a l i d a t e d by the presence of the bulk photovoltaic e f f e c t . The extent to which the photovoltaic e f f e c t i s responsible f o r the process depends on the magnitudes of other f i e l d s present. The experi-mental data i n Chapter 8 demonstrates t h i s point. I t should, however, be noted that the term Kctl does not exactly describe the r e s u l t of the proposed new mechanism when the l i g h t i n t e n s i t y changes r a p i d l y with distance. Thus, the impulse response to a d e l t a function of l i g h t 6(x - a) involves an exponential t a i l a s s o c i -ated with the loss of the i n i t i a l momentum, i n a d d i t i o n to the e f f e c t s of d i f f u s i o n and d r i f t which have been discussed p r e v i o u s l y . Although the mean distance involved i n the i n i t i a l f l i g h t i s only 0.08 nm, according to Glass et al.(1974b), t h i s distance i s too short to be d i r e c t l y meaningful and must imply that most excited electrons do not escape from t h e i r traps. Those that do escape may t r a v e l quite a p p r e c i -able distances on the scale defined by the hologram g r a t i n g . 33 2.9.2 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 Although a new transport mechanism i s believed to be involved, the presence of b u i l t - i n f i e l d s of p y r o e l e c t r i c o r i g i n may also be important i n some circumstances. As was discussed i n Sec. 2.4 c o o l i n g the c r y s t a l s from a higher temperature w i l l create large f i e l d s . The f i e l d s on the surfaces of the c r y s t a l w i l l normally disappear with the passage of time due to stray charges or leakage paths. However, the f i e l d s are so large that i t seems quite p o s s i b l e that i n j e c t i o n or e x t r a c t i o n of electrons should set up space charges wi t h i n the c r y s t a l with the r e s u l t that large f i e l d s are s t i l l present i n the c r y s t a l even a f t e r applying an external short. 2.9.3 D i f f u s i o n As was discussed i n Sec. 2.5.5, a diffusion-dominated process would require that the t o t a l f i e l d i n the c r y s t a l be l e s s than 100 V/cm. This would include the v i r t u a l f i e l d due to the bulk photovoltaic e f f e c t . Glass et'al.(1974b) have used an argument s i m i l a r to that discussed i n Sec. 2.5.5 to claim that d i f f u s i o n currents are n e g l i g i b l e compared to photocurrents. Their evidence i s that space charge f i e l d s of 10"* V/cm can be produced. However, as was discussed i n Sec. 2.9.1, 3 3 the development of space charge f i e l d s l i m i t e d to 10 V/cm to 2 x 10 V/cm appear p o s s i b l e . In t h i s case d i f f u s i o n would play a s i g n i f i c a n t r o l e . Staebler et a l . (1972b,1974a) claim to have stored holograms by d i f f u s i o n . Some of the confusion over the p r e c i s e r o l e of d i f f u s i o n has a r i s e n because holograms are not r e a d i l y stored when the c-axis of the c r y s t a l i s perpendicular to the plane of the two beams used to form the hologram. Examination of the e l e c t r o - o p t i c tensor of LiNbO-34 i l l u s t r a t e s why holograms may not be s t o r e d i n t h i s c o n f i g u r a t i o n . The equation f o r the o p t i c a l i n d i c a t r i x i s <~1 " r 2 2 E 2 + r 1 3 E 3 ) x l 2 + ^2 + r 2 2 E 2 + r 1 3 E 3 ) x 2 2 n n e o + <r±2 + r 3 3 E 3 ) x 3 2 + 2 ( - r 2 2 E l ) x l X 2 n e + 2 ( r 5 1 E 2 ) x 2 x 3 + 2 ( r 5 1 E l ) x 3 x l = 1 (2.32) where and are the e l e c t r i c f i e l d components i n the X j , 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 & are 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 of 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 plane of i n c i d e n c e and normal to the b i s e c t o r of the two beams as shown i n F i g . 2.5(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 Eg. From Eq. 3.32 the major e f f e c t s are a change i n n g p r o p o r t i o n a l to r ^ (30.8 x 10 ^ cm/V)and a change i n n Q p r o p o r t i o n a l to r ^ (8.6 x 10 ^ cm/V). The change i n n Q occurs whether the l i g h t i s propagating i n the x^ or x 2 d i r e c t i o n s . I f the c r y s t a l i s turned through 90° so t h a t the c - a x i s i s normal to the plane of i n c i d e n c e , the f i e l d component t h a t i s c r e a t e d depends i n which c r y s t a l d i r e c t i o n the l i g h t i s propagating. F i g . 2.5(b) shows the case where the g r a t i n g v e c t o r (the d i r e c t i o n normal to the planes of constant r e f r a c t i v e index) i s i n the x 2 d i r e c t i o n . T h i s creates 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 2 2 (3.4 x 10 cm/V) w i l l r e s u l t . The only change i n n g w i l l be caused by the c r o s s term 2 r ^ j E 2 x 2 x 3 which produces a r o t a t i o n of the i n d i c a t r i x . T h i s however has a very small e f f e c t on n £ . To read holograms i n t h i s con-35 36 f i g u r a t i o n the e l e c t r i c vector of the l i g h t would have to be p o l a r i z e d i n the X 2 d i r e c t i o n . The e f f i c i e n c y would be about 100 times l e s s than fo r holograms stored i n the c o n f i g u r a t i o n of F i g . 2.5(a). F i g . 2.5(c) shows the case where the grating vector i s i n the x^ d i r e c t i o n . T h i s creates'a f i e l d E^. In t h i s case there 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 , the only change i n i n d i c a t r i x being a small r o t a t i o n . I t i s evident that the E^ component of the space charge f i e l d has the greatest e f f e c t on the i n d i c e s of r e f r a c t i o n . The f a c t that holograms are not e a s i l y stored when the c-axis of the c r y s t a l i s normal to the planes of incidence i s not a v a l i d argument against d i f f u s i o n as a transport mechanism. Amodei (1971b,1972b) has shown that space charge f i e l d s of 1500 V/cm may be developed by d i f f u s i o n . Chen (1969) reported s t o r i n g holograms i n the c o n f i g u r a t i o n of F i g . 2.5(b) at much reduced e f f i c i e n c y . (Chen di d not s p e c i f i c a l l y s tate the o r i e n -t a t i o n except to say that the c-axis was normal to the plane of i n c i -dence.) Although Amodei believed d i f f u s i o n to be a v a l i d transport mechanism, he was unable to explain why he could not store holograms with the c-axis normal to the plane of incidence. In Chapter 8 more evidence i s presented to support hologram formation both by d i f f u s i o n and by the bulk photovoltaic e f f e c t . 37 CHAPTER 3 ELLIPSOMETRIC PROBE OF THE PHOTOREFRACTIVE EFFECT IN LiNbOg 3.1 Introduction A useful method for probing the photorefractive e f f e c t i n LiNbO^ involves the measurement of the birefringence of a sample at an array of p o s i t i o n s before and a f t e r the optically-induced change i n the i n d i c e s . In t h i s study an ellipsometer was used;to measure the changepin'-biref riagencen- • -In ellipsometry ( F i g . 3.1) the s i g n a l to the detector i s nulled by r o t a t i n g the analyser and p o l a r i z e r with a quarter wave plate set at ±45° azimuth. The changes i n r e l a t i v e phase and amplitude of two orthogonal l i n e a r p o l a r i z a t i o n s are obtained as discussed i n d e t a i l l a t e r i n t h i s chapter. In the adjustable-compensator method used by Chen (1969) and others (Glass 1972, Serreze 1973) ttie f a s t and slow axes of a S o l e i l -Babinet compensator and the c r y s t a l coincide. The analyser and polar-i z e r are crossed at ±45° azimuth and a n u l l i s obtained by adjusting the compensator phase retardation. This method does not allow for dichroism, while ellipsometry does. Ellipsometry has been used extensively f o r the study of f i l m s on surfaces by r e f l e c t e d l i g h t (see Appendix G) but i t does not seem to have been applied to the present problem apart from work i n t h i s laboratory (Wong 1973, Cornish, Moharam and Young 1975a). Automation i s necessary to r e a l i z e the p o t e n t i a l of the method which would other-wise be excessively laborious. In our system, the instrument i s balanced under computer con t r o l on a l i n e or g r i d of points before and a f t e r 38 POLARIZER LASER QWP MOTOR DRIVE CIRCUITS CRYSTAL ANALYSER Pl-DTODETECTOR -<] ENCODER OUTPUT UNIT i INTERFACE VARIABLE GAIN AMPLIFIER I N C R E M E N T A L P L O T T E R I i PDP8E COMPUTER i I TELETYPE Fig. 3.1 Schematic of computer-controlled ellipsometer system. 39 exposure to the la s e r beam which causes the index changes. The o p t i c a l properties which are probed using e l l i p s o m e t r i c or adjustable-compensator techniques d i f f e r from those probed using holographic techniques. From e l l i p s o m e t r i c measurements, the birefringence may be cal c u l a t e d . In holography, changes i n the index for a p a r t i c u l a r p o l a r i z a t i o n are observed rather than the birefringence. 3.2 Theory and Operation of the Ellipsometer The ellipsometer measures the r a t i o (p) of the complex trans-m i t t i v i t i e s f o r l i g h t with the E-vector along the c-axis (T ) and at ri g h t angles to the c-axis (T ). When no multiple r e f l e c t i o n s of l i g h t occur within the sample, p = tanip; exp A = T^/T^ where t. and t„„ are the Fresnel transmission c o e f f i c i e n t s f o r the ITT 2TT a i r / c r y s t a l and c r y s t a l / a i r i n t e r f a c e s i r - l i g h t , s i m i l a r l y f o r t ^ Q and t_ : .6 = 2iTdn / A , where d = sample thickness, n = extraordinary 2q ' • _TT e o' ^ ' e J index (as modified by the e l e c t r o - o p t i c e f f e c t ) , ^ q = vacuum wavelength; and ^ a - i s s i m i l a r l y defined using n Q , the ordinary index. Light from a monochromatic, unpolarized source (He-Ne la s e r i n t h i s case) i s passed through the p o l a r i z e r which produces l i n e a r l y p olarized l i g h t of azimuth P. A f t e r passage through the quarter wave plat e , e l l i p t i c a l l y polarized l i g h t i s incident on the sample. The two orthogonally polarized waves, a f t e r passing through the sample, suffer a r e l a t i v e phase retardation ( A ) and a r e l a t i v e amplitude r e -duction (tan ij) ) . For a measurement, the ellipsometer i s balanced by varying 40 the parameters of the e l l i p t i c p o l a r i z a t i o n of the l i g h t incident upon : the -sample u n t i l the transmitted p o l a r i z a t i o n i s l i n e a r and can be e x t i n -guished when the analyser i s set to azimuth A. The parameter v a r i a t i o n i s achieved by ro t a t i n g the p o l a r i z e r with the quarter wave plate f i x e d at ±45° azimuth. (For a l l measurements i n t h i s t h e s i s , the QWP = -45°.) This method of varying the e l l i p t i c i t y of the l i g h t i s chosen because the c a l c u l a t i o n of A and tan\pfrom the e x t i n c t i o n azimuths of P and A i s greatly s i m p l i f i e d . The convention f o r measuring angles i s as follows. For the normal mode of operation, that i s for measurements made by r e f l e c t i n g l i g h t from the sample, the p o l a r i z e r , analyser and quarter wave plate angles are measured from the plane of incidence. P o s i t i v e angles are measured counter-clockwise when looking towards the l i g h t source. In the:experiments to be described i n t h i s chapter, the l i g h t from the p o l a r i z e r arm of the ellipsometer was normally incident on the c r y s t a l . The o p t i c a l properties of the c r y s t a l were probed by transmitting l i g h t through the c r y s t a l rather than r e f l e c t i n g l i g h t from i t s surface. The c-axis of the c r y s t a l was aligned to be perpendicular to the normally incident beam and at zero azimuth. In determining the r e l a t i o n s h i p s between A and P, and t a n ^ and A i t i s convenient to follow the path of the l i g h t which i s incident on the p o l a r i z e r through to the analyser using the Jones vectors and ma t r i c e s ( S h u r c l i f f 1962). The p o l a r i z e r i n i t i a l l y i s at azimuth P and the f a s t axis of the quarter wave plate(QWP) i s at Q. Referring to Fi g . 3.2, the l i g h t from the p o l a r i z e r , resolved i n the d i r e c t i o n s of the f a s t and slow axes of the QWP can be represented by the normalized Jones vector 41 cos(P - Q) sin(P - Q) (3.2) X-F i g . 3.2 Fast axis of the QWP (F); slow axis of the QWP (S); opt i c axis of the c r y s t a l (X); azimuth of the p o l a r i z e r (P). The r a t i o of the complex t r a n s m i t t i v i t i e s of the QWP may be written as p = T exp(-iA ) c c c = T (cosA - i s i n A ) (3.3) c c c where T i s the r a t i o of the transmittance along the f a s t axis to that c along the slow axis and i s the r e l a t i v e phase retardation. The Jones matrix i s 42 1 0 0 P , (3.4) To rotate the coordinate axis of the l i g h t from the f a s t and slow axes of the QWP to axes p a r a l l e l to the c r y s t a l ' s o p t i c axis and perpendicular to t h i s axis, the counter-rotator matrix cos Q - s i n Q s i n Q cos Q (3.5) i s used. The sample i s simply another b i r e f r i n g e n t plate and can be represented i n a manner s i m i l a r to the QWP, so that p = T exp(iA )= T (cosA + i.sinA ) s s r s s s s (3.6) and the Jones matrix i s given by 1 0 (3.7) The state of p o l a r i z a t i o n of the l i g h t a f t e r passing through the sample i s given by r 1 0 0 a 1 0 cos(P-Q) sin(P-Q) cosQcos(P-Q) - g s i n Q sin(P-Q) p {sinQ cos(P-Q) +p cos Q sin(P-Q)} s c (3.8) For t h i s l i g h t to be extinguished by the analyzer i t must be l i n e a r l y p olarized with an azimuth e, given by tan e = p {sinQcos(P-Q) + p cos Q sin(P-Q)} _s c cosQcos(P-Q) - p c s i n Q sin(P-Q) 43 p {tan Q + p tan(P-Q)} = — °- . (3.9) 1 - p tanQ tan(P-Q) c The analyser must be 90° from e to extinquish the l i g h t , so that tan A = — - — N tan e p tan Q tan(P-Q) - 1 -£ . (3.10) p {tan Q + p tan(P-Q)} Assuming a perfect quarter wave plate(T = 1), with Q = 45°, p = - i c c and tan Q = -1. Eq. 3.10 can be solved f o r pg to give n - 1 - i tan(P+45°) p = p —=. . (3.11) tan A [ l + i tan(P+45 )J Substituting Eq. 3.6 and equating r e a l and imaginary parts y i e l d s • A g = -2P - 90° ± nn, n = 0,1,2,... (3.12) and T = + 1/tan A . (3.13) s Thus the p o l a r i z e r reading i s proportional to the phase change and the tangent of the analyser reading i s proportional to the inverse of the transmittance r a t i o . The r e l a t i o n between the phase change^ and the birefringence of the c r y s t a l i s A = 2jrd (n -n ) . (3.14) s — 7 — e o A T i s re l a t e d to the r a t i o of the absorption c o e f f i c i e n t s along the s fa s t and slow axes of the c r y s t a l . For constant thickness, a change i n P indicates a change i n birefringence and a change i n A indicates a change i n the r a t i o of the two absorption c o e f i c i e n t s along the two p r i n c i p a l d i r e c t i o n s . 44 3.3 S e n s i t i v i t y of the Ellipsometer Using Eqs. 3.12 and 3.14 the s e n s i t i v i t y of the ellipsometer to changes i n birefringence can e a s i l y be computed. If P^ i s the i n i -t i a l p o l a r i z e r reading and P^ the reading a f t e r a change i n birefringence A(n -n ), then e o A ( v n o > = ( 3 - 1 5 ) The s e n s i t i v i t y of the p o l a r i z e r (and analyser) reading i s 0.01° _g corresponding to a change of 3.5 x 10 i n birefringence for a 1.0 mm thick c r y s t a l . However, the r e s o l u t i o n of the measurements i s l i m i t e d by the temperature dependence of the r e f r a c t i v e i n d i c e s . For a 1°C change i n temperature, the change i n the birefringence (at X = .650 nm) i s -4 ^0.5 x 10 (Boyd et a l . 1967). For a 1.0 mm thick c r y s t a l t h i s corre-sponds to 13° change i n the p o l a r i z e r . Errors may a r i s e i n ellipsometry due to imperfect components and imperfect^alignment of the instrument. These errors however, are n e g l i g i b l e compared to the uncertainty introduced by small f l u c t u a t i o n s i n the temperature of the crystal.(See Appendix F.) 3.4 The Automated Ellipsometer The ellipsometer was a Rudolph type 43603-200E, modified for computer con t r o l (PDP 8/E) as indicated i n F i g . 3.1. A Spectra Physics 1 raw He-Ne laser (model 133,unpolarized) was used as a l i g h t source. The detector was a photomultiplier (RCA 8645 tube with a Kepco regulated voltage source). I t was mounted on the end of the analyser arm with a pin hole, a ground glass d i f f u s e r and a 632.8 nm interference f i l t e r 45 (10 nm band pass) used to r e s t r i c t spurious l i g h t signals from i l l u m i n -ating the detector. This allowed the instrument to be used with normal room l i g h t i n g . The Rudolph 546.1 nm quarter wave plate was replaced with a quartz Soleil-Babinet compensator (Gaertner model L-135) mounted on the Rudolph graduated c i r c l e and set for quarter wave retardation at 632.8 nm. The p o l a r i z e r and analyser were driven through anti-backlash gears by stepping motors (IMC Magnetics Corp. #PIN 008-008) with absolute shaft encoders (Decitrak TR 511-CW/D) to read the angles. The s e n s i t i v i t y of the shaft encoders was 0.01 which was one motor step. Balances reproduced with l i t t l e more sc a t t e r . The encoder output unit converted the angles to binary-coded decimal f o r input to the computer v i a the i n t e r f a c e . The i n t e r f a c e was constructed mostly of standard D i g i t a l Equipment Corp. components. The motor drives were c o n t r o l l e d through the i n t e r f a c e . The error s i g n a l from the detector was amplified by a v a r i a b l e gain amp l i f i e r and interfaced with the computer through an analog-to-digital converter (DEC A811, accuracy of 0.1% F.S.). The analog error s i g n a l was displayed on a meter to allow manual n u l l i n g of the ellipsometer. A second set of stepping motors was used to move the c r y s t a l about i n a plane perpendicular to the p o l a r i z e r arm of the ellipsometer. Anti-backlash gears between the motor and stage allowed the sample to be moved i n 0.8 um steps, i f desired. Programs and data were stored on a dual Dectape u n i t . An incremental p l o t e r (Houston model DP-10) allowed graphical output. The programs to co n t r o l the system were written i n a combination of Fortran and assembly language and are l i s t e d i n Appendix H. 46 The balance procedure used was based on the p r i n c i p l e that the l i g h t at the detector varies symmetrically f o r small excursions of the p o l a r i z e r and analyser from t h e i r balance positions (Archer 1962) . The p o l a r i z e r was balanced f i r s t and then the analyser. This was repeated at each balance to reduce- s c a t t e r . The computer determines which way i t must drive the p o l a r i z e r to reduce the error s i g n a l and i t drives the motor u n t i l the error s i g n a l goes through the minimum. It then sums a number of readings (usually 12) a f t e r each step and then reverses, d r i v i n g the motor through the p o s i t i o n of minimum s i g n a l . I t then takes a running sum of readings on the present side of the minimum, adding one and dropping the twelfth previous reading, u n t i l t h i s sum equals the sum taken on the other side of the minimum. The balance point (the mid point between the sums) i s calculated and the p o l a r i z e r driven to that point. The.analyser i s then balanced i n the same manner. This e n t i r e procedure i s completed a second time and then the analyser and p o l a r i z e r readings, as given by the shaft encoders, are stored. To measure the birefringence along, for instance, the c-axis of the c r y s t a l , the ellipsometer i s balanced and then the c r y s t a l i s moved a small distance perpendicular to the probing beam. This proced-ure i s followed repeatedly and a f t e r each ellipsometer balance, the analyser reading,, p o l a r i z e r reading and c r y s t a l p o s i t i o n are stored i n the computer memory. When the memory buffer i s f i l l e d the data i s transfered to the magnetic tape u n i t and the process continues to com-p l e t i o n . During the scan the p o l a r i z e r reading may be p l o t t e d . A f t e r the scan, the data stored on the magnetic tape may be processed and plo t t e d . 47 3.5 Sample Alignment The c r y s t a l s were aligned using the He-Ne las e r of the e l l i p s -ometer. The c r y s t a l s were supplied (see Appendix D) with the edges of the rectangular c r y s t a l s p a r a l l e l to the c r y s t a l axes. The c-axis of the c r y s t a l was aligned to zero azimuth by adjusting the c r y s t a l t i l t to make the appropriate edge p a r a l l e l to the laser beam. To adjust the angle of incidence of the l a s e r beam to 90°, the c r y s t a l was adjusted to cause the beam to be r e f l e c t e d back on i t s e l f . The c r y s t a l was then s l i g h t l y misaligned so that the r e f l e c t e d beam would not enter the p i n hole of the quarter wave pl a t e . This prevented the p o s s i b i l i t y of problems due to multiple r e f l e c t i o n s between the sample and the quarter wave plate (Oldham 1967). 3.6 Temperature Control for Ellipsometer Measurements As was mentioned previously, thermostating the c r y s t a l i s necessary for measurements made on the ellipsometer. ' F i g . 3.3 shows a schematic of the apparatus used. The c r y s t a l was enclosed i n an i n s u l -ated p l a s t i c box. A fan and a proportional temperature c o n t r o l l e r (YSI model 72) with a thermister detector and a f i n e wire heater were used to maintain the temperature to within ±0.02°C. No windows were used, the ellipsometer arms projecting through holes i n the box. A d i g i t a l thermometer (HP model 2802A) was used to measure the absolute temp-erature to approximately ±0.05°C, and another thermister i n a bridge c i r c u i t was used to measure v a r i a t i o n s i n temperature with a s e n s i t i v i t y of 0.01°C. The temperature was maintained a few degrees above room temperature (approximately 29°C). F i g . 3.4 gives an i n d i c a t i o n of the temperature s t a b i l i t y achieved. Since most of the scatter i n e l l i p s o -48 iPOLARIZER \ ARM THERMISTER - SAMPLE HEATER PROPORTIONAL TEMPERATURE CONTROLLER INSULATED BOX FAN F i g . 3.3 Schematic of the apparatus used to thermostat the l i t h i u m niobate c r y s t a l s during ellipsometer measurements. < 0.06 0.04 0.02 0 0 16 24 TIME / minutes 32 40 48 F i g . 3.4 T y p i c a l measurement of the temperature s t a b i l i t y i n s i d e the insulated box. The temperature was monitored with the thermister bridge c i r c u i t . 49 meter readings was due to temperature f l u c t u a t i o n s , better c o n t r o l would extend the s e n s i t i v i t y of the instrument. However the setup was adequate for the present purpose. The effectiveness of temperature con t r o l increases i f the o p t i c a l pathlength.is reduced so that t h i n c r y s t a l s are more sui t a b l e for t h i s type of experiment. F i g . 3.5 shows the e f f e c t of repeatedly balancing the e l l i p -someter with the LiNbO^ c r y s t a l (3mm thick) fixed at one p o s i t i o n , while the temperature slowly changed from 28.0 C to 24.0 C. F i g . 3.6 i s t y p i c a l of the scatter i n the p o l a r i z e r readings. The points are the d i f f e r e n c e i n the p o l a r i z e r readings for two repeated scans along the c-axis of a c r y s t a l (3 mm t h i c k ) . 3.7 The E f f e c t s of M u l t i p l e Internal Reflections 3.7.1 The E f f e c t on the Measurement of the Birefringence When multiple r e f l e c t i o n s of the l i g h t occur between the surfaces of the c r y s t a l , the r e l a t i o n s of the ellipsometry readings to the o p t i c a l properties of the c r y s t a l are not as straightforward as previously shown. If the l i g h t source employed i s a'laser, the coherence length w i l l generally be longer than a few centimeters and therefore always longer than the thickness of the c r y s t a l s examined (which are <_1 cm) . In considering the passage of an o p t i c a l wave through a c r y s t a l , both the amplitude and phase must be accounted f o r . Because a portion of the wave w i l l be r e f l e c t e d each time i t i s incident on a boundary, many waves w i l l be present within the sample, ha l f t r a v e l l i n g i n a d i r e c t i o n other than that of the' incident l i g h t wave. Constructive and destructive interference w i l l occur. The s i t u a t i o n i s analogous to the transmission of l i g h t through t h i n s o l i d films 50 260 r 215 (• 170 ^ 28 26 TEMPERATURE / °C 24 F i g . 3.5 The e f f e c t of temperature on the p o l a r i z e r reading when meas-uring the birefringence i n a 3 mm t h i c k c r y s t a l of LiNbO^. + l r 60 01 •a o < -1 • • • . • • * • • • • • . • * 1 I • • • » 1 0.5 DISTANCE ALONG C AXIS / mm 2 .0 2 . 5 F i g . 3.6 The dif f e r e n c e i n the p o l a r i z e r readings of two scans along the c-axis showing the scatte r i n the readings. 51 (Heavens 1955). For the case where there are no m u l t i p l e r e f l e c t i o n s , Eq. 3.1 holds. However, when m u l t i p l e r e f l e c t i o n s are considered the expres-sions f o r the t r a n s m i t t i v i t i e s , Trr and To, are of the form V 2 e*P(-^> ( 3 A 6 ) T = -1 - r, exp (-216) S u b s t i t u t i n g these i n Eq. 3.1, the r e s u l t obtained i s A (1 - r 2 exp(-2i<5 a)) P = (1 - r ^ 2 exp(-2io^)) (3.17) where A i s the value p would have i f m u l t i p l e r e f l e c t i o n s are absent. Here r and r are F r e s n e l r e f l e c t i o n c o e f f i c i e n t s . From t h i s formula, i t a TT follows that the observed A and the r e l a t i v e phase change (6^ -<5a) are r e l a t e d through tan A=; B tan(6 7 r - S a ) , (3.18) where B = 1 - R R + (R -R ) sin(6 + <5„)/sin(6n-- 6-i.) 1 + R R - (R + R ) c o s ( 6 n +<5ir)/cos(6rT - & v) a i r a T 7 . a 7 r 0 " 2 2 and R = r , R = r . Here B would be u n i t y i f m u l t i p l e r e f l e c t i o n s a a IT TT were absent. The importance of m u l t i p l e r e f l e c t i o n s does not seem to have been noted when usin g t h i s method (or the adjustable-compensator method). Using these methods, data t h a t has been p r e v i o u s l y p u b l i s h e d may be i n e r r o r to some extent (Chen 1969, Serreze et a l . 1973, Glass et a l . 1972, Glass et al.1974b, Wong 1973). F i g . 3.7 shows the change i n A ( v e r t i c a l s c a le) as a f u n c t i o n of change i n b i r e f r i n g e n c e and a l s o of t h i c k n e s s f o r assumed values of the v a r i o u s parameters. (Here n^ and n g were assumed to have changed due to a space charge f i e l d , w i t h F i g . 3.7 Calculated change i n A ( v e r t i c a l ) as a function of (a) change i n birefringence A(n -n ) (calculated appropriate to e l e c t r o - o p t i c e f f e c t with f i e l d as eshown) and (b) thickness. Probe wavelength = 632.8nm. 53 the numerical values of the e l e c t r o - o p t i c c o e f f i c i e n t s from Turner (1966) being used.) For the range shown, the change i n A i s nearly proportional to the change i n the birefringence, but the s e n s i t i v i t y depends on the o p t i c a l thickness of the c r y s t a l . 3.7.2 The E f f e c t of Mul t i p l e Internal Reflections on the  Photorefractive Process The photorefractive process i n l i t h i u m niobate depends on the i n t e n s i t y of the i l l u m i n a t i o n and, i n the i n i t i a l stage of the pro-cess, the r e l a t i o n s h i p i s believed to be l i n e a r . Insight into the e f f e c t of multiple r e f l e c t i o n s can be gained by considering the simple case of a plane, uniform l i g h t wave normally incident on the c r y s t a l . In the absence of multiple r e f l e c t i o n s the i n t e n s i t y w i l l be uniform throughout the c r y s t a l (no absorption). With multiple r e f l e c t i o n s , the amplitude f ( z ) of the e l e c t r i c f i e l d as a function of distance z into the c r y s t a l may be obtained by summing beams. Referring to F i g . 3 . 8 the sum of the beams at a distance z into the sample i s f( z ) = t^ exp(-i27rnz/;y) + t j t ^ exp(-i2irn(2d - z)/X) + t j r r ' exp(-i2Trn(2d + z)/X) •+ . . . ( 3 . 1 9 ) This i s a geometric ser i e s with a sum t, exp(-iz6/d) ( 1 + r e x p ( - 2 i ( l - z/d)6) f (z) = _2 (3.20) 1 - r 2 exp(-216) where T 2 = r \ = r » t n e F r e s n e l r e f l e c t i o n c o e f f i c i e n t of the c r y s t a l / a i r i n t e r f a c e and 6 = 2Trnd/A for normal incidence. 54 F i g . 3.8 Mu l t i p l e r e f l e c t i o n s i n a d i e l e c t r i c slab. E f f e c t i v e l y , a standing wave i s set up between the c r y s t a l surfaces, and the amplitude of the wave var i e s p e r i o d i c a l l y through the c r y s t a l . If the c r y s t a l thickness i s changed s l i g h t l y , the r e l a t i v e phase of the waves launched a f t e r r e f l e c t i o n w i l l be d i f f e r e n t , and the interference of the forward t r a v e l l i n g and backward t r a v e l l i n g waves w i l l be a l t e r e d . If the mean i n t e n s i t y of the l i g h t within the sample i s calcul a t e d , i t i s found to vary p e r i o d i c a l l y with sample thickness as shown i n F i g . 3.9. Because n and n„ are not equal, d i f f e r e n t mean i n t e n s i t i e s e 0 within the c r y s t a l w i l l r e s u l t for TT and o polarized l i g h t f o r the same incident i n t e n s i t y . 3.8 Birefringence Measurements Along the c-axis of the C r y s t a l In i n i t i a l l y probing the c r y s t a l s along the c-axis i t was found that the p o l a r i z e r reading varied considerably. F i g s . 3.10 and 3.11 show scans along two undoped LiNbO^ c r y s t a l s and F i g . 3.12 shows a scan along an Fe-doped (0.015 mole%) c r y s t a l . Considering F i g . 3.11, i t would appear that e i t h e r the 55 N F i g . 3.9 V a r i a t i o n with thickness i n the mean i n t e n s i t y of l i g h t , due to multiple r e f l e c t i o n s . The incident beam i s of un i t i n t e n s i t y , d = thickness, d = 1 mm and X = 441.6 nm. 56 '210T 170 0 2 DISTANCE ALONG C - AXIS (mm) F i g . 3.10 The v a r i a t i o n i n the p o l a r i z e r reading along the c-axis of the undoped c r y s t a l #5 (see Appendix D). 110 DISTANCE ALONG C - AXIS (mm) F i g . 3.11 V a r i a t i o n i n the p o l a r i z e r (P) and analyser (A) readings along the c-axis of the undoped c r y s t a l #4 (see Appendix D). 58 n o r 100 60 £ 9 0 80 70 6 0 0 2 t L DISTANCE ALONG C - AXIS (mm) F i g . 3.12 V a r i a t i o n i n the p o l a r i z e r reading along the c-axis of the Fe-doped c r y s t a l #2 (see Appendix D). 59 birefringence or the thickness are varying p e r i o d i c a l l y along the c-axis ( v a r i a t i o n i n P) and also that the dichroism. i s varying p e r i o d i c a l l y ( v a r i a t i o n i n A). However, when multiple r e f l e c t i o n s are allowed f o r the data i n F i g . 3.11 can be f i t t e d , using Eq. 3.17, assuming a simple gradient i n thickness and extraordinary index, with the ordinary index remaining constant. This i s shown i n F i g . 3.13. The gradient i n t h i c k -ness i s within the tolerances of the o p t i c a l p o l i s h i n g (sides p a r a l l e l to 10 arc seconds) and the gradient i n n^ could have been caused by a gradient i n the non-stoichiometry of the sample as i t was pulled from the melt. I t i s believed that n^ i s dependant on the Li20:Nb20^ r a t i o but that n^ i s not (Bergman et a l . 1968). 3.9 Optically-Induced Birefringence Change due to a One-Dimensional Gaussian Beam 3.9.1 Introduction Wong (1973) has shown that considerable s i m p l i f i c a t i o n of the analysis of the optically-induced index change i s achieved when the problem i s made s p a t i a l l y one-dimensional, and exposure i s mini-mized to reduce saturation e f f e c t s . To investigate the photorefractive e f f e c t , c r y s t a l s were i r r a d i a t e d with a narrow s t r i p of l i g h t from a He-Cd la s e r (X = 441.6nm) that spanned the c r y s t a l as shown i n F i g . 3.14. The diameter of the l a s e r beam was reduced with two microscope objectives and then expanded with a c y l i n d r i c a l lens to give a gaussian p r o f i l e along the c-axis, as shown i n F i g . 3.15. The curve i n F i g . 3.15 was measured with a Gamma S c i e n t i f i c Model 2900 scanning auto-photometer. 60 1051 DISTANCE ALONG C - AXIS (m m ) F i g . 3.13 The c i r c l e s are the p o l a r i z e r readings shown i n F i g . 3.11 The s o l i d curve was calculated with the following assumed parameter^: thickness = 1.00 mm + 1.92 wavelengths/cm,,n = 2.20657 + 3.75 xlO /cm, n = 2.29058. 6 o 61 F i g . 3.15 P r o f i l e of the l i g h t i n t e n s i t y along the c-axis of the c r y s t a l f o r the method of i l l u m i n a t i o n shown i n F i g . 3.14. 62 3.9.2 T h e o r e t i c a l Considerations The shape of the expected birefringence can be determined mathematically for a s i m p l i f i e d model. As w i l l be shown, the e l l i p s o -meter can be used to d i f f e r e n t i a t e between induced index changes due to a d r i f t dominated process and a d i f f u s i o n dominated process. The migration length of the excited electrons can be measured i f i t i s long enough to be resolved by the instrument. For the l i n e a r i z e d model where the space charge f i e l d i s small compared to the t o t a l f i e l d i n the c r y s t a l and the trap occu-pancy i s considered to be only s l i g h t l y perturbed, the rate of genera-t i o n of electrons i s proportional to the l i g h t i n t e n s i t y , and the rate of capture of electrons i s proportional to the free electron concentra-t i o n . This i s analogous to the case discussed i n Chapter.2 except that the 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 i s gaussian rather than s i n u s o i d a l . The equations of Chapter 2 which govern the process are J = -neyE 0 + eD9n/9x (3.21) £ - i f + g ( l i g h t ) - 0 (3.22) a£. = 11 (3.23) "dt ~ 9x 9 E s c _ R ( 3- 2 4) 9x e The current due to the photovoltaic e f f e c t i s formally equi-valent to what would be produced by d r i f t i n an e l e c t r i c f i e l d , as discussed i n Chapter 2. This e f f e c t i s included as part of EQ. For the d r i f t - o n l y case, the space charge f i e l d i s given by E (x) = n(x)eE ty / e . . . sc O n (3.25) ^ 63 For the d i f f u s i o n - o n l y case, the space charge f i e l d i s given by E s c ( x ) = -(eD't/e ) 3n(x) . (3.26) 3x Eq. 3.21 and 3.22 are used to c a l c u l a t e n(x) and dn/dx. For a r b i t r a r y i n t e n s i t y v a r i a t i o n , the number of electrons i n the conduction band, n(x), i s given by the convolution of the i n t e n s i t y pattern, g(x), with the impulse response, h(x), at x = o, for the process being considered (Wong 1973, Young et a l . 1974). For d r i f t , the impulse response i s h(x) = exp(-x/L) (3.27) y E o where u(x) i s the unit step function, L i s the d r i f t length and i s equal to uEQ-r.For d i f f u s i o n , the impulse response at x = 0 i s two decaying exponentials back-to-back: h(x) = h j- exp(-|x|/L') (3.28) where L' = ( D T ) 2 i s the d i f f u s i o n length. The convolution of g ( l i g h t ) with the impulse response i s g(x) * h(x) = fl g(y)h(x-y)dy (3.29) where the gaussian l i g h t i n t e n s i t y function has the form , 2 2, g = g Q e x p ( - a x ) . F i g . 3.16 shows the s p a t i a l v a r i a t i o n of g(x), n(x) and E (x) for the two cases of d r i f t and d i f f u s i o n . The s i t u a t i o n sc i l l u s t r a t e d i s for long migration length. If the migration length were very short, n(x) would be a r e p l i c a of g(x). 64 DRIFT DIFFUSION Fig.3.16 The space charge f i e l d s (E ) developed for the cases of d r i f t and d i f f u s i o n , cansed by the i n t e n s i f y d i s t r i b u t i o n g(x). n(x) i s the s p a t i a l d i s t r i b u t i o n of electrons i n the conduction band during i l l u m i n a t i o n . 65 3.9.3 Experimental Results The r e s u l t s of measurements on a number of c r y s t a l s , both undoped and Fe-doped (0.015 mole %) a l l showed the shape c h a r a c t e r i s t i c of a drift-dominated process. F i g . 3.17 i l l u s t r a t e s a t y p i c a l measure-ment on an undoped c r y s t a l . There was no apparent one-sided exponential t a i l on the curves. In the doped c r y s t a l s , the magnitude of the induced index change was as much as ten times as great as that seen i n undoped c r y s t a l s . F i g . 3.18 shows the r e s u l t s of a measurement on an Fe-doped c r y s t a l . Instead of a sin g l e peak occurring as would be expected, two peaks developed. Possibly, the large change i n the indices was s u f f i c i e n t to appreciably change the o p t i c a l path length of the c r y s t a l . Due to multiple r e l e c t i o n s , the s e n s i t i v i t y of the ellipsometer may have been decreased (see F i g . 3.7) causing AP to decrease i n the region between the two peaks even though A(n - n ) may have increased. In the narrow beam experiments, the shape of the curve i s what would be expected for a process involving d r i f t rather than one involving d i f f u s i o n . The lack of an exponential t a i l on ei t h e r side of the illuminated region indicates that the migration length i s smaller than the r e s o l u t i o n of the instrument (<<100ym). The s p a t i a l r e s o l u t i o n i s l i m i t e d by the diameter of the probing beam since, i n the setup used, the sample could be displaced i n steps of approximately lym. The r e s o l u t i o n might be increased by reducing the probing beam diameter and using deconvolution techniques to extract the information on the s p a t i a l v a r i a t i o n i n the birefringence. This would be useful f or examining materials i n which the electrons were displaced more than 1'.ym. Recently von der Linde et al.(1975a, 1975b) have claimed that photorefractive 66 F i g . 3.17 E l l i p s o m e t r i c scan of the 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 s i n g l e l a s e r beam focussed by a c y l i n d r i c a l lens. Top: Change i n p o l a r i z e r reading A P along c-axis. Bottom; Scan of i n t e n s i t y across l a s e r beam on same h o r i z o n t a l scale as AP curve. 67 0 1 2 3 DISTANCE ALONG C - AXIS /mm F i g . 3.18 E l l i p s o m e t r i c scan of the op 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 i n an Fe-doped (0.015 mole %) c r y s t a l . Top: Change i n the p o l a r i z e r reading along the c-axis. Bottom: Scan of the i n t e n s i t y of the l a s e r beam causing the change. 68 processes i n KTN involve the r e d i s t r i b u t i o n of electrons with a d r i f t length of about 12 ym. 3.9.4 Discussion Ellipsometry i s better suited than adjustable-compensator methods for studying photorefractive e f f e c t s because i n ellipsometry dichroism i s accounted for while i n adjustable-compensator methods i t i s not. In the adjustable-compensator method,-in the absence of multiple r e f l e c t i o n s , f ixed dichroism would r e s u l t i n a f i x e d error i n estimating A, which would :.tend to cancel i n examining changes due to the photorefractive e f f e c t . A change i n dichroism due to the photorefractive e f f e c t would produce an er r o r . M u l t i p l e r e f l e c t i o n s w i l l lead to further errors i n determining A (quite apart from the problem of i n t e r p r e t i n g A ) . From F i g . 3.7 i t can be seen that f o r a c r y s t a l of constant thickness the change i n A i s nearly l i n e a r with the change i n b i r e -fringence. D i f f i c u l t i e s i n i n t e r p r e t i n g A are due to the v a r i a t i o n of the s e n s i t i v i t y of the ellipsometer with the thickness of the sample at.the point being probed. 3.10 Measurements on Crystals Heated i n Li^CO^ 3.10.1 Introduction Measurements were made on c r y s t a l s before and a f t e r heating i n I^CO^ to investigate the e f f e c t s of the treatment on the birefringence and on the photorefractive process. Previously, some e f f e c t s of the treatment on hologram w r i t i n g and on the o p t i c a l absorption had been reported (Staebler et a l . 1974,Phillips et a l . 1974) which indicated 3+ 2+ that the treatment reduced i r o n ions from Fe to Fe . The experiments 69 to be described ind i c a t e that the I^CO^ treatment a f f e c t s the b i r e f r i n -gence of the c r y s t a l by decreasing the value of the extraordinary index. I t i s postulated that t h i s i s due to i n - d i f f u s i o n of Li^O' into the c r y s t a l , which,'in addition to causing i r o n reduction, serves to destroy shallow traps. 3.10.2 Experimental Procedures and Results An undoped c r y s t a l grown from a stoichiometric melt was heated i n a i r to 520°C for 40 hours while packed i n I^CO^ powder. This treatment i s claimed to convert more than 96% of the i r o n impurities to the divalent state(P h i l l i p s et a l . 1974). The c r y s t a l was scanned along the c-axis a f t e r the Li2C0.j treatment. The v a r i a t i o n i n the p o l a r i z e r .".reading i s shown by the lower curve i n F i g . 3.19. The upper curve i s a reproduction of F i g . 3.11 which shows the p o l a r i z e r v a r i a t i o n before the treatment. The pe r i o d i c nature of the readings has been removed and the curve has s h i f t e d downwards. An attempt to f i t the lower curve i n F i g . 3.19 i s shown i n F i g . 3.20. The gradient i n n and the nominal value of n were smaller than the values 6 e e used to f i t the curve measured p r i o r to the I^CO^ treatment. ,A better f i t could probably be achieved by assuming a nonlinear change i n n g along the sample. This would be consistent with uneven d i f f u s i o n of l i t h i u m into the sample along the c-axis. When more heavily doped samples are heated i n I^CO^ t h i s uneven d i f f u s i o n i s r e a d i l y seen. The treatment causes the samples to turn brown and the e f f e c t i s more pronounced near the unpolished edges of the sample. O p t i c a l damage due to a narrow s l i t produced patterns of the same shape as were found f o r the untreated sample. However, the damage at d i f f e r e n t places along the c-axis was more uniform a f t e r treatment. 70 no r 40 I « ' 1 1 > 1 ' I l i 0 2 4 DISTANCE ALONG C - AXIS (mm) F i g . 3.19 The change i n the p o l a r i z e r reading ( c r y s t a l #4) caused by heating i n L^CC^. TOP: before heat treatment (same as F i g . 3.11). BOTTOM: a f t e r heat treatment. DISTANCE ALONG C-AXIS (mm) F i g . 3.20 The c i r c l e s are the p o l a r i z e r readings shown by the bottom curve i n F i g . 3.19. The s o l i d curve was c a l c u l a t e d with the following assumed parameters: thickness = (1.0 mm - 0.19 wavelengths)+ 1.92 wavelengths/cm; n e = 2.20642 + 1.25 x 10-5/cm; n Q = 2.29058. (compare with F i g . 3.13). 71 F i g . 3.21(a) shows the change observed i n p o l a r i z e r readings due to o p t i c a l damage produced by i d e n t i c a l exposures at various places along the c-axis before treatment. This v a r i a b i l i t y was removed by the L^CO^ treatment as shown i n F i g . 3.21(b). The v a r i a t i o n i s believed to in d i c a t e the combined e f f e c t of multiple r e f l e c t i o n s and the gradient i n stoichiometry. Thermal decay of the o p t i c a l damage i n undoped LiNbO^ before and a f t e r L^CO^ treatment was measured by taking repeated scans along the c-axis over a period of a few days. F i g . 3.22 shows how the change i n the p o l a r i z e r reading AP varied with time. The elapsed time from the end of the exposure creating the damage to the measurement of the birefringence at the peak i n AP i s given beside the curves. The curves do not a l l have the same baseline as shown. (This was normalized to i l l u s t r a t e the decay more re a d i l y . ) The sma 11 s h i f t s (P < 1 ) were att r i b u t e d to the temperature not being exactly the same when each scan was taken. E f f e c t i v e l y , a small increase i n temperature lowers the p o l a r i z e r readings. In F i g . 3.23 the logarithm of the change i n AP i s plotted against time and shows an exponential decay with a time constant of approximately 46 hours. The same experiment was repeated for the same undoped c r y s t a l a f t e r the L^CO^ treatment. A f t e r 30 hours, n e g l i g i b l e decay had occurred. 3.10.3 Discussion The L^CO^ treatment introduced by P h i l l i p s and Staebler a f f e c t s not only the absorption of LiNbO^ but also the birefringence. In addition, thermal decay of the o p t i c a l damage was diminished a f t e r 72 20, DISTANCE ALONG C-AXIS (mm) (a) DISTANCE ALONG C-AXIS (mm) (b) F i g . 3.21 Change in polarizer reading due to irradiation of three places along the c-axis with a one-dimensional Gaussian beam giving an average of 110 J/cm at a wavelength of 441.6 nm. (a) before and (b) after lithium carbonate treatment. 73 A r 1 2 3 DISTANCE ALONG C - AXIS (mm) F i g . 3 . 2 2 Thermal decay of o p t i c a l damage i n an undoped l i t h i u m niobate c r y s t a l before heating i n Li2C03. The times i n d i c a t e the elapsed time from the end of i l l u m i n a t i o n to the measurement of the peak i n each curve ( ± 1 min). 74 F i g . 3.23 Logarithm of the change i n AP as a function of time. The experimental points correspond to the peak amplitudes of the curves i n F i g . 3.22. 75 treatment. A decrease i n the concentration of empty traps ( F e J T ) w i l l increase the migration (or d i f f u s i o n ) length of electrons and w i l l , therefore, enhance hologram w r i t i n g , o p t i c a l erasure and thermal decay. This i s e a s i l y seen by noting that the farther electrons move, the greater the e f f e c t i n a l l three cases. The diminished thermal decay seen cannot, therefore, be due to the reduction of i r o n centres. I f , however, the Li2CC>3 treatment served to destroy shal-2+ low traps as well as to reduce i r o n centres to the Fe state,- t h i s would account for a l l the above observations. Electrons would be captured by the shallow traps during hologram w r i t i n g and would con-t r i b u t e to the hologram (or o p t i c a l inhomogeneity), but they would escape more r e a d i l y by thermal a c t i v a t i o n than those trapped i n the i r o n centres. Destruction of the shallow traps would diminish thermal decay. Since i t would increase the distance t r a v e l l e d by free electrons before trapping (as would also occur with the reduction of i r o n centres) i t would therefore a i d i n increasing o p t i c a l w r i t i n g and erasure. This explanation i s consistent with previous reports (e.g. Chen et a l . 1968) that thermal decay occurs i n two stages, an i n i t i a l rapid decay followed by a slower decay. As to why treatment i n I^CO^ should produce these e f f e c t s i t i s well known that o u t - d i f f u s i o n of l i t h i u m (Kaminow et a l . 1973) can be used to produce o p t i c a l wave guides. LiNbO^ can c r y s t a l l i z e i n a non-stoichiometric form, (I^O).^ (Nb 20^) , where \> ranges from 0.48 to 0.50 moles (Carruthers-etal. 1971). I t i s believed that the ordinary index, n Q , i s independent of vt but within the given range the extra- ' ordinary index, n o , increases almost l i n e a r l y • as v decreases (Bergman et 76 a l . 1968). In the production of o p t i c a l wave guiding l a y e r s , i t was suggested (Kaminow et a l . 1973) that L^O was released when the c r y s t a l was heated i n vacuum, thus increasing n . e When LiNbO^ i s heated while packed i n L ^ C O ^ , i t seems reasonable to postulate that Li^O i s di f f u s e d into the c r y s t a l . Intro-2-duction of 0 would destroy oxygen vacancies. I f the oxidation state of the i r o n centres serves to maintain e l e c t r o n e u t r a l i t y then i n t r o -duction of extra L i + would explain the reduction of ir o n centres. The introduction of L^O i s consistent with the decrease of n& required to f i t the data of F i g . 3.19. No explanation f o r the reduction of the i r o n centres was given by P h i l l i p s and Staebler(1974a) but i t i s understood (personal communication) that they had also considered the above explanation. 77 CHAPTER 4 THE USE OF FABRY-PEROT FRINGES TO OBSERVE THE PHOTOREFRACTIVE EFFECT 4.1 I n t r o d u c t i o n M u l t i p l e i n t e r n a l r e f l e c t i o n s of a l a s e r beam between the two o p t i c a l l y p o l i s h e d surfaces of a c r y s t a l produce l i g h t and dark f r i n g e s , depending on the o p t i c a l t h i c k n e s s of the c r y s t a l . The s i t u a t i o n i s s i m i l a r to that found i n a Fabry-Perot e t a l o n (Born and Wolf 1959)• The o p t i c a l path change r e q u i r e d to move a dark f r i n g e to the p o s i t i o n of the next adjacent dark f r i n g e i s A/2 where X i s the wavelength of i l l u m i n a t i o n . The geometry of the f r i n g e s i s an i n d i c a t i o n of the v a r i -a t i o n of the o p t i c a l t hickness of the c r y s t a l . O p t i c a l inhomo-g e n e i t i e s can e a s i l y be seen w i t h t h i s technique. The f r i n g e s can be made to move by hea t i n g the s a r p l e , or by o p t i c a l l y - i n d u c e d r e f r a c t i v e index changes. 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 independently and the magnitude of the o p t i c a l change e s t i -mated. T h i s technique i - e s p e c i a l l y u s e f u l f o r v i s u a l i n s p e c t i o n of the s p a t i a l v a r i a t i o n of o p t i c a l l y - i n d u c e d inhomogeneities. 4.2 Experimental Procedures The f r i n g e s are e a s i l y v i s i b l e to the eye when the r e f l e c -t i o n of an expanded l a s e r beam from the sur f a c e of the c r y s t a l i s viewed. The o p t i c a l arrangement f o r the photographs taken i n t h i s Chapter i s shown i n Figure 4.1. 78 BEAM SPLITTER-BEAM EXPANDER CRYSTAL ]-CAMERA F i g . 4.1 O p t i c a l arrangement f o r taking photographs of the Fabry-Perot interference f r i n g e s . 79 The beam from a la s e r (either He-Ne or argon ion) was s p a t i a l l y f i l t e r e d and expanded to illu m i n a t e the ent i r e c r y s t a l . A beam s p l i t t e r was placed before the c r y s t a l to d e f l e c t the r e f l e c t e d beam towards the camera. 4.3 Experimental Results Photographs of the Fabry-Perot fringes are shown i n F i g . 4.2. For F i g 4.2(a) - 4.2(d), an o s c i l l o s c o p e camera was used with a d i f f u s e r placed where the scope face would usually be. The c r y s t a l was illuminated with an argon ion la s e r ( A = 541.5nm). The two photos of Fi g s . 4.2(e) and 4.2 (f) were taken with a 35mm camera without a lens. The l i g h t source was a He-Ne las e r ( A = 632.8nm). The photos on the l e f t ( F i g . 4.2(a), 4.2(c), 4.2(e) corres-pond to the extraordinary index and those on the r i g h t (Fig. 4.2(b), 4.2(d), 4.2(f)) to the ordinary index. F i g . 4.2(a) and 4.2(b) show the c r y s t a l a f t e r any optically-induced inhomogeneity has been annealed out of the c r y s t a l by heating i t for a few hours at 270°C. The v e r t i c a l band towards the r i g h t end of the photos i s p a r t i a l l y due to f a u l t y p o l i s h i n g of the surface and p a r t i a l l y due to o p t i c a l inhomo-geneities i n the bulk produced during growth. The photos i n F i g . 4.2(c) and 4.2(d) show two e f f e c t s : The t h i n v e r t i c a l l i n e on the l e f t i s the r e s u l t of i r r a d i a t i n g the c r y s t a l with a narrow s t r i p of l i g h t as described i n Chapter 3. The s p a t i a l extent of the index change i s approximately as wide as the beam. In the c e n t r a l area of the c r y s t a l , the fringes have been heavily d i s t o r t e d a f t e r a number of holograms have been stored i n the c r y s t a l . These e f f e c t s are due to the build-up of large dc space charge f i e l d s within the <•> CO Fig. 4.2. Fabry-Perot fringes In an Fa-dcpad crystal shoving optically-induced changes i n the refrectivs indices, ( a ) , (c) (e) vera Bade with extraordinary polarised light and show tha variation in rx&. ( b ) , (d) „ (f) were ssada with ordinary pol-arized light and show tha variation in n . 81 c r y s t a l . The sin u s o i d a l v a r i a t i o n s i n index that produce d i f f r a c t i o n would be too small to be seen. F i g . 4.2(e) and 4.2(f) show o p t i c a l damage due to i r r a d i a t i o n with a s i n g l e beam of c i r c u l a r symmetry. These patterns are analogous to the experimental r e s u l t s of Chen (1969) except that here the v a r i a t i o n i n n e and n^ are seen separately rather than the v a r i a t i o n i n the b i r e -fringence. The damaged portion shown i n F i g . 4.2(f) was approximately the same s i z e as the beam. The damage i n F i g . 4.2(e) extends w e l l beyond the edges of the l i g h t beam. This i s i n contrast to the t h i n l i n e shown i n F i g . 4.2(c) and 4.2(d). Damage i s more apparent for the extraordinary index 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 involved C 1 ^ ) i s about three times as large as for the ordinary case ( r^-j) • The change i n the ordinary index F i g . 4.2(f) w i l l increase with increased exposure and tend toward the s p a t i a l v a r i a t i o n already present i n F i g . 4.2(e). When c r y s t a l s were illuminated with a t h i n s t r i p of l i g h t p a r a l l e l to the c-axis from one c-face to the other, no s p a t i a l change i n the fringes was observed. The l i g h t i n t e n s i t y , i n t h i s case, would vary s p a t i a l l y , perpendicular to the c-axis, and be e s s e n t i a l l y uniform along the c-axis. 4.4 Discussion The f r i n g e changes, produced by the photorefractive e f f e c t , were observed i n a Fe-doped (0.015 mole %) c r y s t a l , 2.5mm thi c k . Similar e f f e c t s were seen i n other Fe-doped c r y s t a l s 1.5mm thick and 10mm thick, however no changes could be seen i n the fr i n g e patterns of an undoped c r y s t a l (3mm t h i c k ) . This indicates that the f i e l d s induced i n undoped c r y s t a l s are much smaller than those induced i n doped c r y s t a l s . 82 An estimate of the magnitude of the space charge f i e l d s i n doped and undoped c r y s t a l s can be obtained from these photos. To move a dark f r i n g e to the p o s i t i o n of the next dark f r i n g e , the optica.! thickness must change by A/2. I f the thickness, d, remains constant then And = A/2 and An = A/2d. For A= 500 nm and d = 2.5 mm, An = 10 -^. -4 -4 In doped c r y s t a l s then, An > 10 and i n undoped c r y s t a l s , An < 10 -4 The space charge f i e l d to give An = 10 i s E = - 2 A n - = 6.0 kV/cm sc 3 r „ n 33 e where n = 2.24 and r„„ = 30.8 x 10~ 1 0 cmV _ 1. e 33 The absence of photorefractive processes when the c r y s t a l was illuminated with a s t r i p of l i g h t p a r a l l e l to the c-axis can be a t t r i b u t e d to the nature of the e l e c t r o - o p t i c tensor as was discussed i n Chapter 2. The same s i t u a t i o n a r i s e s when hologram storage i s attempted with the plane of incidence of the reference and object beams perpendicular to the c-axis. In both cases the l i g h t i n t e n s i t y v a r i e s normal to the c-axis and i s e s s e n t i a l l y constant along the c-axis. In our c r y s t a l s , the o p t i c a l face contained the a-axis and the c-axis, and the l i g h t was transmitted along the b-axis. Space charge f i e l d s created along the a-axis would cause only a small r o t a t i o n of the o p t i c a l i n d i c a t r i x and the r e s u l t i n g change i n index would be very small. 83 CHAPTER 5 EXPERIMENTAL CONSIDERATIONS FOR HOLOGRAM FORMATION 5.1 Elementary Equations To probe the mechanism of hologram storage i n LiNbO^, the most elementary holographic pattern, the sinusoidal grating, was chosen. This prototype hologram configuration i s quite general since any a r b i t r a r y i n t e n s i t y v a r i a t i o n can be separated into s inusoidal patterns through Fourier decomposition. This type of hologram can be formed when two coherent plane waves i n t e r f e r e within the volume of the c r y s t a l . One of these waves i s commonly c a l l e d the reference wave and may be represented by R = Re {r exp(i$ r) exp(icot)} (5.1) The other wave which i s commonly c a l l e d the s i g n a l or object wave may be represented i n a s i m i l a r manner by S = Re{ s exp(i$ g) exp(iut)} (5.2) where w has a si n g l e value which i s equal for both waves, and Re{ } indicates the r e a l part of the complex quantity within the brackets. For convenience, Eq. 5.1 and 5.2 are u s u a l l y divided by exp(iwt) and the Re{ } symbol dropped leaving R = r exp(i$ r) and (5.3) S = s exp(i$ g) . The i n t e n s i t y I, i n the region of interference, i s found by taking the scalar product of the sum of the two amplitude vectors I = (R + S).(R + § ) * = r . r + s.s + r.s exp(i($ -* )} + exp(-i($ )} 84 or I = R + S + 2r.s cos(« - * ) s r (5.4) where R and S are the i n t e n s i t i e s of the i n d i v i d u a l waves. The t h i r d term i n Eq. 5.4 i s the interference term which contains the r e l a t i v e phase information. F i g . 5.1 shows the int e r f e r e n c e of two plane waves. For an angle 20 between the wave normals, the period, & , of the sinu-s o i d a l i n t e n s i t y d i s t r i b u t i o n i s given by 2 £sinO = X (5.5) where x i s the wavelength of the l i g h t i n the medium i n which the l i g h t i s propagating. K x ) —£9 F i g . 5. 1 Interference pattern of two plane waves. To write a hologram i n LiNbO^, the c r y s t a l i s placed i n the region of the interference pattern. The 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 induces a s p a t i a l 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 i c e s of the c r y s t a l which constitutes a volume hologram. To read out the informa-t i o n stored i n the hologram, the c r y s t a l i s illuminated with a plane wave (the reference wave) and the volume d i f f r a c t i o n g r a t i n g s c a t t e r s the l i g h t i n a manner that reconstructs the object or s i g n a l wave used 85 to form the grating. This i s shown i n F i g . 5.2. \—l 1 a - \ F i g . 5. 2 D i f f r a c t i o n of the reference wave by the hologram grating. Maximum d i f f r a c t i o n occurs when Bragg's law (Eq. 5.5) i s s a t i s f i e d . The d i f f r a c t i o n e f f i c i e n c y r\, of the hologram i s the r a t i o of the d i f f r a c t e d i n t e n s i t y to the incident i n t e n s i t y . Kogelnik (1969) has shown that f o r th i c k phase holograms (see Appendix A) 2 n = exp(- ad/cose) s i n (vd) (5.6) where a i s the absorption, d i s the grating thickness, and v= irAn/A o cos8 f o r perpendicular p o l a r i z a t i o n and v = TrAn cos28A o cos6 f o r p a r a l l e l p o l a r i z a t i o n . Here, AQ i s the vacuum wavelength and An i s the amplitude of the s i n u s o i d a l r e f r a c t i v e index gr a t i n g . Eq. 5.6 neglects m u l t i p l e i n t e r n a l r e f l e c t i o n s between the faces of the c r y s t a l . These e f f e c t s are discussed i n the next chapter. 5.2 The O p t i c a l System F i g . 5.3 shows a schematic of the experimental setup used. Light from the l a s e r (either with a Coherent Radiation Model 54 argon ion, or an RCA Model LD2186 He-Cd) was s p l i t with a beam s p l i t t e r i n t o two beams. The transmittance of the beam s p l i t t e r was v a r i a b l e which 86 allowed adjustment of the r e l a t i v e beam powers. Two f i r s t surface mir-rors were used to d i r e c t the two beams to cause them to i n t e r s e c t within the volume of the crystal.-The geometry of the setup was such that the path lengths of the two beams from the beam s p l i t t e r to the c r y s t a l were within 1.0 cm of being equal. This ensured that the path diffe r e n c e was l e s s than the coherence length of the la s e r (10 cm). A s i l i c o n photovoltaic de-: . tector (Alphametrics model dc 1010 with a PI 110 broadband probe) was placed a f t e r the c r y s t a l i n l i n e with the object beam. When the object beam shutter was closed, the energy d i f f r a c t e d from the reference beam toward the detector could be measured, thus allowing the d i f f r a c t i o n e f f i c i e n c y of the hologram to be determined. F i g . 5.4 shows another method used to monitor the d i f f r a c t i o n e f f i c i e n c y . An a n c i l l a r y He-Ne l a s e r was positioned so that the angle of incidence of the beam s a t i s f i e d the Bragg condition of the phase grating produced by the high power argon ion or He-Cd l a s e r s . As the hologram developed, more and more energy would be d i f f r a c t e d from the incident path of the He-Ne beam allowing continuous measurement of the d i f f r a c t i o n e f f i c i e n c y . A low power(2 mw) He-Ne laser (Spectra Physics model 132) was chosen because the photorefractive e f f e c t i s i n e f f i c i e n t with l i g h t of wavelength 632.8 nm as compared with l i g h t of wavelengths less than 520 nm. Each of the above methods has i t s drawbacks experimentally. With the f i r s t method, the hologram formation must be interrupted to read the d i f f r a c t i o n e f f i c i e n c y . During reading, the reference beam w i l l cause some o p t i c a l erasure of the hologram. For c e r t a i n measure-ments, the optical.erasure on readout was reduced by attaching a par-87 F i g . 5.3 Experimental arrangement f o r measuring the d i f f r a c t i o n e f f i c i e n c y of plane wave holograms by i n t e r m i t t e n t l y blocking the S beam and measuring the i n t e n s i t y d i f f r a c t e d from the R beam. F i g . 5.4 A l t e r n a t i v e arrangement f o r measuring the d i f f r a c t i o n e f f i c i e n c y by continuously monitoring the a u x i l i a r y He-Ne beam. 88 t i a l l y s i l v e r e d glass to an electromagnetic shutter. When the object beam shutter was closed, to f a c i l i t a t e a measurement, the reference beam was attenuated by a factor of approximately 100. The second method does not erase the hologram or necessitate i n t e r r u p t i o n of i t s formation but i t does present other problems. If the angle of incidence of the He-Ne beam i s not very close to the Bragg angle, then the d i f f r a c t i o n e f f i c i e n c y of the hologram de-termined with t h i s beam i s much reduced from i t s true value. Not only i s the alignmentvery c r i t i c a l but determination of the accuracy of the alignment i s not an easy task. The f i r s t method does not have t h i s alignment problem since the reading and w r i t i n g beams are the same. Mechanical s t a b i l i t y during hologram formation i s another c r i t i c a l experimental consideration. The highest s p a t i a l frequency being recorded determines the v i b r a t i o n that may be to l e r a t e d . This i s generally of the order of the wavelength of the l i g h t used for r e -cording. The recording medium must not move more than a f r a c t i o n of t h i s distance r e l a t i v e to the f r i n g e pattern being recorded. To keep the recording medium steady i s not a problem, but to keep the fr i n g e pattern stable s p e c i a l precautions are necessary. To keep the f r i n g e pattern steady the o p t i c a l paths of the reference and object beams must remain constant. This means that mechanical v i b r a t i o n s , a c o u s t i c a l and thermal disturbances must be mini-mized. To accomplish t h i s , experiments were performed on an o p t i c a l bench. The o p t i c a l bench had been constructed by epoxying s t e e l s t r i p s to a massive concrete base (2.13 x 1.72 x 0.15 m). The table was supported by two columns of cement blocks. Layers of f e l t were used between each row of blocks to reduce the e f f e c t s of b u i l d i n g v i b r a t i o n s . 89 To reduce thermal and a c o u s t i c a l disturbance, a p l e x i g l a s s cover was used to enclose the components on the table top. The l a s e r was l e f t outside the cover because of the heat it.generated during operation. To check the e f f e c t s of these precautions, a simple i n t e r -ferometer was set up as shown i n F i g . 5.5. The detector was masked so that i t was illuminated by a portion of one brig h t interference fringe. to S3 w H S5 Mm \ LASER 2 4 6 TIME / minutes DETECTOR BRIGHT FRINGE DETECTOR APERTURE 10 F i g . 5.5 TOP; Arrangement of the interferometer used to check s t a b i l i t y , BOTTOM: V a r i a t i o n i n the detector s i g n a l with time. The l i n e with the arrows in d i c a t e s 20 % of the excursion of the s i g n a l when the frin g e s were made to move past the detector aperture. 90 5.3 Hologram Storage With t h i s setup, holograms were e a s i l y formed i n LiNbO^. F i g . 5.6 shows the build-up of the e f f e c t i v e d i f f r a c t i o n e f f i c i e n c y ( r a t i o of d i f f r a c t e d i n t e n s i t y to incident i n t e n s i t y ) , over a period of 4 minutes, of a hologram i n a Fe-doped (0.015 mole %) c r y s t a l . The saturation e f f e c t i s t y p i c a l of a l l holograms found i n t h i s m a t e r i a l . This appears to be caused by the space charge f i e l d reaching a value where i t opposes the process which forms the hologram (Alphonse et a l . 1975, Moharam et a l . 1975, Gaylord pers. comm.). No attempt was made to estimate the reduced d i f f r a c t i o n e f f i c i e n c y due to unstable f r i n g e problems. 30 0 60 120 180 240 TIME / s e c F i g . 5.6 Build-up of the e f f e c t i v e d i f f r a c t i o n e f f i c i e n c y n, ( r a t i o of incident to d i f f r a c t e d beam power) with time i n an Fe-doped (0.015 mole %) c r y s t a l . 91 . Source i Wo J/cm 2 ad ,S (cm 2/J) Exposure to get 1% n (mJ/ cm2) Glass et a l . 1975b Alphonse et a l . 1975 Staebler et a l . 1974a 0.1 0.122 0.225 0.1 1.224 .3 .444 1.22 330 225 81.6 (a) Undoped C r y s t a l 0.0169 9.94 0.0462 0.367 2700 Fe doped (0.015M%) 0.059 9.94 0.083 0.715 1400 Undoped ^  * (1) 0.0448 5.676 0.023 .343 292 Treated * ( i i ) 0.029 0.8819 0.023 1.43 69.9 i n ( i l l ) 0.0149 0.251 0.023 2.58 38.7 L1 2C0 3 (iv) 0.079 0.251 0.023 13.68 7.3 (b) Table 5.1 S e n s i t i v i t y of LiNbO- to hologram storage. (a) Miscellaneous published data; (b) data obtained from t h i s study. * (1) Only the c e n t r a l part of the c r y s t a l was illu m i n a t e d , with 0.0 V/cm applied. ( i i ) Non-uniform i l l u m i n a t i o n of whole c r y s t a l , with 0.0 V/cm applied. ( i i i ) C r y s t a l nearly uniformly illuminated with 0.0 V/cm applied. (iv) C r y s t a l nearly uniformly illuminated with 3 kV/cm applied. Measurements ( i ) to (iv) are described i n Chapter 8. 92 In Table 5.1 (a), the photorefractive s e n s i t i v i t i e s of LiNbO^ for various published data are l i s t e d . Measurements made during the present study are given i n Table 5.1 (b). The change i n r e f r a c t i v e index An during hologram recording i s l i n e a r i n the incident energy density W q = / i dt, i n the i n i t i a l stages. The s e n s i t i v i t y S for small values of d i f f r a c t i o n e f f i c i e n c y n i s • ^ S = r (5.7) W ad o where a i s the absorption and d the c r y s t a l thickness. The (ad) factor normalizes the s e n s i t i v i t y f o r varying amounts of o p t i c a l absorption i n the c r y s t a l s . Tables 5.1 and 5.2 also l i s t the exposure necessary to achieve 1% d i f f r a c t i o n e f f i c i e n c y . There i s a wide v a r i a t i o n among the the d i f f e r e n t measurements. Some of the v a r i a t i o n may be because multiple i n t e r n a l r e f l e c t i o n s were neglected i n the c a l c u l a t i o n s of the d i f f r a c t i o n e f f i c i e n c y (see Chapter 6). Other discrepancies may be i n t r o -duced i f the uniformity of the i l l u m i n a t i o n varied among the measure-ments (see Chapter 8). The most s e n s i t i v e measurement (n = 1% obtained 2 with 7.3 mJ/cm ) was achieved i n an "undoped" c r y s t a l with 3 kV/cm applied across the c—faces. Presumably greater s e n s i t i v i t y could be 2+ attained by increasing the concentration of Fe ions and by applying a larger f i e l d . 93 CHAPTER 6 INFLUENCE OF MULTIPLE INTERNAL REFLECTIONS AND THERMAL EXPANSION ON THE EFFECTIVE DIFFRACTION EFFICIENCY OF HOLOGRAMS IN LiNbOg 6.1 Introduction A cen t r a l question both for engineering ap p l i c a t i o n s and for studying the mechanisms of hologram formation i s how much d i f f r a c t i o n e f f i c i e n c y i s produced i n given c r y s t a l s by exposure to known amounts of l i g h t . I t i s shown i n t h i s chapter that i t i s e s s e n t i a l to take multiple i n t e r n a l r e f l e c t i o n s between the c r y s t a l surfaces into account when measuring d i f f r a c t i o n e f f i c i e n c y . The i n t e n s i t y of the l i g h t that i s d i f f r a c t e d by the hologram grating i s very s e n s i t i v e to the o p t i c a l thickness of the c r y s t a l . Small changes i n temperature such as those produced by the la s e r beams used i n w r i t i n g the hologram, or fl u c t u a t i o n s i n the ambient temperature w i l l cause s i g n i f i c a n t changes i n the o p t i c a l thickness. The e f f e c t s of multiple r e f l e c t i o n s during the w r i t i n g process, although probably important, are not considered. Analysis of the e f f e c t s f o r i l l u m i n a t i o n with a sing l e beam was given i n Chapter 3, but the problem i s more complicated f o r two-beam i n t e r a c t i o n . 6.2 Theory The c a l c u l a t i o n s follow the method developed by Kogelnik (1967) i n connection with some problems involving the d i f f r a c t i o n of l i g h t by u l t r a s o n i c waves. For s i m p l i c i t y , the analysis i s r e s t r i c t e d to the case i n which the planes of constant r e f r a c t i v e index, which constitute the hologram grating,.".ave normal to the o p t i c a l face of the c r y s t a l . 94 Kogelnik's (1969) coupled wave analysis of d i f f r a c t i o n i n thick holograms shows that for a beam of unit power density, po l a r i z e d i n the plane of incidence, incident at the Bragg angle 9^, the energy exchange between the coupled :waves leads to.'.a d i f f r a c t e d wave of amplitude S(d) = § {exp(Y 2d) - e x p ^ d ) } exp (-igd) (6.1) cos SJCYJ. - Y 2 ) where y^, y^ = -(a ± i§.)/cos 0 ^ § = -(Trnj^/A) cos 2(9^^ - T T ) 3 = ( 2 T r n A ) cos 0 , e 1 The c r y s t a l thickness i s given by d, the r e f r a c t i v e index by n g , the amplitude of the index modulation forming the grating by n^, and the absorption of the c r y s t a l by a. This equation assumes that the grating i s surrounded by a medium with the same average r e f r a c t i v e index. For gratings formed i n c r y s t a l s , r e f l e c t i o n s w i l l occur at the boundaries of the grating due to the change i n the index at the c r y s t a l surfaces. F i g . 6.1 shows the i n t e r a c t i o n of the multiply-reflec^t . ted wave with the hologram grating. At the f i r s t boundary, the Fresnel transmission c o e f f i c i e n t t^ gives "the amplitude of the primary wave : . entering the c r y s t a l . A f t e r one t r a v e r s a l of the c r y s t a l , part of the refr a c t e d wave i s r e f l e c t e d and a wavelet of amplitude t^t 2S(d) emerges. Both the primary wave and the ref r a c t e d wave are r e f l e c t e d back into the grating and the i n t e r a c t i o n continues. The next re f r a c t e d wavelet to emerge has traversed the grating three times i n t o t a l and has an 2 amplitude t ^ t 2 r S(3d). Here t 2 i s the Fresnel transmission c o e f f i c i e n t at the second boundary, and r i s the Fresnel r e f l e c t i o n c o e f f i c i e n t f o r beams incident on the surface from within the c r y s t a l . The t o t a l 95 t 1 t 2 S(d) t j _ t 2 r S(3d) F i g . 6.1 M u l t i p l e r e f l e c t i o n s i n a hologram grating. of a l l the d i f fracted wavelets emerging from the c r y s t a l i s S. * = t , t 0 ( S ( d ) + r 2S(3d) + r 4S(5d) + ...) to t 1 2 Since en t 2 = 1 + r , and t j = 1 - r , Eq. 6.2 can be w r i t t S = (1 - R)( S(d) + R S(3d) + R 2S(5d) +...) to t (6.2) (6.3) where R = r = tan (e - 9 X ) tan 2(6 + 0 X) Here e i s the angle at which the beam i s inc i d e n t on the c r y s t a l surface. I f Eq. 6.1 i s inserted i n Eq. 6.3, two geometric s e r i e s are obtained which can be summed to give S ( d ) ( l - R) [ l + R exp{d(Y 2 + Y x - 2IB)}] t o t 1 - R[exp{2d(Y 2 - 13)}+ exp{2d(Y 1 - 16)}] +R2 exp{2d(Y 2 4^-213)} (6.4) 96 F i g . 6.2 E f f e c t i v e d i f f r a c t i o n e f f i c i e n c y of a hologram i n a l i t h i u m niobate c r y s t a l i s p l o t t e d against temperature change. For both curves, n (T X = 2.252, X = 488 nm, d = 3 mm. The s o l i d curve i s c a l c u l a t e d f a r no absorption. The dashed l i n e i s c a l c u l a t e d with ad = 0.28. 97 700r F i g . 6.3 Experimentally obtained r e f l e c t e d i n t e n s i t y of an argon i on la s e r beam incident on a 3 mm t h i c k c r y s t a l of undoped LiNbO^ i s p l o t t e d against time of exposure. T^ he angle of incidence was 15 and the t o t a l beam i n t e n s i t y was 2360 W/m . 98 30„ J21 t i t i i i i j 0.0 ' 1.0 2.0 3.0 4.0 AT (°C) F i g . 6.4 Effective d i f f ract ion eff ic iency of a hologram in Fe-doped LiNbOg is plotted against temperature. The "crosses" are experimentally obtained points. The sol id l ine i s calculated with n (T ) = 2.27351. e o d(T n) = 1.40 mm, X = 441.6 nm, ad = 0.252, T = 31.55°C, and SS* = 0.5. 99 The d i f f r a c t e d wave due to m u l t i p l e r e f l e c t i o n s i s given by S t o t = T ' S ( d > ( 6' 5> where S(d) i s the wave d i f f r a c t e d i n the absence of m u l t i p l e r e f l e c t i o n s . The e f f e c t i v e d i f f r a c t i o n e f f i c i e n c y , n , defined as the r a t i o of the i n t e n s i t i e s of the i n c i d e n t beam e x t e r n a l to the c r y s t a l and the d i f -f r a c t e d beam a f t e r l e a v i n g the c r y s t a l , i s n = SS*TT* (6.6) e With no m u l t i p l e r e f l e c t i o n s , the d i f f r a c t i o n e f f i c i e n c y would j u s t be A ss . A Since the transmittance f a c t o r TT i s a f u n c t i o n of the path l e n g t h , the d i f f r a c t e d F o u r i e r components of a r e a l hologram would be d i f f e r e n t l y a f f e c t e d by m u l t i p l e r e f l e c t i o n s . In the present case however, the e f f e c t of a change i n path l e n g t h can be e a s i l y computed. The e f f e c t of a change i n temperature (T - TQ) can be accounted f o r by w r i t i n g the e x t r a o r d i n a r y index n & and the thickness d as n £ = n E ( T o ) ( l + 6<T-- T Q ) ) ( 6 > ? ) and d = d ( T o ) ( l + a'(T - T Q ) ) . ( 6 > 8 ) -4 o Reported values of the thermal c o e f f i c i e n t s are 6 = 0.392 x 10 / C (Boyd et a l . 1967) at 450 nm and a' = 16.7 x 10~6/°C (Nassau et a l . 1966) Hobden and Warner (1966) a l s o give a value f o r the temperature depen-dence of the r e f r a c t i v e index. The s o l i d l i n e i n F i g . 6.2 shows the v a r i a t i o n i n the e f f e c t i v e d i f f r a c t i o n e f f i c i e n c y w i t h changes i n temperature f o r a l o s s -l e s s d i e l e c t r i c g r a t i n g 3mm t h i c k . The dotted l i n e i s f o r an absorbing g r a t i n g of the same t h i c k n e s s . For a t h i c k e r g r a t i n g the pe r i o d of o s c i l l a t i o n would be sm a l l e r . Very small uniform absorption does not 100 a f f e c t the period of the o s c i l l a t i o n s , but the maximum e f f e c t i v e d i f -f r a c t i o n e f f i c i e n c y and the amplitude of the o s c i l l a t i o n s are smaller for the same phase grating. These r e s u l t s were calculated for a r e f r a c t i v e index grating that would give a d i f f r a c t i o n e f f i c i e n c y of 100% i n the absence of multiple r e f l e c t i o n s and absorption. In some circumstances, multiple r e f l e c t i o n s may a c t u a l l y enhance the d i f f r a c t i o n e f f i c i e n c y . F i g . 6.3 indicates that a grating that d i f f r a c t s 10% of the incident l i g h t with no r e f l e c t i o n s can d i f f r a c t 15% of the l i g h t f o r c e r t a i n o p t i c a l paths. 6.3 Experimental Results Two experiments were performed to determine the e f f e c t s of temperature on the d i f f r a c t i o n e f f i c i e n c y of a simple grating i n a 3mm thick, undoped c r y s t a l of LiNbO^. The f i r s t experiment was to determine i f the beams used to read and write holograms could heat the c r y s t a l s u f f i c i e n t l y to a f f e c t the d i f f r a c t i o n e f f i c i e n c y . F i g . 6.4 shows the r e f l e c t e d i n t e n s i t y vs time for an argon ion las e r (A = 488nm) incident at 15°. The beam was polarized with i t s e l e c t r i c vector i n the plane of incidence and i t s 2 i n t e n s i t y was 2360 W/m . The i n t e n s i t y reflectance of a two-surfaced system i s given by (Heavens 1955) T:12 + 2 r x r 2 cos ( 2 6 ^ + x\ ( 6 > 9 ) 2 2 1 + 2 r x r 2 cos ( 2 6 ^ + ^ r 2 where r ^ and r 2 are the ordinary Fresnel c o e f f i c i e n t s f o r the f i r s t and second surface r e s p e c t i v e l y , ( r 2 = - r ^ f o r t h i s case) "and 6j-= 2 1 0 1 ^ 0 0 3 ^ 101 F i g . 6.4 indicates a change i n the o p t i c a l path length 8^ due to heating. The rate of r i s e of temperature i s i n i t i a l l y f a s t , slowing down as a steady state i s approached. A change i n temperature w i l l a f f e c t r 1 and r 2 through Eq. 6.7, however the e f f e c t i s n e g l i g i b l e compared to the e f f e c t on 8^ ( i . e . the l i g h t must be coherent to s u f f e r an i n t e n s i t y change due to temperature f l u c t u a t i o n s ) . A second experiment was also c a r r i e d out to show that small changes i n the ambient temperature w i l l change the e f f e c t i v e d i f f r a c t i o n e f f i c i e n c y due to multiple r e f l e c t i o n s . A hologram was formed i n an Fe-doped (0.015 mole%) c r y s t a l of LiNb0 3 using two plane waves o r i g i n a t i n g from a RCA He-Cd 15 mW l a s e r (A = 441.6 nm). The r a t i o of the reference to the object beam i n t e n s i t y was 1.2 with an angle of incidence between beams of 27.5°. Both beams were p o l a r i z e d with the e l e c t r i c vector p a r a l l e l to the plane of incidence which also contained the c-axis of the c r y s t a l . The c r y s t a l was placed i n a chamber with the temperature c o n t r o l l e d to better than ±0.02°C. The hologram was formed at 35.64 ±0.02°C u n t i l the e f f e c t i v e d i f f r a c t i o n e f f i c i e n c y was about 25%. A f t e r w r i t i n g the hologram, the e f f e c t i v e d i f f r a c t i o n e f f i c i e n c y was measured by momentarily exposing the c r y s t a l to the reference beam (at reduced i n t e n s i t y ) from time to time. F i g . 6.5 i n d i c a t e s that the hologram decayed s i g n i f i c a n t l y (due to thermal release of electrons from traps) i n the f i r s t t h i r t y minutes, a f t e r which the thermal decay was small enough not to a f f e c t the measurements of i n t e r e s t . A f t e r t h i s i n i t i a l period, the d i f f r a c t e d l i g h t i n t e n s i t y was monitored as the temperature was allowed to slowly f a l l , and the r e s u l t s are shown i n F i g . 6.6. The points are experimental and the s o l i d l i n e it was f i t t e d by varying n e and SS (the absolute d i f f r a c t i o n e f f i c i e n c y i n Eq. 6.6), with the temperature dependence of n and d (from Eqs. 102 6.7 and 6.8) included. The e f f e c t i v e d i f f r a c t i o n e f f i c i e n c y i s c l e a r l y s e n s i t i v e to changes i n temperature as small as 1°C. The e f f e c t i s almost e n t i r e l y due to a change i n the o p t i c a l path length through -3 the c r y s t a l . The angle of r e f r a c t i o n only changed about 10 degrees. This would have n e g l i g i b l e e f f e c t on the Bragg condition. Thermal expansion i n the c - d i r e c t i o n i s eight times smaller than i t i s i n the a or b d i r e c t i o n s (Nassau et a l . 1966) and i s , i n any case, so small as to cause n e g l i g i b l e change i n the grating spacing. In conclusion, i t has been shown that a small change i n temperature within the LiNbO^ c r y s t a l can s i g n i f i c a n t l y change the e f f e c t i v e d i f f r a c t i o n e f f i c i e n c y of a hologram stored i n the c r y s t a l . A temperature increase can be caused by the absorption of moderately intense l a s e r beams used to read and write holograms. The e f f e c t i s due to thermal expansion increasing the o p t i c a l path length of the c r y s t a l . M u l t i p l e i n t e r n a l r e f l e c t i o n s cause the d i f f r a c t e d i n t e n s i t y to o s c i l l a t e as the o p t i c a l thickness changes. 103 - CHAPTER 7 PHOTOCURRENTS IN LITHIUM NIOBATE 7.1 Introduction Photocurrents i n LiNbO^ may be measured i n the absence of an exte r n a l l y applied f i e l d (Chen 1969). As was outlined' i n Sec. 2.6, Glass et al.(1974b ,1975a) have used the bulk photovoltaic e f f e c t to explain t h i s . In t h i s chapter, experimental observations of the photo-current are given which support the theory that photocurrents w i l l flow i n the absence of b u i l t - i n f i e l d s i n the c r y s t a l . The experiments to be described show that the r e l a t i o n between the photocurrent and the radient i n t e n s i t y i s l i n e a r over the range considered. Cooling c r y s t a l s from a temperature that relaxes space charge f i e l d s , with and without a short applied to the c-faces of the c r y s t a l , did not a f f e c t the photocurrent. Photocurrents were measured during hologram formation i n doped and undoped c r y s t a l s . An attempt was made to co r r e l a t e the photocurrent with the induced change i n the r e f r a c t i v e index. 7.2 Experimental Procedure For s i n g l e beam measurements, the c r y s t a l s were illuminated with an argon ion l a s e r ( A = 514.5 nm). The beam was expanded to i l l u m i -nate the whole c r y s t a l . To make e l e c t r i c a l connection to the c r y s t a l s , gold electrodes were evaporated i n the c-faces of the c r y s t a l s over a f l a s h of chromium. The photocurrent was measured with a Keithley 602 electrometer. In the holographic measurements, the c r y s t a l s were illuminated 104 i n the c e n t r a l portion of the c r y s t a l with a beam diameter of 3 mm. The angle between the beams (26) was 30°. The e l e c t r i c vector of each beam and the c-axis of the c r y s t a l were i n the plane of incidence. The photocurrent was measured i n the same way as the singl e beam experiments. Both an argon ion l a s e r ( A = 480 nm) and a He-Cd la s e r ( X= 441.6 nm) were used to write holograms. 7.3 Results When the c r y s t a l s were illuminated, both a p y r o e l e c t r i c current and a photocurrent were measured. F i g . 7.1 shows the short c i r c u i t current measurement on an undoped LiNbO^ c r y s t a l . The i n i t i a l peak i s the p y r o e l e c t r i c contribution to the current caused by beam heating. The steady state current i s the photocurrent. The photocurrent exhibited no decay a f t e r 43 hours of continuous i l l u m i n a t i o n . When the l i g h t was turned o f f , a p y r o e l e c t r i c current of opposite p o l a r i t y was measured as the c r y s t a l cooled. F i g . 7.2 shows the photocurrent for' d i f f e r e n t i n t e n s i t i e s measured on an undoped c r y s t a l . The r e l a t i o n -ship i s c l e a r l y l i n e a r . To test the e f f e c t s of p y r o e l e c t r i c f i e l d s on the photo-current an Fe-doped (0.015 mole %) LiNbO^ was slowly cooled from 375°C with ( i ) the c-faces shorted and ( i i ) with the c-faces open-c i r c u i t e d . In each case the photocurrent was measured a f t e r the c r y s t a l reached room temperature. The r e s u l t s of three experiments are given i n Table 7.1. F i g . 7.3 shows that i n the i n i t i a l stages of hologram form-ic a t i o n, a r c s i n (n 2) was l i n e a r i n exposure for an undoped c r y s t a l . F i g . 7.4 shows the same r e s u l t f o r an iron-doped c r y s t a l . From 105 'LIGHT ON LIGHT OFF F i g . 7.1 Time development of the p r y r o e l e c t r i c and photo currents during i l l u m i n a t i o n and of the p y r o e l e c t r i c current a f t e r the l i g h t i s turned o f f . l . O r INTENSITY / (mW cm - 2) F i g . 7.2 The photocurrent i n an undoped LiNbO^ c r y s t a l for d i f f e r e n t i n t e n s i t i e s . 106 F i g . 7.3 I n i t i a l stage of hologram formation i n an undoped c r y s t a l at two wavelengths. Arcsin(/n) i s proportional to the change i n index. EXPOSURE /(mJ/cm2) F i g . 7.4 I n i t i a l stage of hologram formation i n an Fe-doped(0.015 mole %) c r y s t a l at two wavelenths. 107 Kogelnik's (1969) d e r i v a t i o n of the 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 sinu-s o i d a l grating, a l i n e a r r e l a t i o n s h i p i s expected since the amplitude of the index modulation i s proportional to a r c s i n (i/r\ ) • The i r o n -doped c r y s t a l produced a greater change i n the index than the undoped c r y s t a l f o r the same exposure. As shown i n F i g s . 7.3 and 7.4, the process was more e f f i c i e n t at A = 441.6 nm. SEQUENCE PHOTOCURRENT pA ABSORPTION COEFFICIENT -1 cm I / a pA cm sc 5.4 0.48 11.3 oc 3.6 0.31 11.2 sc 5.8 0.51 11.4 Table 7.1 E f f e c t s of s h o r t - c i r c u i t (sc) and open-circuit (oc) cooling on the photocurrent (A = 514.5 nm) . The absorption was measured on a Cary Spectrophotometer with incoherent l i g h t with corrections made for r e f l e c t i o n s . The c r y s t a l s were cooled from 375°C to 25°C. During these measurements the photocurrent was measured a f t e r the pyro-e l e c t r i c t ransient. The currents are l i s t e d i n Table 7.2. PHOTOCURRENT (pA) CRYSTAL A = 488 nm A = 441.6 nm Doped 0 .224 0.216 Undoped 1 .44 0.842 Table 7.2 Photocurrents measured during hologram formation i n an Fe-doped (0.015 mole %) and undoped c r y s t a l at two d i f f e r e n t wavelengths. 108 7.4 Discussion The measurement of the photocurrent i n LiNbO^ shows that i t does not decay with time when the e n t i r e c r y s t a l i s i l l u m i n a t e d and that i t i s proportional to the i n t e n s i t y of the i l l u m i n a t i o n . Glass et a l . (1974b) have represented the photocurrent density as J = KCtl (7.1) where I i s the i n t e n s i t y , a the absorption and K a constant depending on the impurities i n the c r y s t a l , (see Sec. 2.6). From the slope of F i g . 7.2 -1 -9 and with a = 0.115 cm , K = 1.14 x 10 Acm/W f o r the undoped c r y s t a l —9 used. This i s comparable to Glass et a l . ' s value of < = 3.0 x 10 Acm/W f o r a Fe-doped c r y s t a l . Although a new transport mechanism i s thought to be involved, i t i s not immediately obvious that the e f f e c t s of p y r o e l e c t r i c f i e l d s can be neglected. I t i s w e l l known that i f no free charges were e x t e r n a l l y a v a i l a b l e , LiNbO^ c r y s t a l s cooled from ~200°C with no f i e l d present would develop very large i n t e r n a l and external f i e l d s due to the uncompensated change i n remanent polarization(Amodei e t a l . 1972b). These f i e l d s are so large that i t seems possible that i n j e c t i o n or ex t r a c t i o n of electrons should set up space charges w i t h i n the c r y s t a l with the r e s u l t that large scale f i e l d s remain i n the c r y s t a l even a f t e r applying an external short. Shorting the c-faces of the c r y s t a l during cooling would prevent the development of these space charges. The data i n Table 7.1 shows that the photocurrent was independent of the e l e c t r i c a l condition during co o l i n g . This experiment f a i l e d to confirm the existence of a b u i l t - i n f i e l d . I t suggests that b u i l t - i n f i e l d s of p y r o e l e c t r i c o r i g i n developed over the temperature range encountered here may be neglected and that some other mechanism i s responsible f o r the photocurrent. 109 In attempting to correlate the photocurrent measured during hologram formation with the index modulation required to give the measured diffraction efficiency, only the i n i t i a l region of linear index change was considered. The holograms were considered to be formed by electrons drif t i n g in a f i e l d with the migration length being short compared with the grating spacing. The space charge f i e l d is given by Eq. 2.7 eLg mt E = — cos kx ( 7 . 2 ) sc e where L = E QyT. The change in the refractive index i s given by Eq. 2 .31 3 -n r 0 0 E * e 33 sc Using Eq. 5.5 and Eq. 6.6 the effects of multiple reflections may be allowe for. The measured build-up in the diffraction efficiency i s n e = TT* s inVf ( ^ o s d e C ° S 2 9 ) ( 7 . 4 ) o * h If arcsin( (n e/TT K ) is plotted as a function of time, the slope of the curve can be calculated from Eq. 7.2 to Eq. 7.4 to be 3 ird n cos 26 r_.meg L cos kx S 1"--= '27T~co's/o " 0.5) 0 Eq. 7.5 may be solved for the expression g QL- This may be compared with the value of g QL computed from the photocurrent i n the following way. For a crystal with electrodes separated by a distance I, the average photocurrent measured at the electrodes i s assumed to be given by I p = e g o v f ( 7 . 6 ) 110 where g Q V i s t h e number o f e l e c t r o n s ( e ) g e n e r a t e d p e r s e c o n d i n t h e i l l u m i n a t e d v o l u m e (V) and L / £ i s , on t h e a v e r a g e , t h e f r a c t i o n o f t h e d i s t a n c e e a c h e l e c t r o n t r a v e l s b e t w e e n t h e e l e c t r o d e s . From E q . 7 . 5 and E q . 7 . 6 i n d e p e n d e n t measu remen ts o f g Q L c a n be made. T a b l e 7 . 3 l i s t s t h e v a l u e s o f g L c a l c u l a t e d by t h e two ° o J methods f o r a doped ( 0 . 0 1 5 m o l e % i r o n ) c r y s t a l and a n undoped c r y s t a l . C r y s t a l W a v e l e n g t h (nm) _ / 1 A 1 2 - 2 - 1 g o L / 1 0 m s f r o m p h o t o c u r r e n t f r o m d i f f r a c t i o n e f f i c i e n c y Undoped 488 4 4 1 . 6 9 . 6 2 5 . 6 3 0 . 1 7 3 0 . 6 3 Doped 488 4 4 1 . 6 1 . 7 9 1 . 5 6 0 . 9 6 3 1 . 0 9 - T a b l e 7 . 3 A c o m p a r i s o n o f g L , c a l c u l a t e d f r o m t h e p h o t o c u r r e n t , to g Q L c a l c u l a t e d f r o m 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 c o m p a r i s o n o f g Q L f o r d i f f e r e n t c r y s t a l s and d i f f e r e n t w a v e l e n g t h s c a n n o t b e made b e c a u s e t h e i n t e n s i t y o f i l l u m i n a t i o n was n o t t h e same for a l l t h e m e a s u r e m e n t s . T h e r e a r e a number o f p o s s i b l e r e a s o n s f o r the d i s c r e p a n c y be tween t h e v a l u e s o f g Q L c a l c u l a t e d b y t h e two d i f f e r -ent m e t h o d s . The d i f f r a c t i o n e f f i c i e n c y measurement g i v e s a n e s t i m a t e of the m a g n i t u d e o f t h e i n d e x m o d u l a t i o n . I t d o e s n o t a c c o u n t for any " d c " c h a n g e i n t h e i n d e x w h i c h may be p r o d u c e d . A s w i l l be shown I l l i n the next chapter, the geometry of i l l u m i n a t i o n of the c r y s t a l a f f e c t s the rate at which holograms are formed. In addition, the o p t i c a l and thermal h i s t o r y of the c r y s t a l may be important. The i n c l u s i o n of TT i n Eq. 7.4 to account for multiple r e f l e c t i o n s pro-vides only an approximate c o r r e c t i o n since the o p t i c a l thickness of the c r y s t a l i s not known to a f r a c t i o n of a wavelength. Subsequent to these measurements a method has been developed to circumvent t h i s problem (Moharam, Cornish and Young 1975) . Op t i c a l erasure of the hologram during i t s formation may reduce the rate at which the hologram forms. The o p t i c a l erasure would contribute to the photocurrent but would decrease the index grating. This would cause a discrepancy between g QL measured using these two methods. In conclusion, the photocurrent was larger than would be expected from the hologram measurements. A better c o r r e l a t i o n might be achieved i f some of the problems l i s t e d above -»were taken i n t o account. 112 CHAPTER 8 THE EFFECTS OF INTERNAL AND APPLIED FIELDS ON HOLOGRAMS STORED IN LlNb0 3 8.1 Introduction The a p p l i c a t i o n of an external f i e l d during the w r i t i n g of holo-grams may be an important method of c o n t r o l l i n g the process. I t i s also an a t t r a c t i v e means of i n v e s t i g a t i n g the r e l a t i v e contributions of d i f f u s i o n , i n t e r n a l f i e l d s and the bulk photovoltaic e f f e c t to the photo-r e f r a c t i v e process. The published data on which processes a c t u a l l y occur are i n apparent con t r a d i c t i o n . Staebler et a l . found that e i t h e r sign of applied f i e l d (± 2kV/cm (1972b) and ± lOkV/cm (1974a)) increased the rate of w r i t i n g equally. This was taken as i n d i c a t i n g d i f f u s i o n . On the other hand, Ohmori et a l . (1974) and Yasojima et a l . (1972) found that one sign of f i e l d increased and the other reduced the rate of generation of o p t i c a l damage. This was taken as i n d i c a t i n g that d r i f t i n k 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 was occurring, as proposed by Chen (1969) and the magnitude of the f i e l d was estimated. The r e s u l t s of the previous experiments by Staebler et a l . on the e f f e c t of applied voltage, i n which holograms were found to be written by d i f f u s i o n only, assume a new s i g n i f i c a n c e i n the l i g h t of the new theory, the bulk photovoltaic e f f e c t , since although p y r o e l e c t r i c f i e l d s might be absent i n a p a r t i c u l a r c r y s t a l , i t i s not obvious how the bulk photovoltaic e f f e c t should be inoperative. The experiments to be described show that the r e s u l t s of applied f i e l d experiments depend on the f r a c t i o n of the c r y s t a l illuminated and on the i n t e n s i t y of l i g h t . In addition, during repeated cycles of hologram w r i t i n g and reading, with consequent o p t i c a l erasure, the 113 d i f f ract ion eff ic iency of the holograms depends on the voltage applied during the previous cycles as well as that applied during the current wri t ing. The explanation of the results i s that large-scale "dc" space charge f ie lds (as well as the spat ia l "ac" f ie lds which produce the holograms) are b u i l t - i n by exposure to l igh t . The actual f i e l d in the crystal depends, therefore, on the f i e l d b u i l t - i n during the previous and current exposure, as well as on the applied f i e l d . An experimental complication i s that applied f ie lds of the magnitudes required in these experiments can appreciably affect the opt ical thickness of the crysta l and hence can affect both reading and writing holograms due to multiple internal re f lect ions. It i s well known that exposure to l ight can "build in" a large scale "dc" f i e l d . Although not expl ic i ty taken into account in previous work on the effect of applied voltage, there is no dispute that such f ie lds are created. They are what is observed in compensator ex-periments (e.g. the or ig inal work of Chen (1969)) and in ellipsometer experiments (Chapter 3). A f i r s t point concerns the spat ia l extent of the "dc" space charge f ie lds re lat ive to the il lumination that produces them. As was discussed ear l ier (sec. 2.3) Chen (1969) explained the results of his circular-beam experiments with a dipole-type f i e l d which extended well outside the i l l u m i -nated area. Chen referred to electrons released and retrapped "outside the illuminated area". This phrase has been quoted or similar statements made in several papers (Johnston 1970, Yasojima et a l . 1972, Peterson et a l . 1971, Clarke et a l . 1973). Actual ly , the trapping process i t s e l f i s believed 114 to be independent of the l i g h t i n t e n s i t y . Few electrons are transported outside the illuminated volume and then only f o r short distances. This was shown i n Chapter 3 by experiments i n which the l i g h t beam e f f e c t i v e l y v a r i e d i n i n t e n s i t y only i n one dimension. With t h i s geometry, the f i e l d and hence the change i n bi r e f r i n g e n c e were e s s e n t i a l l y confined to the i r r a d i a t e d volume. The next point concerns the e f f e c t of how uniformly the c r y s t a l i s Illuminated. For complete, uniform i l l u m i n a t i o n of the c r y s t a l (which i s impossible with the two plane wave beams used f o r hologram production) the f i e l d i n the c r y s t a l would be the applied f i e l d plus the " v i r t u a l " f i e l d . With p a r t i a l i l l u m i n a t i o n a "dc" space charge f i e l d w i l l also be present. Thus, with as near as p o s s i b l e the whole c r y s t a l i l l u m i n -ated, i t i s expected that an applied f i e l d equal and opposite to the " v i r t u a l " f i e l d would allow hologram w r i t i n g by d i f f u s i o n only. F i e l d s applied with e i t h e r p o l a r i t y about t h i s value should increase the rate of hologram production because of increased transport length. I f p l o t s of d i f f r a c t i o n e f f i c i e n c y vs. voltage show a minimum which i s not zero then d i f f u s i o n may be s i g n i f i c a n t . 8.2 Experimental Procedures 8.2.1 Sample Preparation and Hologram Measurements The sample used was nominally pure LiNbO^ (sample #4 of Appendix D) heated i n I^CO^ at 520°C f o r 40 hours, a treatment due to P h i l l i p s and Staebler. The absorption was 0.23 cm -* at 514.5 nm, Holo-grams were formed using an argon l a s e r (A = 514.5 nm) p o l a r i z e d p a r a l l e l *The " v i r t u a l " f i e l d (E^) i s the f i e l d due to the bulk photovoltaic e f f e c t as discussed i n Sec. 2.9.1. 115 to the c-axis with an angle of incidence of 10°. With these c r y s t a l s , decay of o p t i c a l damage and of holograms i s n e g l i g i b l e i n the dark (Chapter 3) apparently due to the destruction of shallow traps but holo-grams are r a p i d l y erased by i l l u m i n a t i o n with the reference beam. I t i s poss i b l e and convenient, therefore, to make repeated experiments and to acquire information on r e p r o d u c i b i l t y , which i s absent i n previous work. A l l measurements of the e f f e c t i v e d i f f r a c t i o n e f f i c i e n c y were made by w r i t i n g with two plane waves f o r a s p e c i f i e d exposure and then blocking the s i g n a l beam and monitoring the reference beam u n t i l readout caused the hologram to decay to zero. The magnitude of the e f f e c t i v e d i f f r a c -t i o n e f f i c i e n c y , n, ( r a t i o of d i f f r a c t e d i n t e n s i t y to incident i n t e n s i t y ) was that achieved immediately a f t e r the w r i t i n g and at the commencement of o p t i c a l readout(Fig. 8.1). F i e l d s were applied through aluminum electrodes evaporated on to the c-faces of the c r y s t a l . The convention i s that the d i r e c t i o n of the applied f i e l d i s p o s i t i v e i f dire c t e d towards the -c face of the c r y s t a l . This i s i n the same d i r e c t i o n as the " v i r t u a l " f i e l d due to the photovoltaic e f f e c t . 8.2.2 M u l t i p l e I n t e r n a l R e f l e c t i o n s In the i n i t i a l work a serious problem became apparent which i s not mentioned i n e a r l i e r work on the e f f e c t of applied voltage. This i s that both applied voltage and the large scale (as opposed to s i n u s o i d a l ) space charge f i e l d s produced by exposure to l i g h t modify the o p t i c a l thickness of the c r y s t a l through the e l e c t r o - o p t i c e f f e c t to a s u f f i c i e n t extent that, because of m u l t i p l e i n t e r n a l r e f l e c t i o n s , both the w r i t i n g and reading e f f i c i e n c y are changed by appreciable amounts. This means that, i f t h i s e f f e c t i s neglected, applied voltage could be in t e r p r e t e d 4 3 s« 2 -l u 0 1 * ' I t I t « ' i ' • t 0 10 20 30 40 50 . 60 TIME / sec Fig. 8«1 Measurement of one w r i t e , read-erase c y c l e . The hologram was w r i t t e n f o r 15 sec w i t h a s h u t t e r b l o c k i n g the d e t e c t o r . A s h u t t e r was then used to block the S i g n a l beam and the hologram erasure monitored u n t i l the hologram decayed. The e f f e c t i v e d i f f r a c t i o n e f f i c i e n c y n e given i n other f i g u r e s was that measured at the beginning of read-out i . e . at 15 sec f o r t h i s example. 117 .551 I i i i S .51 5 s i ' » i i > i .45[ • i 1 i l i t i i j i_ - 1 0 1 APPLIED FIELD (kV/cm) F i g . 8 . 2 E f f e c t of applied f i e l d on the transmittance of LiNbOg. The c r y s t a l thickness was 1 cm; the angle of incidence of the l a s e r beam was 1 0 ° with X = 5 1 4 . 5 nm and e l e c t r i c vector p a r a l l e l to c-axis. Top: c r y s t a l a f t e r thermal erasure of damage at 2 7 0 °C. Bottom: the same c r y s t a l a f t e r hologram w r i t i n g . The transmittance i s diminished because some of the l i g h t i s d i f f r a c t e d by the hologram. The phase s h i f t along the h o r i z o n t a l axis i s due to the o p t i c a l l y - i n d u c e d "dc" space charge f i e l d . The bars show the e f f e c t of small,temperature f l u c t u a t i o n s ( < 0 . 2 ° C ) . 118 as having an e f f e c t on the physics of the el e c t r o n transport process when i n f a c t i t was acting only through the e l e c t r o - o p t i c e f f e c t . The importance of multiple i n t e r n a l r e f l e c t i o n s was previously pointed out i n connection with the measurement of d i f f r a c t i o n e f f i c i e n c y (Chapter 6) where they may account f o r some of the c y c l i c v a r i a t i o n s of d i f f r a c t i o n e f f i c i e n c y with exposure which have been reported i n the l i t e r a t u r e . The present e f f e c t has not previously been mentioned i n the l i t e r a t u r e . F i g . 8.2 shows the observed transmittance changes due to applied voltages i n a c r y s t a l 1 cm th i c k . The upper curve i s f o r a fre s h c r y s t a l with no o p t i c a l damage and the lower curve was recorded a f t e r a few holograms had been stored i n the c r y s t a l . The e n t i r e curve i s lower because some of the incident beam used to record the curve was d i f f r a c t e d by the stored hologram. The main points of i n t e r e s t are the change i n transmittance with applied voltage and the phase s h i f t between the two curves caused by the e f f e c t s of space charge f i e l d s induced while w r i t i n g the holograms. The v e r t i c a l bars show the f l u c t u a t i o n s i n transmittance due to small temperature f l u c t u a t i o n s (<0.2°C). One so l u t i o n to the problem of mult i p l e r e f l e c t i o n s , which was t r i e d i n i t i -a l l y , i s to r e s t r i c t measurements to voltages g i v i n g equal path lengths (modulo A/2), i . e . equivalent points i n F i g . 8.2, f o r example minima i n transmittance. The reading and w r i t i n g voltages w i l l change with t o t a l exposure and must be determined by measuring transmittance or re f l e c t a n c e . An a l t e r n a t i v e way around the problem of mu l t i p l e r e f l e c t i o n s , and the method used i n the experiments to be described, i s to use s u f f i c i e n t l y t h i n c r y s t a l s that the change i n o p t i c a l thickness may be neglected. In t h i s case, the e f f e c t i v e d i f f r a c t i o n e f f i c i e n c y s t i l l 119 1 2 3 4 5 SUCCESSIVE WRITE-READ CYCLES F i g . 83 E f f e c t of p r i o r exposure at d i f f e r e n t voltages on hologram w r i t i n g f o r p a r t i a l i l l u m i n a t i o n of the sample. The increasing transient shows 3 successive write-read c y c l e s " a t zero^applied f i e l d a f t e r a 15 sec exposure to the reference beam with +3 kVcm applied. The decreasing transient shows 3 successive write-read cycles a f t e r a 15 sec exposure to the reference beam with -3 kVcm applied. The bars show the standard deviation of 11 holograms formed at zero applied f i e l d . Writing and reading conditions were the same as for the lower curve i n F i g . 8.4 . 120 - 3 - 2 - 1 0 1 2 3 APPLIED VOLTAGE /kV 1/2 F i g . 8.4 Normalized values of a r c s i n n e vs. applied voltage (for e l e c -trodes 1 cm apart ) . The r e l a t i v e area of i l l u m i n a t i o n i s shown by the c i r c l e s and the c r y s t a l face (1 cm square) by the squares. Curve A: the exposure corresponded to point 3 on F i g . 8.5 and the i n t e n s i t y was 8.36 mW/cm . Curve B: the exposure corresponded to point 2 of F i g . 8.5 and the i n t e n s i t y was 44 mW/cm2. Curve C: exposure and i n t e n s i t y , same as B. Curve D: the exposure corresponded to point 1 of F i g . 8.5 and the i n t e n s i t y was 378 mW/cm2. 121 d i f f e r s from the absolute d i f f r a c t i o n e f f i c i e n c y , but the d i f f e r e n c e i s not appreciably changed by the applied voltage or by space charge f i e l d s . 1/2 I t s h a l l l b e assumed that values of a r c s i n n are a measure of the e r e l a t i v e amplitudes of the r e f r a c t i v e index gratings. 8.3 Results The d i f f r a c t i o n e f f i c i e n c y produced by a given exposure was found to depend on the voltage applied during previous exposures as w e l l as that applied during the current exposure. This e f f e c t was more pronounced when only a portion of the c r y s t a l was illuminated, rather than the whole c r y s t a l . The r e s u l t s shown i n F i g . 8.3 were obtained with two beams of 2.8 mm diameter i l l u m i n a t i n g the c e n t r a l part of the 1 cm square face of the c r y s t a l . The increasing transient shows the r e s u l t of wri t i n g at zero voltage (and then reading u n t i l the hologram decayed) three sequentialstimes, a f t e r previous i l l u m i n a t i o n with +3 kV/cm applied. The decreasing transient shows the r e s u l t of the same experiment but with -3 kV/cm applied during the previous exposure. The bars represent the standard deviation of the hologram e f f i c i e n c y of eleven sequential runs performed at zero applied v o l t s . F i g . 8.4 shows the e f f e c t of applied voltage on the r e f r a c t i v e index grating amplitude per u n i t exposure, for four d i f f e r e n t i l l u m i n a t i o n s . The geometry of i l l u m i n a t i o n f o r each curve i s shown by the c i r c l e s and the r e l a t i v e s i z e of the c r y s t a l i s represented by the squares. Before each hologram formed at ± applied voltage, three or four write, read-erase cycles were completed at zero applied voltage i n an attempt to e s t a b l i s h s i m i l a r s t a r t i n g conditions and to check f or fatigue e f f e c t s . Curve A of F i g . 8.4 was measured f or exposures shown by point 3 on 122 F i g . 8.5 and with the beams expanded w e l l beyond the c r y s t a l edges so that the i n t e n s i t y was nearly uniform across the c r y s t a l . For curve B, the beams were expanded to ju s t i l l u m i n a t e the whole c r y s t a l . The exposure for t h i s case corresponded to point 2 on F i g . 8.5. Curve C was measured at the same i n t e n s i t y and exposure as curve B but with only the c e n t r a l portion of the c r y s t a l illuminated. This geometry was the same as that for F i g . 8.3. Curve D was measured with the same geometry as curve C But at higher i n t e n s i t y and exposure as indicated by p o i n t l l on F i g . 8.5. The r e s u l t s i n F i g . 8.4 show that the e f f i c i e n c y of w r i t i n g holograms depends on the geometry of i l l u m i n a t i o n and on the i n t e n s i t y . The curves are not symmetrical about zero applied voltage with the asymmetry being more pronounced at higher i n t e n s i t y and with only part of the c r y s t a l illuminated. In one sequence of experiments, over 150 holograms were wr i t t e n i n one region of the c r y s t a l and read but with no noticeable fatigue i n wr i t i n g e f f i c i e n c y . 8.4 Discussion In order to i l l u s t r a t e the p r i n c i p a l features required to explain the rather complex r e s u l t s described above, a very simple model i s con-sidered i n which the c r y s t a l i s uniformly illuminated by a si n g l e beam over a length L with darkfcregions of length 2 at each side of the illuminated region.' Neglecting d i f f u s i o n , the currents i n the illuminated and dark regions, r e p e c t i v e l y , are J ( l i g h t ) = epnE + K C I I (8.1) and J(dark) = eun E (8.2) o a 123 EXPOSURE/Wsec cm~2 Fig. 8.5 The time development of a r c s i n n e during hologram w r i t i n g f or the conditions of curve D of F i g . 8.4. Up to 4 W sec/cm 2 the curves for the reduced i n t e n s i t i e s (curves A, B and C) i n F i g . 8.4 coincided. The numbers i n d i c a t e the exposures used i n F i g . 8.4. 124 where the free c a r r i e r concentration i n the illuminated region i s n = n + gx and n i s the c a r r i e r concentration i n the dark. The f i e l d o o i n the illuminated region i s E== E + E and the f i e l d i n the dark 3. S C region i s E where E i s the applied f i e l d and E i s the space charge 3. 3- SC f i e l d . The d i s c o n t i n u i t y i n the current density at the edges of the illuminated region produces sheets of spacecharge, Q per unit area, due to trapped electrons such that | 2 . = J ( l i g h t ) - J(dark) . ' (8.3) The i n i t i a l value of Q i s Q q and i s the charge l e f t from previous exposures. The constraint of constant applied voltage V gives (E + E )L + E & = V (8.4) a sc' a where e E g c = - Q and e i s the p e r m i t t i v i t y . Substituting Eq. 8.1 and 8.2 into Eq. 8.3 gives 4^ = ey(n + gx)(E + E ) + x a l - eyn E . (8.5) dt o a sc o a Substituting for E and E i n terms of Q and V, 3- S C dQ_S dt n T o £+Lj . E (A+L-) The s o l u t i o n of Eq. 8.6 i s eyQ + eyjxV + ^ ^ ( 8 > 5 ) Q(t) = {1 - (1- ^ ) exp(-t/t o)} (8.7) where t = o n (J2.+L) ' eygx{l + — - k . Substituting the " v i r t u a l " f i e l d E^ for Kal/eugx, the steady state space 125 c h a r g e f i e l d i s . Q . . V + E (4+L) E (t=») = - = - T t - / T + T N ( 8 . 8 ) H I + ° T ) gTL T h i s s i m p l i f i e d m o d e l shows t h a t t h e " d c " s p a c e c h a r g e f i e l d a d j u s t s e x p o n e n t i a l l y t o w a r d s i t s f i n a l v a l u e w h i c h depends on t h e a p p l i e d v o l t a g e , t h e i n t e n s i t y o f i l l u m i n a t i o n and t h e d a r k c o n d u c t i v i t y . The f i e l d e i t h e r i n c r e a s e s o r d e c r e a s e s f r o m t h e v a l u e l e f t f r o m p r e v i o u s e x p o s u r e s . I f a s y m m e t r i c a l l y p l a c e d , s m o o t h l y v a r y i n g l i g h t i n t e n s i t y ( e . g . G a u s s i a n ) i s a s s u m e d , t h e n , w i t h z e r o d a r k c o n d u c t i v i t y and n e -g l e c t i n g d i f f u s i o n , t h e s t e a d y s t a t e s p a c e c h a r g e f i e l d becomes E s c ( x ) = - ( E a + E v ) ( l - I ( x ) / I ( 0 ) ) ( 8 . 9 ) " w h e r e 1 ( 0 ) i s t h e i n t e n s i t y a t t h e e l e c t r o d e s . T h i s s t e a d y s t a t e s p a c e c h a r g e f i e l d i s i n d e p e n d e n t o f t h e i n t e n s i t y o f l i g h t f o r a g i v e n s p a t i a l d i s t r i b u t i o n . I f t h e G l a s s e t a l . t h e o r y i s c o r r e c t , t h e d e p e n d e n c e f o u n d b y Chen (1969) on i n t e n s i t y i m p l i e s t h a t h i s c r y s t a l s had a p p r e c i -a b l e d a r k c o n d u c t i v i t y . The d e v e l o p m e n t o f t h e s p a c e c h a r g e f i e l d f o r 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 i s i l l u s t r a t e d i n F i g . 8 . 6 , i n a n i d e a l i z e d way f o r t h e f i r s t e x p o s u r e o f a c r y s t a l ( i . e . t h e i n i t i a l c o n d i t i o n i s z e r o s p a c e c h a r g e ) . W i t h t h e g e o m e t r y o f F i g . 8 . 6 ( a ) , t h e f i e l d ( e x -c l u d i n g t h e " v i r t u a l " f i e l d ) i s e x p e c t e d t o d e v e l o p a s shown i n F i g . 8 . 6 ( b ) f o r a n a p p l i e d f i e l d w h i c h a s s i s t s t h e " v i r t u a l " f i e l d , and i n F i g . 8 . 6 ( c ) f o r a n a p p l i e d f i e l d w h i c h j u s t c a n c e l s t h e " v i r t u a l " f i e l d and i n F i g . 8 . 6 ( d ) f o r a n a p p l i e d f i e l d w h i c h o p p o s e s and i s g r e a t e r t h a n t h e " v i r t u a l " f i e l d . D i f f u s i o n h a s b e e n n e g l e c t e d and t h e d a r k c o n d u c t i v i t y 126 c + I i i i — I r i - — i i t 04-(b) 0 (c) Ol E - i i i — i -(d) F i g . 8.6 Idealized i l lus t ra t ion of the development of the "dc" space charge f i e ld ( ) during the f i r s t exposure of a c rys ta l . (a) geometry of i l lumination. (b) for an applied f i e l d which assists the "v i r tua l" f i e l d . (c) for an applied f i e l d which just cancels the "v i r tual" f i e l d , (d) for an applied f i e ld which opposes and i s greater than the "v i r tua l" f i e l d . The applied f i e ld is shown by the sol id l ine ( ). 127 has been taken as zero. For p a r t i a l or non-uniform i l l u m i n a t i o n of the c r y s t a l , i f the s p a t i a l v a r i a t i o n of l i g h t i n t e n s i t y could be made the same f o r erasure and w r i t i n g , and i f the dark conductivity was n e g l i g i b l e , then the steady state space charge f i e l d l e f t by the erasure process should essen-t i a l l y compensate the " v i r t u a l " and applied f i e l d s . The more s p a t i a l l y l i m i t e d the beam, the more exact would be the compensation. The hologram would be written by d i f f u s i o n , not d r i f t (except f o r the feedback e f f e c t of the si n u s o i d a l space charge f i e l d ) and applied voltage should have l i t t l e e f f e c t . With non-negligible dark conductivity the saturation space charge f i e l d would depend on the l i g h t i n t e n s i t y . These considerations account q u a l i t a t i v e l y for the diminished e f f e c t of an applied f i e l d for the experimental conditions of curve D i n F i g . 8.4. For curve C of F i g . 8.4 the reduced i n t e n s i t y may have produced only p a r t i a l compen-satio n of the " v i r t u a l " and applied f i e l d s . b y the space charge f i e l d , thus showing more dependence on the applied f i e l d . The large scale space charge f i e l d s c e r t a i n l y account f o r the e f f e c t s of previously applied voltages. When the steady state voltage i s changed, the previous steady state space charge f i e l d no longer compen-sates for the applied and " v i r t u a l " f i e l d s . A few cycles of erasure and w r i t i n g are required to reach a steady state response, as shown i n F i g . 8.3. From the point of view of applications i n read-write memory systems, the above shows that b e n e f i c i a l r e s u l t s would be obtained by su i t a b l e changes i n applied voltage during w r i t i n g and erasure cycl e s . 128 Curves B and A of F i g . 8.4 show the r e s u l t s of successively more uniform i l l u m i n a t i o n of the c r y s t a l . The wr i t i n g e f f i c i e n c y i s greater, the more nearly uniform the i l l u m i n a t i o n beacuse the "dc" space charge f i e l d which develops to oppose the w r i t i n g process i s not as l a r g e . The voltage which gives minimum e f f i c i e n c y with the c r y s t a l as nearly uniformly illuminated asspossible (so that space charge f i e l d s are minimal) allows the " v i r t u a l " f i e l d to be estimated, since the minimum should correspond to a c a n c e l l a t i o n of the applied and " v i r t u a l " f i e l d s . Values of 0.05 to 1 kV/cm were obtained which are not out of l i n e i n view of the wide range (1.5 and 40 kV/cm) reported by Glass et a l . (1974b, 1975a) for d i f f e r e n t c r y s t a l s . The f a c t that the minimum e f f i c i e n c y of hologram w r i t i n g with as nearly.as possible uniform i l l u m i n a t i o n was not zero i n the above experiments indicates that d i f f u s i o n may be s i g n i f i c a n t . However, some question e x i s t s on t h i s since the s p a t i a l patterns of erasure and w r i t i n g beams are nec e s s a r i l y d i f f e r e n t and the i l l u m i n a t i o n can.be uniformiin neither case. I t i s believed, therefore, that the r e s u l t s are i n general agreement with hologram w r i t i n g due to d r i f t i n applied, space charge and " v i r t u a l " f i e l d s , p o s s i b l y with some contribution from d i f f u s i o n . 129 CHAPTER 9 LUMINESCENCE DUE TO IRON CENTRES 9.1 I n t r o d u c t i o n Luminescence i n l i t h i u m niobate has been observed due to chromium i m p u r i t i e s , (Burns et a l . 1966, Glass 1969, 1973, Hordvik 1972, 1973), but not, apparently, due to i r o n , which i s the important dopant f o r the hologram storage a p p l i c a t i o n s of t h i s m a t e r i a l (Peterson et a l . 1973, Amodei et a l . 1972). A search was made f o r luminescence because i t was thought that t h i s might help towards understanding the process of hologram w r i t i n g , which appears to i n v o l v e a new e l e c t r o n t r a n s p o r t mechanism, s p e c i a l to f e r r o e l e c t r i c s (Glass et a l . 1975b). Absorption of l i g h t produces a photocurrent d i r e c t l y , r a t h e r than foy merely l i b e r a t i n g e l e c t r o n s , which then produce current by d r i f t i n g of d i f f u s i n g . 9.2 Experimental Procedures The samples were e x c i t e d at room temperature using chopped r a d i a t i o n at 325, 488 or 515 nm from a He-Cd or an Argon-ion l a s e r w i t h e l e c t r i c v e c t o r along the c - a x i s . "As an i n d i c a t i o n of the i n t e n s i t y , luminescence was j u s t observable w i t h the naked eye from undoped s p e c i -mens tr e a t e d w i t h l i t h i u m carbonate. The luminescent r a d i a t i o n was c o l l e c t e d w i t h an e l l i p s o i d a l m i r r o r and focussed on the entrance s l i t s of a Perkin-Elmer 98G monochromator as shown i n F i g . 9.1. F i l t e r s were used to prevent l a s e r l i g h t e n t e r i n g the monochromator and a l s o , to e l i m i n a t e incoherent l i g h t from the e x c i t i n g l a s e r beam. The monochro-mator was f i t t e d w i t h a g r a t i n g w i t h 300 lines/mm, blazed at 640 nm i n NOVA-2 COMPUTER and INTERFACE INTEGRATOR i MONOCHROMATOR 3 3 S-20 TTtT" ELLIPSOIDAL MIRROR LOCK-IN AMPLIFIER SAMPLE ITS? CHOPPER LASER F i g . 9.1 Schematic of the apparatus used to measure the photoluminescence i n LiNbO LO O 131 the f i r s t order. The luminescence passing through the monochromator was detected with a cooled S-20 photomultiplier tube and phase-sensitive detector. The spectrometer i s interfaced to a NOVA-2 computer for s i g n a l averaging and analysis of data, (Thewalt, 1975). 9.3 Results and Discussion F i g . 9.2 (top) shows the luminescence spectra observed for a congruent c r y s t a l doped with 0.015 mole % i r o n before and a f t e r heating the c r y s t a l i n a i r to 520°C for 20 h while packed i n L i 2 C 0 3 > A w e l l -defined peak at 770 nm appears following treatment. In F i g . 9.2 (bottom) L i 2 C 0 3 treatment of an "undoped" c r y s t a l i s shown to introduce a small amount of extra luminescence at 770 nm, presumably corresponding to the few ppm of unintentional i r o n present i n t h i s c r y s t a l . I t i s concluded that the peak i s due to the presence of i r o n . I t i s not understood why the background luminescence was so small i n the untreated doped sample. The 2+ L i 2 C 0 3 treatment reduces i r o n centres to the Fe s t a t e . The r e s u l t a n t increase i n o p t i c a l absorption i s shown i n F i g . 9.3 for the c r y s t a l s used. A peak or shoulder appears i n the region of 470 nm following L i 2 C 0 3 treatment. The absorption edge moves to longer wavelengths on adding i r o n and again following L i 2 C 0 3 treatment. The f a c t that the luminescence peak i s so much increased on converting the i r o n to the 2+ Fe state suggests that i t i s due to e x c i t a t i o n of these centres. How-ever, the luminescence at 770 nm was larger with the L i ^ O ^ - t r e a t e d , undoped c r y s t a l than with the untreated, doped c r y s t a l which showed more 2+ absorption at the Fe peak at 470 nm, and, therefore, probably contained 2+ more Fe centres. I t has already been.mentioned . 132 WAVELENGTH (nm) 800 700 600 500 400 n j 1 1 r~ ' PHOTON ENERGY (eV) F i g . 9.2 Photoluminescence spectra of LiNbO, at 300 UK. The "excitation wavelength was 325 nm with a power of about 5 mW. TOP: 0.015 mole % Fe-doped LiNbO^ before and a f t e r annealing treatment i n I^CX^. The v e r t i c a l bars represent the standard deviations f o r the data a f t e r four scans. BOTTOM: Undoped LiNbO-j before and a f t e r annealing treatment i n Li^CX^. Due to the d i f f i c u l t i e s i n reproducing the o p t i c a l alignment f o r d i f f e r e n t samples, the r e l a t i v e i n t e n s i t i e s f o r the four spectra are accurate to only 30 %. In t h i s and l a t e r f i g u r e s no c o r r e c t i o n has been made for the S-20 response of the f i l t e r s . 133 F i g . 9.3 Absorption spectrum f or undoped and f o r 0.015 mole % Fe-doped LiNb0 3 before and a f t e r heating i n L i 2 C 0 3 at 520 C f o r 20 hours as measured with a Cary spectrophotometer. 134 that the treatment destroys shallow traps (perhaps due to oxygen vacan-cies) as well-as reducing the i r o n centres (see Chapter 2-). I t may be that the luminescence, i n f a c t , involves the trapping of electrons by 3+ Fe centres and that the removal of shallow traps means that a higher 3+ proportion of excited electrons are captured by the Fe centres. F i g . 9.4 shows that a c t i v a t i o n using l i g h t of wavelengths t y p i c a l l y used for hologram w r i t i n g also gave the same luminescence 2+ peak. In t h i s case, the Fe centre absorption i n the region of 470 nm should be involved as opposed to the absorption edge t r a n s i t i o n ( P h i l l i p s et a l . 1974a, Clark et a l . 1973. Redfield et a l . 1974). According to Glass et a l . (1974b,1975'a) on absorption of l i g h t electrons are expelled 2+ from Fe centres with momentum i n one d i r e c t i o n along the c-axis. The question a r i s e s as to whether electrons enter the conduction band, or 3+ whether they reach an Fe s i t e d i r e c t l y through intervalence t r a n s f e r . Hush (1967) defines intervalence transfer as "an o p t i c a l t r a n s i t i o n which involves transfer of an electron from one nearly l o c a l i z e d s i t e to an adjacent one, the donor and acceptor being metal ions which pos-sess more than one accessible oxidation state". In the case where an electron i s transferred between a donor and acceptor ion of the same element (homonuclear intervalence transfer) no luminescence i s expected (Hush 1967). This, then would not appear to be the case for o p t i c a l trans-2+ 3+ s i t i o n s i n LiNbO^ involving Fe and Fe ions. Clark et a l . (1973) 2*f* C1 have suggested that intervalence transfer occurs between Fe and Nb ions (heteronuclear intervalence transfer) so that the absorption pro-cess i s represented as F e 2 + ( e ) + N b ^ F e 3 + + Nb 5*^) 13 5--WAVELENGTH (nm) gQO 700 sm PHOTON ENERGY (eV) F i g . 9.4 Photoluminescent spectra of 0.015 mole % Fe-doped LiNbO-j at 300°K f o r two d i f f e r e n t wavelengths of e x c i t a t i o n : 488 nm and 515 nm. The sample was treated i n L^CO-j. The power of the e x c i t a t i o n l i g h t was 0.4 W. The r e l a t i v e i n t e n s i t i e s of the two spectra are accurate to within 10 %. F i l t e r s were used to block the e x c i t i n g l a s e r l i g h t , and also to block any background luminescence. 136 They postulate that electrons are then free to move i n the conduction 3. band made up of the niobium d o r b i t a l s u n t i l they are retrapped by Fe ions. In t h i s case, the luminescence which was observed would occur when the excited electrons were retrapped. In conclusion, the observed luminescence due to i r o n centres would appear to indicate that electrons are d i s t r i b u t e d by means other 2+ 3+ than homonuclear intervalence transfer between Fe and Fe s i t e s . The d e t a i l s of the process which are involved however,are not presently understood. The ..occurrence of homonuclear intervalence transfer of course i s not ruled out, p a r t i c u l a r l y at higher i r o n concentrations. 137 CHAPTER 10 CONCLUSIONS The purpose of t h i s work was to study the mechanisms of the photorefractive e f f e c t to further the understanding of the process; for engineering a p p l i c a t i o n s . The course of the work was b r i e f l y as follows. I n i t i a l l y , the phenomenon was investigated to c l a r i f y the r o l e of d r i f t and d i f f u s i o n as charge transport mechanisms. Later, when the bulk photovoltaic e f f e c t was proposed as a transport mechanism, further ex-periments were c a r r i e d out to investigate t h i s e f f e c t . In addition, studies were made to extend the usefulness of automated ellipsometry i n probing the photorefractive process. The e f f e c t s of applied f i e l d s on the photorefractive e f f e c t , and the e f f e c t s of multiple i n t e r n a l r e f l e c -tions on techniques used to probe t h i s effect,were also investigated. The contributions which were made to the subject may be sum-marized as follows: a) A t h e o r e t i c a l treatment of the development of r e f r a c t i v e index gratings through d r i f t and d i f f u s i o n was made without the r e s t r i c t i o n of short migration length. The p r i n c i p a l r e s u l t was that the e f f i c i e n c y of hologram w r i t i n g increases for increased migration length up to a ce r t a i n l i m i t . I t was also shown that the increased migration length would not l i m i t the r e s o l u t i o n of the recording medium. b) An explanation was provided f o r the apparent contradiction which existed i n the published data from applied f i e l d experiments. I t was shown that the r e s u l t s of such experiments depend on the f i e l d applied during previous exposure to l i g h t as w e l l as on the f i e l d applied during the current experiment. In addition, the measurements depended on the 138 portion of the c r y s t a l illuminated since t h i s affected the magnitude of the space charge f i e l d that could be developed. The r e s u l t s were con-s i s t e n t with holograms formed by d r i f t , d i f f u s i o n and the bulk photo-v o l t a i c e f f e c t . I t was found that e i t h e r sign of the applied f i e l d could increase the e f f i c i e n c y of hologram formation but the e f f e c t was not symmetrical. The asymmetry was a t t r i b u t e d to the photovoltaic e f f e c t . No value of the applied f i e l d would t o t a l l y i n h i b i t hologram formation i n d i c a t i n g that d i f f u s i o n was p a r t i a l l y responsible. c) Further evidence was given for the existence of the bulk photovoltaic e f f e c t . I t was shown that the photocurrent i n l i t h i u m niobate was not p r i m a r i l y due to p y r o e l e c t r i c f i e l d s developed during cooling of the c r y s t a l s . d) The importance of multiple i n t e r n a l r e f l e c t i o n s on the i n t e r p r e t a t i o n of experimental data was recognized. I t was shown that the e f f e c t i v e d i f f r a c t i o n e f f i c i e n c y measured outside the c r y s t a l can vary s i g n i f i c a n t l y from the absolute d i f f r a c t i o n e f f i c i e n c y of the phase grating. Small changes i n temperature such as those produced by exposure to medium inten-s i t y l a s e r beams produce s u f f i c i e n t change i n c r y s t a l thickness to have a s i g n i f i c a n t e f f e c t . M u l t i p l e r e f l e c t i o n s were also shown to be impor-tant i n applied f i e l d experiments and i n the i n t e r p r e t a t i o n of ellipsometry and adjustable-compensator measurements. e) The advantages of using an ellipsometer to probe the birefringence and o p t i c a l l y induced changes i n the birefringence of l i t h i u m niobate c r y s t a l s were investigated. . This was p r a c t i c a b l e since the e l l i p s o -meter was automatedt'through' computer c o n t r o l . In addition to using ellipsometry, i t was shown that large scale changes i n the r e f r a c t i v e indices could be r a p i d l y inspected i f the c r y s t a l was made to act as a 139 Fabry-Perot interferometer. f ) I t was shown that when l i t h i u m niobate c r y s t a l s are heated i n l i t h i u m carbonate, the treatment reduces i r o n impurities , changes the bire f r i n g e n c e of the c r y s t a l and decreases the rate at which o p t i c a l l y induced space charge f i e l d s decay. To explain these r e s u l t s , i t was proposed that the treatment destroys shallow traps. g) Luminescence due to i r o n centres was observed f o r the f i r s t time. The s p e c t r a l behaviour of the luminescence was not understood but would be r e l a t e d to the r e d i s t r i b u t i o n of o p t i c a l l y excited electrons among traps. 10.1 Suggestions f o r Further Research Further i n v e s t i g a t i o n s are required to characterize more completely the mechanisms of the photorefractive e f f e c t . Some of the parameters involved i n charge transport which are not presently known are the quantum e f f i c i e n c y of photo-excitation, the l i f e t i m e of free c a r r i e r s , the capture cross s e c t i o n of traps, and the migration length of free electrons. These parameters are undoubtedly influenced by the concentration of defects and impurities i n the c r y s t a l . A d d i t i o n a l work i s required on the nature of the defects and t h e i r c o n t r o l . I t was suggested i n Chapter 3 that heating l i t h i u m niobate i n l i t h i u m carbonate destroys shallow traps. 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R e d f i e l d , D . and B u r k e , W . J . 1974 J . A p p l . P h y s . 4 5 , 4 5 6 6 . R o i t b e r g , M . B . , N o v i k , V . K . , G a v r i l o v a , N . D . 1969 K r i s t a l l o g r a f i y a 1 4 , 9 3 9 . S a v a g e , A . 1966 J . A p p l . P h y s . 3 7 , 3 0 7 1 . S e r r e z e , H . B . and G o l d n e r , R . B . 1973 A p p l . P h y s . L e t t . 2 2 , 6 2 6 . S h u r c l i f f , "W.A. 1962 " P o l a r i z e d L i g h t " ( H a r v a r d U n i v e r s i t y P r e s s ) . S m i t h , R . G . , F r a s e r , D . B . , D e n t o n , R . T . and R i c h , T . C . 1968 J . A p p l . P h y s . 3 9 , 1 6 0 0 . S t a e b l e r , D . L . and A m o d e i , J . J . 1972a F e r r o e l e c t r i c s 3^, 1 0 7 . S t a e b l e r , D . L . and A m o d e i , J . J . 1972b J . A p p l . P h y s . 4 3 , 1 0 4 2 . S t a e b l e r , D . L . arid P h i l l i p s , W. 1974a A p p l . O p t . 1 3 , 7 8 8 . S t a e b l e r , D . L . and P h i l l i p s , W. 1974b A p p l . P h y s . L e t t . 2 4 , 2 6 8 . S t a e b l e r , D . L . B u r k e y r W - J . Y P h i l l i p s , W. and A m o d e i , J . J . 1975 A p p l . P h y s . L e t t . 2 6 , 1 8 2 . S t e p h e n s o n , N . C . and R o t h , R . S . 1971 A c t a C r y s t . B 2 7 , 1 0 3 7 . S t e w a r t , W . C , M e z r i c h , R . S . , C o s e n t i n o , L . S . , N a g l e , E . M . , W e n d t , F . S . and L o h a n , R . D . 1973 RCA R e v . 3 4 , 3 . S t o r c k , E . and W o l f f , U . 1975 J . A p p l . P h y s . 4 6 , 3 5 0 9 . T h a x t e r , J . B . 1969 A p p l . P h y s . L e t t . 1 5 , 2 1 0 . T h a x t e r , J . B . and K e s t i g i a n , M . 1974 A p p l . O p t . 1 3 , 9 1 3 . T h e w a l t , M . L . W . 1975 M . S c . T h e s i s 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 . T u r n e r , E . H . 1966 A p p l . P h y s . L e t t . J3, 3 0 3 . V a h e y , D.W. 1975 J . A p p l . P h y s . 4 6 , 3 5 1 0 . v a n H e e r d e n , P . J . 1963 A p p l . O p t . _2, 3 9 3 . v o n d e r L i n d e , D . , G l a s s , A . M . and R o d g e r s , - K . F . 1974 A p p l . P h y s . L e t t . 2 5 , 1 5 5 . 146 von d e r L i n d e , D . , G l a s s , A . M . , and R o d g e r s , K . F . 1975 A p p l . P h y s . L e t t . 2 2 . W i l d m a n n , M . 1975 P r o c . I E E E 6 3 , 1 1 6 0 . Wong, W . K . Y . 1973 M . A . S c . T h e s i s 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 . Y a s o j i m a , Y . , O h m o r i , Y . and I n u i s h i , Y . 1972 T e c h . R e p o r t s O s a k i U n i v . 2_2, 5 7 5 . Y e e , K . K . and Y o u n g , L . 1975 A p p l . O p t . 1 4 , 1 3 1 6 . Y o u n g , L . 1960 P r o c . R . S o c . L o n d . A 2 5 8 , 4 9 6 . Y o u n g , L . 1963 J . E l e c t r o c h e m . S o c . 1 1 0 , 5 8 9 . Y o u n g , L . , Wong, W . K . Y . , T h e w a l t , M . L . W . and C o r n i s h , W . D . 1974 A p p l . P h y s . L e t t . 2 4 , 2 6 4 . 147 APPENDIX A FURTHER PROPERTIES OF LITHIUM NIOBATE A.1 Miscellaneous P h y s i c a l Properties The structure of the f e r r o e l e c t r i c c r y s t a l l i t h i u m niobate i s rhombohedral, point group symmetry 3m with a = 0.5499 nm and a = 55° 52' (Nassau et a l . 1966a). At room temperature the..crystalline structure consists of planar sheets of oxygen atoms i n approximately hexagonal close packing. The r e s u l t i n g octahedral i n t e r s t i c e s are one-t h i r d occupied by Nb and one-third by L i with the remainder vacant (Abrahams et a l . 1966a,1966b). A l l the. oxygen octahedra.are d i s t o r t e d and there are two values each for the L i - 0 and Nb-0 distances. Abrahams et a l . conclude that l i t h i u m niobate i s not p r i m a r i l y an i o n i c c r y s t a l but one i n which covalent bonds dominate. The pure c r y s t a l has very l i t t l e absorption from 350 nm to 5um. I t i s a u n i a x i a l c r y s t a l with ordinary and extraordinary r e f r a c t i v e indices at 500 nm reported to be n = 2.34 and n = 2.24 at 25°C. o e The dispersion and temperature dependence of the r e f r a c t i v e indices has been reported (Boyd et a l . 1964,'1967,Miller et a l . 1965,Hobden et a l . 1966,Nelson et a l . 1974). The d i e l e c t r i c constanthas been measured-for - d i r e c t i o n s -perpendicular to the c-axis as 78,and along the c-axis as 32 (Nassau et a l . 1966). High temperature transport processes have been measured by Jorgensen and B a r t l e t t (1969) who found that both i o n i c and e l e c t r o n i c conductivity occur. The e l e c t r i c a l conductivity i s completely i o n i c at 1 atm of oxygen and 1000°K, while for low oxygen p a r t i a l pressures the conductivity at 1000°K becomes completely e l e c t r o n i c . The e l e c t r o n i c conductivity i s proportional to p n *. 148 The electron m o b i l i t y was calculated to be 1.7 cm^/Vsec at 1000°K i n a 50% CO/50% C0 2 atm and exh i b i t s a i'1^ temperature dependence. The i o n i c conductivity i s c a t i o n i c and i s most probably due to transport by l i t h i u m ions. The Curie temperature i s 1210°C and the melting point about 1260°C (Nassau et a l . 1966). At room temper-ature the c r y s t a l i s a stable f e r r o e l e c t r i c . The p y r o e l e c t r i c c o e f f i c i -ent i s 10~ yC/(m deg) i n the region of 100°C (Roitberg et a l . 1970).-The c o e f f i c i e n t s of thermal expansion are 16.7 x 10 -^ per °C i n the a-axis d i r e c t i o n and 2 x 10 ^  per ° C i n the c-axis d i r e c t i o n . A. 2 C r y s t a l Growth Nassau et al.(1966b) have described techniques used to grow si n g l e c r y s t a l s of l i t h i u m niobate. The most common i s the Czochralski technique with an e l e c t r i c f i e l d applied during growth ( F i g . A . l ) . The c r y s t a l i s rotated as i t i s pull e d from the melt. I f no f i e l d i s applied during growth a multi-domain c r y s t a l forms. E i t h e r p o l a r i t y may be applied, however i f p o l a r i t y i s reversed during growth a 180° domain w a l l i s produced. C r y s t a l s which are not poled during growth may be poled afterwards, but only at temperatures above 1C00°C. A.3 Thermal Bleaching and F i x i n g of Holograms i n LiNbO^ O p t i c a l erasure of holograms which usually occurs during read-out can be avoided i f the holograms are f i x e d (Amodei et a l . 1972a, Staebler et a l . 1972a). When a c r y s t a l i n which a hologram has been stored i s heated to 100°C for 20 or 30 minutes and then cooled, i t i s found that the hologram has been bleached. The phase grating may be restored by i l l u m i n a t i n g the c r y s t a l with l i g h t of wavelengths 400 to 500 nm. The 149 • PULLING MECHANISM P t / P t - R h j l THERMOCOUPLE 11 IN GROUNDED I Pt SHIELD /) o-iomA MOLTEN LlNbOj Jl.O CM . A . l A p p a r a t u s f o r t h e C z o c h r a l s k i g r o w t h o f l i t h i u m n i o b a t e an e l e c t r i c f i e l d ( N a s s a u e t a l . 1 9 6 6 b ) . 150 restored grating cannot be o p t i c a l l y bleached. Amodei et a l . have proposed that the hologram i s bleached not by thermally activated electrons which are r e d i s t r i b u t e d uniformly, but by some kind of i o n i c movement which compensates the space charge. When the c r y s t a l i s cooled and then illuminated, o p t i c a l l y excited electrons r e d i s t r i b u t e uniformly. The phase grating reappears due to the i o n i c displacements which are i n s e n s i t i v e to l i g h t . Q u a l i t a t i v e l y , t h i s explanation accounts f o r the experi-mental r e s u l t s . Further i n v e s t i g a t i o n of the mechanisms involved i n ele c t r o n i c and i o n i c transport are required before d e t a i l s of the thermal processes i n l i t h i u m niobate can be f u l l y understood. I f the io n i c motion at lOO^C i s due to l i t h i u m ions, i t may be that the charge compensation i s caused by l i t h i u m ions migrating to vacant oxygen octa-hedral i n t e r s t i c e s normally present i n the c r y s t a l (Abrahams et a l . 1966a). When the c r y s t a l i s cooled these ions are frozen i n t h e i r new s i t e s i n the c r y s t a l l a t t i c e . 151 APPENDIX B ELECTRO-OPTIC BEHAVIOUR OF LITHIUM NIOBATE The propagation of . electromagnetic waves i n an anisotropic d i e l e c t r i c c r y s t a l i s dependent on the propagation and p o l a r i z a t i o n d i r e c t i o n of the wave with respect to the c r y s t a l axes. The d i e l e c t r i c properties at o p t i c a l frequencies are given by D = e e.. E. (B.1) i o i j 3 where D" i s the displacement, E i s the e l e c t r i c f i e l d , e Q the p e r m i t t i v i t y of free space and e_^_. the tensor p e r m i t t i v i t y of the medium. Combining Eq. 1 with Maxwell's equations leads to the conclusion that two waves of d i f f e r e n t v e l o c i t i e s may, i n general, propagate through the c r y s t a l for a given wave normal (Nye 1960). The r e f r a c t i v e indices of the two waves may be obtained 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^ are the p r i n c i p a l d i r e c t i o n s of the p e r m i t t i v i t y tensor, the i n d i c a t r i x i s defined by the equation 2 2 2 X i 3C« " V ^ 2 + ^ = 1 (B.2) n l n 2 n 3 where n^ = TL^ = /e^, n^ =/E~ . I f a s t r a i g h t l i n e i s drawn from the centre of the e l l i p s o i d p a r a l l e l to the wave normal of the propagating wave, then an e l l i p s e may be formed by cleaving the e l l i p s o i d through i t s centre, perpendicular to t h i s l i n e . The semi-axes of. t h i s e l l i p s e define the two d i r e c t i o n s of p o l a r i z a t i o n which may propagate. The indices of r e f r a c t i o n seen by the two propagating waves are then given by the length of the semi-axes. If an e l e c t r i c f i e l d i s present, the r e f r a c t i v e index of the 152 c r y s t a l Is a l t e r e d and the new I n d i c a t r i x i s i n general described by . 1 „ t - S + z i j k E k + R i j k * E k E * + x i x j = 1 ( B - 3 ) i,j;k,£ n where the Indices i,j,k,£ run from 1 to 3. The c o e f f i c i e n t s z.., and ' J ' ' ljk Rijk£ a r e t* i e H n e a r a n < * t n e quadratic e l e c t r o - o p t i c c o e f f i c i e n t s . Contractions i n the indices are usually made as follows: r , •«-*• z,..*, mk ( i j ) k E.,. w, „, where m and n run from 1 to 6 and m i s re l a t e d to (ij)(k£) and R mn ( i j ) and n to (kJl) as follows: 1 + 11, 2 + 22, 3 + 33, 4 + 23, 5 + 13, 6 + 12. Certain systems cannot exhibit 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 with a centre of symmetry) while a l l materials exhibit the quadratic e f f e c t . Lithium niobate 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 considerations require that some of the l i n e a r e l e c t r o -optic c o e f f i c i e n t s are equal and that some are zero as shown by the following matrix (class 3m). 0 0 0 0 r42 - r - r 22 22 :22 0 :42 0 0 13 :13 :33 0 0 0 -10 -10 where (Turner 1966) r 1 3 = 8.6 x 10 cm/volt, r „ = 3.4 x 10 cm/volt, 22 r 4 2 = 28 x 10 1 0 cm/volt, r 3 3 = 30.8 x 10 1 0 cm/volt, 153 A f u r t h e r p r o p e r t y o f L i N b O ^ i s t h a t i t i s a u n i a x i a l c r y s t a l w i t h x ^ c o n s i d e r e d as t h e p o l a r a x i s . H e n c e , t h e i n d i c a t r i x i s a n 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 s e m i - a x e s a r e e q u a l s o t h a t n o = n l = n 2 ' n e = n 3 * The i n d i c a t r i x i s t h u s g i v e n b y ( _ i 2 " r 2 2 E 2 + r 1 3 V X l 2 + ( J T + r 2 2 E 2 + r 1 3 V *2 n n o o + ( J T + r 3 3 E 3 ) X 3 2 + 2 ( " r 2 2 E l } X l X 2 n e + 2 ( r 4 2 E 2 ) x 2 x 3 + 2 ( r 4 2 E 1 ) x g x± = 1 . (B .4 ) From t h i s e q u a t i o n i t c a n b e s e e n 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 a n 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 m a j o r a x e s 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 te rms w o u l d be z e r o , no r o t a t i o n o f t h e p r i n c i p a l a x e s o f t he i n d i c a t r i x o c c u r . I f howeve r E ^ o r E 2 a r e p r e s e n t t h e n a r o t a t i o n o c c u r s . F o r a wave p r o p a g a t i n g i n t h e x 2 d i r e c t i o n , w i t h o n l y E^ p r e -s e n t , t h e e q u a t i o n o f t h e i n d i c a t r i x r e d u c e s t o (-T + r 1 3 V X l 2 + ^ 2 + r 3 3 V X 3 2 = 1 ( B - 5 ) n n o e The e f f e c t o f E„ i s t o i n t r o d u c e c h a n g e s , An and An i n t h e two r e -3 ° o e f r a c t i v e i n d i c e s . M a n i p u l a t i o n o f e q u a t i o n B . 5 shows t h a t 3 3 n r E„ n r 0 0 E_ . o 13 3 , , e . 33 3 A n o = 2 a n d A n e = 2 (B-6) The change i n i n d e x i s t h e n p r o p o r t i o n a l t o t h e f i e l d . 154 APPENDIX C COUPLED WAVE THEORY FOR THICK HOLOGRAM GRATINGS Kogelnik (1969) has treated the Bragg d i f f r a c t i o n of plane waves by t h i c k phase holograms using coupled wave theory. A condensed version of h i s treatment w i l l be o u t l i n e d for the case of p e r f e c t Bragg conditions and for the case where the grating planes are p a r a l l e l to the b i s e c t o r of the two incident beams R and S, as i n d i c a t e d i n F i g . C . l . F i g . C . l Model of a t h i c k hologram grating of thickness d, grating spacing I and Bragg angle, i n the medium, 9. The analysis assumes monochromatic l i g h t to be i n c i d e n t on a phase holo-gram grating of thickness d at an angle 0 p o l a r i z e d perpendicular to the plane of incidence. Only two waves are assumed to be present: the r e -ference wave R and the s i g n a l wave S. These two waves are the only waves obeying the Bragg condition. The assumption l i m i t s the a n a l y s i s to t h i c k 155 holograms. (Storck and Wolff (1975) have shown that, i n the case of low d i f f r a c t i o n e f f i c i e n c y , the analysis i s also v a l i d f o r t h i n holograms). Wave propagation i n the grating i s described by V 2E.+ K 2E = 0 (C.l) The propagation constant i s rel a t e d to the r e l a t i v e d i e l e c t r i c constant e , and the conductivity a, by 2 co2 K = — e - iioua (C . 2 ) c where c i s the v e l o c i t y of l i g h t , y i s the permeability of the medium and u the angular frequency of the e l e c t r i c f i e l d E(x,z). The f r i n g e s of the phase grating r e s u l t from a s p a t i a l modulation of e : e = • ? + E , cos kx (C.3) o 1 where i s the amplitude of the s p a t i a l modulation and E q i s the average d i e l e c t r i c constant. (The s p a t i a l modulation in.the conductivity i s assumed to be n e g l i g i b l e ) . The grating vector k, i n the case under consideration has only an x-component and i s given by k = - y (C.4) where JI i s the grating period and i s r e l a t e d to the wavelength of l i g h t X by X = 2 £ s i n 0 . Eqs.C.2 and C.3 can be combined to give K 2 = B 2 - 2 i a B + 2 K ' B ( e + i k x + e " i k x ) (C . 5 ) where B i s the average propagation constant and a the average absorption constant: ± 2TTE 2 . o yea 6 = —'—5 a = JL (C.6) X 2e 2 o 156 The coupling constant K' i s defined as .-.ire, K . = (C.7) 2 X E The coupling constant describes the coupling between the reference and the s i g n a l wave. Op t i c a l media may be characterized by t h e i r r e f r a c t i v e index n, when the following conditions are met. 2Trn » a; n » n 1 (C.8) Here n i s the average r e f r a c t i v e index and n^ the amplitude of the s p a t i a l modulation. These conditions w i l l be assumed to be met, i n which case Trn, 3 = 2Trn K = (C .3) A ' A The s p a t i a l modulation i n the r e f r a c t i v e index,forms a grating which couples the two waves R and S and causes an exchange of energy be-tween them. The complex amplitudes of these two waves, R(z) and S(z) vary along z as a r e s u l t of the energy interchange and because of absorp-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 grating i s E (x,z) = R(z) expC-iP^ • x) + S(z) exp(-i£ • x) (CIO) where x = P = s i n 6 6 cos 0 s i n ( - 0 ) 0 cos(-0)J To solve the coupled wave equations, Eq. C l , C.5 and C.10 are combined. - i / -ip.x , -i£.x".;„ By comparing terms with equal exponentials (e and e ) The r e s u l t s of t h i s analysis may be applied to p a r a l l e l p o l a r i z a t i o n by s u b s t i t u t i n g K " for K'where = K cos 20 157 we a r r i v e at R" - 2iR'g cos G - 2ia6R + 2K ' £ S =0 ( C . l l ) and S " - 2 i S ' B cos 9 - 2ia|3S + ( S 2 - a 2)S + 2 K ' B R = 0 (C. 12) 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 with respect to z. I f the i n t e r a c t i o n between the R and S beams i s slow then the R'1 and S'' terms may be neglected. E q . C . l l and C.12 can be rewritten i n the form R' cos 0 + aR = f l ic's (C.13) S' cos 6 + aS = -iKR (C.14) P h y s i c a l l y , the R and S waves change t h e i r amplitude along z due to coupling to the other wave ( K R , K'S) and due to absorption (aR, aS). The general s o l u t i o n to the coupled wave equations i s R(z) = r 1 exp(a 1z) + exp(a 2z) (C.15) S(z) = expfo^z) + s 2 exp(a 2z) (C.16) The constants r ^ and s^ depend on the boundary conditions. The constants o± may be obtained by s u b s t i t u t i n g Eq. C-15 and C.16 into the coupled wave equations. The so l u t i o n i s a, , = (a ± 12K) (C.17) 1,2 cos8 To f i n d r ^ and s^ the boundary conditions for transmission hologram are introduced. The reference wave R i s assumed to s t a r t with u n i t ampli-tude at z = 0. As i t propagates through the phase grating i t decays as i t couples energy into S which i s assumed to be zero at z = 0. The boundary conditions are R(0) = 1, S(0) = 0 (C18) Solving for r ^ and s^ and s u b s t i t u t i n g i n Eq.C.16 the amplitude of the si g n a l wave as i t leaves the grating i s given by 158 S ( d ) = c o s 1 ^ ) ( e X p ( V } - 6 X p ( Y l d ) > ( C ' 1 9 ) where y^ 2 - -(a ± l<)/cos 9. The d i f f r a c t i o n e f f i c i e n c y of the g r a t i n g n i s defined as n = SS* . (C.20) In the present case Eq. C.20 reduces to n = exp( 2 ^ 7 1 ) s i n 2 v d (C.21) v cos 0 cos 29 where v = Trn^/X cos 9 f o r perpendicular p o l a r i z a t i o n and v = 7rn^ ^ C Q S Q f o r p a r a l l e l p o l a r i z a t i o n . 159 APPENDIX D SOURCES OF THE LITHIUM NIOBATE CRYSTALS The c r y s t a l s used i n t h i s study were obtained from C r y s t a l Technology_ Inc., Mountain View, C a l i f o r n i a and from Harshaw Chemical Company, Solon Ohio. Table D.l l i s t s the nominal dimensions, o r i e n t a t i o n and impurity doping of the c r y s t a l s . C r y stals 1, 2 and 3 were cut from C r y s t a l Technology boule #10-375 which was grown along the c-axis. C r y s t a l 4 was cut from boule #10-286 which was grown along the b-axis. The f i f t h c r y s t a l was purchased from Harshaw. A l l the c r y s t a l s were grown by the Czochralski technique. The composition of the melt from which the c r y s t a l s were grown i s given i n Table D . l . A. stoichiometric melt contains more.Li than does a con-gruent melt (Redfield et a l . 1974). In our c r y s t a l s , the stoichiometric melt was 49.0 mole% L i o 0 while the congruent melt was 48.6 mole % L i 9 Q . C r y s t a l Dimensions (mm) Polished face Iron-doping (mole %) Composition of the melt a b c 1 10 10 10 a 0.015 congruent 2 10 2.5 20 b 0.015 congruent 3 10 1.5 20 b 0.015 congruent 4 10 1.0 10 b undoped stoichiometric 5 15 3 20 b undoped congruent Table D.l Lithium niobate c r y s t a l s used i n t h i s study. 160 APPENDIX E THE APPLICATION OF LITHIUM NIOBATE IN A HOLOGRAPHIC MEMORY SYSTEM Read-write o p t i c a l memories based on hologram storage i n an erasable medium are of i n t e r e s t because they o f f e r the p o s s i b i l i t y of large storage capacity with l i t t l e or no mechanical motion and high speed random a c c e s s i b i l i t y . Access to data would be on a page-by-page basis with possibly 10 5 b i t s read or written i n p a r a l l e l . The basic components for a holographic memory are shown i n F i g . E . l . An array of l i g h t valves composes the page of data to be stored. For w r i t i n g , the s i g n a l and reference beams are d i r e c t e d to the appropriate p o s i t i o n on the holo-graphic storage medium by beam d e f l e c t o r s . For reading, the reference beam would be d i f f r a c t e d on to a sensor array by the hologram. Erasure could occur simultaneously during the reading or w r i t i n g cycles or could be accomplished separately. The page composer and sensor arrays would be e l e c t r i c a l l y addressable by the computer. Possib l e devices that could be used to d e f l e c t the beam are acousto-optic and e l e c t r o - o p t i c d e f l e c t o r s . The acc-isto-optic d e f l e c t o r i s the simpler of the two but i t i s slower. The e l e c t r o - o p t i c d e f l e c t o r i s more complicated and presently requires high operating voltages. Both these devices can be made with LiNbO^ c r y s t a l s (Korpel et a l . 1966, Chen 1970). The system i s arranged so that a l l the stored holograms d i f f r a c t the reference beam on to a s i n g l e detector array. The detector may be an array of photodiodes produced as an integrated c i r c u i t . Economically i t may be more f e a s i b l e to make up the detector array with 161 X - Y DEFLECTORS LASER BEAM SPLITTER MIRROR R E A D - O U T ARRAY MIRROR STORAGE MEDIUM (a) DEFLECTED LASER BEAM LiNb03 CRYSTAL PHOTODETECTOR ARRAY (b) F i g . E . l . (a) Schematic of a read, w r i t e , erase i n - s i t u holographic o p t i c a l memory. In t h i s c o n f i g u r a t i o n , the page composer i s used i n r e f l e c t i o n . Page composers may also be used i n transmission, (b) Schematic of readout i n a page-organized holographic memory. 162 a number of small integrated c i r c u i t s . The l a s e r power required i s determined by the number of elements i n a page and the e f f i c i e n c y of the o p t i c a l system. For instance, i n a system i n which the holographic d i f f r a c t i o n e f f i c i e n c y was 10 % and the transmission e f f i c i e n c y of a l l other components was 20%-then only 2% of the l a s e r power would reach the detector array. I f t h i s 4 -6 was then shared between 10 detector elements only about 10 % of the lase r power would reach each element. For a detector that required 1 pJ of l i g h t , a reading speed of ly's per hologram would require a la s e r with a power of 1 W. The page composer envisioned i n t h i s type of system would consist of an array of elements that could be switched between a trans-parent condition and an opaque condition to correspond to ones and zeros. A number of d i f f e r e n t types of page composers have been investigated but only prototypes have been b u i l t . Some experimental holographic systems have used nematic l i q u i d c r y s t a l s (Stewart et a l . 1973, d'Auria et a l . 1974). These require a buffer memory and at le a s t one e l e c t r i c a l connection for each b i t . In addition, they are inherently slow, t y p i c a l l y r e q u i r i n g several milliseconds to switch states. F e r r o e l e c t r i c ceramics such as PLZT have also been considered f o r use i n page composers. Transparent electrode s t r i p s are deposited on each face of a s l i c e with p a r a l l e l s t r i p s on the one side being orthogonal to those on the other side. Coincident addressing changes the birefringence between the crossed electrodes. For an N x N array of elements only 2N e l e c t r i c a l connections are required. Although the devices are fas t e r than l i q u i d c r y s t a l s , at present the PLZT fatigues both e l e c t r i c a l l y and o p t i c a l l y . 163 Carlsen (1974) has proposed an a l t e r n a t i v e method f o r record-ing the d i g i t a l data. In h i s system data b i t s would be stored sequen-t i a l l y as they a r r i v e d from the computer thus a l l e v i a t i n g the need f o r a large page composer. The advantages of p a r a l l e l input are l o s t unless the data to be stored i n one page i s prearranged. Otherwise the data stored i n any one page would be so diverse that there would not be much advantage i n r e t r i e v i n g i t a l l i n a p a r a l l e l output mode. With random access sequential input, the computer could tag the data f o r storage i n a given page. In t h i s manner, p a r a l l e l output of r e l a t e d data could be achieved. Carlsen proposed that each b i t i n a page be stored with a d i f f e r e n t angle with a l l b i t s i n a page superimposed upon one another to y i e l d a m u l t i p l e exposure hologram. Each page would be stored i n a d i f f e r e n t l o c a t i o n i n the recording medium. In using m u l t i p l e exposure holograms, the need may a r i s e to s e l e c t i v e l y erase some of the super-imposed b i t s but not others. Huignard et a l . (1975) have demonstrated that t h i s i s p o s s i b l e . Using LiNbO^ as a storage medium, they f i r s t stored a number of holograms i n superposition. To erase a given hologram they recorded the same object again but with the reference wave s h i f t e d i n phase by u. This produced a complimentary s p a t i a l modulation i n the r e f r a c t i v e index thus eli m i n a t i n g that p a r t i c u l a r hologram. From a p r a c t i c a l point of view t h i s method would not be very u s e f u l since extreme s t a b i l i t y would be required. I t i s of i n t e r e s t to compare holographic memory systems with other systems to determine what r o l e they might play. Kiemle (1974) has shown that holographic memories using a s i n g l e detector array and g a s i n g l e hologram plate are l i m i t e d i n storage capacity to about 10 164 b i t s . New memory techniques such as charge-coupled devices and magnetic bubble domains may be capable of t h i s capacity range and may provide cheaper s o u l t i o n s . Conventional recording technologies such as magnetic drums and disks w i l l probably be improved. Development of holographic memories therefore should s t r i v e to complement these and other technologies because i t i s u n l i k e l y that they w i l l completely replace them. I t i s envisioned that holographic memories w i l l be able to provide c a p a c i t i e s comparable to those of magnetic tape storage systems, but with much shorter access times. 8 The capacity l i m i t mentioned above, of 10 b i t s f or holographic memories using a s i n g l e detector array and recording p l a t e , a r i s e s because the angle at which the detector array can be i l l u m i n a t e d i s l i m i t e d . This determines the maximum s i z e of the recording p l a t e . Kiemle (1974) however has investigated the concept of using modules which each contain a page composer, hologram storage medium and detector. The memory system would have one l a s e r , one x-y beam d e f l e c t o r and a number of passive beam d i v i d e r s . This concept would allow random access to v i r t u a l l y unlimited capacity. In Table E . l a comparison i s made of some memory systems. MEMORY TYPE CAPACITY ( b i t s ) RANDOM ACCESS TIME DENSITY REFERENCE ( S e m i c o n d u c t o r 1 0 9 - 1 0 1 0 < 1 ms 3 2000 b i t s / c m Hodges (1975) C o r e io 6 1 y s •-• • '• ' Rajchman (1970) D i s k 1 0 8 100 ms 5 2 1 .4x10 b i t s / c m M a t i c k (1972) M a g n e t i c Tape ( IBM TBM Sys tem) 1 0 1 2 - 1 0 1 3 > 5 s 5 2 2 . 4 x 1 0 b i t s / c m Wi ldmann (1975) IBM 3850 S y s t e m i o 1 2 15 s 6 o 2 . 5 x 1 0 b i t s / c m H a r r i s e t a l . ( 1 9 7 5 ) M a g n e t i c B u b b l e s 1 0 6 - 1 0 8 . 5 - 5 ms 16 x l O 6 b i t s / c m 2 Bobeck e t a l . ( 1 9 7 5 ) H o l o g r a p h i c M e m o r i e s > i o 1 3 1 y s - 1 ms 1 0 8 b i t s / c m 2 K i e m l e (1974) T a b l e E . l A c o m p a r i s o n o f some computer memory s y s t e m s . 166 ' APPENDIX F ELLIPSOMETER ALIGNMENT The ellipsometer i s a geometrical instrument which depends on the r e l a t i v e alignment of i t s components f o r i t s accuracy. E r r o r s may a r i s e from zero errors i n the azimuth scales of the p o l a r i z e r , quarter wave pl a t e and analyzer, from a zero error i n the angle of incidence scale and from imperfections i n the o p t i c a l elements, e s p e c i a l l y the quarter wave p l a t e . The alignment method used was that described by Aspnes and Studna (1971). They pointed out that the use of a transparent r e f l e c t i n g surface eliminated the e f f e c t of f i r s t order e l l i p t i c i t i e s i n the p o l a r i z e r and analyser. These e f f e c t s can produce errors i n the alignment. The procedure of a t y p i c a l alignment was as follows. The l i g h t source (He-Ne Laser) was adjusted with the analyser arm i n the straight-through p o s i t i o n . With the quarter wave p l a t e i r i s stopped down to i t s smallest p o s i t i o n , and with no aperture i n the analyser arm, the d i r e c t i o n of the l a s e r beam was adjusted to give a maximum s i g n a l from the detector. Next the angle of incidence zero error was checked. Apertures were inserted i n both ends of the analyser arm to define the axis of t h i s arm. The analyser arm was then moved i n O.of steps about 90° and the angle of incidence was pl o t t e d vs. the detector s i g n a l . The r e s u l t i n g curve was symmetrical about i t s maximum and peaked at 89.97°, i n d i c a t i n g an error of -0.03° i n the s c a l e ( F i g . F . l ) . The zero errors i n the p o l a r i z e r and analyser scales were checked with the quarter wave pl a t e removed, and with the angle of incidence at 70°. An o p t i c a l l y f l a t quartz slab was used as a r e f l e c t o r . 167 to u to u to o O Pi o H U W H W J I I L i i .1 i i i i 89.9 .95 90. ANGLE OF INCIDENCE / deg .05 F i g . F . l Zero c o r r e c t i o n f o r the angle of incidence scale. 2.0;92 .8 .4 .2h 1.0;91 0 90 .2 I -cj—o" X / X X / X / .6 P / deg .8 1.0 91 F i g . F.2 Corrections to the analyser and p o l a r i z e r s c a l e s , (o-o-o) b a l -ancing the analyser near 0? for set values of P near 90°(outer scales f o r abscissa and ord i n a t e ) . (x-x-x) balancing the p o l a r i z e r near 0° f o r set values of A near 90° (inner scales f o r abscissa and or d i n a t e ) . The corre c t c o r r e c t i o n f a c t o r s are: A = -0.69?,P = -1.85°. 168 The slab was thick enough to exclude the r e f l e c t i o n from i t s near surface from entering the analyser arm. With both P and A set to approximately 90°, the p o s i t i o n of the slab was adjusted for a maximum s i g n a l from the detector. This procedure ensures that the front surface of the r e f l e c t o r l i e s on the axis of r o t a t i o n of the P and A arms, and that the surface around, and l o n g i t u d i n a l axes of the P and A arms a l l l i e i n one plane, thus e s t a b l i s h i n g the plane of incidence. o The analyser was then nulled near A = 0 for set values of P about 90° to obtain a s t r a i g h t l i n e p l o t of A vs. P . Then the o o p o l a r i z e r was nulled for P near 0 for set values of A about 90 , to obtain another s t r a i g h t l i n e . These l i n e s were plotted with a -TT/2 s h i f t i n the ordinate and +TT/2 s h i f t i n the abscissa of the second curve, as shown i n Fig-E,".^.. At the point of i n t e r s e c t i o n the trans-mission axes of the analyser and p o l a r i z e r are either p a r a l l e l or perpendicular to the plane of incidence. The c o r r e c t i o n i n the two scales was made by s e t t i n g the p o l a r i z e r and analyser to the i n t e r -o section values and then r o t a t i n g the shaft encoders to read 90.00 or 0.00°. To determine the zero error i n the quarter wave plate scale, o o the p o l a r i z e r was set to 0.00 and the analyser set to 90.00 . The quarter wave plate rotated about 0.00° to locate the p o s i t i o n of minimum si g n a l at the detector. When the s i g n a l i s a minimum, the f i r s t axis of the quarter wave plate i s p a r a l l e l to the p o l a r i z e r trans-mission axis and so the error i n the quarter wave plate scale can be found .(Fig. F.3). I o •MINIMUM 1.0 QUARTER WAVE PLATE ANGLE /deg"' 2.0 F i g . F.3 Zero c o r r e c t i o n for the quarter wave plate scale. 170 For a l l of the measurements made with the ellipsometer, the quarter wave plate was positioned at -45° . An i d e a l quarter wave plate would have a r e l a t i v e phase retardation, A , of 90.00° and a transmittance r a t i o , T , of 1.0° Small deviations from these c values can produce large errors i n the ellipsometry readings and so i t i s advisable to have as perfect a quarter wave plate as possi b l e . To adjust A c and T , the analyser arm was fi x e d i n the straight-through p o s i t i o n with P at 0.00°, A at 90.00°, and the quarter wave plate at 315.00°. The tuning micrometer screw on the S o l e i l -Babinet compensator was turned and the positions which gave e x t i n c t i o n were noted. These positions correspond to retardations of 0, 2 T , 4TT. . .. Quarter wave retardation ( i . e . mr/2) i s found by adding onequarter of the differ e n c e between e x t i n c t i o n settings to any one of the se t t i n g s . : The f i n a l s e t t i n g of the quarter wave plate was accomplished using the ellipsometer readings of a c a r e f u l l y aligned, Inconel-coated glass s l i d e . For a p e r f e c t l y aligned instrument, the ellipsometry readings i n zones 1 and 3 w i l l give the same values of and A . The micrometer on the compensator was adjusted to minimize the spread i n t h e m's and A's. I t was found that the p o s i t i o n * The r e l a t i o n s between zones 1 and 3 are: - A = 90 + 2P, 135 > P > -45 Zone 1 1 1 i|> = A± 90 > A± > 0 A = 2P 3 - 90 225 > P 3 > 45 Z o n e 3 y =180 - A 3 180 > A 3 > 90 McCrackin(1963) gives a d e s c r i p t i o n of a l l the zones, -45 1711' of minimum di f f e r e n c e i n ijj was not the same as the p o s i t i o n of minimum differ e n c e i n A s . A serie s of readings were taken f or d i f f e r e n t p o s i t i o n s and using the program of McCrackin(1969) A and T were c c cal c u l a t e d . The s e t t i n g chosen gave A =90.0 77° and T = 0.99959. APPENDIX G * ELLIPSOMETRIC INVESTIGATION OF THE ELECTRO-OPTIC AND ELECTROSTRICTIVE EFFECTS IN ANODIC T a ^ FILMS (Cornish & Young 1973) G. 1 Introduction Ellipsometry may be used to detect and measure separately the small changes i n r e f r a c t i v e index and thickness which occur i n t h i n d i e l e c t r i c f i l m s when a f i e l d i s applied. The occurrence of birefringence i s also detectable. The e l l i p s o m e t r i c technique consists i n measuring the r e l a t i v e phase and amplitude changes i n the s and p components of l i g h t on r e f l e c t i o n from the f i l m . The s and p components have the e l e c t r i c vector perpendicular and p a r a l l e l to the plane of incidence r e s p e c t i v e l y . I f R^ and R g are the complex r e f l e c t i v i t i e s for the two components, one obtains ^ and A where R /R = (tan i|> )exp(iA). In the present work, a PDP8-E computer p s was used to perform the ellipsometer balancing procedure as w e l l as to record currents, voltages and elapsed time. In previous work on these f i l m s , Holden and Ullman(1967,1969) discovered the modulation of i n t e n s i t y r e f l e c t i o n on applying a.c. f i e l d s to the f i l m s using a monochromator with a l o c k - i n a m p l i f i e r detection technique. They explained t h e i r r e s u l t s i n terms of t h i c k -ness modulation. Frova and Migliorato(1968,1969)attributed the e f f e c t to r e f r a c t i v e index changes and compared t h e i r c o e f f i c i e n t s to those of oxygen-octahedra f e r r o e l e c t r i c c r y s t a l s . Ord, Hopper and Wang(1972) were the f i r s t to apply e l l i p s o -metry to t h i s problem. They were able to show that the thickness increased and the index decreased on increasing an applied f i e l d . I t had previously been mistakenly assumed that the f i l m s would become * This appendix represents a continuation and refinement of work sub-mitted for the M.A.Sc. degree and led to the p u b l i c a t i o n l i s t e d . The method developed has since been applied to Nb-Oj. by Yee and Young(1975) . 173 t h i n n e r . These a u t h o r s r e p o r t e d t h e i r d a t a on t he b a s i s o f a l i n e a r d e p e n d e n c e o n f i e l d , s i n c e o n l y a n a r r o w r a n g e o f f i e l d was u s e d , s u c h a s t o g i v e f i l m g r o w t h ( i o n i c c o n d u c t i o n ) a t a r a t e w h i c h c o u l d be t r a c k e d b y t h e i r a u t o m a t e d e l l i p s o m e t e r . T h e y t r e a t e d t h e e f f e c t o n t h e b a s i s o f an i s o t r o p i c i n d e x c h a n g e b u t i n u n p u b l i s h e d w o r k , Hopper and De Smet h a v e c o n s i d e r e d t h e q u e s t i o n o f a n i s o t r o p y w i t h a f i e l d a p p l i e d ( M . A . H o p p e r , p e r s o n a l c o m m u n i c a t i o n ) . The e f f e c t s i n q u e s t i o n a r e o f p o s s i b l e i n t e r e s t i n a p p l i -c a t i o n s s u c h a s i n t e g r a t e d o p t i c a l c i r c u i t s , whe re t h e y c o u l d be u s e d i n f a b r i c a t i n g m o d u l a t o r s ( F r o v a and M i g l i o a t o ( 1 9 6 8 , 1 9 6 9 ) . They a r e a l s o o f i n t e r e s t s i n c e t h e y g i v e i n f o r m a t i o n on t h e s t r u c t u r a l c h a n g e s i n t h e s e f i l m s , f o r e x a m p l e , on t h e c h a n g e s w h i c h l e a d t o s t r o n g h i s t o r y e f f e c t s i n t h e i o n i c c o n d u c t i o n p r o c e s s w h i c h o c c u r s i n f i l m g r o w t h . A l s o , a q u a d r a t i c e l e c t r o s t r i c t i o n e f f e c t was s u g g e s t e d (Young 1963 ) a s one s o u r c e o f t h e q u a d r a t i c f i e l d t e r m s i n t h e a c t i -v a t i o n e n e r g i e s f o r i o n i c m o t i o n (Young 1960 ) . I t may r e a s o n a b l y be assumed t h a t t h e f i l m s , w h i c h h a v e i n o r m a l l y b e e n c o n s i d e r e d t o be a m o r p h o u s , c o n s i s t o f a d i s o r d e r e d f o r m of one o f t h e v a r i e t i e s o f c r y s t a l l i n e T a 2 0 ^ w h i c h S t e p h e n s o n and R o t h h a v e r e c e n t l y i n v e s t i g a t e d i n a s e r i e s o f p a p e r s ( e . g . 1971 ) . T h e s e s t r u c t u r e s a p p e a r t o be c o m p l i c a t e d s e q u e n c e s o f h e r r i n g b o n e d c h a i n s of f u s e d p e n t a g o n a l b i p y r a m i d s o r d i s t o r t e d o c t a h e d r a l b i p y r a m i d s . I t h a s been s u p p o s e d t h a t t h e h i s t o r y e f f e c t s i n t h e i o n i c c o n d u c t i o n p r o c e s s i n d i c a t e b o t h p o i n t d e f e c t c o n c e n t r a t i o n c h a n g e s and g e n e r a l d i s t o r t i o n o f t h e l a t t i c e . H o w e v e r , t h e r e i s c o n t r o v e r s y o v e r t h e e x p l a n a t i o n o f t h o s e e f f e c t s ( D e l l ' O c a , P u l f r e y and Y o u n g 1972) ; D ignam 1972 ). 174 G.2 Experimental Procedures... The automated ellipsometer was s i m i l a r i n general design to that used by Ord et al.(1972). A Rudolf (type 43603-200E) ellipsometer was modified by the addition of stepping motor drives (I.M.C. Magnetic Corp. type 008-008) on the analyser and p o l a r i z e r . Anti-backlash gears (W.M. Berg,._Inc.) were used, with one motor step corresponding to a 0.01° r o t a t i o n of the o p t i c a l elements. Software a c c e l e r a t i o n of the motors was employed. One d i f f e r e n c e from the design of Ord et a l . was that absolute, brush-like shaft encoders (Theta Instrument Company) were used to read the analyser and p o l a r i z e r angles. A f t e r each ellipsometer balance, the changes i n the p o l a r i z e r and analyser angles as indicated by the shaft encoders were compared with the estimate of those angles as calculated by the computer by a l g e b r a i c a l l y summing output pulses sent to the motors. An error warning was printed i n d i c a t i n g any discrepancies, such as would occur, for example, i f a motor did not respond vproperly to every pulse. A 1 mW He-Ne las e r source at 632.8nm was used. A Soleil-Babinet compensator (Gaertner Corp.) was used as the quarter wave plate. A photomultiplier (type RCA 931, l a t e r RCA 8645) was used as detector. It was preceded by a narrow band o p t i c a l f i l t e r , thus allowing ordinary room i l l u m i n a t i o n . The i n t e r f a c e to a PDP8-E computer was constructed using standard D i g i t a l Corp. components. I t provided d i g i t a l and analogue inputs, a clock and also relay controls to i n i t i a t e current flow. Balance time was not minimized but was t y p i c a l l y 2.2s. The ellipsometer was aligned using the method of McCrackin, Passaglia, Stromberg & S t e i n -berg (1963). In s i t u measurements were made with a t r i a n g u l a r prism c e l l 175 made by j o i n i n g o p t i c a l glass f l a t s with epoxy r e s i n . The angle of incidence (63.46°) was fi x e d by the need for normal incidence on the c e l l faces. The s o l u t i o n was 0.2M sulphuric acid and was co n t r o l l e d at 298K. The cathode was p l a t i n i z e d platinum. Tantalum specimens (Materials Research Corp.) were sing l e c r y s t a l s l i c e s electropolished i n 10% by volume 48% by mass HF i n 98% by mass H 2S0 4-Measurements were made at i n t e r v a l s i n two zones f or c a l i b r a -t i o n purposes. In tracking, one zone only was used. C e l l window errors were corrected f o r . G'. 3 Results Since the films are s o l i d , and, also, f o r that matter, since they are attached to a s o l i d surface, i t i s to be expected that they would become anisotropic ( b i r e f r i g e n t ) on applying a f i e l d . One would expect that the fil m s would become u n i a x i a l with the optic axis perpendicular to the f i l m surface. Since the films are grown by the a p p l i c a t i o n of high f i e l d s - that i s i n the anisotropic state and, furthermore, since the growth process i s i t s e l f d i r e c t i o n a l , i n that the ions t r a v e l normal to the f i l m , one might even expect that, on removal of the f i e l d , some anisotropy of the same type might remain frozen-in. Despite the f a i l u r e to detect structure i n these f i l m s i n d i f f r a c t i o n experiments therefore, they need not be completely f r e e of microstructure or be completely i s o t r o p i c . The ellipsometry angles ty, A for homogeneous i s o t r o p i c , non-absorbing f i l m s follow a closed loop (at l e a s t , modulo 2TT i n A) . With -anisotropy of the expected kind, the r e f r a c t i v e index would be represented by an o p t i c a l i n d i c a t r i x which was an e l l i p s o i d of r o t a t i o n 176 about the f i l m normal. If so, the s - l i g h t responds to n Q the radius of the c i r c u l a r p r i n c i p a l section. The p l i g h t responds to n^, where i f <()^  i s the angle of incidence and n^ the index of the ambient, the angle of r e f r a c t i o n fy^ i s given by n^ s i n <j>^  = n^ s i n §^ with the i n d i -c a t r i x determining a r e l a t i o n between and n^; 2 2 2 2 2 2 n s i n cb0/n + n cos d>0/n = 1. p T 2 e p T 2 o The r e f l e c t i v i t i e s R and R are then c a l c u l a b l e i n the usual s p way except that d i f f e r e n t indices n^ and n^ are used. The r e s u l t i s that, for an anisotropic f i l m , the data s p i r a l either upwards or downwards (depending on the sign of the birefringence) i n the , A domain instead of following a closed loop. The problem of ellipsometry with anisotropic f i l m s was considered recently by Engelsen (1971). EigV G.l shows the lower part of ^, A domain for f i l m s which were grown successively to greater thicknesses. The ellipsometer was balanced during the growth process (with the f i e l d on) and again during periodic interruptions (with the f i e l d removed). With the f i e l d removed, the data l i e on a s i n g l e curve for successive cycles of growth (see legend for o p t i c a l constants). The data for zero f i e l d f i t a curve for an i s o t r o p i c f i l m with constants as shown i n the f i g u r e caption. With the f i e l d on, the curve traces a higher path on each cycle. The data for constant applied f i e l d f i t computed curves for an anisotropic f i l m . Thus, the data are consistent with the f i l m s being i s o t r o p i c , homogeneous and non-absorbing within some l i m i t s with the f i e l d o f f , but becoming markedly anisotropic (though s t i l l homogeneous through t h e i r thickness) when the f i e l d was on. 17.7 F i g . G . l Lower part of ip,A domain f o r increasing thickness of f i l m s on tantalum up to three c y c l e s . Lower curve (dashed) i s computed for an i s o t r o p i c oxide and the experimental points are for zero f i e l d : B , f i r s t c y c l e ; © , second c y c l e ; A , t h i r d c y c l e . Upper three curves ( s o l i d l i n e s ) represent an ani s o t r o p i c f i l m with experimental points f o r f i e l d a pplied: • , f i r s t c y c l e ; o >second c y c l e ; A , t h i r d c y c l e . O p t i c a l constants: thantalum n-ik = 2.46-i2.573; i s o t r o p i c oxide, n n = 2.195; ani s o t r o p i c oxide, n -n = -0.090; n -n = 8(n -n ), with $-1.6. r ' o n ' e n o n F i g . G.2 Upper part of if>, A domain. Key as f o r F i g . G . l . 178 Films grown i n the present e l e c t r o l y t e do not show detectably the e f f e c t s seen for f i l m s grown i n phosphoric a c i d which consist of two layers (Dell'Oca & Young 1970). The outer layer has a lower index due to incorporation of phosphate; t h i s layer i s believed to grow due to metal ion motion as opposed to oxygen ion motion. This e f f e c t , however, gives a quite d i f f e r e n c t kind of s p i r a l l i n g of the ^, A p l o t . F i g . G.2 shows s i m i l a r r e s u l t s for the top portion of the I curves, confirming the behaviour demonstrated by F i g . G.l.. To proceed further i t was assumed that i f n and n are e o the changes produced by a f i e l d E and i f i s the index of the oxide with no f i e l d applied, then 8 = A n /A n = (n -n ) / (n -n ) e o n e n o i s a constant independent of E. The method of separately determining changes of thickness and of index i s i l l u s t r a t e d by F i g . G.3. This shows, on the if) , A plane, contours of equal thickness and equal n Q , ca l c u l a t e d f or our angle of incidence and assuming B = 1.6 as determined by obtaining the best f i t of computed curves and experimental data i n F i g . G.l and G.2. Also shown are experimental data for a given 2 -1 f i l m to which a sequence of f i e l d s was applied up to 5.07 x 10 MV m The decrease i n index and the increase i n thickness can be read o f f i n F i g . G.3 for each value of the f i e l d . In the previous work by Ord et al.(1972) the films were assumed i s o t r o p i c , i . e . i n c a l c u l a t i n g s i m i l a r charts the value of B was taken as 1. The data for a s i n g l e f i l m do not exclude such treatment. The anisotropy i s detected and measured only by growing the f i l m so as to cover more than one c y c l e . 179 n=2.1947 - \ zero field I 2.1930 W/deg F i g . G.3 Contours ( ) of constant index n (with 3= 1.6) and contours ( ) of constant f i l m thickness on tantalum with substrate constants as i n F i g . G . l . The experimental points are for a f i l m on tantalum with a range of f i e l d applied as shown and maintained u n t i l steady-state changes were established. The f i e l d s were such that neg-l i g i b l e ion current was produced and hence n e g l i g i b l e growth. 180 APPENDIX H PROGRAMS USED TO CONTROL THE ELLIPSOMETER SYSTEM H.l Introduction There were seven main programs used to c o n t r o l the automated ellipsometer and to c o l l e c t data. The programs were written i n a com-bination of Fortran,and assembly language to be compatible with the 0S8 operating system used by the PDP-8E computer. Using the programs l i s t e d i n Table H . l , the p o l a r i z e r and analyser readings could be accumulated while the c r y s t a l was scanned through the probing beamr. The data was stored on the Dectape Unit for output v i a the teletype or the incremental p l o t t e r . Table H.2 l i s t s the subroutines used by the main programs. The most frequently used programs were c a l l a b l e from a keyboard monitor program (SKEY,SV). The functions that could be performed from the keyboard monitor were ( i ) step the xy motors which moved the c r y s t a l through the probing beam; ( i i ) p o s i t i o n the pen of the xy p l o t t e r ; ( i i i ) balance the ellipsometer; and (iv) scan the c r y s t a l h o r i z o n t a l l y and record the p o l a r i z e r angle, analyser angle, and p o s i t i o n of the c r y s t a l . A f t e r f i n i s h i n g a command, the program returned to the key-board monitor. Other programs (DIFFA.SV and.DIFFP.SV) were used to subtract either the analyser or p o l a r i z e r readings of two scans along the c r y s t a l and to p l o t the diff e r e n c e on the xy p l o t t e r . The two programs ANAL.SV and POLA.SV plotted the analyzer and p o l a r i z e r readings from a si n g l e scan. DATA.SV was used to type out the contents of a f i l e where the data from one scan was stored. MAIN PROGRAMS SUBROUTINES CALLED FUNCTION 1. S K E Y . S V SRUN, S B E N , S S T E P , SPLTXY S X Y P E N , SMOT2, S B E l , SRDA SRPHOT, SREV, STEPN K e y b o a r d m o n i t o r p rog ram 2 . P L O T . S V SAXES Draws a x e s and c a l c u l a t e s s c a l e f a c t o r s f o r p l o t t i n g 3 . P O L A . S V S A X E S , SDEGR,SPLTXY P l o t s p 6 1 a r . i z e r . : r e a d i n g s : f r o n r o n e f i l e and t y p e s r e s u l t s 4 . A N A L . S V S A X E S , SDEGR, SPLTXY P l o t s a n a l y s e r r e a d i n g s ".from 'one f i l e and t y p e s r e s u l t s 5 . D I F F A . S V S A X E S , SDEGR, SPLTXY. S u b t r a c t s a n a l y s e r r e a d i n g s f r o m two f i l e s , p l o t s and t y p e s t h e d i f f e r e n c e 6 . D I F F P . S V S A X E S , SDEGR, SPLTXY S u b t r a c t s p o l a r i z e r r e a d i n g s f r o m two f i l e s , p l o t s and t y p e s t h e d i f f e r e n c e 7 . DATA.SV SDEGR Types c o n t e n t s o f a f i l e on t h e t e l e t y p e Table H.l Main programs used to c o l l e c t data and control the automated ellipsometer. F I L E CALLING NAME FUNCTION 1. SBE1 BE I W r i t e s P and A on t h e t e l e t y p e 2 . STEPN STEPN S t e p s t h e d e s i g n a t e d mo to r 10 t i m e s 3 . SREV REV R e v e r s e s t h e d i r e c t i o n o f t h e d e s i g n a t e d mo to r 4 . SSTEP STEP S t o p s t h e d e s i g n a t e d mo to r once 5 . SRUN RUN S c a n s t h e c r y s t a l and r e c o r d s t h e p o s i t i o n s P and A 6 . SMOT2 M0T2 Moves XY m o t o r s t o r e p o s i t i o n t h e c r y s t a l 7 . SAXES A X E S ( I Y R , I X R , IYMUL, IXMUL) Draws a s e t o f a x e s . IYR and IXR a r e t h e l e n g t h s o f t h e Y and X a x e s . IYMUL and IXMUL a r e t h e s c a l i n g f a c t o r s c a l c u l a t e d 8 . SBEN B E ( P , A) B a l a n c e s t h e e l l i p s o m e t e r once 9 . SDEGR D E G R ( I L , I H , ANGLES) C o n v e r t s d o u b l e p r e c i s i o n o c t a l number ( I L , IH) t o d e c i m a l number (ANGLES) 10 . SPLTXY P L T X Y ( I , X , IY ) P l o t s one p o i n t on XY p l o t t e r 1 1 . SXYPEN XYPEN Moves t h e p l o t t e r pen t o a d e s i g n a t e d l o c a t i o n 1 2 . SRDA R D A ( I L . I H , ANGLES) Reads t h e s p e c i f i e d s h a f t e n c o d e r and r e t u r n s a n g l e a s a d o u b l e p r e c i s i o n o c t a l number ( I L . I H ) and a s a d e c i m a l number (ANGLES) FILE CALLING NAME FUNCTION 13. SRPHOT PvPHOTO (ERR) Reads the error sign a l from the detector of the ellipsometer Table H.2 Subroutines used by the main programs l i s t e d i n Table H.l H.2 Main Programs 1. SKEY.SV C K E Y B O A R D COMMAND P R O G R A M ' COMMON I D I P * 11 A L , I I A H , I I P L J I I P H S '<5T#CLA R F A P C ! j ? ) ! 5 ! ^ T 2 F C M A T < ' : ' A2> I F < IN<?T-133 )4j3, 4 3 C A L L 8F.1 4 COVTIV'.'E IF( TMCT_c;.c;a ) 5 , t pi / • . 5 C A L L MOTS 10 C O N T I N U E I F C I N S T - l 17.1 7 C A L L hljrvj : 6 C O V T I N ' L E IF<IMST-1036>9,8»9 <? C A L L YYPF.N 9 C O N T I N U E 5 J M P S T EMI 2. POLT.SV C . ' r>"A" AXES AMD CLACULATE SCALE FACTOR? W P I T f ( 1 , 1 ) 1 FORMAT C "7.-EP0 PLOTTER ' ) READC UHJIY' • ' 2, FORMATC " Y ^ANGE IM DEGRFF.S = 'I3> READC1,3)IXR . 3 FORMATC' X "flMfiE = 'I3> CALL AXFSClYR,T.X?.,IYMTrL,IXMt)L> ; CALL EXIT END 185 3. POLA.SV C PLOT POLARIZER TRADINGS FOTIND IN F I L E ENTERED C .SR OPT ION: IF S«11 = 1 5 TYPE OUT RESULTS ALSO. DIMF.-VSIO'J I A H S M ) . IARCRPIO) DIMENSION D P ( P 0 0 ) , I X ( 2 P 1 ) DPH=-370. ' DPL=3 7P. .READ*1,1)FILF.l 1. v FORMAT*"ENTER 1ST DATA FILK NAME: "A6) READ<1,3)\' • - , 3 FORMAT ( ' MO. OF DATA LTVFS='I3) C XY RLOTTFR INITIALISATION .WRITE* 1 , 1 / 4 ) ' .14 FORMAT* "PLOTTER READY? PRESS COMT. SRI 1=01 FOR AXES • ) S HLT • ' • _ IXA=« IYY=0 CALL !OPEN* * DTA1 * > F I L E 1 ) RFAD(4,Zi) C I A l C.J), I AP C J ) , I A3, IA4, IXC J ) , IAS, J=1,N) 4 ' FO"MAT < 6A3) DO ft. 1 = 1 ,'M IAL=IA1CI) IAH=IA!?CI) CALL DEfJR C I AL, I AH, D A ) DPCI)=PA IFCDPH-DPCI))7,8,8 7 DPH=DP(I) R CO^TIMtiH I F ( D P L - D P C I ) ) 6 , 5 , 5 5 ' .. DPL=DP<I) 6 CONTINUE '•TRITE* 1 , 1 S)DPH, DPL 15. FORMAT*"PLOT LIMITS: YHIGH="F6.2" YLOV="F6•2 ) RFAP* 1, 18)rPH,pD!J . . 18 FORMAT*"SET YHIGH=*FS.9," VLOV="F6.?) IYM=TFIxcr.PH-.pPL) • ' IXM=IXCN) C PLOT AXES AND SET SCALE A3 = FL0AT *IYM)/10• '•'RITE* 1 ,9 ) A3 9 FORMAT* "EACH Y DIVISION ='F'4.2' DEGREES') . CALL AXF^dVM, IXM, tYMflL, IXMfJL) YM"L=FLOAT*IYMUL) C SCALE DATA AND PLOT- POIMTS DO 10 1=1,M 1X1=IX*I)*IXMUL PP=(DP(I )-DPL)*10.*YM(jL IPP=IFIX(PP) IDX=IXI-IX> IDY=IPP-IYA IXA=IXI . IYA=IPP 10 CALL PLTXY(IDX,IDY) "RITE*1,11) 11 FORMAT*"TYPE RESULTS? YES: SF11=01; PRESS COMT") S HLT S CLA » . . • S 740* S AMD <K1 S S7A CLA S . JMS TYRE CALL EXIT S TYPE,P l.'RITF* 1, 1P)< IX( I ),DP( I ), 1=1,N) 12 FORMAT(13,F8•3) S . JMP I TYPE •' END . ' P L O T A N A L Y S E R R E A D I N G S F O U N D I N F I L E E N T E R E D S R O P T I O N : I F S R 1 1 = 1 ; T Y P E O U T R E S U L T S A L S O . D I M E N S I O N I A 1 C 2 0 0 ) , I A S C S 0 0 ) ' D I M E N S I O N D P C S 0 0 ) , I X C 8 0 0 ) D P H = - 3 7 0 . D P L = 3 ~ 7 0 . R E A D C 1 , 1 ) F I L E 1 F O R M A T C ' E N T E R 1 S T D A T A F I L E N A M E : , ' A 6 > R E A D C 1 , 3 ) N F O P M A T ( ' N O . O F D A T A L I N E S * ' 1 3 ) - ... X Y P L O T T E R I N I T I A L I S A T I O N W . R I T E C 1* 1 4 ) ' ' F O R M A T ( ' P L O T T E R R E A D Y ? P R E S S C O N T . . S R I 1 = 01 F O R A X E S ' ) H L T . I X A = 0 I X Y = 0 •• " C A L L I O P E N C * D T A 1 ' , F I L E 1 ) . R E A D C 4 , 4 ) C I A 3 , I A 4 , I A 1 C J ) , I A S C J ) , I X C J ) , I A 5 , J = 1 » N ) " F O R M A T C 6 A S ) DO IS I = 1 , N I A L = I A 1 C I ) I A H = I A 2 C I ) C A L L P E G R C I A L , I A H , P A ) '. . • D P C I ) = 3 A 0 . - P A I F C D P H - D P C I ) ) 7 , 8 , 8 • D P H = D P C I ) C O N T I N U E I F C D P L - D P C I ) ) 6 , 5 J 5 D P L = D ? C I ) C O N T I N U E ' '.•'RITEC l , ' l 5 > D P H , D P L F O R M A T C ' P L O T L I M I T S : Y H I G H = ' F 6 . 2 ' Y L 0 W = ' F 6 . 2 ) R E A D C 1 , 1 H ) D P H , D P L F O R M A T C ' S E T Y H I G H = ' F 6 • 2 » ' YL0W='F6.8) I Y M = I F I X C D P H - D P L ) I K M = . I X C N ) P L O T A X E S A N D S E T S C A L E A 3 - F L 0 A T C I Y M ) / 1 0 • <.'!R I T E C 1 , 9 ) A 3 F O R M A T C ' E A C H Y D I V I S I O N = ' F 4 . 2 ' D E G R E E S ' ) C A L L A X E S C I Y M , I X W , I Y M U L , I X M U L ) Y M U L = F L O A T C I Y M U L ) S C A L E D A T A A N D P L O T P O I N T S -DO 1 o. I = 1 , N I X I = I X C I ) * I X M U L P P = C D P C I ) - D P L ) * 1 0 . * Y M U L I P P = I F I X C P P ) I D X = I X I - I X A . . I D Y = I P P - I Y A I X A = I X I I Y A = I P P C A L L P L T X Y C I D X , I D Y ) W P I T E C 1 , 1 1 ) . F O R M A T C ' T Y P E R E S U L T S ? . Y E S : S R 1 1 = 0 1 5 P R E S S C O N T ' ) H L T ' . . C L A 7 4 0 6 A N D C K 1 S 7 A C L A , J M S T Y P E C A L L E X I T . '' . T Y P E , 0 V R I T E C 1 , I S ) C I X C I ) , D P C I ) , 1 = 1 , N ) F O R M A T C I 3 , F 8 . 2 ) . J M P I T Y R E E N D • . , ' • 187 criRTRftOT A\JL.' R*"APT \ir;c FO'A'P IM THE P F I L E S ENTR ED PLOT THE PI E"E\'CF .»« . PI STANCE ALUM 'i CRYSTAL C O OPTir»\j. tr " 1 1 = 1: T Y p r Ofy qjrcrjLTS A L c O . D I M E N S i Q V i a t (omn). rA,>f9n"i),inicP''iri)»iPP(?rip) DIMENSION C c ( P " i > » IXCPCIP) nCrf=-T7n. DPL=37P. RFAPC1.1)rIL=M K0PN1T( • v \ i T F c i « T DATA F I L E NAME: ' A* ) R « n ( 1 , P ) F I L F P F '|C».AT( 'f.vT« P\'D PAT A F I L E \JAMF8 ' A* > PPAPC 1 , -> ) V ITOPv;AT( • MO . OF DATA L I N FS = ' I 3 ) •YV PLOTTER J \i I T T AL I 7 t 0\J ' " > ! T P ( | , | Z l ) Vf)RMAf( 'PLOTTER RFAPY? DKtrce CONT. SB 1 1=01 FOR AXES') HLT l v n = T IXY = o CALL !ncr.\'( • PTA1 ' . F l LF 1 > REAP{ /), /O ( I A3 . I A'i, I A I ( J ) , I APC J ) , I * ( j >. I A^, J=-l » N ) r . l P i j u T C « A P ) CALL 1 O D V-v< • PTA 1 ' , K I LEP ) B V U n C f l . / i ) < l AT, T A/i . t * t ( ,J) , t 'IPC .J) . I A<5. I A«, J = t , N) FINP MA" AMD MIN <7AL"E? OF PR PO « I = 1» M l A L = i a i C I ) TAH=IAP(M CALL PE<~.R( I AL. T AH. PA) IP'.= ir<i<n inH=I^P<1) CALL PFRC ( IBt,» I"H, P'U P D C I » - D A - P M I FC PP( I )--\r . )P ! , pn, p,^  ; . T D T T V ( 1 , J o > t * ( T >, PRC I ) REAPC1,93)PP(I) FORMAT < • S ET D EL A = ' F f • P ) COMTIN"E IF<PR'-t-r c( I ) )7.5(»n p D H = r i D(I) -C O N T I -IFCPP!.-PPt I ) X , ^. S p p t . , = r>P( I ) CONTIV'E w n ? ( i , i m p o H . r P L FORMAT ( ' PLOT L'MITC? YH!T4='F*.P' YLOM= • f * . ? ) REAP( I » 3 s 1 P.PH. PPL POCM A T C ' C RT V H I ^ H : ' t e n , • vt_QW= » F f . ? ) I Y V = i u i X C P R H - t R L ) PLOT AXE 4 1 AND SCALE .»1=FLOATC i v - ^ j / i p i . W P l T E t l , 0 ) ^ T FnoviAT (• c-ACH v PPH«I'VM -s'F/i.P' DWfcF.S ' > CALL AVFS ( I YM t I XM , I YM'JL. I X*T!L ) Vv!Tt!. = i.-|jnAT( IYM"D SCALE PATA A\!p .PLOT POINTS PO 1 'A I = 1 , N • IV! = I Y C I ) * I xyi!L pp- ( r,P< I ) - PPL ) * 1 0 • + YM"L I P P = I F I X ( P P ) IDX=IYt-I XA trtv=IDP-TYA IXA=IXI 188 IVA=IPD \6 c.M.i* P L T X V C i n v , \VY, V R I T F ( l . l l ) \\. FOR/ATf'TYRR R " " L T S ? YES: S R . l = f l l 8 PRESS COMT') K L T s CLA 7/>'*«. 4\!P (HI 5 57« CLA S j y s TVPF \ CALL FX I T s TVPF.o ' MR I T F f 1 . I P ) C txc I ). TP<I )•1 = 1»MV 12 POP MAT ( P > F ^ i P ) s J M P I TYPF END 6. DIFFP.SV C cnnT'-ACT POL. RFADT NGS FOUMD IN THE P F I L E S ENTERED C PLOT THE DIFFERENCE DISTANCE ALONG CRYSTAL C SP OPTIONS I F <!P11=1J TYPE OUT RESULTS ALSO. DIMTCJS I ON lftl(P«l».)» I A-2( P.V..B) , I P l CP'/.".), I ™S ( 2553 > DIMENSION DR<S0E1>» IK<8?-0) DPH=-37S. D DL = 3 7*). READ< 1.- D F I L E 1 1 FORMAT< ' SN'TF." 1ST DATA FILE. NAME: * Afi) T»5M5< 1, P. ) F I L E R p, FORMAT ('ENTER 2ND DATA F I L E NAME: ' A6> READ<1,3)N 3 FORMATC 'NO• OF DATA LINES='!3) C XV PLOTTER INITIALISATION '•'PITEf 1, 14) 1/1 FORMAT C 'PLOTTER READY? D P E S S CONT. SRI 1=01 FOP AXES') S HLT IXA=0 * IXY=0 . ' CALL I O R E N C ' D T A 1 ' , F I L F 1 ) F.EADC.4, 4) C IA1 C.J), IA2C J ) , IA3, IA/l, IXC J ) , I A5, J= 1 , N ) 4 FORMAT(*AS) CALL IOPENC'DTA1',FILES) P E A D ( C IB1 C-J), ISC< J ) , I "3, I A/|, I A 5, I AC, J=1,M) C FIND MAX AND MIN VALUES OF DP DO 6 1=1,M IAL=1A1(I) IAH=IA?.CI) CALL DEG'3 ( I AL, I AK, PA) -I B L = I B 1 ( I ) IBH=IBP.<I) CALL DEGRCIBL,IBH,PB) DPC1)=PA-PB IFCDPH-DPC I ) )7, F., FS 7 D P H = D R ( I ) 8 CONTINUE IF C D P L - D P C I ) ) 6 , 5, 5 5 DRL = r.R<n 6 CONTTNO; VRITEC1,15)DPH, DPL 15 FORMATC'PLOT L I M I T S : YH1GH='F6.P* YL0V='F6.8> PEADC1,30)DPH, D°L • . ' 30 FORMATC ' SET YHI 0H= ' F6 . 2, ' SET YLO'>'= ' F6 . 2 ) . IVM=I FIX(DPH-DPL) IXM=IXCN) C PLOT AXES AND SET SCALE A3 = FL0AT(I VM)/1' TI. 189 VRITEC1L9)A3 9 . FORMAT ('EACH Y DIVISION ='F4.2 • DEGREES') CALL AXESCIYM,IXM,IYMUL,IXMUL) YMUL-FLOATCIYMUL) . ' C SCALE DATA AND RLOT POINTS D O Iff 1 = 1, M I X I = IX ( I ) * I Vi'i' !L PR=CDPCI )-DPL)*10.*YMUL IPP=I F I X ( P P ) IDX=IXI-IXA IDY=I?p-lYA IXA=Ixt . I Y A = I ? C 10 HALL C L T V y c I P X , I D Y ) '•'p I T r'" ( 1 » ! 1 ) 11 F O R M A T C ' TYPE RESULTS? YES : S P l l = 3 1 J PRESS CONT ' ) S HLT S CLA c. AND (HI c S7 A CLA S JMS TYPE CALL EXIT R TYPE, P> W P I T F . ( 1 , 1 P ) ( I X ( I ) , D P ( I ) , I = 1,N) IP FORMAT I I 3 , F « . P ) c, JMP I TYPE - . • END i 7. DATA.SV C READ CONTENTS OF F.N AM E AND TYPE ON TTY 0 ANGLES ARE CONVERTED TO DECIMAL FOR O U T P U T DIMENSION P ( 2 ~ 0 ) , ACSPfO DIMENSION IPLCP^H), I P H C P P U , I AL ( P ? - ) » lAH(P.OO), IYC200) READC1,13)FMAME 13 FORMAT<•ENTER DATA F I L E NAME: •Afi) P.EADC1,1)N 1 FOPMATv'NO. OF L I N E IN F I L E = ' I 3 ) RF.ADC 1, 14)N1,N2 14 FORMATC'LIST LINES »I3' T O '13) • CALL IOPENC 'DTA1 ',FN.AME) D O 3 I = 1, N READ(/i,S) IPLC I ), IPHC I ), I ALC I ), IAHC I ), IY( I ), IX . '3 CONTINUE ' " S FORMAT<6AP) D O 4 I=N1,N2 I A l = I P L C I ) IA2=IPHCI) C A L L D E G R C I A 1 , I A P , A M G ) PCI)=ANO IA1 = 1 A L C I ) I A P = I A H C I ) CALL DEGR C 7 A1,IA2,ANG) AC I ) = 360•-ANG A CONTINUE WRITEC1,!!?) 10 FORMATC ' P A Y X') D O 11 I=N1,N2 V P I T E C 1 , 1 2 ) P C I ) , A C I ) , IYC I ), I X 12 FORMATCF.'J.P- IX,F6.?., IX, 13, I X , 13) 11 CONTINUE CALL EXIT E N D 190 H.3 Subroutines Called by the Main Programs 1 . S B E 1 BALANCE ELLIPSOMETER AND PRINT P&A. SUBROUTINE BEI CALL BE<P,A> VRITEC l . D P . f l FORMATC 'P=•F6.8,8X, 'A='F6«S) RETURN END 2. STEPN STEP Y MOTOR N TIMES TO ACCOUNT FOR GEAR REDUCTION SUBROUTINE STEPN CLA DO 1 1=1,10 CALL STEP TAD WAIT DCA w . ' WT, ISZ V • J M p r.TI* CONTINUE RETURN WAIT,5000 M, 0 END 3. SREV s s s s s s s •s s s s RF.UERRF. MOTOR DIRECTION SUBROUTINE REV COMMON IDIR CLA \IDIR ' <K5 TAD AND CMA AMD DC A TAD AND TAD TAD DC A \IDIR RETURN CHDIP,P! END <K5 CHDIR \ IDIR CK6772 (Kiono CHDIR SSTEP c S T E P MOTOR ONCE S U B R O U T I N E S T E P COMMON I DIP, s , C L A s TAD M D I R •s • AND C K 1 0 0 0 s S7A C L A s JMP ACCST s TAD A C C S T P ' 5 S.7.A C L A r c JMP ACCNTU S JM3 S T E P I RETURN 5 A C C S T , I AC S S7 A s JMP A C C S f s TAD ( K 1 0 0 0 s . CM A c AND \ I D I R s DC A M D I R s TAD CK-P.0 s DC A A C C S T P s TAD ( K 5 7 0 0 s. DCA WTIME s ACCNTU,JMS S T E P 1 s TAD " T I M E s TAD CK1Q0 s DCA NT I ME s TAD " T I M E s B, I AC s sr. A s JMP B ' s ISZ A C C S T P RETURN RETURN s A C C S T P , 0 s •. " T I M E , 0-s STEP1.0 s 6334 s CLA s TAD UT s C, I AC s S2A s JMP C 5 JMP I ST E P I s WT,7000 END 192 SCAN THE CRYSTAL . SUJIROUTI NE RIIN COMMON I D l n , I IAL,I I AH,I I PL,I IFH DIMENSION IA L C 1O O ) , I A H C 1 0 0 ) , I PLC 10 0 ) , I P H C 1 0 0 ) , I Y C 1 0 0 ) READC1,1)1IX,I IY FORMAT C'START IMG POSITION: x='I 3,' • Y=•I 3) READC1,2)M1,I MCI FORMAT C'1 ST SEC: M='I3,' I N C = ' I 2 ) . READC1,3)N2, INC2 FORMATC•2ND SEC: M= * I 3, ' INC='I 2) XY PLOTTER INITIALISATION IXA=fl I YA=G 6503 • • . READC 1,30)1 X M M l . , I YM'TL FORMATC'PLOTTER: IXM'.FL= ' 12, ' IYMUL='I2) READC1,33)P1 FORMATC'SET Y-ZEPO='F6.2) READC1,32)FNAMS FORMATC ' DATA. STORAGE F I L E C6 LETT NAME) = 'A6> WRITEC1,31) FORMATC 'SR AND PLOTTER SET? PRESS CONT') HLT YM!!L= FLOAT C IYMI7L) CALL 00RENC'DTA1",FNAME) KK=100 J= l ' • . CALL DECP,A) I A L C J ) = I I A L IAHCJ)=IIAH , I P L C J ) = I I P L IPHCJ)=IIPH IYCJ)--0 JMS PLOT N=N1 INC=INC1 JMS SCAN N=N2 • ' INC=INC2 JMS SCAN N=N1 ' . \ INC=INC1 JMS SCAN KK=J • y JMS TAPE CALL OCLOSE WRITEC1,10)I I X , I I Y , P , A .. FORMATC'FINISHED: x='I3,' Y='I3,' P='F6.2,' Y='F6.2//) RETURN SCAN,0 DO i>. I = 1, N J=J+1 DO 7 12=1,INC , CLA TAD CK60 6332 CALL STEPN IIY=IIY+INC I Y C J ) ~ I I Y CALL BECP,A) I A L C J ) = I I A L -193 .1AHC J) = 11 AH IPLCJ)=IIPL IPHCJ)=IIPH C IS ? R 0 9 = 1 , PLOT CI IY,P) S 7604 S AMD CK/i '• " . , S srA CLA S JMS PLOT C IF SP 1 1 = I t-'RITE P,.A C IF S P 1 0 = 1 , H A L T * . . " S' • CLA S 7604 -S • AMD <K3 S PCA \ I S " I K C I S " P S . ? n Sfl VRITEC 1, 81 )?,.A, I IY . . . 2 1 F0PMATCPF7.2,14) . • GO T0C95,83,83) ISP 23 CONTINUE S HLT C IF S R 0 5 = 1 , RETURN S 7604 S AMD CK100 S S2A CLA RETURN 2 5 CONTIN"E C AFTER 1 " I 0 READINGS, OUTFUT TO TAPE I F < 1 W 0 - J J 5 , 5 , 4 5 CONTINUE S JMS TAPE 4 CONTINUE 5 JMP I SCAN S TAPE, 0 DO 9 K = 1 * K K 9 WITEC 4 ,8 ) IPLCK) , IFHCK) , IAL (K) , IAHCK) , IYCK) , IIX 8 FORMAiC 6AP) J = 0 S JMP I TAPE S PLOT,0 IYI=II V*IXMUL . P P = < P - P 1 ) * 1 0 . * Y « T . T L IPP=IFIXCFP) IDX=JYI-IXA , . IDY=IPP-IYA IXA=7YI IYA=IPP CALL PLTXYCIDX,IDY) S JMP I PLOT END SM0T2 C MOVE X , Y MOTORS FROM THE KEY BOARD SUBROUTINE MOT?. COMMON I D I R I D I F = 'D S DCA " A I T R E A D C 1 , 1 ) M Y t FORMATC ^ = ' 1 4 ) • R E A D C 1 , P ) N X . 2 FORMATC ' X = • I/i) C MOVE Y MOTOR I F C M Y > 3 , 4 , 5 3 'N=IABSC"-JY) S CLA S TAD CK/lO S J M S . M O V E Y 4 GO TO 6 5 N=NY S CLA S TAD CX60 • o 5 JMS MOWEY C MOVE X MOTOR 6 I F < N X ) 7 , H , 9 7 N=IARSCNX> S . CLA S ' TAD CKP.Ofl S JMS MOVE 8 RETURN 9 N=NX S CLA , ' S TAD CK300 S JMS MO»E RETURN S . '• MOVERS S 6 3 3 « S CLA DO 10 1 = 1 , N C A L L S T E P S WT, 1ST. '-'AIT S ' J M P 10 . CONTINUE S JMP I MOVE S " A I T , 0 S • M 0 V E Y , 0 S 6332 S CLA DO 11 I = 1 , M 11 C A L L STE°N .S JMP I MOVEY END 195 7. SAXES • C DBA'.' A X E S A N D C L A C U L A T E S C A L E F A C T O R S S U 3 R 0 U T I N E A X E S C I Y R , I X R , I Y ' ^ I J I X M U L ) S 6 5 0 0 ' S C L A C L L S 6 5 0 6 I Y D = I Y P * 1 0 * I M A X = 2 0 0 0 IMAY= I 7 0 0 I X M ' J L = I N A X / ' I X R I Y M U L = I MAY /1 Y P <-'RITF.( 1 , 4 ) IX;.;T;L, I Y M U L 4 . F O R M A T ? ' I XM'.1.= * 1 4 , * I Y M U L = ' 1 4 ) 5 C L A S 7 6 0 4 . • S A N D CK1 S SZA C L A S " J M S A X I S R E T U R N S A X I S , 0 M O C E = 0 I P = I Y R I M U L = I Y M " L 1 5 I M C = I R * I M U L / 1 0 DO 1 0 1 = 1 , 1 0 DO 11 J = 1 , I M C S T A D L D D N Y S J M S X L A T F . 1 1 C O N T I N U E DO I P N = l , 3 0 S T A D L U D E * S J M C X L A T E 1 2 C O N T I N U E DO 1 3 .K=l,20 S T A D L D D " X S J M S X L A T E 1 3 C O N T I N U E 1 0 C O N T I N U E I R = I R * I M U L DO 1 4 I = 1 > I P S T A D L U D S Y S J M S * L A T E • 1 4 C O N T I N U E I F ( M O D E ) 1 6 , 1 6 , 1 7 1 6 M 0 D E = 1 I R = I X R I M U L = I X M U L S C L A S T A D L D D E X S D C A L D D N Y . S T A D L U D N Y S D C A L U D E X S • T A D L D D S Y ' S D C A L D D U X S T A D L " D " X S DCA L U D S Y G O T O . 1 5 1 7 C O N T I N U E S J M P I A X I S ' S L D D E X , I P S' L U D E X , 1 1 S L D D U X , * S L U D ' J X , 5 ? L D D N Y , P P S L U D N Y # 2 1 S L D D S Y * 4 2 S L U D S Y , 4 1 S X L A T E , 0 S X A * 6 5 0 1 S JMP X A S 6 5 0 6 S C L A C L L S JMP I X L A T E . S I N C T . 0 • END 196 8. SBEN C BALANCE ELLIPSOMETER ONCE. SUBROUTINE BE CP,A) COMMON IDIR,I IAL,I I AH,I I PL,I IPH DIMENSION ESC20) IBAL=0 31 IBAL=IBAL+1 S CLA S TAD CK3 S JMS SETM S JMS BAL S CLA S TAD (K4O10 S JMS SETM . S JMS BAL GO T0C31,32)I3AL 38 CONTINUE CALL RDACI IAL,I I AH,A) S CLA S 6338 A=360.-A CALL PDA CI I PL,1 1PH,P) RETURN C C BALANCE,SPECIFIED UNIT 5 BAL,0 NSTEP=0 MINF=0 . ' • • W=16 t CONTINUE S JMS SUM SUM1=S :9. CONTINUE 5 JMS SUM SUM8=S C IF MINF=1» APPROACHING A MIN IF<MINF)3,3,4 C - REVERSE MOTOR IF NOT APPROACHING A MIN 3 IF(SUM2-SUM1>5,5,6 6 CALL REV MINF=1 GO TO 1 4 IFCSUM2-SUMU5i7,7 C PASSED THRU MIN? GO TO 7 5 SUM1=SUM2 GO TO 2 7 CONTINUE S JMS SUM SUM1=S , CALL REV J=2*N DO 8 1=1,J CALL STEP 8 NSTEP=NSTEP+1 C BUILD SUM2 UNTIL IT EQUALS SUM1 SUM2=0 DO. 9 1 = 1, N CALL PPHOTOCERR) CALL STEP NSTEP=NSTEP+1 ESC I) = EPP 9 SUM8=SUM2+EPR C DROP LAST AND ADD NEW READING IN CIPCULAR BUFFER 12 1 = 1 10 CALL STEP v NSTEP=MSTEP+1 CALL RPHOTOCF.RR) SUM2 = SUMP.+EPP-ES< I ) F.St I > = F.PP IF<SUM9-snMi)l 1,13*13 1 = 1 + 1 N1=N+1 IF( I-.Ml ) 10, 13, IP GO TO 10 J=MSTEP/2 CALL PEV DO 14 I = 1 , J CALL STEP J M P I n.AL S ETM,0 TAD (K1000 633?. DCA M D I R JMP I SETM SUM* 0 S = 0. DO 40 IS=1»M CALL RPHOTO C ERR) CALL STEP S=S+ERR JMP I SUM END CONCERT READINGS TO ANGLES SUBROUTINE DEGR(IL*IH*ANGLE) CLA TAD I M L DCA ANGL TAD I \ I H DCA ANGH TAD ANGL AND (K17 DCA M A I TAD ANGL RTR RTR AND CK17 DCA \ I A 9 TAD ANGL RTL RTL RAL AND CK17 ' DCA M.A3 TAD ANGH AND CK17 EC A M A 4 " TAD ANGH ' RTR RTR AND <K17 DCA \ I A 5 A =.01*FLOATCIA1)+.1*FL0ATCIA2)+FL0ATCIA3> ANGLE=A+10.=-FLOAT< IA4)+100.*FLOATCIA5> RETURN ANGL* 0 ANGH,0 , END 198 10. SPLTXY C PLOT ONE POINT SUBROUTINE PLTXYCIX,IY> IDX=IX IDY=IY IF CIDX) 5,6,7 5 CONTINUE S PA,TAD LUDWX S JMS XLATE S IS7, \IDX S JMP PA GO TO 6 7 IDX=-IDX S PB,TAD LUDEX S JMS XLATE S ISZ \IDX 5 JMP PB 6 CONTINUE IF ( IDY) 8,9, 10 8 CONTINUE S PC,TAD LUDSY S JMS XLATE S 1SZ \IDY S •JMP PC GO TO 9 10 IDY=-IDY S PD/TAD LUDNY • S ' JMS XLATE S ISZ \IDY S JMP PD 9 CONTINUE S JMS WAIT S 6505 S JMS WAIT S 6503 RETURN S XLATE,0 S XA,6501 S JMP XA S 6506 S CLA CLL S JMP I XLATE • S LUDWX,0 5 S LUDEX,11 S LUDSY,41 S LUDNY,21 S WAIT,0 S XB,6501 S JMP XB / S 6502 S JMP i WAIT END 11. SXYPEN C MOUE THf X V D I . O T T " o r \ | FPOM TTY SUBROUTINE XYPEN PEATH 1 , 1 ) I X , J v 1 ' F0"vtAT< ' XIMC= YINC= ' I4> CALL PLTXYCIX,IY) RETURN END 199 12. SRDA C READ THE SHAFT ENCODER SPECIFIED SUBROUTINE RDA <IL,IH,ANGLE) S CLA CLL s fisos S DCA ANGH S 6304 S DCA ANGL S TAD ANGL S AND C K 1 7 S D C A M A I S TAD ANGL" S RTR S RTR S AND < K 1 7 S. DCA M A 2 S ' TAD ANGL S RTL S RTL-S RAL S A N D - ( K l 7 S DCA \IA3 S TAD ANGH S AND (Kl7 S D C A M A 4 . S TAD ANGH S RTR S RTR S A N D < K 1 7 S DCA M A S A = • 01 * F L O A T ( I A l ) + . l * F L O A T C I A S ) + FLOAT CIA3) ANGLE=A+10.*FLOATCIA4)+100.*FLOATCIA5) S TAD ANGL S ' DCA I M L •S TAD ANGH S DCA I M H RETURN S ANGL,0 S , ANGH,0 END 13. SRPHOT C R E A D T H E E R R O R S I R N A L S U B R O U T I N E R P H O T O <ERR> «5 C L A S T A D (K17 S 6 3 2 3 . S A , 6 3 ? ! f JIVJP A S 6 3 P 4 S CMA .S D C A M E R R ERR = F L O A T . ( I E R R ) RET'TRN .. E N D 

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