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Voltage induced optical waveguide modulators in lithium niobate 1989

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VOLTAGE INDUCED OPTICAL WAVEGUIDE MODULATORS IN LITHIUM NIOBATE by NICOLAS AUGUST FLEMING JAEGER B.Sc.E.E., The U n i v e r s i t y o f the P a c i f i c , 1981 M.A.Sc, The U n i v e r s i t y o f B r i t i s h Columbia, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept t h i s t h e s i s as conforming t o the r e q u i r e d standards THE UNIVERSITY OF BRITISH COLUMBIA March 1989 © N i c o l a s August Fleming Jaeger, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. 1 further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) A b s t r a c t Two ty p e s o f o p t i c a l modulator were s t u d i e d , both o f the v o l t a g e induced o p t i c a l waveguide type f i r s t proposed and demonstrated by Channin i n 1971. An o p t i c a l waveguide was c r e a t e d i n an e l e c t r o o p t i c s u b s t r a t e by a p p l y i n g v o l t a g e between two e l e c t r o d e s d e p o s i t e d on the s u b s t r a t e . Channin used wide e l e c t r o d e spacings which r e s u l t e d i n l a r g e o p e r a t i n g v o l t a g e s b e i n g n e c e s s a r y . The d e v i c e s d i s c u s s e d i n t h i s t h e s i s had much s m a l l e r e l e c t r o d e s p a c i n g s and t h e r e f o r e o p e r a t e d a t much reduced v o l t a g e s . They are o f the p l a n a r and r i d g e t y p e s . In the p l a n a r type the e l e c t r o d e s were d e p o s i t e d on t o p o f a p l a n a r s u b s t r a t e and i n t h e r i d g e type a r i d g e o f e l e c t r o o p t i c m a t e r i a l s e p a r a t e d t h i c k e l e c t r o d e s . The t h e o r y o f o p e r a t i o n f o r both types o f ^ o T " . r o was developed and they were modeled, f a b r i c a t e d , and t e s t e d . N u m e r i c a l l y d e r i v e d r e s u l t s were o b t a i n e d f o r l i g h t w i t h wavelengths o f 442 and 633 nm which showed t h a t the confinement o f t h e l i g h t i n c r e a s e d w i t h i n c r e a s i n g v o l t a g e , d e c r e a s i n g gap width, and d e c r e a s i n g wavelength. The t h e o r y was f u r t h e r developed t o i n v e s t i g a t e the performance o f the d e v i c e as a v o l t a g e - c o n t r o l l e d l i n k i n g waveguide between two o p t i c a l f i b e r s . The optimum c o u p l i n g e f f i c i e n c y , as a f u n c t i o n o f v o l t a g e and i n t e r e l e c t r o d e gap width, from o p t i c a l f i b e r s t o both types o f d e v i c e was c a l c u l a t e d i n terms o f t h e model. Key asp e c t s o f the t h e o r y were — i i i - c o nfirmed by the measurements made on the f a b r i c a t e d d e v i c e s . A p l a n a r device was used as a f r o n t - e n d s w i t c h between a l a s e r and an o p t i c a l f i b e r u s i n g a V-groove etched i n s i l i c o n t o a l i g n the v o l t a g e induced waveguide w i t h the f i b e r . One problem was a decay phenomenon i n which the induced waveguide disappeared over a p e r i o d of time dur i n g which a constant v o l t a g e was a p p l i e d t o the e l e c t r o d e s . T h i s was b e l i e v e d t o be due t o the p h o t o r e f r a c t i v e e f f e c t . I t was found t h a t the device would recover upon the a p p l i c a t i o n o f a f l y - b a c k c y c l e . - i v - T able o f Contents Page A b s t r a c t i i T a b l e o f Contents i v L i s t o f F i g u r e s v i i Acknowledgements x i i Chapter 1 INTRODUCTION 1 Chapter 2 THEORY 12 2.1 I n t r o d u c t i o n 12 2.2 The E l e c t r o o p t i c E f f e c t 14 2.3 The Conformal Mappings 18 2.3.1 The P l a n a r Device 19 2.3.1.1 The Device S t r u c t u r e 19 2.3.1.2 The E l e c t r i c F i e l d D i s t r i b u t i o n s . . . 21 2.3.2 The Ridge Device 2 9 2.3.2.1 The Device S t r u c t u r e . . . . . . . . . 29 2.3.2.2 The E l e c t r i c F i e l d D i s t r i b u t i o n s . . . 33 2.4 The V a r i a t i o n a l Method 41 2.5 The Hermite-Gaussian Approximations 49 2.5.1 The O p t i c a l F i b e r 51 2.5.2 T h e V o l t a g e I n d u c e d O p t i c a l W a v e g u i d e . . . 5 8 2.6 The C o u p l i n g C o e f f i c i e n t 65 Chapter 3 FABRICATION 74 3.1 I n t r o d u c t i o n . 74 —v- 3.2 The VIOWM 75 3.2.1 The P l a n a r VIOWM 77 3.2.2 The Ridge VIOWM 79 3.2.3 C u t t i n g and P o l i s h i n g . . 88 3.3 The S i l i c o n V-grooves 91 3.4 D e v i c e / O p t i c a l F i b e r Alignment 98 Chapter 4 RESULTS 105 4.1 I n t r o d u c t i o n . 105 4.2 The P l a n a r VIOWM 106 4.2.1 C a l c u l a t e d R e s u l t s 107 4.2.2 Measured R e s u l t s 121 4.3 The Ridge VIOWM . . 135 4.3.1 C a l c u l a t e d R e s u l t s 135 4.3.2 Measured R e s u l t s . . . * . 147 4.4 The Fron t - E n d Switch. . . . . . . . . . . . . . 156 4.5 D i s c u s s i o n . . . . . . . . 164 Chapter 5 Summary, C o n c l u s i o n s , and Suggestions f o r F u r t h e r Work . . . . . . . . . . . . . . . . 168 5.1 I n t r o d u c t i o n . , 168 5.2 Summary * 168 5.3 C o n c l u s i o n s 170 5.4 Suggestions f o r F u r t h e r Work 171 Appendix A THE ELECTROOPTIC EFFECT 174 A . l I n t r o d u c t i o n 174 A. 2 The R e l a t i v e D i e l e c t r i c Impermeability Tensor . 175 - v i - A.3 The E l e c t r o o p t i c E f f e c t 176 A. 4 E q u a t i o n s 2.1 and 2.2 179 Appendix B STATIONARY FORMULAS 184 B. l I n t r o d u c t i o n . 184 B.2 E q u a t i o n 2.10 184 B. 3 The P r o p a g a t i o n Constant 186 Appendix C THE COUPLING COEFFICIENT 189 C l I n t r o d u c t i o n 189 C. 2 The Normalized Amplitudes a f and a v 190 C.3 The Numerator o f the C o u p l i n g C o e f f i c i e n t . . . 192 C.4 The Denominator o f the C o u p l i n g C o e f f i c i e n t . . 195 C.5 E q u a t i o n 2.17 197 Ref e r e n c e s . 198 - v i i - L i s t o f F i g u r e s 1.1 The p l a n a r VIOWM i n c r o s s s e c t i o n . Here l i g h t i s p r o p a g a t i n g out o f the page toward the r e a d e r . * . 5 1.2 The r i d g e VIOWM i n c r o s s s e c t i o n . Here l i g h t i s p r o p a g a t i n g out o f the page toward.the r e a d e r . . . 6 1.3 The VIOWM a c t i n g as a l i n k i n g waveguide between two o p t i c a l f i b e r s 10 2.1 The p l a n a r VIOWM s t r u c t u r e 20 2.2 An i n t e r m e d i a t e model o f the p l a n a r VIOWM s t r u c t u r e 22 2.3 The model used t o analyze t he p l a n a r VIOWM s t r u c t u r e . . . . . . . . . . . . . .•* . . . . . . 23 2.4 The W and S-planes 25 2.5 A p l o t o f E y ( y , z ) f o r t he p l a n a r VIOWM . . . . . . 30 2.6 A p l o t o f E z ( y , z ) f o r t he p l a n a r VIOWM 31 2.7 The r i d g e VIOWM s t r u c t u r e 32 2.8 The model used t o analyze t he r i d g e waveguide VIOWM s t r u c t u r e 34 2.9 The (a) £, (b) W, and (c) S-planes 36 2.10 A p l o t o f E y ( y , z ) f o r t he r i d g e VIOWM. Here the p l o t has been c u t a l o n g t he l i n e z = 0 showing the f i e l d f o r z £ 0 on l y 42 2.11 A p l o t o f E z ( y , z ) f o r t he r i d g e VIOWM. Here the p l o t has been cut al o n g t h e l i n e z = 0 showing t h e f i e l d f o r z £ 0 on l y . 43 2.12 The r e f r a c t i v e index d i s t r i b u t i o n o f a f i b e r w i t h a s t e p index p r o f i l e 52 2.13 A Gaussian o p t i c a l f i e l d d i s t r i b u t i o n i n which w y f = w z f 5 6 2.14 A Gaussian o p t i c a l f i e l d d i s t r i b u t i o n i n which Wyf = w z f/2 . 57 - v i i i - 2.15 A Hermite-Gaussian o p t i c a l f i e l d d i s t r i b u t i o n i n which Wyv = w z v 61 2.16 A Hermite-Gaussian o p t i c a l f i e l d d i s t r i b u t i o n i n which Wy^ = w z v/2 62 2.17 A p l o t o f Bv 2 v e r s u s and w z v f o r a VIOWM w i t h a 4 um i n t e r e l e c t r o d e gap w i t h 50.0V a p p l i e d t o the e l e c t r o d e s f o r XQ = 442 nm . . . . . . . . . . 66 2.18 The development o f the o p t i c a l f i e l d f o r a VIOWM w i t h a 4 um i n t e r e l e c t r o d e gap f o r (a) 10.0, (b) 30.0, and (c) 50.0V a p p l i e d t o the e l e c t r o d e s f o r XQ = 442 nm 67 2.19 The i n t e r f a c e between the o p t i c a l f i b e r and the VIOWM. 71 3.1 The i n t e r e l e c t r o d e gap o f a VIOWM. Here the gap i s 4 um wide . 80 3.2 SEM p i c t u r e o f a r i d g e e t c h e d i n LiNb03- The s c a l e o f the upper p i c t u r e i s 5 times t h a t o f the lower p i c t u r e 83 3.3 SEM p i c t u r e showing t h a t t he r i d g e i s about 7.5 um wide. -". . . . . . . . . . . . . . . . . . . . . 84 3.4 P r o f i l o l m e t e r output showing t h a t the h e i g h t o f the r i d g e i s 4 urn 85 3.5 The aluminum e l e c t r o d e s o f a r i d g e VIOWM formed by t h e s e l f - a l i g n e d t echnique 87 3.6 The p o l i s h i n g j i g : t he main body ( r i g h t ) and t h e p o l i s h i n g p l a t e ( l e f t ) . . . . . ; . '* . . . . . . 89 3.7 The endface o f two r i d g e VIOWMs, epo x i e d t o g e t h e r , a f t e r a 1 um alumina p o l i s h 92 3.8 A p l a n a r VIOWM w i t h the ends p o l i s h e d 93 3.9 The p o l i s h e d end o f a p l a n a r VIOWM where the i n t e r e l e c t r o d e gap i s seen t o run p e r p e n d i c u l a r t o t h e endface 94 3.10 The V-groove f a b r i c a t i o n p r o c e s s 99 3.11 The V-groove a r r a y , f i b e r , VIOWM, p i n , and probes d u r i n g t h e alignment procedure . . . . . . . . . . 102 - i x - 3.12 The V-groove a r r a y , f i b e r , VIOWM, probes, and i n p u t o b j e c t i v e a f t e r t he alignment and permanent bonding procedure 103 4.1 A t o p o g r a p h i c a l p l o t o f f o r XQ = 442 nm. . . . 108 4.2 A t o p o g r a p h i c a l p l o t o f w z v f o r KQ = 442 nm. . . . 109 4.3 A t o p o g r a p h i c a l p l o t o f Wy^. f o r XQ = 633 nm. . . . 110 4.4 A t o p o g r a p h i c a l p l o t o f w z v f o r XQ = 633 nm. . . . I l l 4.5 The c o u p l i n g c o e f f i c i e n t T v s . v o l t a g e f o r g = 2 jun where t h e c o u p l i n g at 30 V i s maximized . . . . 114 4.6 The c o u p l i n g c o e f f i c i e n t T v s . v o l t a g e f o r g = 2 Hm where the c o u p l i n g a t 50 V i s maximized . . . . 115 4.7 The c o u p l i n g c o e f f i c i e n t T v s . v o l t a g e f o r g = 4 nm where t h e c o u p l i n g a t 30 V i s maximized . . . . 116 4.8 The c o u p l i n g c o e f f i c i e n t T v s . v o l t a g e f o r g = 4 nm where t h e c o u p l i n g at 50 V i s maximized . . . . 117 4.9 The optimum power t r a n s f e r T^ as a f u n c t i o n o f a p p l i e d v o l t a g e and i n t e r e l e c t r o d e gap width . . . 119 4.10 The power t r a n s f e r T 4 between two o p t i c a l f i b e r s where t h e c o u p l i n g a t 30 V i s maximized. . . . . . 122 4.11 The power t r a n s f e r T 4 between two o p t i c a l f i b e r s where t h e c o u p l i n g a t 50 V i s maximized. . . . . 123 4.12 The b a s i c l a b o r a t o r y apparatus used t o make measurements on VIOWMs . . . . . . . . . ... . . . 125 4.13 The p o l a r i z e d output o f the o p t i c a l f i b e r . . . . . 126 4.14 The output o f a VIOWM w i t h a 4 nm i n t e r e l e c t r o d e gap f o r a 70 V peak-to-peak t r i a n g l e wave a p p l i e d t o t he e l e c t r o d e s . . . 128 4.15 The output o f a VIOWM w i t h a 4 um i n t e r e l e c t r o d e gap f o r a 100 V peak-to-peak t r i a n g l e wave a p p l i e d t o t h e e l e c t r o d e s 12 9 4.16 The output o f a VIOWM w i t h a 4 um i n t e r e l e c t r o d e gap f o r a 130 V peak-to-peak t r i a n g l e wave a p p l i e d t o the e l e c t r o d e s • • • • • 1 3 0 4.17 A comparison o f t h e t h e o r e t i c a l and measured r e s u l t s —x- f o r a p l a n a r d e v i c e 133 4.18 An a l t e r n a t e l a b o r a t o r y apparatus setup. . . . . . 134 4.19 The output o f a l o n g VIOWM w i t h a 70 V peak-to-peak t r i a n g l e wave a p p l i e d t o the e l e c t r o d e s f o r KQ = 633 nm . . . * . . . 136 4.20 A t o p o g r a p h i c a l p l o t o f Wy^. f o r a r i d g e h e i g h t 0.5 times the i n t e r e l e c t r o d e gap width where X Q = 442 nm . . . . . . . . . 139 4.21 A t o p o g r a p h i c a l p l o t o f f o r a r i d g e h e i g h t 1.0 times t h e i n t e r e l e c t r o d e gap width where X0 = 442 nm 140 4.22 A t o p o g r a p h i c a l p l o t o f w v v f o r a r i d g e h e i g h t 1.5 times t h e i n t e r e l e c t r o d e gap width where XQ = 442 rim . . . . . . . . . . . . . . 141 4.23 A t o p o g r a p h i c a l p l o t o f w z v f o r a r i d g e h e i g h t 0.5 times the i n t e r e l e c t r o d e gap width where XQ = 442 nm . . . . . . . . . . . 142 4.24 A t o p o g r a p h i c a l p l o t o f w z v f o r a r i d g e h e i g h t 1.0 times the i n t e r e l e c t r o d e gap width where XQ = 442 nm ;. 143 4.25 A t o p o g r a p h i c a l p l o t o f w z v f o r a r i d g e h e i g h t 1.5 times t h e i n t e r e l e c t r o d e gap width where XQ = 442 nm . . . . . . . . . . . . . . . . . . . . . . 144 4.26 The c o u p l i n g c o e f f i c i e n t T v s . v o l t a g e f o r g = 7 um f o r a h a l f - h e i g h t r i d g e where the c o u p l i n g at 30 V i s maximized. . . . . . . . . . . . . . . . . 145 4*27 The c o u p l i n g c o e f f i c i e n t T v s . v o l t a g e f o r g = 7 um f o r a h a l f - h e i g h t r i d g e where t h e c o u p l i n g at 50 V i s maximized. . . . . . . . . . . . . . . . . 146 4.28 The optimum power t r a n s f e r T •as a f u n c t i o n o f a p p l i e d v o l t a g e and i n t e r e l e c t r o d e gap width f o r a r i d g e 0.5 times the gap width 148 4.29 The optimum power t r a n s f e r T as a f u n c t i o n o f a p p l i e d v o l t a g e and i n t e r e l e c t r o d e gap width f o r a r i d g e 1.0 times the gap width 147 4.30 The optimum power t r a n s f e r as a f u n c t i o n o f a p p l i e d v o l t a g e and i n t e r e l e c t r o d e gap width f o r a r i d g e 1.5 times t h e gap width 150 - x i - 4.31 The power t r a n s f e r T between two o p t i c a l f i b e r s where t h e c o u p l i n g a t 30 V i s maximized f o r the r i d g e d e v i c e 151 4.32 The power t r a n s f e r T 4 between two o p t i c a l f i b e r s where t h e c o u p l i n g a t 50 V i s maximized f o r the r i d g e d e v i c e . . . . 152 4.33 The output o f a h a l f - h e i g h t VIOWM w i t h a 7.5 um i n t e r e l e c t r o d e gap f o r a 100 V peak-to-peak t r i a n g l e wave a p p l i e d t o the e l e c t r o d e s where XQ = 442nm . 154 4.34 A comparison o f t h e t h e o r e t i c a l and measured r e s u l t s f o r an h a l f - h e i g h t r i d g e . 155 4.35 The output o f a h a l f - h e i g h t VIOWM w i t h a 7.5 um i n t e r e l e c t r o d e gap f o r a 120 V peak-to-peak t r i a n g l e wave a p p l i e d t o the e l e c t r o d e s where XQ = 442nm 157 4.36 The output o f a h a l f - h e i g h t VIOWM f o r a ±20 V square wave a p p l i e d t o the e l e c t r o d e s . . . . . . . 158 4.37 The output o f an o p t i c a l f i b e r w i t h a VIOWM a c t i n g as a f r o n t - e n d s w i t c h f o r a ±50 V square wave. Here t h e switc h e d o p t i c a l power. i s - 240 jiW. . . . 160 4.38 The decay o f t h e output o f a VIOWM as a f u n c t i o n - x i i - Acknowledgments T h i s p a r t i s easy. My p a r e n t s have been t h e g r e a t e s t h e l p t o me thoughout t h i s work. Without t h e i r support and guidance I doubt t h a t I ever would have reached t h i s p o i n t . They deserve and r e c e i v e my deepest l o v e , thanks, and a p p r e c i a t i o n . Next t o my p a r e n t s Dr. L. Young deserves my g r a t i t u d e and thanks. Without h i s guidance and support I s u r e l y would never have found an i n t e r e s t i n i n t e g r a t e d o p t i c s , an a r e a o f r e s e a r c h t h a t I t h o r o u g h l y enjoy. T h i s has been a most important c o n t r i b u t i o n t o my l i f e . Now t h i n g s get a l i t t l e more d i f f i c u l t . While s e v e r a l p e o p l e s t a n d out i n my mind as having made my l i f e as a graduate student t o l e r a b l e , even e n j o y a b l e , i t i s h a r d t o say who has c o n t r i b u t e d what. There are those who have d i r e c t l y c o n t r i b u t e d t o my r e s e a r c h e i t h e r through a c t i o n s or t h r o u g h d i s c u s s i o n s . There are a l s o those who, by b e i n g a v a i l a b l e as f r i e n d s , made the h a r d e s t times somewhat e a s i e r and t h e r e b y made i t p o s s i b l e t o keep on going. When I t r y t o s o r t out t h e one group from t h e o t h e r I f i n d t h a t the f r i n g e s b e g i n t o b l u r and many o f the same f a c e s appear i n my minds eye on e i t h e r s i d e . When I t r y t o d i f f e r e n t i a t e t hose who s h o u l d be mentioned from those who sho u l d not I f i n d a department f u l l o f shoulds and an empty column o f s h o u l d n o t s . I can t h i n k o f no one who has not touched me i n some way t h a t has e f f e c t e d who I am and what I have done. - x i i i - Yet i f I thank one and a l l and mention no one i n p a r t i c u l a r I w i l l not be b e i n g t r u e t o those who were most t r u e t o me. I have, t h e r e f o r e , compiled a l i s t o f those people who, f o r one reason o r another w i l l always be p a r t o f my memory o f what graduate s t u d i e s t r u l y was. I am sure t h a t the f a c e s o f t h e s e people w i l l l i n g e r i n my memory l o n g a f t e r the t r i a l s and t r i b u l a t i o n s are f o r g o t t e n , l o n g e r even than the s u b j e c t matter o f my t h e s e s . I cannot l i s t e v e r y t h i n g t h a t each o f t h e f o l l o w i n g people has done f o r me nor can I a t t a c h a weight t o t h e importance o f my i n t e r a c t i o n s w i t h them, I am sure t h a t each one o f them knows, I would j u s t l i k e t o g i v e them a l l my s p e c i a l thanks: I. Abdel-Motaleb, B. A h l b o r n , N. B e a u l i e u , M. Beddose, F. Berry, E. Bonn, D. B o s h i e r , K. Brindamour, J . C l a r k , A. Choi, D. Daines, J . Dindo, S. Dindo, H. Dommel, R. Donaldson, C. Dumont, D. F l e t c h e r , B. G u i d i c i , D. Hui, C. Jaeger, N. Jaeger, R. Jankowski, E. J u l l , H. Kato, M. Kharadly, A. MacKenzie, P. Matz, D. M i c h e l s o n , C. N e s b i t t , C. Passmore, W. Passmore, A. P r i n c e , G. Schmidt, C. S h e f f i e l d , L. Snider, C. Sudhakar, P. Townsley, J . Weber, L. Wedepohl, and M. Wvong. (And t h a t i s the s h o r t l i s t ! ) . To t h o se not l i s t e d above, thank you. - 1 - C h a p t e r 1 INTRODUCTION I n t e g r a t e d o p t i c s i s t h e name g i v e n t o t h e a n a l y s i s , d e s i g n , study, and a p p l i c a t i o n o f d e v i c e s made u s i n g o p t i c a l components t h a t a re f a b r i c a t e d by t h e same methods as, and ar e s i m i l a r i n s c a l e t o , i n t e g r a t e d e l e c t r o n i c components. One o f t h e most commonly used m a t e r i a l s f o r t h e f a b r i c a t i o n o f i n t e g r a t e d o p t i c components and systems i s LiNb03. The k i n d s o f components t h a t can be f a b r i c a t e d i n LiNb03 i n c l u d e waveguides [1,2], modulators [3-7], p o l a r i z a t i o n c o n v e r t e r s [8,9], and frequency c o n v e r t e r s [10,11]. T h i s t h e s i s i s concerned w i t h the f a b r i c a t i o n o f v o l t a g e i n d u c e d o p t i c a l waveguide modulators (VIOWM) i n LiNb03- The v o l t a g e induced o p t i c a l waveguide was f i r s t i n t r o d u c e d by Channin i n 1971 [12]. Channin's d e v i c e had a s e r i o u s l i m i t a t i o n i n t h a t i t needed an minimum o p e r a t i n g v o l t a g e o f 300V. Channin's work was f o l l o w e d by t h a t o f S o r e f e t a l . [13] which a l s o needed such l a r g e v o l t a g e s . The apparent need f o r such l a r g e v o l t a g e s caused t h e v o l t a g e -2- i n d u c e d o p t i c a l waveguide t o be d i s r e g a r d e d and oth e r d e v i c e s t o r e c e i v e more a t t e n t i o n . In t h i s t h e s i s i t i s shown that, t h e v o l t a g e l e v e l s can be reduced by at l e a s t an o r d e r o f magnitude. Baumert e t a l . [14] f a b r i c a t e d and t e s t e d v o l t a g e i n d u c e d o p t i c a l waveguides i n KNbC-3 (which has a l a r g e r e l e c t r o o p t i c c o e f f i c i e n t * than LiNbC>3) w i t h o n - o f f r a t i o s o f 12dB f o r 35V a p p l i e d . These d e v i c e s had l a r g e e l e c t r o d e spacings, l a r g e c a p a c i t a n c e s , and l a r g e o p t i c a l f i e l d d i s t r i b u t i o n s . Our d e v i c e s were a b l e t o achieve e q u a l l y good c h a r a c t e r i s t i c s at lower v o l t a g e s and had much lower c a p a c i t a n c e s . Savatinova e t al... [15] have demonstrated an e l e c t r i c a l l y induced Ti:LiNb03 s t r i p - w a v e g u i d e . In t h e i r d e v i c e the o p t i c a l c h a r a c t e r i s t i c s o f a p l a n a r waveguide, formed by m o d i f y i n g the s u r f a c e l a y e r o f the LiNb03 by the i n - d i f f u s i o n o f T i , a re changed by a p p l y i n g v o l t a g e t o e l e c t r o d e s on t h e s u r f a c e o f the waveguide. L i g h t i s s e l e c t i v e l y c o u p l e d i n t o and out o f a p a r t i c u l a r mode u s i n g p r i s m c o u p l e r s . Regions o f the output i n which t h e r e i s a change i n t h e l i g h t i n t e n s i t y can then be i n t e r r o g a t e d . They r e p o r t a 9V o p e r a t i n g v o l t a g e w i t h a p o s s i b l e 95% modulation. The d e v i c e s s t u d i e d i n t h i s t h e s i s d i f f e r from th e d e v i c e o f Savatinova e t a l . i n t h a t t h e r e i s no The e l e c t r o o p t i c c o e f f i c i e n t i s discussed i n appendix A. -3- d i f f u s i o n and t h a t c o u p l i n g i s ac h i e v e d by the more d i r e c t method o f '"butt-coupling" [16] or " e n d - f i r e - c o u p l i n g " [17] which removes t h e need f o r prisms. Prisms are bu l k y and make f o r an o p t i c a l system needing somewhat p r e c i s e a l i g n m e n t . A l s o i n our d e v i c e s the o p t i c a l f i e l d d i s t r i b u t i o n s o f the guided modes can be c o n t r o l l e d so as t o be w e l l matched t o the f i e l d d i s t r i b u t i o n s i n s i n g l e mode f i b e r s . As s i n g l e mode f i b e r i s b e i n g used more and more f r e q u e n t l y t h i s i s an important advantage as regards c o u p l i n g e f f i c i e n c i e s and p o s s i b l e a p p l i c a t i o n s . The a b i l i t y t o b u t t - c o u p l e t o our d e v i c e w i l l have f u r t h e r advantages as f a r as the s i z e o f the d e v i c e and the p a c k a g i n g o f a d e v i c e w i t h f i b e r p i g t a i l s a t t a c h e d i s concerned. Another d e v i c e t h a t has been i n v e s t i g a t e d by Kawabe e t a l . [18] c o n s i s t s o f a p l a n a r d i f f u s e d T i waveguide i n LiNbC«3 i n which a 420 nm h i g h r i d g e has been etched. When v o l t a g e i s a p p l i e d t o e l e c t r o d e s on e i t h e r s i d e o f the r i d g e t h e o p t i c a l c h a r a c t e r i s t i c s o f t h e s u b s t r a t e are a l t e r e d and th e g u i d e d wave i s r a d i a t e d i n t o the b u l k o f t h e c r y s t a l . Kawabe e t a l . r e p o r t an e x t i n c t i o n c o e f f i c i e n t -19dB f o r ±10V (a 20 v o l t d i f f e r e n c e ) f o r the E z mode on a 3 mm long d e v i c e . We have been ab l e t o achieve s i m i l a r r e s u l t s w i t h out t h e need f o r e i t h e r a T i i n d i f f u s i o n or a r i d g e by u s i n g l o n g d e v i c e s ( i n l o n g d e v i c e s the c o u p l i n g from the i n p u t f i b e r t o t h e output f i b e r due t o b u l k modes i s s m a l l ) . A l s o -4- where t h e throughput o f t h e i r d e v i c e seems t o s a t u r a t e ours c o n t i n u e s t o i n c r e a s e f o r v o l t a g e d i f f e r e n c e s beyond 20V. F u r t h e r problems t h a t were encountered by Kawabe et a l . i n c l u d e d s u r f a c e waveguiding and the p r o p a g a t i o n o f modes f o r which th e v o l t a g e s r e q u i r e d were about 3 times as l a r g e . They d i d not comment on the e f f e c t o f b u l k mode c o u p l i n g i n t h e i r d e v i c e . In t h i s t h e s i s two types o f VIOWM are s t u d i e d ; the f i r s t i s p l a n a r and the second has a r i d g e . The f i r s t i s easy t o make but t h e second has added advantages. I t w i l l be shown t h a t t h e i r performance i s as good or b e t t e r than th e performance o f t h e o t h e r d e v i c e s o f the v o l t a g e induced genre which have been d i s c u s s e d above. They are both s t r i c t l y o f the v o l t a g e induced waveguide type i n t h a t the s u r f a c e o f the s u b s t r a t e has not been m o d i f i e d i n o r d e r t h a t i t be p r e d i s p o s e d t o the g u i d i n g o f o p t i c a l waves. In t h e p l a n a r d e v i c e ( f i g u r e 1.1) two metal e l e c t r o d e s , s e p a r a t e d by a narrow gap, are i s o l a t e d from a s i n g l e c r y s t a l LiNb03 s u b s t r a t e by a t h i n l a y e r o f Si02« The second d e v i c e ( f i g u r e 1.2) d i f f e r s from the p l a n a r i n t h a t a r i d g e i s i n c l u d e d between the e l e c t r o d e s which i d e a l l y a r e o f the same h e i g h t as the r i d g e . A p p l i c a t i o n o f v o l t a g e t o the e l e c t r o d e s e s t a b l i s h e s an e l e c t r i c f i e l d i n the s u b s t r a t e . The s u b s t r a t e i s a c r y s t a l -5- A i r M e t a l E l e c t r o d e \ /- n t e r e l e c t r o d e G a p _7\ M e t a l E l e c t r o d e N / V///////. O p t i c a B u f f e r L a y e r W a v e g u i d i ng R e g i o n o—> z E l e c t r o o p t i c v S u b s t r a t e y F i g u r e 1.1: The p l a n a r VIOWM i n c r o s s s e c t i o n . Here l i g h t i s p r o p a g a t i n g out o f the page toward the reader. -6- A i r E l e c t r o d e \ R i d g e E l ec t r o d e / O p t t e a l B u f f e r L a y e r W a v e g u i d i n g R e g i o n E l e c t r o o p t i c S u b s t r a t e 0—> z F i g u r e 1.2: The r i d g e VIOWM i n c r o s s s e c t i o n . Here l i g h t i s p r o p a g a t i n g out o f the page toward the reader. -7- which e x h i b i t s t he l i n e a r e l e c t r o o p t i c e f f e c t . T h e r e f o r e the e l e c t r i c f i e l d a l t e r s t he r e f r a c t i v e i n d i c e s o f the s u b s t r a t e . The f i e l d i s l a r g e s t i n the r e g i o n s c l o s e t o the e l e c t r o d e edges as w e l l as i n the immediate r e g i o n o f the i n t e r e l e c t r o d e gap hence as the v o l t a g e i s i n c r e a s e d the o p t i c a l wave i s i n c r e a s i n g l y c o n f i n e d t o t h a t r e g i o n . * The SiC-2 l a y e r i s i n c l u d e d t o a c t as an o p t i c a l b u f f e r l a y e r i s o l a t i n g t h e o p t i c a l f i e l d from the metal e l e c t r o d e s . While t h e o p e r a t i o n o f the d e v i c e i s c o n c e p t u a l l y s t r a i g h t f o r ward t o study and d e s i g n VIOWMs i t was nece s s a r y t o develop a method o f a n a l y s i s . Our method c o n s i s t e d o f u s i n g conformal mapping tech n i q u e s t o c a l c u l a t e the r e f r a c t i v e index d i s t r i b u t i o n e s t a b l i s h e d i n a VIOWM f o r a p a r t i c u l a r v o l t a g e . Then f o l l o w i n g t he approach used by Marcuse f o r a n a l y z i n g s t e p index f i b e r s [19], we used a v a r i a t i o n a l t e c h n i q u e t o f i n d t he o p t i c a l f i e l d d i s t r i b u t i o n s f o r waveguides w i t h t h e s e r e f r a c t i v e index d i s t r i b u t i o n s . In t h i s method a p p r o p r i a t e t r i a l f u n c t i o n s w i t h v a r i a b l e parameters are assumed t o approximate the o p t i c a l f i e l d d i s t r i b u t i o n s . We have used Hermite-Gaussian See appendix A f o r a d i s c u s s i o n of 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 . As w i l l be seen i n s e c t i o n 2.2 the v o l t a g e must have the c o r r e c t s i g n , r e l a t i v e t o the c r y s t a l axes, f o r the r e f r a c t i v e index t o i n c r e a s e otherwise the r e f r a c t i v e index w i l l be decreased and l i g h t w i l l be r a d i a t e d out of t h a t r e g i o n . -8- f u n c t i o n s as the t r i a l f u n c t i o n s . The v a r i a b l e parameters are determined f o r these f u n c t i o n s by making use o f the s t a t i o n a r y n a t u r e o f the e i g e n v a l u e s o f the s c a l a r wave e q u a t i o n . The width parameters v a r y w i t h v o l t a g e a l l o w i n g one t o p r e d i c t t h e development o f o p t i c a l f i e l d d i s t r i b u t i o n s w i t h i n c r e a s i n g v o l t a g e . When d e s i g n i n g VIOWMs i n LiNb03 the c o n t r o l l a b l e parameters i n c l u d e : t he s u b s t r a t e o r i e n t a t i o n , the wavelength o f the l i g h t t o be guided, the o p t i c a l b u f f e r l a y e r m a t e r i a l , t he o p t i c a l b u f f e r l a y e r t h i c k n e s s , t he i n t e r e l e c t r o d e gap o r i e n t a t i o n , the i n t e r e l e c t r o d e gap width, t h e o p e r a t i n g v o l t a g e , the r i d g e h e i g h t , and when b e i n g used w i t h f i b e r s t he f i b e r l o c a t i o n and i f p o s s i b l e the w i d t h parameters o f the f i b e r mode. One a p p l i c a t i o n o f a VIOWM i s as a v o l t a g e c o n t r o l l e d l i n k i n g waveguide between two o p t i c a l f i b e r s ( f i g u r e 1.3) i n which o p t i c a l power i s t r a n s f e r r e d from one f i b e r t o t h e oth e r v i a t h e VIOWM. In t h i s a p p l i c a t i o n i t i s not s u f f i c i e n t t o know how t h e o p t i c a l f i e l d d i s t r i b u t i o n i n the VIOWM develops o n l y but one must a l s o know how the development e f f e c t s t he c o u p l i n g o f l i g h t t o and from the f i b e r s . A c l o s e d form e x p r e s s i o n f o r the c o u p l i n g c o e f f i c i e n t f o r b u t t - c o u p l i n g between an o p t i c a l f i b e r and a VIOWM i s developed i n t h i s t h e s i s . Thus a complete model f o r t h e study and d e s i g n o f VIOWMs i n t h i s a p p l i c a t i o n has been developed. The model al l o w e d us t o p r e d i c t t h e modus -9- o p e r a n d i o f the VIOWM as a l i n k i n g waveguide f o r s e v e r a l a p p l i c a t i o n s . F o r i n s t a n c e i t i s shown t h a t the VIOWM can be used as a d i g i t a l s w i t c h or as a l i n e a r modulator. When b e i n g used as a d i g i t a l s w i t c h t he change i n o p t i c a l power t r a n s f e r can be maximized f o r a g i v e n v o l t a g e l e v e l or the s e n s i t i v i t y o f the d e v i c e t o s m a l l v a r i a t i o n s i n i n p u t v o l t a g e l e v e l may be minimized. The t h e o r y o f o p e r a t i o n o f VIOWMs was confirmed by f a b r i c a t i n g and t e s t i n g s e v e r a l d e v i c e s . These were o f both t h e p l a n a r and r i d g e t y p e s . The method o f f a b r i c a t i o n i s d e s c r i b e d i n cha p t e r 3. A l s o a r r a y s o f V-grooves [20] were e t c h e d i n s i l i c o n s u b s t r a t e s f o r the alignment o f o p t i c a l f i b e r s w i t h VIOWMs i n a f l i p - c h i p arrangement [21]. T h e i r f a b r i c a t i o n i s d e s c r i b e d as w e l l , as i s the alignment t e c h n i q u e and t h e method o f e u t e c t i c bonding between e l e c t r o d e s on t h e V-groove s u b s t r a t e s and on t h e e l e c t r o d e s on t h e VIOWMs. Te s t s were made on the d e v i c e s . T h e i r v o l t a g e response was measured. The c o u p l i n g c o e f f i c i e n t s between VIOWMs and o p t i c a l f i b e r s were measured. The d e v i c e s were t e s t e d as d i g i t a l modulators. I t was shown t h a t the output power c o u l d be modulated i n a l i n e a r f a s h i o n by the a p p l i e d v o l t a g e and t h a t the c o u p l i n g e f f i c i e n c y c o u l d be op t i m i z e d f o r a p a r t i c u l a r v o l t a g e f o r a p a r t i c u l a r l o c a t i o n o f the i n p u t o p t i c a l f i e l d d i s t r i b u t i o n , both these phenomena were p r e d i c t e d by t h e t h e o r y . Another p r e d i c t i o n was t h a t a ^ 0 . -11- •"turn-on" v o l t a g e e x i s t s f o r r i d g e waveguides at which l i g h t r a p i d l y becomes c o n f i n e d t o the r i d g e . T h i s e f f e c t was a l s o e x p e r i m e n t a l l y confirmed. A d e v i c e w i t h an o p t i c a l f i b e r a t t a c h e d , u s i n g an a r r a y o f s i l i c o n V-grooves, was a l s o f a b r i c a t e d and t e s t e d . T h i s d e v i c e was e n v i s i o n e d t o be used as a f r o n t - e n d s w i t c h f o r the o p t i c a l system o f an o p t i c a l image r e c o r d e r such as the FIRE 9000 produced by the l o c a l company MacDonald D e t t w i l e r and A s s o c i a t e s o f Richmond, B r i t i s h Columbia, Canada. A d.c. decay phenomenon was observed which i s b e l i e v e d t o be due t o t h e p h o t o r e f r a c t i v e e f f e c t [22]. I t was found t h a t t h i s c o u l d be compensated f o r by the a p p l i c a t i o n o f a n e g a t i v e w f l y - b a c k " v o l t a g e t o the e l e c t r o d e s o f the VIOWM. The a p p l i c a t i o n o f a n e g a t i v e v o l t a g e was a l s o found t o be u s e f u l on s h o r t d e v i c e s as i t c r e a t e d a s o r t o f w a n t i - waveguide" out o f which l i g h t i n b u l k modes was r e f r a c t e d enhancing t h e e x t i n c t i o n r a t i o o f the d e v i c e . -12- C h a p t e r 2 THEORY 2.1 I n t r o d u c t i o n The t h e o r y o f t h e VIOWM i s c o n t a i n e d i n f i v e s e c t i o n s o f t h i s c h a p t e r e n t i t l e d : The E l e c t r o o p t i c E f f e c t , The Conformal Mappings, The V a r i a t i o n a l Method, The Hermite-Gaussian Approximations, and The Coupling C o e f f i c i e n t . In t h e s e c t i o n on the e l e c t r o o p t i c e f f e c t e x p r e s s i o n s f o r t h e change i n the r e f r a c t i v e index are d e r i v e d and f o r t h e r o t a t i o n o f t h e i n d i c a t r i x about the X - a x i s o f the s u b s t r a t e . Both are g i v e n i n terms o f the a p p l i e d e l e c t r i c In the coordinate system used here the x, y, and z-axes are coincident with the X, Y, and Z-axes of the LiNbC>3 substrate, r e s p e c t i v e l y . They correspond to the x^, X £ , and x3-ay.es i n appendix B of Nye [23] pp. 276-288, i . e . , x| | x i | | a, y| |x2-Lxi, and z| IX3I |c. The r e s u l t of t h i s i s that the x-direction becomes the d i r e c t i o n of propagation therefore the symbol, p, w i l l designate the propagation constant i n the x - d i r e c t i o n . -13- f i e l d . I t i s assumed t h a t Y-cut LiNbC>3 i s the e l e c t r o o p t i c s u b s t r a t e . T h i s corresponds t o the o r i e n t a t i o n t h a t was used i n t h e f a b r i c a t i o n and t e s t i n g o f the ex p e r i m e n t a l d e v i c e s d e s c r i b e d i n the f o l l o w i n g c h a p t e r s . The a p p l i e d e l e c t r i c f i e l d i s assumed t o have Y and Z components o n l y . A g a i n t h i s corresponds t o the case i n our experiments. The e l e c t r i c f i e l d d i s t r i b u t i o n i n s i d e the e l e c t r o o p t i c s u b s t r a t e needs t o be a n a l y z e d i n order t o p r e d i c t the b e h a v i o r o f t h e d e v i c e s a t v a r i o u s v o l t a g e s . In the conformal mappings s e c t i o n both the p l a n a r and r i d g e d e v i c e t y p e s are modeled and the e l e c t r i c f i e l d d i s t r i b u t i o n s e s t a b l i s h e d i n them are d e r i v e d . In t h e v a r i a t i o n a l method s e c t i o n a technique f o r ap p r o x i m a t i n g the e l e c t r o m a g n e t i c f i e l d d i s t r i b u t i o n o f an o p t i c a l wave p r o p a g a t i n g i n a v o l t a g e induced o p t i c a l waveguide, u s i n g t h e c a l c u l u s o f v a r i a t i o n s , i s p r e s e n t e d . In t h e s e c t i o n on t h e Hermite-Gaussian approximation t h e reasons f o r c h o o s i n g t h e s e t r i a l f u n c t i o n s are p r e s e n t e d . A l s o i n c l u d e d i n t h i s s e c t i o n i s a d i s c u s s i o n j u s t i f y i n g t h e use o f a Gaussian approximation f o r the o p t i c a l f i e l d d i s t r i b u t i o n i n a single-mode o p t i c a l f i b e r . In t h e c o u p l i n g c o e f f i c i e n t s e c t i o n a c l o s e d form a n a l y t i c e x p r e s s i o n f o r the c o u p l i n g c o e f f i c i e n t i s d e r i v e d f o r c o u p l i n g between an o p t i c a l f i b e r and a v o l t a g e induced o p t i c a l waveguide. T h i s e x p r e s s i o n i s d e r i v e d u s i n g the approximate o p t i c a l f i e l d d i s t r i b u t i o n s assumed f o r both the -14- o p t i c a l f i b e r and the v o l t a g e induced o p t i c a l waveguide. I t i s assumed t h a t t h e ends o f the two waveguides can be p o l i s h e d t o such a h i g h degree t h a t they can be brought i n t o i n t i m a t e c o n t a c t and t h a t power i s ""butt-coupled" from one waveguide t o the o t h e r . 2.2 T h e E l e c t r o o p t i c E f f e c t In t h i s s e c t i o n we w i l l o b t a i n the r e l a t i o n s h i p between the change i n t h e r e f r a c t i v e index and the a p p l i e d e l e c t r i c f i e l d . Both the magnitudes o f the p r i n c i p a l axes and the o r i e n t a t i o n o f t h e i n d i c a t r i x are a f f e c t e d . The change i n o r i e n t a t i o n i s shown t o be s m a l l f o r the f i e l d s used and can be i g n o r e d . However the change i n the magnitude o f the p r i n c i p a l axes c o n t a i n s a q u a d r a t i c term t h a t may be as much as 10% o f t h e change due t o the l i n e a r term, f o r t h e f i e l d s used, and i s r e t a i n e d . The mathematical formalism f o r the e l e c t r o o p t i c e f f e c t i s i n c l u d e d i n appendix A. The l a r g e s t l i n e a r e l e c t r o o p t i c c o e f f i c i e n t o f LiNb03 i s r33 [24] and i t i s c l e a r l y d e s i r a b l e t h a t advantage be t aken o f i t . Y-cut LiNbC"3 was chosen t o take advantage of r33. The d e v i c e s were f a b r i c a t e d so t h a t the i n t e r e l e c t r o d e gap runs p a r a l l e l t o the X - a x i s . T h i s ensured t h a t the d i s t r i b u t i o n o f the e l e c t r i c f i e l d component p a r a l l e l t o the Z-axis o f the s u b s t r a t e , E z , was symmetric w i t h r e s p e c t t o -15- t h e XY-plane. The d i s t r i b u t i o n o f the e l e c t r i c f i e l d component p a r a l l e l t o the Y-axis o f the s u b s t r a t e , E v , was anti- s y m m e t r i c t o the XY-plane. In what f o l l o w s an equ a t i o n g i v i n g t h e change i n the e x t r a o r d i n a r y r e f r a c t i v e index, An e, i n terms o f the two e l e c t r i c f i e l d components Ey and E z i s d e r i v e d . The g e n e r a l e q u a t i o n r e l a t i n g the change i n the o p t i c a l i n d i c a t r i x t o an e l e c t r i c f i e l d i s g i v e n i n appendix A. When a f i e l d i s a p p l i e d t o an e l e c t r o o p t i c c r y s t a l a change i s i n c u r r e d i n t h e i m p e r m e a b i l i t y . One can c o n s t r u c t the new i m p e r m e a b i l i t y t e n s o r i n terms o f 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 and the e l e c t r i c f i e l d components. When the s u b s t r a t e i s LiNb03 and the e l e c t r i c f i e l d i s e n t i r e l y i n the y z - p l a n e t h e n B4 = r^^y i s t n e o n l y nonzero o f f - d i a g o n a l element. B4 w i l l have an e f f e c t o n l y i n t h e yz- p l a n e and t h e d i s t o r t i o n i n the i n d i c a t r i x , i n t h i s p l a n e , can be found u s i n g the Mohr C i r c l e * . The d i s t o r t i o n w i l l , i n g e n e r a l , c o n s i s t o f changes i n both the magnitudes and o r i e n t a t i o n s o f the p r i n c i p a l axes. The change i n o r i e n t a t i o n can be expressed as an angle o f r o t a t i o n , ©, w i t h r e s p e c t t o the o r i g i n a l axes. F o l l o w i n g the procedure i n Nye** one can o b t a i n Nye [23] pp. 43-47. ** I b i d . -16- 2r , , E 42 y tan(2e) = (2.1) - 2 - 2 n - n + r _ , E - r o r > E - r » _ E e o 33 z 22 y 23 z w h i c h i s d e r i v e d i n a p p e n d i x A . S i n c e t h e e l e c t r o o p t i c c o e f f i c i e n t s a r e s m a l l t h e l a s t t h r e e t e r m s i n t h e d e n o m i n a t o r c a n be i g n o r e d . We c a n a p p r o x i m a t e t h e a n g l e o f r o t a t i o n o f t h e i n d i c a t r i x i n t h e Y Z - p l a n e b y 42 y -2 -2 n - n e o The h i g h e s t v a l u e s f o r E y were i n t h e r e g i o n s n e a r t h e e l e c t r o d e e d g e s f o r t h e t h i n e l e c t r o d e s a n d n e a r t h e c o r n e r s o f t h e t h i c k e l e c t r o d e s . The peak v a l u e s o f E y a r e on t h e o r d e r o f 10^ V/m a n d w i l l c a u s e a r o t a t i o n o f l e s s t h a n 1°. T h e r e f o r e t h e r o t a t i o n i s i g n o r e d . A i l t h e g u i d e d o p t i c a l modes o f t h i s d e v i c e w i l l p r o p a g a t e i n t h e x - d i r e c t i o n . However t h e y w i l l n o t h a v e t h e same p r o p a g a t i o n c o n s t a n t . T h i s i s due t o t h e l a r g e a n i s t r o p y o f LiNbC>3. The g u i d e d modes c a n be d i v i d e d i n t o two b a s i c t y p e s TE a n d TM l i k e modes. A TE l i k e mode i s one i n w h i c h t h e e l e c t r i c f i e l d o f t h e mode i s p r e d o m i n a n t l y p o l a r i z e d p a r a l l e l t o t h e Z - a x i s o f t h e s u b s t r a t e a n d a TM l i k e mode i s one i n w h i c h t h e e l e c t r i c f i e l d i s p r e d o m i n a n t l y p o l a r i z e d i n t h e X Y - p l a n e . T h e s e modes a r e a n a l o g o u s t o t h e E X p q a n d E V p q modes o f r e c t a n g u l a r c h a n n e l -17- waveguides d e s c r i b e d by M a r c a t i l i [25] (where t h e s u p e r s c r i p t s x and y r e f e r t o the c o o r d i n a t e system used by M a r c a t i l i ) . Due t o the d i f f e r e n c e i n 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 r 3 3 = 30.8xlO" 1 2m/V and r 2 3 = 8.6xlO" 1 2m/V [24] the change i n t h e r e f r a c t i v e index "seen" by a TE l i k e mode w i l l be approximately 3 times t h a t seen by a TM l i k e mode. The lowest o r d e r TE l i k e mode w i l l be the most important o f the modes p r o p a g a t i n g i n the VIOWM f o r the v o l t a g e s used. T h e r e f o r e i t w i l l be assumed throughout t h i s t h e s i s t h a t t he mode b e i n g c o u p l e d i n t o , o r out of, and p r o p a g a t i n g i n the VIOWM i s a TE l i k e mode. P o l a r i z a t i o n - p r e s e r v i n g f i b e r c o u l d be used t o ensure t h a t t h i s i s the case. F o r TE l i k e modes the waveguide c r e a t e d by the a p p l i c a t i o n o f v o l t a g e t o the e l e c t r o d e s o f a VIOWM i n v o l v e s t h e e x t r a o r d i n a r y r e f r a c t i v e index, n e , o f LiNb03. I t i s the change i n t h e c o r r e s p o n d i n g p r i n c i p a l a x i s o f t h e o p t i c a l i n d i c a t r i x due t o the a p p l i e d e l e c t r i c f i e l d t h a t causes the waveguide t o be c r e a t e d . The change i n the p r i n c i p a l a x i s c o n s i s t s o f both a change i n magnitude and a change i n o r i e n t a t i o n . However, s i n c e the r o t a t i o n i s s m a l l we have i g n o r e d i t and have t r e a t e d t he change as a change i n t h e magnitude o f e x t r a o r d i n a r y r e f r a c t i v e index o n l y . The change An e(y,z) i s g i v e n by l e r 33 Ez<y' z ) n e { r 4 2 E y ( y ' 2 ) } 2 (2.2) A n e ( y , z) - - 2 -18- which i s a l s o d e r i v e d i n appendix A. The f i r s t term on the r i g h t hand s i d e o f e q u a t i o n 2.2 i s the change i n the z- d i r e c t i o n alone and t h e second term i s the added change i n th e magnitude o f the p r i n c i p a l a x i s . We have r e t a i n e d the second term because i t may be as much as 10% o f the f i r s t t erm i n t h e r e g i o n s near the e l e c t r o d e edges or c o r n e r s even though t h e r o t a t i o n i s s m a l l . The conformal mappings t h a t were used t o f i n d t he a p p l i e d f i e l d d i s t r i b u t i o n s , E z ( y , z ) and E y ( y , z ) , are d i s c u s s e d i n the f o l l o w i n g s u b s e c t i o n s f o r both t h e p l a n a r and r i d g e d e v i c e s . F i r s t , however, we must d e r i v e the L a p l a c i a n f o r the o r i e n t a t i o n o f the s u b s t r a t e used by a t r a n s f o r m a t i o n o f v a r i a b l e s . T h i s i s necessary due t o the a n i s o t r o p y o f LiNbC>3. The same t r a n s f o r m a t i o n i s good f o r both t h e p l a n a r and r i d g e d e v i c e s . P o i s s o n ' s e q u a t i o n w i t h r e s p e c t t o the YZ-plane o f LiNbC>3 i s g i v e n by 2.3 The C o n f o r m a l M a p p i n g s V D = e + e = 0 y z which by the t r a n s f o r m a t i o n o f v a r i a b l e s -19- ' e ^ 1/2 y' = y (2.3a) e v y ) and z' = z , (2.3b) where e v and e z are the Y and Z-components (e a and e c) o f the d i a g o n a l i z e d p e r m i t t i v i t y t e n s o r r e s p e c t i v e l y , can be w r i t t e n as t h e L a p l a c i a n 2 2 d v 3 v + = o 2 2 dy' dz' (2.4) 2.3.1 The Planar Device 2.3.1.1 The Device Structure The p l a n a r VIOWM ( f i g u r e 2.1) c o n s i s t s o f an e l e c t r o o p t i c s u b s t r a t e w i t h two t h i n metal e l e c t r o d e s , s e p a r a t e d by a narrow i n t e r e l e c t r o d e gap, d e p o s i t e d on one o f i t s f a c e s . A t h i n o p t i c a l b u f f e r l a y e r i s i n c l u d e d between t h e s u b s t r a t e and the e l e c t r o d e s t o reduce t r a n s m i s s i o n l o s s e s due t o i n t e r a c t i o n s between the o p t i c a l f i e l d and t h e e l e c t r o d e s . The metal e l e c t r o d e s and the o p t i c a l b u f f e r l a y e r are t h i n w i t h r e s p e c t t o the i n t e r e l e c t r o d e gap width and t h e i r -20- Me t a I Electrode ntereIect rode Gap Metal Electrode _r<. . .̂„„„.™,v...vw,..„„™,w™.„ Opt i ca Buffer Layer X 0—> z y Substrate F i g u r e 2.1: The p l a n a r VIOWM s t r u c t u r e . -21- t h i c k n e s s may be i g n o r e d i n modeling the d e v i c e . The a c t u a l s t r u c t u r e was r e p l a c e d by the s t r u c t u r e shown i n f i g u r e 2.2 i n which two c o p l a n a r e l e c t r o d e s o f i n f i n i t e s i m a l t h i c k n e s s are s i t u a t e d d i r e c t l y atop the s u b s t r a t e . T h i s s t r u c t u r e may i n t u r n be analyzed, and the e l e c t r i c f i e l d d i s t r i b u t i o n i n the s u b s t r a t e c a l c u l a t e d , by s u b s t i t u t i n g one i n which the e l e c t r o d e s are embedded i n an a n i s o t r o p i c s u b s t r a t e as i n f i g u r e 2.3. T h i s second s u b s t i t u t i o n w i l l g i v e exact r e s u l t s f o r t h e a n i s o t r o p i c h a l f o f the s t r u c t u r e as i t has e q u i v a l e n t boundary c o n d i t i o n s t o those o f the a n i s o t r o p i c r e g i o n d e p i c t e d i n f i g u r e 2.2. The boundary c o n d i t i o n f o r the normal component o f the a p p l i e d f i e l d i n the i n t e r e l e c t r o d e gap i n the p l a ne o f the e l e c t r o d e s i s E y = o . y=0 f o r b o t h s t r u c t u r e s . 2.3.1.2 The E l e c t r i c F i e l d Distributions The conformal mapping c o n s i s t s o f mapping the s t r i p 0 < u < JT i n the W-plane t o the r e g i o n e x t e r i o r t o a hyperbola, w i t h i t s f o c i a t ±g/2, i n the S-plane. T h i s mapping i s a c h i e v e d by the mapping S = -(g/2)cos(W).* F i g u r e 2.4 Ramo et a l . [26] s e c t i o n 7.6 pp. 331-340. - 2 2 - X A i r V y •> z E l e c t r o d e n t e r e I e c t r o d e G a p E l e c t r o d e A n i s o t r o p i c S u b s t r a t e F i g u r e 2.2: An i n t e r m e d i a t e model o f t h e p l a n a r VIOWM s t r u c t u r e . -23- Anisotropic Material E l e c t r o d e E l e c t r o d e n t ere Iect rode Gap X o — > z Anisotropic  y Material F i g u r e 2.3: The model used t o analyze the p l a n a r VIOWM s t r u c t u r e . -24- d e p i c t s the W and S-planes. The c u r r e n t problem i s concerned w i t h the l i m i t i n g case i n which the f o c i are c o i n c i d e n t w i t h t h e ends o f the e l e c t r o d e s r e s u l t i n g i n the e l e c t r o d e s b e i n g r e p r e s e n t e d by the l i n e s |o| 2: g/2. I f the l i n e s u = 0 and u = JC are assumed t o be e l e c t r o d e s at ground and V 0 v o l t s r e s p e c t i v e l y then the p o t e n t i a l f u n c t i o n i n the s t r i p i s g i v e n by v o V ( u , v ) = u . K The e l e c t r i c f i e l d components i n the S-plane, E c and E^, are g i v e n i n terms o f the p o t e n t i a l f u n c t i o n by o Jtdo V du V dv o o cs rukn jcdo where the Cauchy-Riemann c o n d i t i o n du/da> = — dv/do has been used. Here du/dc and dv/do, are the r e a l and imaginary p a r t s o f dW/dS, dW du 3v dS do da r e s p e c t i v e l y . d V ( u , v ) E = - a do and d V ( u , v ) -25- V 1\ W - p I a n e ..1 ^ S - p l a n e g 2 g 2 F i g u r e 2.4: The W and S-planes. -26- The i n v e r s e mapping o f S = -(g/2)cos(W), g i v i n g W as a f u n c t i o n o f S, i s W = n - c o s - 1 ( 2 S / g ) . T h e r e f o r e t h e d e r i v a t i v e , dW/dS , equals { ( g / 2 ) 2 - S 2 } " 1 / 2 or dw ds I 2 ) 2 2 — o + oo — ±2ooo 1/2 which a f t e r rearrangement g i v e s V cos o _ tan 2 -1 2am 2 2 2 { (g/2) - o + os j v 2 , 2 2 — a + O B , - 2 2 + ia a 1/4 (2.5a) and V sin' o _ tan 2 -1 2<T09 2 2 2 (g/2) - a + a ) I 2 ) -,2 2 2 O + oo ± A  2 2 + 4o OB 1/4 (2.5b) O b v i o u s l y the p o s i t i v e c - a x i s s h o u l d correspond t o the p o s i t i v e z ' - a x i s . T h e r e f o r e i f we wish the p o s i t i v e ©-axis t o correspond t o the p o s i t i v e y ' - a x i s t h i s f i x e s the p o s i t i v e x - a x i s t o be o r i e n t e d so as t o p o i n t i n t o the paper -27- i n f i g u r e 2.2. F i n a l l y on s u b s t i t u t i o n o f equations 2.3a and b i n t o 2.5a and b one o b t a i n s V cos o — t a n 2 -1 1/2 2 z ( e z / e y ) y I ( g / 2 ) 2 - z 2 + ( e z / e y ) y 2 ) r / \ 2 < - •> 2 g e c 2 z 2 , 2 z 2 K — - z + — y + 4z — y • , 2 , 1 «y J v. y ' 1/4 (2.6a) and V s i n o — t a n 2 -1 1/2 2z(e /e ) y z y ( g / 2 ) 2 - z 2 + ( e z / e y ) y 2 J 2 2 < - > g e e 2 z 2 . 2 z 2 — - z + — y + 4z y - , 2 , 1 £ y \ 1 «y J 1/4 (2.6b) S p u t t e r e d s i l i c o n d i o x i d e has a much s m a l l e r r e f r a c t i v e i n d e x t h a n LiNbC>3 (approximately 1.46 as compared t o 2.20 f o r n e a t XQ = 633nm). F o r us i t was a good c h o i c e as the o p t i c a l b u f f e r l a y e r m a t e r i a l as i t was a v a i l a b l e i n the s o l i d s t a t e l a b o r a t o r y here at U.B.C. and i s easy t o work w i t h . U s i n g i t we can approximate the decay constant f o r t h e evanescent f i e l d , normal t o the s u r f a c e by { ( 2 r c A 0 ) ( 2 . 2 0 2 - 1.46 2) 1/ 2} which i s about 16/um t h i s means t h a t t h e f i e l d decays t o about 4% o f i t s v a l u e at the s u r f a c e i n 2000A. An even b e t t e r r e s u l t i s o b t a i n e d at X D = -28- 442nm. However, at low f r e q u e n c i e s the d i e l e c t r i c constant o f SiC-2 i s s i g n i f i c a n t l y lower than e i t h e r Ky = ey/e Q or K z = e z/e 0 f o r t h e LiNbC-3 (approximately 4 as compared t o 43 and 28 r e s p e c t i v e l y ) . T h i s d i f f e r e n c e i n the r e l a t i v e p e r m i t t i v i t i e s o f the two m a t e r i a l s a f f e c t s the e l e c t r i c f i e l d d i s t r i b u t i o n i n the d e v i c e . I t reduces the a p p l i e d f i e l d i n t h e h i g h p e r m i t t i v i t y r e g i o n as compared t o the case where the e l e c t r o d e s are p l a c e d d i r e c t l y i n c o n t a c t w i t h t h e LiNbC-3. T h e r e f o r e i t was important t h a t the b u f f e r l a y e r be kept t h i n so t h a t the e f f e c t was s m a l l . In the c a l c u l a t i o n s made f o r t h i s t h e s i s i t was assumed t h a t the b u f f e r l a y e r t h i c k n e s s was 5% o f the i n t e r e l e c t r o d e gap width- and was t r e a t e d as a mere o f f s e t from the e l e c t r o d e s i n c a l c u l a t i n g the e l e c t r i c f i e l d d i s t r i b u t i o n s . In o t h e r words th e e l e c t r i c f i e l d d i s t r i b u t i o n s were c a l c u l a t e d as though t h e e l e c t r o d e s were d i r e c t l y i n c o n t a c t w i t h t h e s u b s t r a t e t h e n t h e s u r f a c e o f the induced waveguide was t a k e n t o be at a d i s t a n c e o f 5% o f the gap width away from t h e e l e c t r o d e s and i n t o the s u b s t r a t e . The a p p l i e d f i e l d d i s t r i b u t i o n s Ey(y,z) and E z ( y , z ) are p l o t t e d i n f i g u r e s 2.5 and 2.6 r e s p e c t i v e l y . In both o f t h e s e p l o t s i t i s assumed t h a t t h e v o l t a g e , V, a p p l i e d a c r o s s the e l e c t r o d e s i s n e g a t i v e * so t h a t E z ( y , z ) i s p o s i t i v e . Although the y- The IRE Standards on P i e z o e l e c t r i c C r y s t a l s i s used f o r t h e s i g n c o n v e n t i o n here. See L i n e s and G l a s s [27] p. 147 . -2 9- component o f the a p p l i e d f i e l d i s seen t o be o f the same or d e r o f magnitude as the z-component i n the r e g i o n s near o t h e e l e c t r o d e edges the term ( r 4 2 E y ( y ^ 2 ) } i n e q u a t i o n 2.2 i s v e r y s m a l l f o r the f i e l d s t r e n g t h s used. 2.3.2 The Ridge Device 2.3.2.1 The Device Structure While th e p l a n a r VIOWM works t o c r e a t e a l i n k i n g waveguide between the i n p u t and the output f a c e s o f t h e d e v i c e i t i s obvious, by i n s p e c t i n g the s t r u c t u r e shown i n f i g u r e 2.2, t h a t i t does not make the b e s t use o f the l a r g e f i e l d i n the i n t e r e l e c t r o d e gap r e g i o n . In o r d e r t o make the d e v i c e more e f f i c i e n t a r i d g e o f the e l e c t r o o p t i c medium can be i n c l u d e d between t h i c k e l e c t r o d e s as i n f i g u r e 2.7. I f t h e r i d g e i s h i g h enough the normal component o f the a p p l i e d f i e l d at the s u r f a c e , Ey, w i l l be r e l a t i v e l y s m a l l and t h e t a n g e n t i a l component, E z , w i l l be e s s e n t i a l l y u n i f o r m . I t can be seen t h a t Ey w i l l be s m a l l by comparing the normal component o f the f i e l d on both s i d e s o f the boundary. I f E y a i s the normal component o f the f i e l d i n a i r a t the boundary and E y S i s t h a t i n the s u b s t r a t e then E y s = ( e a i r / e s u b s t r a t e > E y a o r E y s - 0.02E y a. I t f o l l o w s then -30- F i g u r e 2.5: A p l o t o f E v ( y , z ) f o r the p l a n a r VIOWM. -31- F i g u r e 2.6: A p l o t o f E z ( y , z ) f o r the p l a n a r VIOWM. -32- A l r X 0 > Z y n t e r e I e c t r o d e G a p Me t o l E l e c t r o d e M e t a l E l e c t r o d e , ^ s L R i d g e | 1 : A n i s o t r o p i c ^ S u b s t r a t e I F i g u r e 2.7: The r i d g e VIOWM s t r u c t u r e . -33- * t h a t i f the r i d g e i s h i g h enough t h a t we can approximate the a c t u a l s t r u c t u r e by t h a t shown i n f i g u r e 2.8. The s t r u c t u r e d e p i c t e d i n f i g u r e 2.8 i s s i m i l a r t o t h a t commonly used f o r c a l c u l a t i n g the f r i n g i n g f i e l d i n a p a r a l l e l p l a t e c a p a c i t o r [28] w i t h the d i f f e r e n c e t h a t the upper e l e c t r o d e makes a 90° angle r a t h e r than f o l d i n g back on i t s e l f . In t h e case o f the p a r a l l e l p l a t e c a p a c i t o r the f r i n g i n g e f f e c t s q u i c k l y d i s a p p e a r as one moves away from the edge, i n t o the c a p a c i t o r , and the f i e l d becomes un i f o r m and normal t o the e l e c t r o d e s . I t i s one o f the g o a l s o f t h i s s e c t i o n t o show t h a t t h i s i s a l s o the case f o r the s t r u c t u r e o f f i g u r e 2.8. 2.3.2.2 The E l e c t r i c F i e l d Distributions In o r d e r t o f i n d the a p p l i e d f i e l d d i s t r i b u t i o n s i n the s t r u c t u r e under a n a l y s i s i t i s n ecessary t o do two mappings. F i r s t t h e s t r i p 0 < T| < n i n the £-plane, f i g u r e 2.9a, i s mapped t o t h e upper h a l f p l a ne o f the i n t e r m e d i a t e W-plane, f i g u r e 2.9b, and second the upper h a l f p l a ne o f the W-plane i s mapped t o the unshaded r e g i o n o f the S-plane, f i g u r e 2.9c. I t i s s u f f i c i e n t t o analyze the s t r u c t u r e shown i n High enough b e i n g when E v s » 0. -34- L - E l e c t r o d e s F i g u r e 2.8: The model u s e d t o a n a l y z e t h e r i d g e w a v e g u i d e VIOWM s t r u c t u r e . -35- f i g u r e 2.9c r a t h e r than t h a t shown i n f i g u r e 2.8 due t o the symmetry o f the two s t r u c t u r e s . The mapping used t o map the £-plane t o the W-plane i s W = e^ and a S c h w a r t z - C h r i s t o f f e l t r a n s f o r m i s used t o map the W-plane t o the S-plane. The S c h w a r t z - C h r i s t o f f e l t r a n s f o r m i s found by i n t e g r a t i n g the d e r i v a t i v e ds (w + l ) 1/2 dw w which g i v e s s - K { 2 ( W + l ) 1 / 2 + In ( w + i ) 1 / 2  - i ( w + l ) 1 / 2  + i + C = K | 2 ( u + l + i v ) i / 2  + In ( u + l + i v ) 1 / 2  - 1 ( u + l + i v ) 1 / 2  + 1 + C (2.7) The complex c o n s t a n t s , K and C, are e v a l u a t e d by f i r s t l e t t i n g u -» -1 on the l i n e v = 0, which maps t o the p o i n t i g / 2 . By d o i n g t h i s e q u a t i o n 2.7 becomes i _ = (K., + i K , ) l n ( - l ) + (C, + i C , ) = iK.,* - K,»t + C, + i C , 2 e q u a t i n g the r e a l and imaginary p a r t s o f which g i v e s -36- V T C- p I a n e — > f ( a ) i V W - p l a n e : u -1 ( b ) CO i S - p I a n e g/2 ( c ) F i g u r e 2.9: The (a) (b) W, and (c) S-plahes. -37- g K l = - -  C2 2K and c l K 2 " " it and by l e t t i n g u -> 0 + on v = 0, which maps t o S = eq u a t i o n 2.7 becomes - ~ - <KX + i K 2 ) ( 2 + ln(0)) + ( C 1 + i C 2 ) t h e imaginary p a r t o f which i s 0 = K 2 (2 - ~) + C 2 which i m p l i e s t h a t K2 must be equal t o zero and t h e r e f o r e so must C i and C2 so t h a t one o b t a i n s g K = 2K and c = 0 . As i t t u r n s out the term (u + 1 + i v ) ^ 2 appears c o n t i n u a l l y through out the s o l u t i o n s f o r the a p p l i e d e l e c t r i c f i e l d d i s t r i b u t i o n s . T h e r e f o r e we w i l l f i r s t f i n d • • *i ft • an e x p r e s s i o n f o r i t i n the form Me . T h i s g i v e s -38- M = { ( u + l ) 2  + v 2 } 1 / 4 and 9 = _ t a n 2 -1 I u + 1 J R e w r i t i n g e q u a t i o n 2.7 one o b t a i n s g s = _ 2K -[ 2Mcos (6) + _ | l n ( M 2  - 2Mcos (6) + 1) - l n (M 2  + 2 M c o s (6) + D ] + i 2 M s i n ( 9 ) + t a n -1 ( M c o s (9) I, M c o s (9) - 1 ) ( MCOS (9) - t a n -1 k M c o s (9) + 1 ) or 2 M c o s ( 9 ) + _ [ l n ( M 2 - 2 M c o s (9) + 1 ) - l n (M 2 + 2Mcos (9) + 1) (2.9a) and 2K 2 M s i n ( 9 ) + t a n -1 M c o s (9) k MCOS (9) - 1 , - t a n -1 M c o s (9) "| k M c o s (9) + 1 ) . . (2.9b) E q u a t i o n s 2.9a and b g i v e the c o o r d i n a t e s , o and ©, i n the S-plane i n terms o f the c o o r d i n a t e s , u and v,' i n the W- pl a n e . In the next paragraph e x p r e s s i o n s f o r the e l e c t r i c f i e l d components, E a and E^, i n the S-plane i n terms o f the W-plane c o o r d i n a t e s , u and v w i l l be d e r i v e d . -39- I t i s convenient t o c o n s i d e r t h a t one o f the e l e c t r o d e s i s h e l d a t zero v o l t s w i t h V 0 v o l t s a p p l i e d t o the othe r e l e c t r o d e . I t i s p o s s i b l e t o use the symmetry o f the e l e c t r o d e c o n f i g u r a t i o n t o see t h a t i f the model o f f i g u r e 2 . 8 i s used t o analyze the e l e c t r o d e c o n f i g u r a t i o n and i f i t i s assumed t h a t t he upper e l e c t r o d e i s at + V 0 v o l t s then the lower e l e c t r o d e must be at + V 0 / 2 v o l t s . In the £-plane t h i s s i t u a t i o n corresponds t o the ̂ - a x i s b e i n g at + V Q / 2 v o l t s and th e l i n e T| = w b e i n g at +V Q v o l t s . The p o t e n t i a l f u n c t i o n i n t h e s t r i p between the ̂ - a x i s and the l i n e T\ = n w i l l be denoted, V(^,TI) , and i s g i v e n by v V($,TI) = — -n . 2K t h e e l e c t r i c f i e l d components are then g i v e n by dv($,Ti) vQdn E a and do 2ndo vodn vod$ ^ CD a>a> 2nd(o 2ndo where the Cauchy-Riemann c o n d i t i o n aT\/d(o = dZ,/da has been used. The p a r t i a l d e r i v a t i v e s dt,/do and dn/da, are the r e a l and imaginary p a r t s o f the d e r i v a t i v e d^/dS, -40- dS da do r e s p e c t i v e l y . Thus the r e l a t i o n s Im 2K dS J and Re 2K L dS a r e o b t a i n e d . The d e r i v a t i v e d£/dS i s g i v e n by df, dW d ln(W) dS dW dS dW dS L dW -1 w = (W+l)- 1 / 2 = M-^- 1 9 W (W+l) 1/2 which g i v e s s i n (0) 2nM and E = - C O S (9) CO 2nM From f i g u r e s 2.8 and 2.9c i t i s obvious t h a t the o-axis w i l l c o r r e s p o n d t o the y ' - a x i s o f the d e v i c e and t h a t the co-axis w i l l c o r r e s p o n d t o the z ' - a x i s . T h e r e f o r e i f we wish -41- t o f i n d t h e e l e c t r i c f i e l d i n the yz-plane, u s i n g the above method, t h e t r a n s f o r m a t i o n s o f the equations 2.3a and b must be a p p l i e d a f t e r t h e conformal mapping. F i g u r e s 2.10 and 2.11 show the d i s t r i b u t i o n s o f E y ( y , z ) and E z ( y , z ) i n t h e s u b s t r a t e . An o f f s e t from the e l e c t r o d e s o f 5% o f t h e gap width has been assumed t o account f o r the o p t i c a l b u f f e r l a y e r . Again, as i n s e c t i o n 2.3.1.2, i t i s assumed t h a t the v o l t a g e a p p l i e d a c r o s s the e l e c t r o d e s i s n e g a t i v e i . e . V 0 i s n e g a t i v e . Although the y-component o f the a p p l i e d f i e l d i s o f the same order o f magnitude as the z-component i n t h e r e g i o n s around the c o r n e r i n the e l e c t r o d e t h e square o f the product r4£Ey, i n e q u a t i o n 2.2, i s v e r y s m a l l f o r the f i e l d s t r e n g t h s used. By i n s p e c t i o n o f f i g u r e s 2.10 and 2.11 i t i s apparent t h a t f o r r i d g e h e i g h t s o f g/2 or g r e a t e r the y-component o f the a p p l i e d f i e l d i s much s m a l l e r than the z-component. T h e r e f o r e t h i s model o f t h e r i d g e VIOWM can be used f o r r i d g e s g/2 h i g h o r h i g h e r . 2.4 T h e V a r i a t i o n a l M e t h o d In o r d e r t o e x a c t l y determine the d i s t r i b u t i o n o f the o p t i c a l f i e l d o f an o p t i c a l waveguide wi t h a r e f r a c t i v e index d i s t r i b u t i o n t h a t v a r i e s i n two dimensions i t i s ne c e s s a r y t o be a b l e f i r s t t o c o n s t r u c t the a p p r o p r i a t e wave -42- F i g u r e 2.10: A p l o t o f E y ( y , z ) f o r the r i d g e VIOWM. Here the p l o t has been cut along the l i n e z = 0 showing the f i e l d f o r z < 0 o n l y . -43- F i g u r e 2.11: A p l o t o f E z ( y , z ) f o r t h e r i d g e VIOWM. H e r e t h e p l o t h a s b e e n c u t a l o n g t h e l i n e z = 0 s h o w i n g t h e f i e l d f o r z < 0 o n l y . -44- e q u a t i o n and second t o s o l v e i t e x a c t l y f o r the o p t i c a l f i e l d d i s t r i b u t i o n . Indeed t h e r e e x i s t c e r t a i n r e f r a c t i v e index d i s t r i b u t i o n s f o r which an exact approach i s p o s s i b l e [29-31]. I t i s , however, f a r more common t h a t w h i l e a s o l u t i o n o b v i o u s l y e x i s t s , as i s evidenced by the f a c t t h a t a c e r t a i n waveguide guides o p t i c a l waves, the problem o f f i n d i n g i t e x a c t l y i s i n t r a c t a b l e . Hence approximate methods are needed f o r o b t a i n i n g the o p t i c a l f i e l d d i s t r i b u t i o n s i n o p t i c a l waveguides w i t h a r b i t r a r y r e f r a c t i v e index d i s t r i b u t i o n s and over a p e r i o d o f time th e s e have been developed. The two main approximate methods t h a t have emerged are the e f f e c t i v e r e f r a c t i v e index method and the f i n i t e element method. The development o f both methods began ca. 1969. The concept o f t h e e f f e c t i v e d i e l e c t r i c c o n s t a n t * was f i r s t i n t r o d u c e d by Knox and T o u l i o s i n 1970 [32] as an e x t e n s i o n t o t h e method o f M a r c a t i l i [25]. Hocker and Burns [33] used t o i t f i n d t h e p r o p a g a t i o n c o n s t a n t s f o r a r b i t r a r y waveguides. O p t i c a l f i e l d d i s t r i b u t i o n s can be o b t a i n e d u s i n g t h e p r o p a g a t i o n c o n s t a n t s and any o f a number o f n u m e r i c a l t e c h n i q u e s f o r s o l v i n g second order d i f f e r e n t i a l e q u a t i o n s , f o r example the method o f Runge-Kutta [34]. The a p p l i c a t i o n o f the f i n i t e element method t o s o l v i n g * The term e f f e c t i v e r e f r a c t i v e index was appl ied somewhat l a t e r but the only d i f fe rence i s that the e f f e c t i v e d i e l e c t r i c constant i s def ined to be e r e = p 2 / k 0 2 and the e f f e c t i v e r e f r a c t i v e index i s def ined to be n ^ f f = p /k 0 . -45- e l e c t r o m a g n e t i c waveguiding problems began w i t h Ahmed and Daly's paper [35]. L a t e r i t was a p p l i e d t o a r b i t r a r i l y shaped inhomogeneous o p t i c a l waveguides by Yeh e t a l . [36], More r e c e n t l y Koshiba e t a l . have used the f i n i t e element method t o an a l y z e a n i s o t r o p i c o p t i c a l waveguides [37]. Other t e c h n i q u e s based on, f o r example, expansion i n terms o f c i r c u l a r harmonics [38], the WKB method* [39], Green's f u n c t i o n s [40], or f i n i t e d i f f e r e n c e methods [41] have been a p p l i e d t o o p t i c a l waveguiding problems but t h e f i n i t e element method and the e f f e c t i v e r e f r a c t i v e index method have r e c e i v e d by f a r the most a t t e n t i o n . V a r i a t i o n a l methods have l o n g been a p p l i e d t o e l e c t r o m a g n e t i c waveguiding problems i n metal c l a d A A waveguides. The v a r i a t i o n a l method has a l s o been used t o o b t a i n approximate width parameters f o r the o p t i c a l f i e l d l i c t r i b u t i o n s and p r o p a g a t i o n c o n s t a n t s o f c i r c u l a r c o r e A A A o p t i c a l f i b e r s . In f a c t the f i n i t e element method i s i t s e l f a v a r i a t i o n a l t e chnique [44] . Here the v a r i a t i o n a l The e i g e n v a l u e e q u a t i o n d e r i v e d i n the WKB method can be used t o f i n d t h e e f f e c t i v e r e f r a c t i v e index and i s i n f a c t the same as t h a t d e r i v e d by Hocker and Burns [33] u s i n g a wave v e c t o r (ray) approach. T h e r e f o r e i t i s not n e c e s s a r i l y a s e p a r a t e method but may i n f a c t be c o n s i d e r e d as p a r t of the e f f e c t i v e r e f r a c t i v e index method. H a r r i n g t o n [42] chapter 7 pp. 317-380. Marcuse [19] pp. 339-347 or Okoshi [43] pp. 114-121. -46- method has been extended t o d e r i v e the o p t i c a l f i e l d d i s t r i b u t i o n s o f the fundamental mode o f a channel waveguide w i t h an a r b i t r a r y r e f r a c t i v e index p r o f i l e . In the v a r i a t i o n a l method an e x p r e s s i o n f o r the e i g e n v a l u e s o f the Euler-Lagrange e q u a t i o n o f a s t a t i o n a r y i n t e g r a l o f a f u n c t i o n a l i s d e r i v e d . In t h i s a p p l i c a t i o n t h e E uler-Lagrange e q u a t i o n i s the s c a l a r wave e q u a t i o n . U s i n g the i n t e g r a l o f the f u n c t i o n a l an e x p r e s s i o n f o r B 2 (the e i g e n v a l u e ) , o f a p a r t i c u l a r mode o f t h e waveguide, i s o b t a i n e d . Furthermore i t i s known t h a t the e i g e n v a l u e s are s t a t i o n a r y v a l u e s . Thus by p l o t t i n g p 2 as a f u n c t i o n o f c e r t a i n v a r i a b l e parameters o f a f u n c t i o n chosen t o approximate t h e o p t i c a l f i e l d d i s t r i b u t i o n those v a l u e s o f the parameters which make p s t a t i o n a r y may be found. In s e c t i o n 2.5 the c h o i c e o f Hermite-Gaussian f u n c t i o n s as the approximate o p t i c a l f i e l d d i s t r i b u t i o n w i l l be motiv a t e d . The Hermite-Gaussian f u n c t i o n s used have two v a r i a b l e parameters t h a t determine the t r a n s v e r s e e x t e n t o f the o p t i c a l f i e l d d i s t r i b u t i o n . Here the equations r e l e v a n t t o the v a r i a t i o n a l method are developed f o r the exact s o l u t i o n . The g e n e r a l form o f the s c a l a r wave e q u a t i o n can be expressed i n t h e form 2 2 V f cv + v (y, z) y = 0 , -47- where „2 32 -.2 3  2  a  2 y dz I f t h e above wave e q u a t i o n i s m u l t i p l i e d by y and the r e s u l t i s i n t e g r a t e d over the e n t i r e y z - p l a n e * i t g i v e s OO OO f f f 2 2 2 j J \|/ V^v + v (y, z) v • dydz «= 0 and i f Green's Theorem i s a p p l i e d t o the f i r s t term o f t h i s i n t e g r a l one gets OO OO OO O" J J ¥ V 2 V dydz - - J { —oo —OO 2 2 — + — > ay > , az , dydz + J y — ds C dn o f which t h e l a s t term on t h e r i g h t hand s i d e i s equal t o ze r o . F i n a l l y t h e i n t e g r a l -I I 2 + , ay , , az , 2 2 - v (y,z)v dydz «= 0 (2.10) i s o b t a i n e d . The domain of integration for a d i e l e c t r i c o p t i c a l waveguide i s the entire transverse plane. This i s due to the boundary condition for the guided modes of these waveguides which forces the o p t i c a l f i e l d to be zero at i n f i n i t y i f the guided power i s to remain f i n i t e . Yariv and Yeh [45] chapter 11 pp. 405-503. I t remains t o be shown t h a t e q u a t i o n 2.10 i s i n f a c t s t a t i o n a r y and t h a t i t i s a minimum. In appendix B the p r o o f i s p r e s e n t e d t h a t e q u a t i o n 2.10 i s s t a t i o n a r y and t h a t t h e g e n e r a l form o f the s c a l a r wave equation, g i v e n above, i s t h e Euler-Lagrange e q u a t i o n . * The wave e q u a t i o n f o r the exact o p t i c a l f i e l d d i s t r i b u t i o n , <l>e(x,y,z) (the s u b s c r i p t e i s f o r e x a c t ) , i s o b t a i n e d from Maxwell's e q u a t i o n s . In t h i s work the s c a l a r wave e q u a t i o n f o r an o p t i c a l waveguide i n which the r e f r a c t i v e index v a r i e s i n two dimensions, i n the yz-plane, i s 2 2 3 *e d *e 2 2 dy dz d 2 l n [ n 2 ( y , z ) ] d l n ^ ] d l n [ n 2 ( y , z ) ] 2 2 2 + + n <y,z)Jc - p z z o dz dz dz where * e ( x , y , z ) = e - 1P x4> e (y, z) , n z ( y , z ) i s the r e f r a c t i v e index d i s t r i b u t i o n , and k Q i s t h e f r e e space wavenumber ( k 0 = 2K/\0). From the above e q u a t i o n the f u n c t i o n v 2 ( y , z ) , i n t h e g e n e r a l form o f the wave e q u a t i o n (page 46), i s g i v e n by d 2 l n [ n 2 ( y , z ) ] dln[4> ] d l n [ n 2 ( y , z ) ] 2 2 2 2 v <y,z) = + + n z ( y , z ) k o - p . dz 2 dz dz Fox [46] c h a p t e r 3 pp. 59-79. -49- However, i n appendix B i t i s shown t h a t e q u a t i o n 2.10 i s s t a t i o n a r y and the g e n e r a l form o f the s c a l a r wave e q u a t i o n i s t h e Euler-Lagrange e q u a t i o n p r o v i d e d t h a t 6v(y,z) = 0. In t h e above e q u a t i o n t h i s i s o b v i o u s l y not so. In s e c t i o n 2.5.2 an approximation t o v ( y , z ) , based on the approximate o p t i c a l f i e l d s d i s t r i b u t i o n s , w i l l be i n t r o d u c e d t h a t i s independent o f 4>e and which i s then i n v a r i a n t and which w i l l be t h e f u n c t i o n t h a t i s a c t u a l l y used. In f a c t i f one had a p r i o r i knowledge o f the f u n c t i o n <)>e then the above e x p r e s s i o n c o u l d a l s o be t r e a t e d as b e i n g i n v a r i a n t . In t h e f o l l o w i n g s e c t i o n the approximate f u n c t i o n s f o r t h e o p t i c a l f i e l d d i s t r i b u t i o n s i n the o p t i c a l f i b e r and the VIOWM w i l l be g i v e n and e q u a t i o n 2.10 and the above eq u a t i o n w i l l be p r e s e n t e d i n terms o f the approximations. 2 .5 T h e H e r m i t e - G a u s s i a n A p p r o x i m a t i o n s In t h i s work approximations t o the o p t i c a l f i e l d d i s t r i b u t i o n s o f the waveguides employed w i l l be used. The approximation c o n s i s t s o f choosing f u n c t i o n s t h a t are s i m i l a r t o those t h a t are expected t o be s o l u t i o n s t o the s c a l a r wave e q u a t i o n . The c h o i c e o f the approximating f u n c t i o n s s h o u l d be based on a knowledge of the c h a r a c t e r i s t i c s o f s o l u t i o n s t o s i m i l a r problems where the exact s o l u t i o n i s known. In the f o l l o w i n g s u b s e c t i o n s a -50- d i s c u s s i o n o f the c h o i c e o f each o f the approximate f u n c t i o n s w i l l be g i v e n i n terms o f i t s a p p l i c a t i o n t o the waveguide used. One o f the main g o a l s o f t h i s work i s t o show t h a t the VIOWM has an a p p l i c a t i o n as a l i n k i n g waveguide between two o p t i c a l f i b e r s (or o t h e r waveguides). I t i s t h e r e f o r e n e c e s s a r y t o be a b l e t o p r e d i c t the e f f i c i e n c y o f c o u p l i n g between the two waveguide types when they are brought t o g e t h e r i n a b u t t - c o u p l i n g arrangement. I f the o p t i c a l f i e l d d i s t r i b u t i o n s i n both waveguide types can be expressed i n terms o f Hermite-Gaussian f u n c t i o n s i t i s p o s s i b l e t o o b t a i n a c l o s e d form a n a l y t i c s o l u t i o n f o r the c o u p l i n g c o e f f i c i e n t . As w i l l be shown i n s e c t i o n 2.6 the c o u p l i n g c o e f f i c i e n t i s o b t a i n e d i n terms o f the o v e r l a p i n t e g r a l o f the two f i e l d d i s t r i b u t i o n s . The c o n t r i b u t i o n t o the o v e r l a p i n t e g r a l made by the evanescent f i e l d s i s o b v i o u s l y s m a l l . T h e r e f o r e the f a c t t h a t the evanescent f i e l d o f the approximate f u n c t i o n s w i l l be o f the form e x p ( - r 2 ) r a t h e r than o f the form exp(-r) w i l l have l i t t l e e f f e c t . * Thus i t i s seen t h a t i f the approximate o p t i c a l f i e l d d i s t r i b u t i o n s are t o be used t o c a l c u l a t e the c o u p l i n g c o e f f i c i e n t s f o r b u t t - c o u p l e d waveguides t h a t i t i s most important t h a t the * F o r t h i s reason Hermite-Gaussian f u n c t i o n s s h o u l d not be used t o approximate o p t i c a l f i e l d d i s t r i b u t i o n s i n evanescent f i e l d c o u p l i n g problems. However, i n the c u r r e n t problem they are e n t i r e l y a p p r o p r i a t e . approximations be c l o s e t o the a c t u a l d i s t r i b u t i o n s i n the r e g i o n s where the f i e l d s are l a r g e s t . 2.5.1 The Optical Fiber I t has l o n g been known t h a t a h i g h p e r m i t t i v i t y if d i e l e c t r i c r o d can a c t as a waveguide and s o l u t i o n s t o the wave e q u a t i o n have been o b t a i n e d f o r rods w i t h v a r i o u s c r o s s s e c t i o n s i n c l u d i n g c i r c u l a r [29], e l l i p t i c a l [30], and r e c t a n g u l a r [25]. T y p i c a l l y o p t i c a l f i b e r s have r e f r a c t i v e index p r o f i l e s t h a t a re e i t h e r c i r c u l a r l y symmetric or e l l i p t i c a l l y symmetric i n the t r a n s v e r s e plane, i . e . a plane normal t o t h e d i r e c t i o n o f p r o p a g a t i o n . A u s e f u l approximation t o the r e f r a c t i v e index d i s t r i b u t i o n o f many f i b e r s i s t h e " s t e p index" p r o f i l e . The st e p index p r o f i l e i s one i n which t h e r e f r a c t i v e index o f the core i s assumed t o have one v a l u e , n c o , and t h a t i n the c l a d d i n g i s assumed t o have another v a l u e , n c i , such t h a t n c o > nc±. F i g u r e 2.12 i l l u s t r a t e s t h e r e f r a c t i v e index p r o f i l e o f a s t e p index f i b e r . Few, i f any, o f the commercially a v a i l a b l e o p t i c a l f i b e r s have a st e p index p r o f i l e . The p r o f i l e s o f th e s e f i b e r s have r e f r a c t i v e index d i s t r i b u t i o n s t h a t are determined by the method o f f a b r i c a t i o n . See Okoshi [43] ch a p t e r 1 pp. 1-16 f o r an h i s t o r i c a l development. -52- F i b e r C l a d d i n g R e f r o d i ve I n d e x : Pi F i b e r C o r e R e f r a c t i v e I n d e x : Pi C O E x t e r n a l M e d i u m R e f r a c t i ve I n d e x : Pi o n ( r ) n -L CO n 4- 0 F i g u r e 2.12: The r e f r a c t i v e index d i s t r i b u t i o n o f a f i b e r w i t h s t e p index p r o f i l e . -53- ' While exact s o l u t i o n s e x i s t f o r both c i r c u l a r and e l l i p t i c a l c r o s s s e c t i o n f i b e r s w i t h a s t e p index p r o f i l e [29,30] t h e y are i n terms o f r a t h e r c o m p l i c a t e d h i g h e r f u n c t i o n s . F o r example the c i r c u l a r core s t e p index f i b e r has s o l u t i o n s t h a t are i n terms o f B e s s e l f u n c t i o n s i n the core and i n terms o f m o d i f i e d K e l v i n f u n c t i o n s i n the c l a d d i n g . * While such s o l u t i o n s form a complete s e t o f b a s i s f u n c t i o n s u s e f u l i n d e s c r i b i n g the o p t i c a l f i e l d d i s t r i b u t i o n s i n l a r g e core multi-mode f i b e r s and f o r c a l c u l a t i n g the p r o p a g a t i o n c o n s t a n t s o f the v a r i o u s modes, thus a l l o w i n g the study o f o p t i c a l f i b e r s i n a g e n e r a l t h e o r e t i c a l sense, they are o f l i t t l e p r a c t i c a l use when d e a l i n g w i t h the l i g h t emanating from a p a r t i c u l a r s i n g l e - mode o p t i c a l f i b e r . O f t e n i t i s e a s i e r t o work w i t h an approximation t o the exact s o l u t i o n . Gaussian** approximations are p a r t i c u l a r l y u s e f u l . They are mathematical s i m p l e r t o use h a v i n g the O A A A form e x p ( - r ^ ) . They are known t o propagate i n f r e e space and so t h e p r o p a g a t i o n c h a r a c t e r i s t i c s (the d i f f r a c t i o n c h a r a c t e r i s t i c s ) o f a beam launched i n t o f r e e space from a Marcuse [19] chapter 8 pp. 286-347. A Gaussian f u n c t i o n i s a Hermite-Gaussian f u n c t i o n m u l t i p l i e d by the Hermite p o l y n o m i a l H G which i s equal t o 1. Verdeyen [47] chapter 3 pp. 53-69. -54- f i b e r , say as p a r t o f an o p t i c a l system, i s p r e d i c t a b l e . The c o u p l i n g e f f i c i e n c y o f a focused l a s e r beam t o a guided Gaussian mode i s e a s i l y c a l c u l a t e d [48]. Furthermore the G a ussian p r o f i l e i s found t o be the s o l u t i o n t o the wave e q u a t i o n f o r f i b e r s w i t h a q u a d r a t i c r e f r a c t i v e index p r o f i l e . * I t i s a l s o demonstrable t h a t the Gaussian approximation i s i n f a c t a good approximation t o the o p t i c a l f i e l d d i s t r i b u t i o n s encountered i n single-mode o p t i c a l f i b e r s . Comparison's o f t h e o r e t i c a l l y p r e d i c t e d p r o f i l e s f o r f i b e r s have been compared t o the Gaussian f u n c t i o n [41] and have i n d i c a t e d a c l o s e c o r r e l a t i o n between the two. A l s o comparisons between measured power d i s t r i b u t i o n s emanating from a c t u a l f i b e r s and the Gaussian f u n c t i o n have been made [49,50] a g a i n showing a c o n v i n c i n g c o r r e l a t i o n . The o p t i c a l f i e l d d i s t r i b u t i o n o f a c i r c u l a r l y symmetric f i b e r may i n g e n e r a l be seen as b e i n g composed o f two degenerate, o r t h o g o n a l p o l a r i z a t i o n s . * * The c h o i c e o f t h e d i r e c t i o n s o f p o l a r i z a t i o n i s a r b i t r a r y f o r t h e s e two modes, t h e r e f o r e i n t h i s work i t w i l l be assumed t h a t one p o l a r i z a t i o n i s c o i n c i d e n t w i t h the y - a x i s and one w i t h the z - a x i s . In the case o f p o l a r i z a t i o n - p r e s e r v i n g f i b e r the two modes are no l o n g e r degenerate s t i l l i t w i l l be assumed * I b i d . ** Okoshi [43] c h a p t e r 4 pp. 48-81. -55- t h a t t h e f i b e r i s o r i e n t e d so t h a t , as above, one p o l a r i z a t i o n i s c o i n c i d e n t w i t h the y - a x i s and one w i t h the z - a x i s . S i n c e t h e e l e c t r i c f i e l d o f the o p t i c a l d i s t r i b u t i o n i n a VIOWM i s p o l a r i z e d p a r a l l e l t o the z- a x i s o n l y the o p t i c a l f i e l d d i s t r i b u t i o n i n t h e f i b e r t h a t i s a l s o p o l a r i z e d p a r a l l e l t o t h e z - a x i s need be c o n s i d e r e d . The o p t i c a l f i e l d i n t h e f i b e r can thus be approximated by the e x p r e s s i o n - i p f x <Kf(x,y,z) = e 4>f(y,z) (2.11) where t h e s u b s c r i p t f stands f o r f i b e r and -<y 2/w y f + * 2 / » * f > / 2 where a f i s t h e no r m a l i z e d amplitude and W y f and w z f are the wid t h parameters i n the y and z - d i r e c t i o n s r e s p e c t i v e l y . The n o r m a l i z e d amplitude i s a cons t a n t t h a t ensures t h a t t he mode c a r r i e s u n i t power and w i l l i s d i s c u s s e d f u r t h e r i n appendix C. F i g u r e 2.13 shows a Gaussian o p t i c a l f i e l d d i s t r i b u t i o n i n which W y f = w z f and f i g u r e 2.14 shows one i n which W y f = w z f/2. -56- F i g u r e 2.13: A Gaussian o p t i c a l f i e l d d i s t r i b u t i o n i n which w y f = w z f - -57- F i g u r e 2.14: A G a u s s i a n o p t i c a l f i e l d d i s t r i b u t i o n i n w h i c h w y f = w z f / 2 - -58- 2.5.2 The Voltage Induced Optical Waveguide Based on t h e m a t e r i a l p r e s e n t e d i n s e c t i o n s 2.2 and 2.3 i t can be seen t h a t the r e f r a c t i v e index d i s t r i b u t i o n w i l l be r a t h e r complex f o r both the p l a n a r and r i d g e VIOWM. Sin c e i t i s v e r y u n l i k e l y t h a t an exact s o l u t i o n t o the wave e q u a t i o n c o u l d be found f o r a waveguide w i t h such a r e f r a c t i v e index d i s t r i b u t i o n the v a r i a t i o n a l method w i l l be used. In t h i s s e c t i o n the c h o i c e o f Hermite-Gaussian f u n c t i o n s t o approximate the a c t u a l o p t i c a l f i e l d d i s t r i b u t i o n s i s j u s t i f i e d . To b e g i n with, c e r t a i n o b s e r v a t i o n s can be made about the r e f r a c t i v e index d i s t r i b u t i o n and a knowledge o f the o p t i c a l f i e l d d i s t r i b u t i o n s i n s i m i l a r waveguides can be used: 1) The r e f r a c t i v e index d i s t r i b u t i o n i s symmetric w i t h r e s p e c t t o t h e xy-plane, 2) The d i f f e r e n c e i n the r e f r a c t i v e index o f the s u b s t r a t e and the o p t i c a l b u f f e r l a y e r i s l a r g e , 3) And f o r a c e r t a i n range o f a p p l i e d v o l t a g e s the change induc e d i n the r e f r a c t i v e index o f the s u b s t r a t e i s l a r g e r i n the i n t e r e l e c t r o d e gap r e g i o n than i n the " s u r r o u n d i n g " r e g i o n and the change induced i n the -59- s u r r o u n d i n g r e g i o n decreases m o n o t o n i c a l l y t o zero at i n f i n i t y . * From the above o b s e r v a t i o n s c e r t a i n c h a r a c t e r i s t i c s o f a bound mode can be a n t i c i p a t e d . 1) The o p t i c a l f i e l d d i s t r i b u t i o n w i l l be e i t h e r symmetric or anti-symmetric w i t h r e s p e c t t o the xy- plane, 2) The shape o f the o p t i c a l f i e l d d i s t r i b u t i o n w i l l be such t h a t i t i s v e r y s m a l l at the boundary between the s u b s t r a t e and the b u f f e r l a y e r as compared t o i t s peak v a l u e f u r t h e r i n t o the s u b s t r a t e , 3) Most o f the power w i l l be c o n f i n e d t o the i n t e r e l e c t r o d e gap r e g i o n and w i l l decrease m o n o t o n i c a l l y t o zero at i n f i n i t y i n the sur r o u n d i n g r e g i o n . Hence t r i a l f u n c t i o n s must be chosen which w i l l have s i m i l a r c h a r a c t e r i s t i c s . In o r d e r t o understand the modus operandi o f the VIOWM onl y the lowest order mode need be c o n s i d e r e d . The lowest o r d e r mode w i l l be the most h i g h l y c o n f i n e d mode f o r the The " s u r r o u n d i n g " r e g i o n i s d e f i n e d here as t h a t r e g i o n i n which t h e r e f r a c t i v e index decrease i s monotonic. -60- lowest o p e r a t i n g v o l t a g e and w i l l have the h i g h e s t degree o f c o u p l i n g t o or from the f i b e r i f the f i b e r i s w e l l p o s i t i o n e d . The approximate o p t i c a l f i e l d d i s t r i b u t i o n f o r the VIOWM w i l l be d e s i g n a t e d <I>v(y,z), s u b s c r i p t v f o r VIOWM, and i s g i v e n by -iB x e * v ( y , z ) ; y ̂  o (2.12) ; y < 0 where Bv i s the p r o p a g a t i o n constant o f the mode and yv 2 2 2 2 -<y /w + * / * )/2 yv zv where a v i s the n o r m a l i z e d amplitude, Wy^. and w 2 v are the it width parameters xn the y and z - d i r e c t i o n s r e s p e c t i v e l y . T - K O r!orraali.zed amplitude i s d i s c u s s e d f u r t h e r i n appendix C. F i g u r e 2.15 shows a Hermite-Gaussian o p t i c a l f i e l d d i s t r i b u t i o n i n which w. yv = w zv and f i g u r e 2.16 shows one i n which W y V = w z v/2. As can be seen from f i g u r e s 2.15 and 2.16 the Hermite- Gaussian approximations t o the f i e l d d i s t r i b u t i o n s meet each o f t he d e s i r e d c h a r a c t e r i s t i c s : they are symmetric w i t h r e s p e c t t o the xy-plane, they are s m a l l , zero i n f a c t , at As can be seen tfv(y,z) has the form (y)exp (-y 2) i n the y - d i r e c t i o n , f o r p o s i t i v e y, and Hg ( z ) e x p ( - z 2 ) i n the z- d i r e c t i o n , f o r a l l z, hence the name Hermite-Gaussian. -61- F i g u r e 2.15: A Hermite-Gaussian o p t i c a l f i e l d d i s t r i b u t i o n i n which w,.,, = w_,,. -62- F i g u r e 2.16: A Hermite-Gaussian o p t i c a l f i e l d d i s t r i b u t i o n i n which w v v = w z v/2. -63- the i n t e r f a c e between the s u b s t r a t e and the o p t i c a l b u f f e r l a y e r , and t h e evanescent f i e l d goes t o zero m o n o t o n i c a l l y at i n f i n i t y . In s e c t i o n 2.4 a l l the r e s u l t s were d e r i v e d i n terms o f the e x a c t f i e l d d i s t r i b u t i o n . Now the equations are p r e s e n t e d i n terms o f the approximate f u n c t i o n s . To b e g i n w i t h t h e approximation t o v (y,z) i s g i v e n i n terms o f the approximate o p t i c a l f i e l d d i s t r i b u t i o n by 2 2 2 3 l n [ n (y,z)] z dln[n (y,z)] 2 2 2 2 v v ( y , z ) - + n z ( y , z ) k o - p v . dz W dz z v which g i v e s t h e s t a t i o n a r y i n t e g r a l f o r t h i s approximation 2 -/ J —o  —OO 2 + , ay , , az , 2 2 d y d z = 0 (2.13) which becomes / J ( a* 1 2 2 V + 2/ w 2 - v v ( y , z ) < ( . v , ay , v az t d y d z (2.14) i n terms o f the approximate f u n c t i o n . The f i r s t two terms o f the i n t e g r a l I v can be e v a l u a t e d e x a c t l y and are g i v e n by -64- I I ( } v I ay J 1.5 dydz = I J 4> dydz v yv and J I v { dz ) dydz 0.5 J J 2 6 dydz v S i n c e I i s a minimum I v must p r o v i d e an upper bound on I ; i . e . I v S I . T h e r e f o r e i f 2.13 i s r e w r i t t e n t o p r o v i d e an e q u a t i o n f o r Bv >v r 2 + 2 • > 8« . 8 2ln[n 2<y,z)] z Bin [n 2 (y, z) ] + n2( y /z)k2 dz ' "zv 3 2 dydz v dydz t h e n r e w r i t i n g e q u a t i o n 2.14 i n a s i m i l a r f a s h i o n p r o v i d e s a lower bound on J J 1.5 0.5 -~ 3 2ln[n 2(y,z)] z 31n[n 2 (y, z) ] + r.2(y,z))c2 "zv • v dydz 2 2 yv zv / ( * v dydz S i n c e the p r o p a g a t i o n constant i s a s t a t i o n a r y v a l u e the above e q u a t i o n can be used t o f i n d the width parameters w- -65- and w z v. P r o o f t h a t ^ i s a s t a t i o n a r y v a l u e i s p r o v i d e d i n appendix B. By v a r y i n g the width parameters a g r i d o f v a l u e s f o r fry can be c r e a t e d and the v a l u e s o f Wy^. and w z v f o r which J ^ 2 i s s t a t i o n a r y may be found g r a p h i c a l l y . F i g u r e 2.17 shows a p l o t o f v e r s u s Wy^. and w z v f o r a p l a n a r VIOWM w i t h a 4um i n t e r e l e c t r o d e gap w i t h 50.0V a p p l i e d t o t h e e l e c t r o d e s and where XQ = 442nm, n e = 2.2884, n 0 = 2.371, r 3 3 = 30.8xl0 - 1 2m/V, and r 4 2 = 28.0xl0 - 1 2m/V. F i g u r e 2.18 shows the development o f the o p t i c a l f i e l d f o r t h e VIOWM w i t h the 4̂ m i n t e r e l e c t r o d e gap w i t h i n c r e a s i n g v o l t a g e , f o r a p p l i e d v o l t a g e s o f 10.0, 30.0, and 50.0 v o l t s , as p r e d i c t e d by the v a r i a t i o n a l method. 2.6 The C o u p l i n g C o e f f i c i e n t Hermite-Gaussian f u n c t i o n s have t h e advantage, as the chosen approximate o p t i c a l f i e l d d i s t r i b u t i o n s , t h a t they are c o n v e n i e n t t o manipulate m a t h e m a t i c a l l y . In the f o l l o w i n g , t h e c o u p l i n g c o e f f i c i e n t s are determined by expanding t h e o p t i c a l f i e l d s i n the two b u t t - c o u p l e d waveguides i n terms o f both guided and r a d i a t i o n modes. T h i s approach has been used by Burns and M i l t o n [51] t o study the c o n v e r s i o n o f modes i n s e p a r a t i n g waveguides and by Hunsperger e t a l . [52] t o study b u t t - c o u p l i n g between s o l i d s t a t e l a s e r s and s u r f a c e waveguides. -66- Maxlmum (2 .88 ,2 .38 ,1058.693) F i g u r e 2.17: A p l o t o f v e r s u s w „ v a n d w z v f o r a VIOWM w i t h a 4(im i n t e r e l e c t r o d e gap w i t h 50.0V a p p l i e d t o t h e e l e c t r o d e s f o r \ 0 = 442nm. - 6 7 - F i g u r e 2.18: The d e v e l o p m e n t o f t h e o p t i c a l f i e l d f o r a VIOWM w i t h a 4\im i n t e r e l e c t r o d e gap f o r (a) 10.0, (b) 30.0, and (c) 50.0V a p p l i e d t o t h e e l e c t r o d e s f o r ̂ Q = 442nm. -68- The c o n t i n u i t y o f the t a n g e n t i a l f i e l d components at t h e i n t e r f a c e between the o p t i c a l f i b e r and the VIOWM i s c o n s i d e r e d . I f both waveguides are assumed t o be designed f o r single-mode o p e r a t i o n one can w r i t e «>,. + R0>, + E _ = T<D + E. (2.15a) f f r e f v t r a n s f o r the e l e c t r i c f i e l d and p ,«>.- + Rp £«> £ + H , = Tp «> + H. (2.15b) K f f f r e f v v t r a n s f o r the magnetic f i e l d f o r c o u p l i n g from the o p t i c a l f i b e r t o the VIOWM. Here R and T are the r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s , r e s p e c t i v e l y , and E r e f and E t r a n g are t h e r e f l e c t e d and t r a n s m i t t e d e l e c t r i c f i e l d d i s t r i b u t i o n s o f the r a d i a t e d modes and s i m i l a r l y H r e f and H t r a n s a r e *-he r e f l e c t e d and t r a n s m i t t e d magnetic f i e l d d i s t r i b u t i o n s o f the r a d i a t e d modes. To s o l v e f o r the c o u p l i n g between the two waveguides the assumption i s made t h a t the power i n the r e f l e c t e d r a d i a t e d modes i s n e g l i g i b l e and t h a t the major p a r t o f the r e f l e c t e d power i s i n the back t r a v e l i n g guided mode. Each o f the r emaining terms, remaining a f t e r the terms r e p r e s e n t i n g t h e r e f l e c t e d r a d i a t e d mode have been dropped, i s m u l t i p l i e d by i n e q u a t i o n 2.15a and i s m u l t i p l i e d by P v ^ * i n e q u a t i o n 2.15b and both are i n t e g r a t e d over the boundary plane, the i n t e r f a c e between the o p t i c a l f i b e r and t h e VIOWM, which i s assumed t o be the p l a ne x = 0. The two -69- r e s u l t i n g e q u a t i o n s are then s u b t r a c t e d , one from the other, l e a v i n g P v ( P f - P v ) { J »*f dydz - RP v(P f + p v ) { J *%>f dydz = 0 —oo — oo —OO — OO and s o l v i n g f o r R g i v e s p , - p n' - n' R = = - T (2.16) P, + p n' + n' where n f ' and n v ' are the e f f e c t i v e r e f r a c t i v e i n d i c e s o f the o p t i c a l f i b e r and the VIOWM, r e s p e c t i v e l y , and r i s the 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 f o r plane waves at normal i n c i d e n c e . To j u s t i f y n e g l e c t i n g the power i n the r e f l e c t e d r a d i a t e d modes one must look at the r e s u l t t h a t e q u a t i o n 2.16 r e p r e s e n t s . In e f f e c t the assumption i s t h a t the r e f l e c t e d power, r^, i n the case o f o p t i c a l waveguides i s e s s e n t i a l l y the same as t h a t f o r plane waves a t normal i n c i d e n c e . The p l a n e wave s o l u t i o n i s , i n f a c t , the l i m i t i n g case f o r weakly guided modes. In o t h e r words as the v a r i a t i o n s i n the r e f r a c t i v e index d i s t r i b u t i o n s o f the waveguides d i s a p p e a r on both s i d e s o f the boundary p l a n e the o p t i c a l f i e l d d i s t r i b u t i o n s extend t o become plane waves and the amount o f r e f l e c t e d power i s e x a c t l y equal t o . The modes o f the waveguides c o n s i d e r e d i n t h i s work are weakly guided, i n t h a t the e x t e n t s o f the o p t i c a l f i e l d -70- d i s t r i b u t i o n s are much g r e a t e r than a wavelength, and the amount o f power r e f l e c t e d i s s m a l l , » 5% o f the t o t a l i n c i d e n t power, t h e r e f o r e the approximation s h o u l d l e a d t o rea s o n a b l e r e s u l t s . U s i n g equations 2.15a and b the t r a n s m i s s i o n c o e f f i c i e n t can be s o l v e d f o r . F i r s t 2.15a i s m u l t i p l i e d by {^PfO-y* and 2.15b i s m u l t i p l i e d by Pv* v* and the r e s u l t i n g e q u a t i o n s are i n t e g r a t e d over the boundary plane x = 0. T h i s time, however, the equations are added r e s u l t i n g i n oo oo oo oo 2B VB f J J ©%> f dydz - T P V O F + P V) J . J «>X dyd 2 —oo —oo —oo —oo which a f t e r some rearrangement g i v e s - - f y 2 a f P f J / -~ 0 2 2 2 2 2 2 2 2 e - { y 2 / - ^ + z 2 / K 2 v + ( y - a ) 2 / w 2 f + <z - b , 2 / w 2 f } / 2 ^ T - «Sr<Pf + Pv> J / —. 0 yv ( y 2 / w 2 + z 2 / w 2 ) yv « v ' d y d z where t h e v a r i a b l e s a and b l o c a t e the c e n t e r o f the o p t i c a l f i b e r r e l a t i v e t o the c e n t e r o f the s u r f a c e o f the waveguiding r e g i o n o f t h e VIOWM, see f i g u r e 2.19. The r a t i o a f / a v i s determined by n o r m a l i z i n g the power i n each mode and i s g i v e n by* See appendix C. -71- Opt i CO I F i b e r [ c o r e Subst r o t e - i Sur f oce VIOWM Wove - gu id e H n t e r f o c e Edge View Wavegu i de 11 o Z HVIOWM V - g r oove End View F i g u r e 2.19: The i n t e r f a c e between the o p t i c a l f i b e r and the VIOWM. -72- - o o 0 ' y I 2 / 2/ 2 x 2/ 2 ̂ —(y /w + z /w ) y v z v e dydz yv . 2 , 2 , 2 , 2 , -(y /w f + z / « z f ) e dydz 1 2 The e x p r e s s i o n f o r the t r a n s m i s s i o n c o e f f i c i e n t can be r e w r i t t e n * as . / 7 / 7 -<a2/w2 + b 2/w 2 )/2 + b2£l / 4 w * 2(pj) ) 1 / 2 n n 1 / 2 e y f z f 2 z f r f r v y z 1/2 1/2 f v yv zv yf zf yv 1/2 1/2 a 2 V 4 W y f f , aQ w e J J erfc| y 1 + (2.17) 2w yf where , 2 2 2w w , yv y f 2 ^ 2 w + w _ yv yf and * See appendix C. -73- 2 ^ 2 zv zf A s i m i l a r l i n e o f re a s o n i n g y i e l d s an eq u a t i o n t h a t i s i d e n t i c a l t o eq u a t i o n 2.17 f o r l i g h t c oupled from a VIOWM t o an o p t i c a l f i b e r . As i s e v i d e n t from e q u a t i o n 2.17 the l o c a t i o n o f t h e f i b e r r e l a t i v e t o t h e i n p u t o f a VIOWM i s important i n de t e r m i n i n g t h e c o u p l i n g . I t i s obvious t h a t i n the z- d i r e c t i o n t h e b e s t c o u p l i n g i s o b t a i n e d when b = 0 as t h i s w i l l g i v e t h e b e s t o v e r l a p between any two Gaussian d i s t r i b u t i o n s independent o f t h e i r r e s p e c t i v e width parameters. On the ot h e r hand f i n d i n g the b e s t l o c a t i o n i n the y - d i r e c t i o n w i l l depend on the o p e r a t i n g v o l t a g e and mode o f o p e r a t i o n , see chapter 4. -74- C h a p t e r 3 FABR I C A T I O N 3.1 I n t r o d u c t i o n In t h i s c h a p t e r the f a b r i c a t i o n o f both p l a n a r and r i d g e VIOWMs i n l i t h i u m n i o b a t e , i n c l u d i n g c u t t i n g and p o l i s h i n g , and o f V-grooves i n s i l i c o n i s d e s c r i b e d . A l s o i n c l u d e d i s a s e c t i o n on the alignment o f the VIOWMs and o p t i c a l f i b e r s u s i n g t h e V-grooves. The f a b r i c a t i o n and alignment i n f o r m a t i o n i s c o n t a i n e d i n t h r e e s e c t i o n s e n t i t l e d : The VTOWM, The S i l i c o n V-groove, and D e v i c e / O p t i c a l F i b e r Alignment. The VIOWM s e c t i o n begins by d i s c u s s i n g i s s u e s t h a t are common t o t h e f a b r i c a t i o n o f both p l a n a r and r i d g e d e v i c e s . Then t h e f a b r i c a t i o n t e c h n i q u e s t h a t are unique t o each d e v i c e t y p e a re c o n t a i n e d i n t h e i r own s u b s e c t i o n s . The -75- c u t t i n g and p o l i s h i n g o f two types o f d e v i c e i s c o n t a i n e d i n a s e p a r a t e s u b s e c t i o n . The s i l i c o n V-groove s e c t i o n i s d e d i c a t e d t o a d e s c r i p t i o n o f the a n i s o t r o p i c e t c h i n g o f s i l i c o n t o produce V-grooves. V-grooves e t c h e d i n s i l i c o n can be used t o form a s t a b l e method o f c o u p l i n g l i g h t i n t o and out o f i n t e g r a t e d o p t i c d e v i c e s by a f l i p - c h i p t e chnique [21]. The d e v i c e / o p t i c a l f i b e r alignment s e c t i o n d e s c r i b e s the alignment o f a VIOWM and an o p t i c a l f i b e r u s i n g an a r r a y o f V-grooves. The combination o f a VIOWM and an o p t i c a l f i b e r on t h e V-groove a r r a y i s used as an o p t i c a l f r o n t - e n d ^ s w i t c h . Measurements on the sw i t c h are p r e s e n t e d i n chapter 4. 3.2 T h e VIOWM The i n i t i a l p r e p a r a t i o n o f the LiNb03 wafers was the same f o r b o t h p l a n a r and r i d g e d e v i c e s as was t h e f i n a l c u t t i n g and p o l i s h i n g stage i n the f a b r i c a t i o n . The d e v i c e s were f a b r i c a t e d on Y-cut* LiNb03 wafers o b t a i n e d from C r y s t a l Technology Inc., P a l o A l t o , Ca. The wafers were 3 i n c h e s i n diameter and 0.04 inches t h i c k . In order t o make These s u b s t r a t e s are cut and p o l i s h e d so t h a t the Y-axis o f t h e c r y s t a l i s normal t o the o p t i c a l q u a l i t y s u r f a c e . -76- b e t t e r use o f the l a r g e wafers they were d i v i d e d i n t o q u a r t e r s by f i r s t c o v e r i n g the e n t i r e o p t i c a l l y p o l i s h e d f a c e o f t h e wafer w i t h a coat o f A l about l\xm t h i c k e v a p o r a t e d u s i n g a c o n v e n t i o n a l d i f f u s i o n pumped vacuum system (made by C a r l Hermann Assoc.) capable o f h o l d i n g 3 s o u r c e s . The A l p r o t e c t e d the o p t i c a l l y p o l i s h e d s u r f a c e from s c r a t c h e s d u r i n g the c u t t i n g p r o c e s s and p r o v i d e d a means o f u n d e r c u t t i n g any i n e r t d i r t t h a t had been p i c k e d - up. The c u t t i n g was done on a h i g h speed diamond saw. The A l l a y e r was removed by e t c h i n g i n a 1:1 s o l u t i o n o f p h o s p h o r i c a c i d and D l (de-ionized) water at 50°C. The wafers were then c l e a n e d by immersion i n a 1% s o l u t i o n o f Alconox f o r 5 min. f o l l o w e d by a 10 min. r i n s e i n D l water, a 5 min. soak i n b u f f e r e d HF f o l l o w e d by a 10 min. r i n s e i n D l water, and a 5 min. soak w i t h u l t r a s o n i c a g i t a t i o n i n b o i l i n g methanol. The sample was heated w i t h t h e methanol then b o t h were t r a n s f e r r e d t o the a g i t a t o r i n o r d e r t o a v o i d e d thermal shock s h a t t e r i n g the LiNb03. A f t e r removal from the b o i l i n g methanol the wafers were blown d r y w i t h n i t r o g e n immediately p r i o r t o b e i n g loaded i n t o a P e r k i n - E l m e r S p u t t e r i n g System Model 3140. The p a t t e r n i n g o f the waveguides on the LiNbC>3 s u b s t r a t e s was done u s i n g p h o t o l i t h o g r a p h y . The mask used was made by S i e r r a c i n , Santa C l a r a , Ca. On the mask were l o n g s t r i p e s v a r y i n g i n l e n g t h from 1 t o 23 mm and i n width from 4 t o 10 nm. P h o t o r e s i s t was p a t t e r n e d on the sample so -77- t h a t the l o n g e r dimension o f the s t r i p e s would run p a r a l l e l t o the X - a x i s o f the c r y s t a l . Thus the waveguides would run p a r a l l e l t o the XY-plane o f the c r y s t a l and the e l e c t r i c f i e l d , c r e a t e d by a p p l i c a t i o n o f v o l t a g e t o the e l e c t r o d e s , would have a component p a r a l l e l t o the Z - a x i s . 3.2.1 The Planar VIOWM A 200 nm o p t i c a l b u f f e r l a y e r o f SiC>2 was RF s p u t t e r e d ( i n the P e r k i n - E l m e r s p u t t e r i n g system) onto the o p t i c a l l y p o l i s h e d f a c e o f a s e c t i o n o f the LiNbC-3 wafer. The t a r g e t was n o m i n a l l y o f 99.95% p u r i t y . The d e p o s i t i o n was preceded by a 1 h r . p r e c l e a n o f the t a r g e t w i t h a s h u t t e r p l a c e d between the t a r g e t and the wafer. The atmosphere i n the chamber was 18 mTorr o f A r and 5 mTorr 0 2. The s p u t t e r i n g was done at 100 W forward and about 2 W r e f l e c t e d power f o r 0.5 h r . The r e f r a c t i v e index o f SiC"2 i s about 1.5 and i t t h e r e f o r e p r o v i d e s a good o p t i c a l b u f f e r l a y e r on LiNbC>3 which has a r e f r a c t i v e index o f about 2.3. See s e c t i o n 2.3.1.2. The wafer was removed from the s p u t t e r i n g system and p a t t e r n e d w i t h p h o t o r e s i s t so as t o a l l o w the i n t e r e l e c t r o d e gap t o be formed u s i n g a l i f t - o f f t echnique [53]. The p h o t o r e s i s t p a t t e r n i n g was done as f o l l o w s : the p h o t o r e s i s t * was a p p l i e d i n a p h o t o r e s i s t s p i n n e r at 4000 rpm f o r 25 s e c ; a prebake was performed at 95°C f o r 25 min.; t h e p h o t o r e s i s t was exposed t o 320 nm r a d i a t i o n (UV) w i t h a power d e n s i t y o f 25 mW/cm2 f o r 40 s e c ; the p h o t o r e s i s t was then soaked i n chlorobenzene f o r 2.5 min. [53]; a post-bake was performed f o r 25 min. at 95°C; the p h o t o r e s i s t was developed i n a 1:1 s o l u t i o n o f S h i p l e y MF-312 d e v e l o p e r and D l water; the f i n a l s t e p was r i n s i n g i n f l o w i n g D l water. By t h i s method a l o n g and narrow p h o t o r e s i s t r i d g e was formed w i t h some u n d e r c u t t i n g so t h a t t h e i n t e r e l e c t r o d e gap c o u l d be f a b r i c a t e d u s i n g l i f t - o f f . The e l e c t r o d e s o f the p l a n a r VIOWM were designed so as t o be j o i n e d t o s i m i l a r e l e c t r o d e s on the S i V-grooves by a e u t e c t i c bond. T h e r e f o r e they were made out o f AuGe w i t h a e u t e c t i c temperature o f 363°C. However, the adhesion o f AuGe on Si02 i s poor and i t was necessary t o i n c l u d e a t h i n T i l a y e r between the two. Thus the e l e c t r o d e s were f a b r i c a t e d i n t h e C a r l Herrmann system, w i t h one T i and one AuGe source, by d e p o s i t i n g 50 nm T i on the Si02 and then 300 nm AuGe on t h e T i . * * S h i p l e y ' s S1400-30. Ghandhi [54] p. 59. -79- Once t h e metal had been d e p o s i t e d the p h o t o r e s i s t was removed by immersion i n a sequence o f s o l v e n t s f o r 5 min. each: hot (95°C) M i c r o s t r i p * * , hot acetone, and 2-propanol. F i g u r e 3.1 shows the i n t e r e l e c t r o d e gap o f a VIOWM. Here t h e gap i s 4 wide. 3.2.2 The Ridge VIOWM In a r i d g e VIOWM the r i d g e s must be formed f i r s t . E t c h i n g LiNb03 has been a c h i e v e d by s e v e r a l authors by s e v e r a l d i f f e r e n t methods i n c l u d i n g : argon i o n m i l l i n g [55], plasma e t c h i n g i n Freon plasmas u s i n g C F 4 [56], r e a c t i v e ion-beam e t c h i n g u s i n g C H F 3 [57], and r e a c t i v e i o n e t c h i n g and u s i n g C F 4 and C C I 2 F 2 [58]. In t h i s work the LiNb03 s u b s t r a t e was s p u t t e r e t c h e d i n an argon plasma. A f t e r t h e i n i t i a l c l e a n i n g t he wafer was p l a c e d i n the P e r k i n s - E l m e r s p u t t e r i n g system w i t h a T i source. The sou r c e was p r e c l e a n e d , by s p u t t e r e t c h i n g , w i t h a s h u t t e r between t h e t a r g e t and the sample u n t i l t he metal was b e i n g d e p o s i t e d . The plasma had a d i s t i n c t i v e b l u e t i n t when T i was b e i n g d e p o s i t e d , p r i o r t o which i t i s u n i f o r m l y p i n k . The p r e c l e a n u s u a l l y t a k e s between 1 and 2 h r . A p r o d u c t of P h i l i p A. Hunt Chemical C o r p o r a t i o n , West P a t e r s o n , N. J . -80- F i g u r e 3.1: The i n t e r e l e c t r o d e gap o f a VIOWM. Here t h e gap i s 4 pint w i d e . -81- The T i was d e p o s i t e d w i t h 100 W forward and about 2 W r e f l e c t e d power which gave a d e p o s i t i o n r a t e o f about 40 A/min. The d e p o s i t i o n was co n t i n u e d f o r 1.5 h r . g i v i n g a metal l a y e r about 500 nm t h i c k . The T i was formed i n t o s t r i p e s by a p p l y i n g a p h o t o r e s i s t mask t o p r o t e c t those areas t h a t were not t o be etch e d . The u n p r o t e c t e d T i was removed i n a C F 4 plasma i n a Plasma-Therm system. The chamber p r e s s u r e was 500 mTorr and t h e forward power was 100 W w i t h 2 W r e f l e c t e d . The e t c h time v a r i e d w i t h T i t h i c k n e s s and was u s u a l l y between 30 and 45 min. The p h o t o r e s i s t was a p p l i e d as d e s c r i b e d above f o r the p l a n a r VIOWM except t h a t the chlorobenzene soak was omit t e d . The remaining T i s t r i p e s were then o x i d i z e d i n a M i n i B r u t e oven at 600°C w i t h a 1 1/min. flow o f O2 f o r 12 hr . The oxide p r o v i d e d the mask f o r the s p u t t e r e t c h o f the LiNb03> The s p u t t e r e t c h was done i n the Perkin-Elmer w i t h 100 W forward and about 2 W r e f l e c t e d power at 18 mTorr. Here t o o t h e time was determined by the t h i c k n e s s o f the o x i d i z e d T i . As t h e T i has a d i f f e r e n t e t c h r a t e than the LiNb03, about h a l f , t he time was d e c i d e d upon by m o n i t o r i n g the d i f f e r e n t i a l e t c h r a t e . Before p l a c i n g the sample i n the A M u l t i v e r s i o n Plasma, R e a c t i v e Ion E t c h and Plasma D e p o s i t i o n System Model PK-1250PE/RIE/PD. -82- s p u t t e r e r t h e t h i c k n e s s o f the o x i d i z e d T i was measured on a Tencor Alpha-Step 200 p r o f i l o m e t e r . A f t e r the f i r s t 4 h r s . o f s p u t t e r e t c h i n g the sample was removed from the s p u t t e r e r and the new h e i g h t was measured. The sample was then r e t u r n e d t o the s p u t t e r e r and the e t c h i n g was co n t i n u e d f o r another 2 h r s . Then i t was removed and the h e i g h t was again measured. T h i s p r o c e s s was repeated u n t i l t h e r e was no i n c r e a s e i n t h e h e i g h t o f the r i d g e s . I f t h e h e i g h t o f the r i d g e was i n s u f f i c i e n t then a new l a y e r o f T i was d e p o s i t e d , p a t t e r n e d and o x i d i z e d and the r i d g e h e i g h t was i n c r e a s e d by f u r t h e r s p u t t e r e t c h i n g . T y p i c a l l y an i n i t i a l T i l a y e r o f 0.5 Mm r e s u l t e d i n an i n c r e a s e i n t h e r i d g e h e i g h t o f about 1.5 jim. Once t h e d e s i r e d r i d g e h e i g h t was ac h i e v e d and o p t i c a l b u f f e r l a y e r o f Si02 was d e p o s i t e d as d e s c r i b e d above f o r th e p l a n a r VIOWMs. F i g u r e 3.2 i s a scanning e l e c t r o n microscope (SEM) p i c t u r e o f a s p u t t e r e t c h e d r i d g e i n LiNb03- The width o f the r i d g e can be o b t a i n e d from f i g u r e 3.3 where i t i s seen t o be about 7.5 |im. F i g u r e 3.4 i s a photocopy o f the p r o f i l o m e t e r output o f the r i d g e shown i n f i g u r e s 3.2 and 3.3. The h e i g h t o f t h e r i d g e i s seen t o be 4 p . The e l e c t r o d e s f o r the r i d g e VIOWM were aluminum. They were s p u t t e r d e p o s i t e d w i t h a b i a s a p p l i e d t o the sample t a b l e i n an argon plasma at 18 mTorr. The b i a s causes some s p u t t e r - c l e a n i n g o f the sample's s u r f a c e which removes -83- F i g u r e 3.2: SEM p i c t u r e of a r i d g e etched i n LiNbC>3 • T n e s c a l e of the upper p i c t u r e i s 5 times t h a t o f the lower p i c t u r e . -84- -85- I fcft i 40 30!: 20 I. ! I /....! i J:;:: .tit 0 26 4b * ttr F i g u r e 3.4: P r o f i l o m e t e r o u t p u t s h o w i n g t h a t t h e h e i g h t o f t h e r i d g e i s 4 pun -86- adsorbed gases and which i n t u r n i n c r e a s e s the adhesion o f the aluminum. The forward power was 100 W and the r e f l e c t e d power was about 2 W. The t a r g e t was p r e c l e a n e d f o r an hour w i t h a s h u t t e r p l a c e d between the sample and the t a r g e t (with t h e b i a s a p p l i e d t o the sample t a b l e f o r o n l y the l a s t few m i n u t e s ) . Then the s h u t t e r was removed and A l was d e p o s i t e d f o r 2 h r s . The d e p o s i t e d metal l a y e r was between 1 and 2 pm t h i c k . The e l e c t r o d e s were formed by a s e l f - a l i g n e d t e c h n i q u e . P h o t o r e s i s t was spun onto the metal c o a t e d sample ( S h i p l e y ' s 1400-30 at 2000 rpm) at a low speed t o g i v e a t h i c k l a y e r i n the v a l l e y s between the r i d g e s and a t h i n l a y e r on the tops o f the r i d g e s . The p h o t o r e s i s t was h a r d baked f o r an hour at 130°C and e t c h e d i n the Plasma- Therm i n an oxygen atmosphere. The chamber p r e s s u r e was 320 mTorr and t h e forward power was 100 W and the r e f l e c t e d power was 2 W. E t c h i n g was c o n t i n u e d f o r about 50 min u n t i l the t h i n p h o t o r e s i s t was removed from the t o p s o f t h e r i d g e s forming a narrow, ~ 4 |im, gap i n the p h o t o r e s i s t . The sample was t h e n p l a c e d i n the Perkin-Elmer system and s p u t t e r e t c h e d i n an 18 mTorr Ar plasma u n t i l the aluminum a l o n g t h e t o p s o f the r i d g e s was removed (1/2 h r . l o n g e r than i t took t o d e p o s i t the A l ) . F i g u r e 3.5 i s a p i c t u r e of the aluminum e l e c t r o d e s f a b r i c a t e d by the s e l f - a l i g n e d technique d e s c r i b e d above. The c e n t r a l dark l i n e r u n ning from the top t o the bottom of the f i g u r e i s t h e t o p o f the LiNb03 r i d g e . -87 F i g u r e 3 . 5 : The aluminum e l e c t r o d e s o f a r i d g e VIOWM formed by t h e s e l f - a l i g n e d t e c h n i q u e . -88- 3.2.3 Cutting and Polishing The i n d i v i d u a l VIOWMs were cut from the s u b s t r a t e u s i n g e i t h e r a h i g h speed diamond saw or a wire saw. The c h o i c e o f t h e c u t t i n g method depended on the d e s i r e d y i e l d . When u s i n g the diamond saw the edge damage can be severe so t h a t t h e waveguides must be cut i n t o about 3 mm l o n g s e c t i o n s . U s i n g the wire saw the waveguides may be cut i n t o 2 mm lo n g s e c t i o n s . In bot h cases the f i n a l d e v i c e i s t o be about 1 mm l o n g . The l e n g t h o f the d e v i c e was kept s h o r t t o minimize the l o s s e s due t o a b s o r p t i o n , s c a t t e r i n g and o p t i c a l f i e l d / m e t a l e l e c t r o d e i n t e r a c t i o n s . The drawback o f a s h o r t d e v i c e i s t h a t t h e r e i s a r e l a t i v e l y l a r g e amount, o f c o u p l i n g due t o b u l k modes. Although a d e v i c e about 1 i n c h l o n g was f a b r i c a t e d most were between 1.6 mm and 0.8 mm. The p o l i s h i n g j i g c o n s i s t e d o f two p a r t s ; t h e main body and t h e p o l i s h i n g p l a t e . The main body had a l a r g e c e n t e r screw t h a t c o u l d be lowered t o p r o v i d e a backstop f o r the c r y s t a l b e i n g p o l i s h e d and a s e t screw t o h o l d t h e c e n t e r screw i n p l a c e . The p o l i s h i n g p l a t e c o n s i s t e d o f a s t a i n l e s s s t e e l p l a t e about 1 mm t h i c k (the t h i c k n e s s decreased w i t h e x c e s s i v e g r i n d i n g and p o l i s h i n g ) w i t h a 2 mm x 10 mm s l o t c ut i n t o i t s c e n t e r . F i g u r e 3.6 shows the p o l i s h i n g j i g : t h e main body and the p o l i s h i n g p l a t e . -89- F i g u r e 3.6: The p o l i s h i n g j i g : t h e main body ( r i g h t ) and t h e p o l i s h i n g p l a t e ( l e f t ) . -90- The samples were p o l i s h e d two a t a time. The f a c e s o f th e two b i t s o f c r y s t a l c o n t a i n i n g the VIOWMs were epoxied t o g e t h e r u s i n g a 5 minute epoxy. The sandwich thus formed was p r e s s e d t o g e t h e r t o remove excess epoxy from between the two. I t i s important t o reduce the t h i c k n e s s o f the epoxy l a y e r so as t o reduce the edge damage t h a t r e s u l t s d u r i n g t h e g r i n d i n g p r o c e s s . When t h e epoxy between the two samples had had time t o s e t t h e sandwich was epoxied i n t o the s l o t i n the p o l i s h i n g p l a t e o f the p o l i s h i n g j i g so t h a t i t p r o t r u d e d e q u a l l y from b o t h s i d e s . The p r o t r u d i n g LiNb03 was then ground, u s i n g 600 g r i t g r i n d i n g paper, u n t i l i t extended about a q u a r t e r o f a m i l l i m e t e r out from e i t h e r s i d e o f the p o l i s h i n g p l a t e . A second g r i n d i s done u s i n g 800 g r i t compound. P o l i s h i n g was done u s i n g a 1 nm alumina s l u r r y u n t i l t h e edge damage was removed. The edge o f the c r y s t a l was c o n t i n u a l l y monitored, u s i n g a microscope, t o determine when the edge damage has been removed. The f i n a l p o l i s h was done u s i n g a 0.05 alumina s l u r r y . F i g u r e 3.7 shows the p o l i s h e d end fa c e o f such a sandwich f o r two r i d g e waveguide samples f o l l o w i n g the 1 pm alumina p o l i s h . The l a r g e l i g h t areas at the top and bottom o f t h e p i c t u r e are the LiNbC>3 s u b s t r a t e s , the dark c e n t r a l r e g i o n i s the epoxy, the t h i n b r i g h t s t r i p e s are the aluminum e l e c t r o d e s . The c e n t r a l bump i s the LiNb03 r i d g e . -91- The r i d g e h e i g h t i s about 0.4 times the width i n t h i s f i g u r e . One can c l e a r l y see t h a t t h e aluminum e l e c t r o d e s c l i m b up t h e s i d e s o f the r i d g e . A l s o the r e s u l t s o f edge damage can be seen on the upper s u b s t r a t e . Once bot h f a c e s o f both VIOWMs have been p o l i s h e d they were removed from the p o l i s h i n g p l a t e by soaking them i n hot M i c r o s t r i p f o r about 5 min. Once they had been removed from the p o l i s h i n g p l a t e they were again soaked i n hot M i c r o s t r i p u n t i l t h e y came apar t , about 1 h r . F i g u r e 3.8 shows a p l a n a r d e v i c e t h a t had been p o l i s h e d on b o t h ends th e i n t e r e l e c t r o d e gap i s seen t o run from one end t o t h e o t h e r . T h i s d e v i c e was about 1.6 mm l o n g . F i g u r e 3.9 shows the end o f the d e v i c e where the i n t e r e l e c t r o d e gap was p e r p e n d i c u l a r t o the p o l i s h e d end f a c e . F i n a l l y t h e excess LiNb03 c o u l d be removed from e i t h e r s i d e o f t h e d e v i c e by mounting i t on a g l a s s microscope s l i d e and c u t t i n g i t u s i n g a low speed diamond saw. 3.3 T h e S i l i c o n V - g r o o v e s The use o f V-grooves a n i s o t r o p i c a l l y etched i n S i s u b s t r a t e s [59,60] f o r the alignment of o p t i c a l f i b e r s w i t h i n - d i f f u s e d waveguides i n LiNb03 p r o v i d e s a p r a c t i c a l method o f a c h i e v i n g e f f i c i e n t b u t t - c o u p l i n g . The " f l i p - c h i p " -92- F i g u r e 3 . 7 : The endface o f two r i d g e VIOWMs, e p o x i e d t o g e t h e r , a f t e r a l p a l u m i n a p o l i s h . - 9 3 F i g u r e 3 . 8 : A p l a n a r VIOWM w i t h t h e ends p o l i s h e d . -94- F i g u r e 3.9: The p o l i s h e d end o f a p l a n a r VIOWM where the i n t e r e l e c t r o d e gap i s seen t o run p e r p e n d i c u l a r t o t h e e n d f a c e . -95- method o f c o u p l i n g between o p t i c a l f i b e r s and Ti:LiNbC>3 waveguides was f i r s t r e p o r t e d by Hsu and M i l t o n i n 1976 [21]. In t h i s method a V-groove i s etched i n a (100) p-type s i l i c o n wafer. The depth o f the V-groove i s c o n t r o l l e d so t h a t an o p t i c a l f i b e r l y i n g i n the V-groove has i t s core p o s i t i o n e d a t a h e i g h t above the unetched s u r f a c e o f the wafer t h a t w i l l r e s u l t i n e f f i c i e n t c o u p l i n g o f l i g h t t o an i n t e g r a t e d o p t i c d e v i c e . The depth c o n t r o l i s o b t a i n e d by t a k i n g advantage o f the a n i s o t r o p y o f the e t c h r a t e o f S i which i s about 35 times g r e a t e r i n t h e (100) d i r e c t i o n than i t i s i n t h e (111) d i r e c t i o n f o r the etchant d e s c r i b e d below. E l e c t r o d e s are formed on the unetched p o r t i o n s o f th e wafer t o p r o v i d e e l e c t r i c a l c o n t a c t w i t h the e l e c t r o d e s o f t h e i n t e g r a t e d o p t i c s d e v i c e which i s p l a c e d "upside down" on t h e wafer c o n t a i n i n g the V-grooves. In t h i s way th e o p t i c a l f i b e r and i n t e g r a t e d o p t i c s d e v i c e are l o c a t e d so as t o f a c i l i t a t e a h i g h degree o f o v e r l a p between the o p t i c a l f i e l d s i n b o t h waveguides when they are b u t t e d t o g e t h e r . The v a r i a b l e s c r i t i c a l t o the v e r t i c a l alignment o f an o p t i c a l f i b e r and an i n t e g r a t e d waveguide, by t h i s method, are t h e o u t e r diameter o f the f i b e r , t he c o r e - c l a d d i n g c o n c e n t r i c i t y and the V-groove depth. I t would be necessary t o c o n t r o l each o f thes e v a r i a b l e s to.sub-micron t o l e r a n c e s ( l e s s t han 1% i n each case) i f one were t o p r e d i c t a b l y -96- o b t a i n a c e r t a i n degree o f c o u p l i n g , t h e r e f o r e i t i s d e s i r a b l e t o have a method o f v a r y i n g the h e i g h t o f the f i b e r above t h e s u r f a c e o f the S i wafer. Sheem and G i a l l o r e n z i [61] accomplished t h i s u s i n g a t a p e r e d f i b e r i n a second, deeper V-groove. In t h i s method the h e i g h t o f the f i b e r b e i n g a l i g n e d was c o n t r o l l e d by s l i d i n g the second t a p e r e d f i b e r i n the deeper V-groove beneath i t . In the p r e s e n t work an a r r a y o f 13 V-grooves was used each o f which i s o f a d i f f e r e n t depth, i n c r e a s i n g by 1 nm from one groove t o the next. The f a b r i c a t i o n o f the V-grooves c o n s i s t s o f 6 b a s i c s t e p s : o x i d e growth, e l e c t r o d e f a b r i c a t i o n , oxide s p u t t e r i n g , V-groove window f a b r i c a t i o n , V-groove e t c h i n g , and oxide removal. In o r d e r t o get r e p e a t a b l e r e s u l t s i t i s n e c e s s a r y t o s t a r t w i t h a c l e a n wafer. T h e r e f o r e the f a b r i c a t i o n was preceded by c l e a n i n g t h e wafer u s i n g the w e l l known RCA p r o c e s s [62] . A t h e r m a l oxide was grown on the wafer f o r two reasons. P r i m a r i l y i t i s not a t t a c k e d by the etchant used t o c r e a t e t h e V-grooves, t h e r e f o r e i t can a c t as the V-groove mask f i x i n g t h e w i d t h o f the windows and thereby the depth o f the V-grooves, and s e c o n d a r i l y i t serves t o i s o l a t e the e l e c t r o d e s e l e c t r i c a l l y from the s u b s t r a t e . The oxide was grown t o a t h i c k n e s s o f 500 nm. To do t h i s an oven was heated t o 1100°C w i t h an 0 2 flow o f 1 1/min. The s l i c e s -97- were i n t r o d u c e d t o the oven and allowed t o heat up f o r 5 min. A flow o f 1.6 1/min. o f H2 was s t a r t e d g i v i n g an O2+H2 "wet" atmosphere. Wet growth i s approximately an or d e r o f magnitude f a s t e r t h a t dry growth ( 0 2 only) [63]. A f t e r 80 min. t h e H 2 flow was stopped and 5 min. a f t e r t h a t the O2 flow was a l s o stopped. There f o l l o w e d a 20 min flow o f N2 at 1 1/min. The e l e c t r o d e s on the V-grooves were formed u s i n g a l i f t - o f f t e c h n i q u e t h a t was i d e n t i c a l t o t h a t used t o form the e l e c t r o d e s o f the p l a n a r VIOWM ( s e c t i o n 3.2.1). The reason t h a t t h e e l e c t r o d e s were formed at t h i s stage was because i f the V-grooves were formed f i r s t t he a p p l i c a t i o n o f p h o t o r e s i s t was uneven i n the i n t e r g r o o v e r e g i o n s making the l i f t - o f f procedure u n c e r t a i n . A second l a y e r o f Si02 was d e p o s i t e d at t h i s p o i n t t o p r o t e c t t h e e l e c t r o d e s from t h e V-groove e t c h a n t . T h i s l a y e r was d e p o s i t e d i n the Per k i n s - E l m e r s p u t t e r i n g system and was about 700 nm t h i c k . Windows were etc h e d i n the Si02- P h o t o r e s i s t was a p p l i e d t o cover those areas t h a t were not t o be etch e d ( i n c l u d i n g t he back o f the waf e r ) . The Si02 was then etched i n b u f f e r e d HF f o r 20 min. f o l l o w e d by a r i n s e i n DI water. The HF e t c h was in t e n d e d t o be a b i t t oo s h o r t t o completely e t c h t h e grown oxide. T h i s was t o a v o i d over e t c h i n g which would r e s u l t i n widening the V-grooves. The f i n a l l a y e r was removed by a 20 min. C F 4 plasma e t c h performed as d e s c r i b e d -98- i n s e c t i o n 3.2.2 f o r the T i e t c h . F i n a l l y the p h o t o r e s i s t vr?>.s removed. The V-grooves were etched i n a s o l u t i o n o f KOH (20% by wei g h t ) , DI water (64%) and 2-propanol (16%). The s o l u t i o n was heated t o 85°C i n a beaker w i t h a condenser. The wafer was immersed f o r 85 min. The wafer was then removed and was blown d r y w i t h N 2. The s p u t t e r e d oxide l a y e r p r o t e c t i n g the e l e c t r o d e s was then removed by e t c h i n g f o r about 5 min. i n b u f f e r e d HF. T h i s was f o l l o w e d by a r i n s e i n DI water and blow d r y i n g w i t h N 2. F i g u r e 3.10 d e p i c t s the e n t i r e V-groove f a b r i c a t i o n p r o c e s s . S i n c e t h e d e v i c e was t o a c t as a f r o n t - e n d switch, i n which i t i s p l a c e d between a l a s e r and an o p t i c a l f i b e r , c o n t r o l l i n g t h e l i g h t c o u p l e d from t h e l a s e r t o the f i b e r by the a p p l i c a t i o n o f v o l t a g e t o i t s e l e c t r o d e s , t h e V-grooves were d e s i g n e d so t h a t the VIOWM would s l i g h t l y overhang one end o f t h e S i wafer w i t h the V-groove on the o t h e r s i d e . 3.4 D e v i c e / O p t i c a l F i b e r A l i g n m e n t The c o u p l i n g between a VIOWM and an o p t i c a l f i b e r was d i s c u s s e d i n s e c t i o n 2.6. I t was p o i n t e d out t h a t the t h e o r e t i c a l development o f the c o u p l i n g c o e f f i c i e n t was -99- (a) Prepare Clean Si Vafer (b) Thermal Oxide Growth (c) Metal Deposition and Patterning (d) Sputtered Oxide Deposition (e) Oxide Patterning and Etch (f) V-groove Etch (g) Protective Oxide Removal Legend: S i 2 Metal F i g u r e 3.10: The V-groove f a b r i c a t i o n p r o c e s s . -100- i n s e n s i t i v e t o whether l i g h t was coupled from the VIOWM t o the f i b e r o r from the f i b e r t o the VIOWM. T h e r e f o r e i t was p o s s i b l e t o a l i g n t he d e v i c e w i t h the l i g h t p r o p a g a t i n g i n the r e v e r s e d i r e c t i o n from t h a t i n t e n d e d f o r the a c t u a l o p e r a t i o n i . e . w i t h l i g h t from the f i b e r c o u p l e d i n t o t he VIOWM. F i r s t a l e n g t h o f o p t i c a l f i b e r was pre p a r e d by removing the p l a s t i c c o a t i n g from both ends and then c l e a v i n g them. The output o f a l a s e r was then f o c u s e d on one o f t h e c l e a v e d end fa c e s w h i l e m o n i t o r i n g the output at the o t h e r . C l a d d i n g modes were removed by a s e r i e s o f t i g h t , about 1 i n c h i n diameter, loops i n the f i b e r . In the case t h a t the f i b e r was multimode, h a v i n g been designed f o r monomode o p e r a t i o n at a lo n g e r wavelength, these loops a l s o s e r v e d t o s t r i p t he h i g h e r order guided modes. The back, o f the sample VIOWM was cemented t o a p i n which was a t t a c h e d t o the boom o f a crane mounted on a two a x i s p i e z o e l e c t r i c m i c r o p o s i t i o n e r . The sample hung upside down w i t h i t s e l e c t r o d e s f a c i n g t he f l o o r . The S i wafer c o n t a i n i n g the V-grooves was mounted r i g h t s i d e up, so t h a t i t s e l e c t r o d e s were f a c i n g t he c e i l i n g , on a stage t h a t c o u l d be heated t o the e u t e c t i c temperature o f AuGe. The s u r f a c e s o f the wafer and the VIOWM were a l i g n e d by p l a c i n g t he two i n c o n t a c t and h e a t i n g the stage j u s t enough t o s o f t e n the cement h o l d i n g the VIOWM t o the p i n . -101- The output end o f the o p t i c a l f i b e r was l a i d i n a V- groove and a s m a l l weight was p l a c e d atop i t h o l d i t i n p l a c e . The sample VIOWM was p o s i t i o n e d so as t o be i n c o n t a c t w i t h t he end o f the f i b e r . A s i g n a l was a p p l i e d t o the e l e c t r o d e s on the wafer c o n t a i n i n g t he V-grooves and the sample's p o s i t i o n was a d j u s t e d u n t i l i t was f e l t t h a t t he be s t c o u p l i n g f o r t h a t groove had been o b t a i n e d . I f i t was nec e s s a r y t h e s u r f a c e s were r e a l i g n e d by h e a t i n g f o l l o w i n g r o t a t i o n s . T h i s procedure was repe a t e d from groove t o groove u n t i l t h e groove g i v i n g t he b e s t c o u p l i n g was found. Once t h e b e s t c o u p l i n g was o b t a i n e d a permanent bond was formed by h e a t i n g t he stage t o t h e e u t e c t i c temperature o f AuGe, 363°C. Dur i n g t he h e a t i n g c y c l e t he cement between the VIOWM and the p i n was o x i d i z e d and the two were e a s i l y s e p a r a t e d a f t e r t h e stage had c o o l e d . The bond was r e i n f o r c e d by t h e a p p l i c a t i o n o f d r o p l e t s o f c y a n o - a c r y l a t e g l u e t o b o t h s i d e s o f the c r y s t a l . The f i b e r was a l s o bonded i n p l a c e by a p p l y i n g a drop o f the same glu e on t h e f i b e r a t t h e f a r end o f the V-groove away from t h e b u t t c o u p l e . F i g u r e 3.11 shows the V-groove a r r a y , f i b e r , VIOWM, p i n and probes d u r i n g t he alignment procedure. F i g u r e 3.12 shows the V-groove a r r a y , f i b e r , VIOWM, probes and i n p u t o b j e c t i v e a f t e r the alignment and permanent bonding procedure. F i g u r e 3.11: The V - g r o o v e a r r a y , f i b e r , VIOWM, p i n , and p r o b e s d u r i n g t h e a l i g n m e n t p r o c e d u r e . -103- F i g u r e 3.12: The V-groove a r r a y , f i b e r , VIOWM, prob e s , and i n p u t o b j e c t i v e a f t e r t h e a l i g n m e n t and permanent bo n d i n g p r o c e d u r e . -104- The c r y s t a l s shown i n f i g u r e s 3.11 and 3.12 are about 1 mm on a s i d e . -105- C h a p t e r 4 RESULTS 4.1 I n t r o d u c t i o n The r e s u l t s , b o t h c a l c u l a t e d from the model and measured, f o r the p l a n a r VIOWM are p r e s e n t e d i n s e c t i o n 4.2 and t h e r e s u l t s f o r the r i d g e VIOWM are p r e s e n t e d i n s e c t i o n 4.3. S e c t i o n 4.4 c o n t a i n s the r e s u l t s o f measurements on the VIOWM permanently mounted on the S i V-grooves w i t h an o p t i c a l f i b e r a t t a c h e d . F i n a l l y s e c t i o n 4.5 p r o v i d e s a d i s c u s s i o n o f t h e v a r i o u s r e s u l t s o b t a i n e d . To p r e d i c t t h e b e h a v i o r o f the VIOWM d e v i c e w i t h o p t i c a l f i b e r s i t was assumed t h a t the f i b e r s have c i r c u l a r l y symmetric o p t i c a l f i e l d d i s t r i b u t i o n s and t h a t the nominal w i d t h parameters were w v f = wzf = 1.5 nm.* This was close to the value given to us by the engineers at MacDonald Dettwiler and Assoc. for the width of the mode of the o p t i c a l f i b e r they use i n the FIRE 900 0 b/w o p t i c a l image recorder. They measured the h a l f width at 1/e of the peak power to be - 2 nm. -106- Other f i b e r s c o u l d , o f course, support modes w i t h d i f f e r e n t mode widths. The n u m e r i c a l i n t e g r a t i o n s were done u s i n g a d a p t i v e t e c h n i q u e s f o r which the accuracy c o u l d be s p e c i f i e d . Turbo P a s c a l was used on an IBM AT compatible w i t h an 80287 math co p r o c e s s o r . The i n t e g r a t i o n r o u t i n e s were taken from the Turbo P a s c a l Numerical Methods Toolbox. The accuracy was s p e c i f i e d t o be t o e i g h t s i g n i f i c a n t f i g u r e s . The accuracy o f the r o u t i n e s were t e s t e d on Hermite-Gaussian f u n c t i o n s and found t o be w i t h i n the s p e c i f i e d t o l e r a n c e . The l i m i t s o f i n t e g r a t i o n were s e t t o 5 times the width parameter o f t h e t r i a l f u n c t i o n i n both the y and z - d i r e c t i o n s . T h i s ensured t h a t the e r r o r i n c u r r e d by n e g l e c t i n g t h a t p a r t o f the i n t e g r a l t h a t was o u t s i d e o f t h e l i m i t s o f i n t e g r a t i o n would be l e s s than the t o l e r a n c e . 4 .2 The P l a n a r VIOWM The c a l c u l a t e d r e s u l t s from the model f o r t h e p l a n a r VIOWM are p r e s e n t e d i n s e c t i o n 4.2.1 and the measured r e s u l t s a re p r e s e n t e d i n s e c t i o n 4.2.2. 4.2.1 Calculated Results F i g u r e s 4.1 and 4.2 are t o p o g r a p h i c a l p l o t s o f t h e w i d t h parameters, Wy-^ and w z v, as f u n c t i o n s o f the a p p l i e d v o l t a g e and i n t e r e l e c t r o d e gap width f o r a VIOWM f o r l i g h t w i t h a wavelength o f 442 nm. F i g u r e s 4.3 and 4.4 are p l o t s o f t h e w i d t h parameters f o r 633nm r a d i a t i o n . These r e s u l t s i n d i c a t e t h a t t h e width parameters decrease w i t h i n c r e a s i n g v o l t a g e , w i t h d e c r e a s i n g gap width and w i t h d e c r e a s i n g wavelength. In o t h e r words the o p t i c a l f i e l d d i s t r i b u t i o n o f a g u i d e d mode becomes more h i g h l y c o n f i n e d t o the h i g h r e f r a c t i v e index r e g i o n i n the v i c i n i t y o f the i n t e r e l e c t r o d e gap as one a p p l i e s an i n c r e a s i n g v o l t a g e t o t h e e l e c t r o d e s and t h i s e f f e c t i s g r e a t e r f o r s m a l l e r gaps as w e l l as a t s h o r t e r wavelengths. The f a c t t h a t i n c r e a s i n g v o l t a g e w i l l r e s u l t i n h i g h e r o p t i c a l confinement i s t o be expected. As the change i n the r e f r a c t i v e index d i s t r i b u t i o n due t o the a p p l i c a t i o n o f v o l t a g e t o t h e e l e c t r o d e s becomes l a r g e r the t u r n i n g p o i n t o f t h e mode moves c l o s e r t o the i n t e r e l e c t r o d e gap. T h i s i s because th e change i n d i r e c t i o n o f a ray i s p r o p o r t i o n a l t o t h e g r a d i e n t o f the r e f r a c t i v e index d i s t r i b u t i o n [19] and t h e change i n the r e f r a c t i v e index i s d i r e c t l y p r o p o r t i o n a l t o t h e a p p l i e d v o l t a g e . T h i s argument i s e q u a l l y w e l l a p p l i e d t o t h e i n c r e a s e i n confinement w i t h d e c r e a s i n g -108- C o n . t o u r s o f c o n s t a n t w y v F i g u r e 4.1: A t o p o g r a p h i c a l p l o t of w y v f o r XQ = 442 nm. -109- C o n t o u r s o f c o n s t a n t w 2 3 4 5 6 G a p W i d t h (/bin) F i g u r e 4.2: A t o p o g r a p h i c a l p l o t o f w z v f o r XQ = 442 nm. -110- C o n t o u r s o f c o n s t a n t w y v G a p W i d t h (yun ) F i g u r e 4.3: A t o p o g r a p h i c a l p l o t of w y v f o r J i Q = 633 nm. - I l l - C o n t o u r s o f c o n s t a n t w Z V CD CP D o > CD Q_ CL < 5 0 4 0 3 0 — 20 1 0 • 10 • 12 14- 2 3 6 • 10- •12- 14- 6 G a p W i d t h (/un) F i g u r e 4.4: A t o p o g r a p h i c a l p l o t of w z v f o r XQ = 633 nm. -112- i n t e r e l e c t r o d e gap width f o r t he e l e c t r i c f i e l d i n the s u b s t r a t e i s i n v e r s e l y p r o p o r t i o n a l t o the gap width. A l s o t h e change i n t h e r e f r a c t i v e index d i s t r i b u t i o n i s d i r e c t l y p r o p o r t i o n a l t o the cube o f the r e f r a c t i v e index o f the s u b s t r a t e which i n LiNbC-3 tends t o i n c r e a s e as one approaches t h e a b s o r p t i o n edge [64]. The l o c a t i o n o f the f i b e r r e l a t i v e t o the inp u t o f the VIOWM i s an important f a c t o r i n de t e r m i n i n g the c o u p l i n g c o e f f i c i e n t . Here, i n any p r a c t i c a l s w i t c h i n g a p p l i c a t i o n t h e l o c a t i o n o f the o p t i c a l f i b e r w i l l depend on t h e i n t e n d e d a p p l i c a t i o n and the range o f a p p l i e d v o l t a g e s . The optimum p o s i t i o n f o r t he f i b e r i n the z - d i r e c t i o n w i l l always be a t b = 0. However, the l o c a t i o n o f the f i b e r i n th e y - d i r e c t i o n must be determined by the a p p l i c a t i o n . In t h e case t h a t t he int e n d e d a p p l i c a t i o n o f the VIOWM i s as a d i g i t a l s w i t c h s i t u a t e d between two single-mode o p t i c a l f i b e r s one might want t o ensure t h a t t he maximum amount o f power i s t r a n s f e r r e d from the in p u t f i b e r t o the output f i b e r f o r a p a r t i c u l a r a p p l i e d v o l t a g e or one might want t o ensure t h a t t h e output power i s i n s e n s i t i v e t o s l i g h t v a r i a t i o n s i n the a p p l i e d v o l t a g e . Both o f thes e s i t u a t i o n s can be ac h i e v e d by j u d i c i o u s l y c hoosing the f i b e r l o c a t i o n . In f i g u r e s 4.5 and 4.6 are p l o t t e d t he c o u p l i n g c o e f f i c i e n t T as a f u n c t i o n o f a p p l i e d v o l t a g e . F o r these f i g u r e s i t has been assumed t h a t t he o p t i c a l f i b e r s are -113- l o c a t e d so as t o maximize the power t r a n s f e r f o r a p p l i e d v o l t a g e s o f 30 V and 50 V r e s p e c t i v e l y . In t h i s case the VIOWM i s assumed t o have a 2 nm i n t e r e l e c t r o d e gap and the wavelength o f t h e l i g h t i s taken t o be 442 nm. I t i s apparent from both f i g u r e s t h a t a g r e a t e r degree o f c o u p l i n g w i l l o c c u r a t v o l t a g e s g r e a t e r than e i t h e r 30 V, i n f i g u r e 4.5, o r 50 V, i n f i g u r e 4.6, however the c o u p l i n g a c h i e v e d a t t h e s e v o l t a g e s are the g r e a t e s t t h a t may be ac h i e v e d f o r t h e s e v o l t a g e s . In f i g u r e s 4.7 and 4.8 i s p l o t t e d the c o u p l i n g c o e f f i c i e n t T as a f u n c t i o n o f v o l t a g e f o r t he o p t i c a l f i b e r s p o s i t i o n e d so t h a t t he maximum c o u p l i n g t h a t can be o b t a i n e d f o r 30 V and 50 V a p p l i e d t o the e l e c t r o d e s f o r an i n t e r e l e c t r o d e gap width o f 4 nm and 442 \xm l i g h t . In f i g u r e 4.5 the a c t u a l peak v a l u e o f the c o u p l i n g o c c u r s a t - 37 V r a t h e r than 30 V. T h i s i n d i c a t e s t h a t i f th e d e v i c e were o p e r a t e d at 37 V then the c o u p l i n g c o e f f i c i e n t would v a r y by o n l y a few pe r c e n t f o r a l a r g e r change i n t h e a p p l i e d v o l t a g e . A l s o f i g u r e s 4.5 and 4.7 show t h a t f o r a p p l i e d v o l t a g e s o f - 3 7 V and 43 V r e s p e c t i v e l y t he c o u p l i n g c o e f f i c i e n t T i s s t a t i o n a r y . The c o u p l i n g c o e f f i c i e n t remains w i t h i n 5% o f i t s maximum v a l u e f o r changes exceeding ±7 v o l t s i n f i g u r e 4.5 and f o r even g r e a t e r v a r i a t i o n s i n f i g u r e 4.7. The e f f e c t d e s c r i b e d above would a l s o be u s e f u l i n r e n d e r i n g t h e VIOWM l e s s s e n s i t i v e t o changes i n t h e F i g u r e 4.5: The c o u p l i n g c o e f f i c i e n t T v s . v o l t a g e f o r g |im where the c o u p l i n g at 30 V i s maximized. -115- F i g u r e 4.6: The c o u p l i n g c o e f f i c i e n t T v s . v o l t a g e f o r g = 2 jim where the c o u p l i n g at 50 V i s maximized. F i g u r e 4.7: The c o u p l i n g c o e f f i c i e n t T vs. v o l t a g e f o r g |im where the c o u p l i n g at 30 V i s maximized. -117- F i g u r e 4.8: The c o u p l i n g c o e f f i c i e n t T v s . v o l t a g e f o r g = 4 um where the c o u p l i n g at 50 V i s maximized. -118- r e f r a c t i v e index p r o f i l e brought about by the p h o t o r e f r a c t i v e e f f e c t . In f i g u r e 4.9 are p l o t t e d t he optimum f r a c t i o n o f power t r a n s f e r t h a t can be o b t a i n e d f o r a p l a n a r VIOWM f o r l i g h t w i t h a wavelength o f 442 nm as a f u n c t i o n o f both v o l t a g e and i n t e r e l e c t r o d e gap width. The f r a c t i o n o f power t r a n s f e r i s g i v e n by T . From t h i s f i g u r e one can see t h a t i t i s p o s s i b l e t o o b t a i n -3dB power t r a n s f e r between an o p t i c a l f i b e r t h a t supports a mode w i t h width parameters W y f = w Zy = 1.5 |im and a VIOWM w i t h an a p p l i e d v o l t a g e o f only 20 V. E x t r a p o l a t i n g t he curves p r e s e n t e d i n f i g u r e 4.9 shows t h a t a t s m a l l e r i n t e r e l e c t r o d e gap widths i t would be p o s s i b l e t o o b t a i n even b e t t e r power t r a n s f e r . C a l c u l a t i o n s f o r i n t e r e l e c t r o d e gap widths l e s s than 2 yaa were not performed because a b u f f e r l a y e r t h i c k n e s s o f 5% o f the i n t e r e l e c t r o d e gap width would be l e s s than 1000 A which would not r e s u l t i n s u f f i c i e n t a t t e n u a t i o n o f the evanescent f i e l d o f t h e guid e d modes. F i n a l l y t h e use o f the VIOWM as a s m a l l s i g n a l l i n e a r modulator between two o p t i c a l f i b e r s i s c o n s i d e r e d . I f the power i s t o be modulated i n a l i n e a r f a s h i o n then the power i n t h e output f i b e r , Pout' w i l l be' r e l a t e d t o the power i n the i n p u t f i b e r , P i n , by * See c h a p t e r 2 s e c t i o n 2.3.1.2. C o n t o u r s o f c o n s t a n t T F i g u r e 4.9: The optimum power t r a n s f e r T as a f u n c t i o n o a p p l i e d v o l t a g e and i n t e r e l e c t r o d e gap width. -120- P = TT T .P. out xn out xn where T 2 i n i s t h e f r a c t i o n o f power t r a n s f e r t o the VIOWM at the i n p u t and s i m i l a r l y T 2 o u t i s the f r a c t i o n o f power t r a n s f e r from t h e VIOWM at the output. The i n p u t and output f i b e r s c o u l d , o f course, be l o c a t e d independently. F o r example one c o u l d be l o c a t e d so t h a t t he v a r i a t i o n i n the power t r a n s f e r was minimized f o r a p a r t i c u l a r v o l t a g e and the o t h e r c o u l d be l o c a t e d at the p o i n t o f s t e e p e s t descent f o r t he same v o l t a g e . Another p o s s i b i l i t y c o u l d be t o l o c a t e b o t h o f the f i b e r s t o have the same amount o f power t r a n s f e r at t h e same v o l t a g e i n which case the r e l a t i o n between t h e output power and the i n p u t power becomes p . - T 4 P . out i n • where T i s t h e c o u p l i n g c o e f f i c i e n t at e i t h e r the i n p u t or the output. In f i g u r e s 4.10 and 4.11 are p l o t t e d T 4 f o r the cases i n which both o p t i c a l f i b e r s are l o c a t e d so as t o ac h i e v e t h e optimum c o u p l i n g c o e f f i c i e n t f o r a VIOWM w i t h a 2 nm i n t e r e l e c t r o d e gap f o r a wavelength o f 442 nm when 30 V and 50 V a p p l i e d t o the e l e c t r o d e s r e s p e c t i v e l y . As can be seen b o t h f i g u r e s i n d i c a t e t h a t t h e r e i s a l i n e a r r e g i o n i n the t r a n s f e r c u r v e s . In f i g u r e 4.10 the l i n e a r r e g i o n appears t o extend from an a p p l i e d v o l t a g e o f - 22 V t o - 26 V i n which the f r a c t i o n o f the power t r a n s f e r r e d goes from - 0.25 t o ~ 0.55. In t h i s case the a p p r o p r i a t e b i a s v o l t a g e -121- would be - 24 V w i t h a dynamic range o f 4 V. S i m i l a r l y i n f i g u r e 4.11 the l i n e a r range extends from - 25 V t o - 35 V and the f r a c t i o n o f power t r a n s f e r r e d goes from - 0.2 t o ~ 0.6. Here an a p p r o p r i a t e b i a s v o l t a g e would be - 30 V w i t h a dynamic range o f 10 V. 4.2.2 Measured Results The apparatus ( f i g u r e 4.12) c o n s i s t e d o f a l a s e r , a 40x o b j e c t i v e , an o p t i c a l f i b e r , the VIOWM under t e s t , a lOx o b j e c t i v e ( a c t i n g as a p r o j e c t o r ) , a p o l a r i z e r , a chopper, and an a p e r t u r e and a d e t e c t o r . The l i g h t , K0 = 442 nm, was c o u p l e d i n t o t h e f i b e r , w i t h c l e a v e d ends, u s i n g t h e 4Ox o b j e c t i v e . The l i g h t was p o l a r i z e d i n two p l a n e s at r i g h t a n g l e s . The f i b e r was wrapped on a s p o o l . I t was l o n g enough f o r any c l a d d i n g modes t o be s t r i p p e d . A loop, about 1 i n c h i n diameter, was put i n the f i b e r j u s t b e f o r e the output end t o remove any l o o s e l y guided modes. T h i s g i v e s an output spot c o n s i s t i n g p r i m a r i l y o f t h e lowest order H E ^ mode o f the f i b e r . The p o l a r i z a t i o n s t a t e at the output o f the f i b e r was not known; however, upon m o n i t o r i n g the output o f the f i b e r w i t h a p o l a r i z e r p l a c e d between the output and a d e t e c t o r the power was seen t o v a r y over time by o n l y a few p e r c e n t ( f i g u r e 4.13). The f i b e r was mounted on a t h r e e a x i s m i c r o p o s i t i o n e r w i t h r o t a t i o n i n the v e r t i c a l and -122- A p p l i e d V o I t a g e F i g u r e 4.10: The power t r a n s f e r T between two o p t i c a l f i b e r s where the c o u p l i n g at 30 V i s maximized. -123- F i g u r e 4.11: The power t r a n s f e r T between two o p t i c a l f i b e r s where the c o u p l i n g at 50 V i s maximized. -124- h o r i z o n t a l p l a n e s and thus c o u l d be a l i g n e d w i t h t he i n p u t f a c e o f t h e VIOWM under t e s t . The stage h o l d i n g the VIOWM was r i g i d . The output o b j e c t i v e was a l s o mounted on a t h r e e a x i s m i c r o p o s i t i o n e r . The p o l a r i z e r was p l a c e d between the output o b j e c t i v e and the ap e r t u r e as was the chopper. When l i g h t a t the output o f a VIOWM i s focus e d upon an ap e r t u r e t h e power t r a n s f e r r e d t o the d e t e c t o r w i l l depend on the shape and s i z e o f the ap e r t u r e and the shape and s i z e o f t h e p r o j e c t e d o p t i c a l f i e l d d i s t r i b u t i o n a t t h e a p e r t u r e . I f t he a p e r t u r e i s c i r c u l a r then the o v e r l a p i s t h a t between t h e o p t i c a l f i e l d d i s t r i b u t i o n o f the guided mode and the r e c t ( r / 2 c ) f u n c t i o n . S i n c e t he t r a n s m i s s i o n o f the r e c t f u n c t i o n i s 1 i n the r e g i o n o f the t o t a l t r a n s m i s s i o n and 0 otherwise i t s e f f e c t i s t o s e t l i m i t s on the r e g i o n o f the p r o j e c t e d image t h a t i s i n t e r r o g a t e d . I f the a p e r t u r e i s l a r g e enough i t w i l l pass the m a j o r i t y o f the power i n the p r o j e c t e d image o f the o p t i c a l f i e l d d i s t r i b u t i o n f o r s u f f i c i e n t l y h i g h v o l t a g e s and i t s e f f e c t on t h e measurements may re a s o n a b l y be i g n o r e d . I t i s , however, i n c l u d e d as i t b l o c k s much o f the power i n the bu l k modes which would otherwise be c o l l e c t e d by the d e t e c t o r . S t i l l some o f t h e power i n the bu l k modes i s c o l l e c t e d r e s u l t i n g A r e c t f u n c t i o n i s a two di m e n s i o n a l step f u n c t i o n . The r e c t f u n c t i o n i s equal t o 1 f o r r £ c and 0 otherwise where c i s the r a d i u s o f the a p e r t u r e . See C o l l i e r , B u r c k h a r d t , and L i n [65] p. 94. T h i s f u n c t i o n i s a l s o r e f e r r e d t o as the c i r c ( r ) f u n c t i o n by Goodman [66] p. 14. -125- 40x Ob j e c i i ve / M l c r o p o s i t l o n e r s F i g u r e 4.12: The b a s i c l a b o r a t o r y apparatus used t o make measurements on VIOWMs. -126- l i g h t l e v e l 10 o to u to c dark l e v e l 7 n i n u t e s F i g u r e 4.13: The p o l a r i z e d output of the o p t i c a l f i b e r . -127- i n nonzero c o u p l i n g when zero v o l t s are a p p l i e d t o the VICVTMs e l e c t r o d e s . The d e t e c t o r was a was a LeCroy Fibercom Analog R e c e i v e r FAR-4HS. I t s g a i n was a d j u s t e d so t h a t 100 nW of i n p u t power corresponded t o 1 V at the output. B e f o r e t a k i n g any measurements the f i b e r was p o s i t i o n e d so t h a t i t s output spot passed though the bu l k o f the c r y s t a l and was fo c u s e d on t h e d e t e c t o r a f t e r p a s s i n g through the p o l a r i z e r and chopper. The peak v a l u e o f the power was then recorded. T h i s " s t r a i g h t through" power c o u l d then be used t o c a l c u l a t e t he c o u p l i n g e f f i c i e n c y t o the waveguide. A l l o f t h e t e s t s were performed on VIOWMs w i t h a 4fim i n t e r e l e c t r o d e gap. In f i g u r e s 4.14, 4.15, and 4.16 a 500Hz t r i a n g l e wave, w i t h z e r o o f f s e t v o l t a g e , was a p p l i e d t o the e l e c t r o d e s o f a VIOWM, n o m i n a l l y 1 mm l o n g . t he peak-to-peak v o l t a g e s a p p l i e d t o the e l e c t r o d e s are 70, 100, and 130 v o l t s r e s p e c t i v e l y . F i g u r e 4.14 demonstrates the shape o f the t r a n s f e r curves at low v o l t a g e s as w e l l as the advantages o f a p p l y i n g a n e g a t i v e v o l t a g e t o the VIOWM i n the o f f s t a t e . In t h i s case -35V needed t o be a p p l i e d t o reduce the o p t i c a l throughput t o a minimum. The ne g a t i v e p o r t i o n o f the i n p u t s i g n a l i s u s e f u l f o r two reasons; f i r s t i t a c t s t o c r e a t e an "anti-waveguide" and second i t h e l p s the VIOWM r e c o v e r from changes i n t h e r e f r a c t i v e index d i s t r i b u t i o n b e l i e v e d t o be -128- F i g u r e 4.14: The o u t p u t o f a VIOWM w i t h a 4 pm i n t e r e l e c t r o d e gap f o r a 70 V p e a k - t o - p e a k t r i a n g l e wave a p p l i e d t o t h e e l e c t r o d e s . - 1 2 9 - F i g u r e 4 . 1 5 : The output o f a VIOWM w i t h a 4 um i n t e r e l e c t r o d e gap f o r a 100 V peak-to-peak t r i a n g l e wave a p p l i e d t o t h e e l e c t r o d e s . -130- F i g u r e 4.16: The output of a VIOWM with a 4 \xm interelectrode gap for a 130 V peak-to-peak t r i a n g l e wav applied to the electrodes. -131- due t o the p h o t o r e f r a c t i v e e f f e c t [67]. By the term a n t i - waveguide i s meant a low r e f r a c t i v e index r e g i o n out o f which l i g h t p r o p a g a t i n g i n the b u l k modes i s s t e e r e d c r e a t i n g a dark spot i n the i n t e r e l e c t r o d e r e g i o n . Upon the a p p l i c a t i o n o f the n e g a t i v e v o l t a g e the dark spot was l o c a t e d e x a c t l y where the output spot was l o c a t e d upon the a p p l i c a t i o n o f the p o s i t i v e v o l t a g e . Thus the e x t i n c t i o n r a t i o was enhanced. The mechanism f o r the change i n the r e f r a c t i v e index d i s t r i b u t i o n i s d i s c u s s e d i n more d e t a i l below. F i g u r e 4.15 demonstrates t h a t i n c r e a s i n g the a p p l i e d v o l t a g e does, i n f a c t , cause the power t r a n s f e r t o s a t u r a t e , as p r e d i c t e d i n the theory, furthermore f i g u r e 4.16 demonstrates t h a t i n c r e a s i n g the v o l t a g e f u r t h e r w i l l cause a r e d u c t i o n i n the c o u p l i n g e f f i c i e n c y . The maximum c o u p l i n g e f f i c i e n c y here was determined t o be about -4dB. T h i s c a l c u l a t i o n i s o b t a i n e d as 10 times the l o g a r i t h m o f t h e output o p t i c a l power from i t s peak v a l u e t o i t s v a l u e w i t h 0 V a p p l i e d d i v i d e d by the s t r a i g h t through power. In t h i s way a t t e n u a t i o n s and r e f l e c t i o n s f o r both the output and s t r a i g h t through powers would be the same w i t h the e x c e p t i o n t h a t t h e output power would be s c a t t e r e d by i n t e r a c t i o n s w i t h the LiNb03/SiC>2 i n t e r f a c e . However these i n t e r a c t i o n s would be s m a l l based on the measurements of about -1 dB/cm l o s s f o r d i f f u s e d waveguides which a l s o have s c a t t e r i n g at t h i s i n t e r f a c e [68]. -132- In f i g u r e 4.17 the t h e o r e t i c a l c o u p l i n g c o e f f i c i e n t T and t h e measured da t a are compared. The t h e o r e t i c a l curve i s f o r a p l a n a r d e v i c e h a v i n g a 4 um i n t e r e l e c t r o d e gap w i t h t h e c o u p l i n g o p t i m i z e d f o r 50.0 V. The measured data i s from f i g u r e 4.16 and has been s c a l e d by a f a c t o r o f about 2.0 t o f i t t h e t h e o r e t i c a l p r e d i c t i o n s a t 0.0 and 50.0 V. A f a c t o r o f t h i s o r d e r i s t o be expected due t o a s l i g h t f i b e r t o d e v i c e misalignment. The shapes o f the t h e o r e t i c a l and measured curves a re i n good agreement except t h a t t h e r e i s a s l i g h t d i s c r e p a n c y a t low v o l t a g e which i s b e l i e v e d t o be due t o b u l k mode c o u p l i n g . Other experiments were performed u s i n g t h e apparatus shown i n f i g u r e 4.18. In t h i s case p o l a r i z e d l i g h t was e n d - f i r e c o u p l e d i n t o t h e VIOWM u s i n g a 4Ox microscope o b j e c t i v e and t h e output was p r o j e c t e d onto a d e t e c t o r (an A l p h a m e t r i c s model dclOlO u s i n g a model 1110s wide spectrum head), through an a p e r t u r e , as b e f o r e . The wavelength o f the l i g h t was 633 nm and t h e VIOWM had a 4 um i n t e r e l e c t r o d e gap but t h i s time i t was about an i n c h l o n g . F i g u r e 4.19 shows the output when a 0.5 Hz 70 V peak- to-peak t r i a n g l e wave w i t h zero o f f s e t v o l t a g e was a p p l i e d t o t h e e l e c t r o d e s . The t o t a l output power was low (21 aW peak) t o reduce the p h o t o r e f r a c t i v e e f f e c t . The l e n g t h o f th e VIOWM ensured t h a t the bu l k mode c o u p l i n g would be s m a l l . The i n p u t c o u p l i n g was such t h a t the peak output o c c u r r e d w i t h about 30 V a p p l i e d . T h i s f i g u r e demonstrates, -133- — Theory © Measured 1.0 -r 0.8 -- 0.6 -- T 2 0.4 -- 0.2 + 0.0 4^ - 0.0 10.0 20.0 30.0 40.0 50.0 Appl ied Vo l tage F i g u r e 4.17: A comparison of the t h e o r e t i c a l and measured r e s u l t s f o r a p l a n a r d e v i c e . -134- VIOV/M 4 0 x Ob j e c t i ve L o s e r l O x O b j e c t i v e C h o p p e r A p e r t ur e \ • D e i e c t o r P o Io r i z e r M i c r o p o s i i i o n e r s F i g u r e 4.18: An a l t e r n a t e l a b o r a t o r y apparatus setup. -135- i n t h e absence o f t h e p h o t o r e f r a c t i v e e f f e c t and b u l k mode c o u p l i n g , t h e v o l t a g e c o n t r o l l e d c o u p l i n g b e h a v i o r o f the VIOWM. The e x t i n c t i o n r a t i o was 23dB. At even lower o p t i c a l power l e v e l s l i n e a r modulation o f t h i s d e v i c e was demonstrated by a p p l y i n g a 15 V b i a s t o the e l e c t r o d e s . 4.3 T he R i d g e VIOWM The c a l c u l a t e d r e s u l t s from the model f o r the p l a n a r VIOWM are p r e s e n t e d i n s e c t i o n 4.3.1 and the measured r e s u l t s a re p r e s e n t e d i n s e c t i o n 4.3.2. 4.3.1 Calculated Results The e f f e c t o f r i d g e h e i g h t on the o p e r a t i o n o f the VIOWM as rega r d s t h e confinement o f the guided mode as a f u n c t i o n o f r i d g e width and v o l t a g e i s s t u d i e d i n t h i s s e c t i o n . I t i s important t o understand the e f f e c t o f r i d g e h e i g h t on the o p e r a t i o n o f VIOWMs. I t w i l l be shown t h a t t h e c o u p l i n g t o an o p t i c a l f i b e r , at a p a r t i c u l a r o p e r a t i n g v o l t a g e , w i l l depend on t h i s parameter as w i l l t he "turn-on v o l t a g e . Here the turn-on v o l t a g e r e f e r s t o the v o l t a g e below which t h e mode becomes detached from the r i d g e . At v o l t a g e below t h e tur n - o n v o l t a g e the width parameters o f the mode -136- I n p u t S i g n a l ( 5 0 V / d i v ) O u t p u t S i g n a ( 2 1/xW p e a k ) ^ - g n d D a r k L e ve T i n e B a s e ( 0 . 5 s / d i v ) F i g u r e 4.19: The output of a long VIOWM w i t h a 70 V peak-to-peak t r i a n g l e wave a p p l i e d t o the e l e c t r o d e s f o r XQ = 663 nm. -137- i n c r e a s e r a p i d l y however the r i d g e VIOWM w i l l c o n t i n u e t o ac t as a waveguide f o r any p o s i t i v e a p p l i e d v o l t a g e . One can see t h i s by c o n s i d e r i n g what the r e f r a c t i v e index d i s t r i b u t i o n o f t h e VIOWM looks l i k e f a r away from the gap. U s i n g the te c h n i q u e o f conformal mapping f o r two co p l a n a r e l e c t r o d e s s e p a r a t e d by an i n f i n i t e s i m a l l y s m a l l gap, w i t h a v o l t a g e V Q a p p l i e d between t o the e l e c t o r d e s , one f i n d s t h a t t h e e l e c t r i c f i e l d p a r a l l e l t o the Z-axis i s g i v e n by V QCOS (e) E z ( y , z ) - - _ K p where p = [ (e z/e y) y 2 + z 2 ] 1 / 2 and 8 = t a n - 1 [ (e z/e y) 1 / 2 y / z ] . From e q u a t i o n 2.2 one sees t h a t the r e f r a c t i v e index d i s t r i b u t i o n w i l l drop o f f as 1/r f o r any p o s i t i v e v o l t a g e (one can i g n o r e t h e c o n t r i b u t i o n o f Ey as i t w i l l be small) as one moves away from the gap. Usi n g e i t h e r t he WKB method [39] o r the wave-vector method o f Hocker and Burns [33] one f i n d s t h a t a ray t r a v e l l i n g a l o n g a r a d i a l l i n e w i l l have a t u r n i n g p o i n t f o r any p o s i t i v e v o l t a g e . * C a l c u l a t e d r e s u l t s are p r e s e n t e d f o r r i d g e VIOWMs w i t h r i d g e h e i g h t s 0.5, 1.0, and 1.5 times the gap width. These t o g e t h e r w i t h the r e s u l t s f o r the p l a n a r d e v i c e complete our a n a l y s i s o f the e f f e c t s o f the v a r i a b l e parameters f o r Jaeger e t a l . [69] pp. 6-7. -138- VIOWMs f o r which t r e n d s can be p r e d i c t e d and the modus op e r a n d i e x p l a i n e d . F i g u r e s 4.20, 4.21 and 4.22 are t o p o g r a p h i c a l p l o t s o f th e W y V width parameters as f u n c t i o n s o f a p p l i e d v o l t a g e and i n t e r e l e c t r o d e gap width f o r r i d g e VIOWMs w i t h h e i g h t s o f 0.5, 1.0, and 1.5 times the gap width r e s p e c t i v e l y f o r X0 = 442 nm. F i g u r e s 4.23, 4.24 and 4.25 are the co r r e s p o n d i n g p l o t s f o r t h e w z v width parameters. The shaded r e g i o n s on the graphs i n d i c a t e where the mode i s detached from the r i d g e . In f i g u r e s 4.26 and 4.27 are p l o t t e d t he c o u p l i n g c o e f f i c i e n t s as f u n c t i o n s o f the a p p l i e d v o l t a g e f o r a VIOWM w i t h a 7 um i n t e r e l e c t r o d e gap and a h a l f - h e i g h t (0.5 times gap width) r i d g e . F o r these f i g u r e s i t has been assumed t h a t t h e o p t i c a l f i b e r s are l o c a t e d so as t o maximize the power t r a n s f e r f o r a p p l i e d v o l t a g e s o f 30 V and 50 V r e s p e c t i v e l y . Again, as f o r t h e p l a n a r d e v i c e , one sees t h a t a g r e a t e r degree o f c o u p l i n g w i l l occur a t v o l t a g e s g r e a t e r t han e i t h e r 30 V, i n f i g u r e 4.26, or 50 V, i n f i g u r e 4.27, however the degree o f c o u p l i n g at thes e v o l t a g e s i s the g r e a t e s t t h a t may be achieved. In f i g u r e s 4.28, 4.29 and 4.30 are p l o t t e d the optimum f r a c t i o n o f power t r a n s f e r , T , t h a t can be o b t a i n e d f o r r i d g e VIOWMs w i t h h e i g h t s o f 0.5, 1.0, and 1.5 times the gap wid t h r e s p e c t i v e l y , f o r X0 = 442 nm, as f u n c t i o n s o f both v o l t a g e and i n t e r e l e c t r o d e gap width. From f i g u r e 4.28 one -139- C o n t o u r s o f c o n s t a n t w y v CD CP O O > "O CD C L < G a p W i d t h ( / xn ) F i g u r e 4.20: A t o p o g r a p h i c a l 0.5 times the i n t e r e l e c t r o d e p l o t o f w y v f o r a r i d g e h e i g h t gap width where = 442 nm. -140- Contours of constant w 4 5 6 7 8 9 Gap W i d t h (/an) F i g u r e 4.21: A topographical 1.0 times the interelectrode plot of W y V for a ridge height gap width where XQ = 4 42 nm. -141- G a p W i d t h ( / xn ) F i g u r e 4.22: A t o p o g r a p h i c a l p l o t , o f w v v f o r 1.5 times the i n t e r e l e c t r o d e gap width where a r i d g e h e i g h t A,0 = 442 nm. -142- C o n t o u r s o f c o n s t a n t w z v F i g u r e 4.23: A t o p o g r a p h i c a l 0.5 times the i n t e r e l e c t r o d e p l o t o f w z v f o r a r i d g e h e i g h t gap width where XQ = 442 nm. C o n t o u r s o f c o n s t a n t w z F i g u r e 4.24: A t o p o g r a p h i c a l p l o t of w z v f o r a r i d g e h e i g h t 1.0 times the i n t e r e l e c t r o d e gap width where \ 0  = 442 nm. -144- C o n t o u r s o f c o n s t a n t w Z V CD CP o o > CD C L C L < 5 0 4 0 — 00 6 3 0 - 2 0 — 1 O H 0 ^ Pi- <£> ro I O - L Q JU..- - A - 1 / W 'V" 2 ^ ' ' " . '-/V; ' Ms wmmmmmm r 5 7 8 Gap W i d t h (/u,n) F i g u r e 4.25: A t o p o g r a p h i c a l p l o t o f w z v f o r a r i d g e h e i g h t 1.5 times the i n t e r e l e c t r o d e gap width where K0 = 442 nm. F i g u r e 4.26: The c o u p l i n g c o e f f i c i e n t T vs. v o l t a g e f o r g 7 um f o r a h a l f - h e i g h t r i d g e where the c o u p l i n g at 30 V i s maximized. F i g u r e 4.27: The c o u p l i n g c o e f f i c i e n t T vs. v o l t a g e f o r g 7 |jm f o r a h a l f - h e i g h t r i d g e where the c o u p l i n g at 50 V i s maximized. -147- can see t h a t i t i s p o s s i b l e t o o b t a i n -3dB power t r a n s f e r between an o p t i c a l f i b e r t h a t supports a mode w i t h width parameters w v f = w 2y = 1.50 nm and a h a l f - h e i g h t VIOWM w i t h an a p p l i e d v o l t a g e o f o n l y - 15 V. A l s o from f i g u r e 4.29 one can see t h a t the same c o u p l i n g can be o b t a i n e d f o r - 12 V and from f i g u r e 4.30 f o r - 11.5 V. Ag a i n we have c o n s i d e r e d t h a t t he r i d g e VIOWM c o u l d be used as a s m a l l s i g n a l l i n e a r modulator l o c a t e d between two o p t i c a l f i b e r s . In f i g u r e s 4.31 and 4.32 are p l o t t e d T^ f o r th e cases i n which b o t h o p t i c a l f i b e r s are l o c a t e d so as t o ac h i e v e the optimum c o u p l i n g f o r a h a l f - h e i g h t r i d g e VIOWM w i t h a 7 nm i n t e r e l e c t r o d e gap f o r a wavelength o f 442 nm when 30 V and 50 V a p p l i e d t o the e l e c t r o d e s r e s p e c t i v e l y . A g a i n b o t h f i g u r e s show s m a l l l i n e a r r e g i o n s but n e i t h e r f i g u r e i n d i c a t e s t he same p o t e n t i a l as a l i n e a r modulator as uoes f i g u r e 4.11 f o r a p l a n a r d e v i c e w i t h a 2nm i n t e r e l e c t r o d e gap. 4.3.2 Measured Results F i g u r e 4.33 shows a 500 Hz, 100 V peak-to-peak t r i a n g l e wave, w i t h zero o f f s e t v o l t a g e , a p p l i e d t o a h a l f - h e i g h t r i d g e VIOWM w i t h a f i b e r at the inp u t as d e s c r i b e d i n s e c t i o n 4.2.2. The base o f the r i d g e i s about 7.5 yon wide and t h e d e v i c e was no m i n a l l y 1 mm lo n g . The knee i n the -148- Contours of constant G a p W i d t h i/ubn) F i g u r e 4.28: The optimum power t r a n s f e r T^ as a f u n c t i o n o f a p p l i e d v o l t a g e and i n t e r e l e c t r o d e gap width f o r a r i d g e 0.5 times the gap width. -149- C o n t o u r s o f c o n s t a n t T 4 5 6 7 8 9 Gap W i d t h (/an) F i g u r e 4.29: The optimum power t r a n s f e r T 2 as a f u n c t i o n o a p p l i e d v o l t a g e and i n t e r e l e c t r o d e gap w i d t h f o r a r i d g e 1 t i m e s t h e gap w i d t h . -150- Contours of constant T 4 5 6 7 8 9 G a p W i d t h (/xn) F i g u r e 4 .30: The optimum power t r a n s f e r T as a f u n c t i o n o f a p p l i e d v o l t a g e and i n t e r e l e c t r o d e gap width f o r a r i d g e 1.5 times the gap width. -151- 1.0 -I 0.8 - 0.6 - 0.4 - 0.2 - 0.0 -J 1 . 1 . 1 0 10 20 30 40 50 A p p l i e d V o l t a g e F i g u r e 4.31: The power t r a n s f e r T between two o p t i c a l f i b e r s where the c o u p l i n g at 30 V i s maximized f o r the r i d g e d e v i c e . -152- A p p l i e d V o I t a g e F i g u r e 4.32: The power t r a n s f e r ,T^ between two o p t i c a l f i b e r s where the c o u p l i n g at 50 V i s maximized f o r the r i d e v i c e . -153- output can c l e a r l y be seen f o r an a p p l i e d v o l t a g e o f about 20 V. T h i s e f f e c t i s b e l i e v e d t o be due t o a r a p i d change i n t h e w i d t h parameters o f the fundamental mode o f the VIOWM as i t becomes more h i g h l y c o n f i n e d t o the r e g i o n o f the r i d g e . In f i g u r e 4.34 t h e t h e o r e t i c a l c o u p l i n g c o e f f i c i e n t T 2 and t h e measured d a t a are compared. The t h e o r e t i c a l curve i s f o r a r e c t a n g u l a r h a l f - h e i g h t r i d g e h a ving a 7 p i n t e r e l e c t r o d e gap w i t h t h e c o u p l i n g o p t i m i z e d f o r 40.0 V. The measured d a t a i s from f i g u r e 4.33 and has been s c a l e d by a f a c t o r o f about 2.2 t o f i t the t h e o r e t i c a l p r e d i c t i o n s at 0.0 and 50.0 V. A f a c t o r o f t h i s o r d e r i s t o be expected due t o a s l i g h t f i b e r t o d e v i c e misalignment and s c a t t e r i n g a t t h e r i d g e w a l l s which were not i d e a l l y smooth. The measurements show a knee a t about 18.0 V i n good agreement •v?ith t h e t h e o r e t i c a l t u r n - o n v o l t a g e . The shape o f t h e curves above 18.0 V do not agree w e l l . T h i s i s b e l i e v e d t o be due t o t h e d i f f e r e n c e i n the shapes o f t h e r e c t a n g u l a r r i d g e s s t u d i e d i n the t h e o r y and t h e r i d g e w i t h s l a n t e d w a l l s t h a t was f a b r i c a t e d . F i g u r e 4.35 shows a 500 Hz, 120 V peak-to-peak t r i a n g l e wave, w i t h z e r o o f f s e t v o l t a g e , a p p l i e d t o a h a l f - h e i g h t r i d g e VIOWM ag a i n w i t h a f i b e r at the i n p u t . Again the base o f t h e r i d g e i s about 7.5 nm and the d e v i c e i s n o m i n a l l y 1 mm l o n g . From t h i s f i g u r e one can again see t h a t t he c o u p l i n g reaches a maximum and then decreases. The maximum - 1 5 4 - F i g u r e 4 . 3 3 : The output o f a h a l f - h e i g h t VIOWM w i t h a 7.5 i n t e r e l e c t r o d e gap f o r a 100 V peak-to-peak t r i a n g l e wave a p p l i e d t o t h e e l e c t r o d e s where A.0 = 442 nm. -155- T 2 1.0 0.8 - 0.6 0.4 - 0.2 0.0 — Theory © Measured _ L l _ 1 1 1 i rvs i — L L _ / _ '\ L L - - _ _ L jL _ _ i / 1 t . ! / 1 L l_ / / L _ 1 /_ 1 1 1 1 _ 1 & ^ — I U -w" 1 1 1 1 1 1 1 1 1 0.0 10.0 20.0 30.0 40.0 50.0 Appl ied Vo l tage F i g u r e 4.34: A comparison o f the t h e o r e t i c a l and measured r e s u l t s f o r a h a l f - h e i g h t r i d g e . -156- c o u p l i n g e f f i c i e n c y here was determined t o be about -4.5dB. Again, as i n s e c t i o n 4.2.2, t h i s c a l c u l a t i o n i s o b t a i n e d as 10 times t h e l o g a r i t h m o f the output o p t i c a l power from i t s peak v a l u e t o i t s v a l u e w i t h 0 V a p p l i e d d i v i d e d by the s t r a i g h t through power. T h i s was not compensated f o r the e f f e c t s o f s c a t t e r i n g at the LiNbC>3/Si02 i n t e r f a c e although, as can be seen i n f i g u r e 3.2, the w a l l s o f the waveguide are f a r rougher than the s u r f a c e and w i l l p r o b a b l y r e s u l t i n much more s c a t t e r i n g than f o r a p l a n a r d e v i c e . I t i s b e l i e v e d , t h e r e f o r e , t h a t t he c o u p l i n g e f f i e c i e n c y at the i n p u t i s b e t t e r than t h i s r e s u l t i n d i c a t e s . F i g u r e 4.36 shows a 500Hz, ±20V square wave a p p l i e d t o the e l e c t r o d e s o f a h a l f - h e i g h t VIOWM. The peak output power i s seen t o be about 50 pW and the minimum output i s seen t o be about 10 jiW c o r r e s p o n d i n g t o an e x t i n c t i o n r a t i o o f about 7dB. 4.4 T he F r o n t - E n d S w i t c h In t h i s s e c t i o n are p r e s e n t e d the r e s u l t s o f measurements on a VIOWM t h a t was permanently a l i g n e d w i t h an o p t i c a l f i b e r i n a f l i p - c h i p arrangement on an S i wafer w i t h e t c h e d V-grooves. The v o l t a g e s i g n a l was a p p l i e d v i a probes t o t h e e l e c t r o d e s on the S i wafer. Here K0 = 442 nm. -157- F i g u r e 4.35: The o u t p u t o f a h a l f - h e i g h t VIOWM w i t h a 7.5 um i n t e r e l e c t r o d e gap f o r a 120 V peak-to-peak t r i a n g l e wave a p p l i e d t o t h e e l e c t r o d e s where \ Q = 442 nm. -158- F i g u r e 4.36: The output of a half-height VIOWM for a ±20 V square wave applied to the electrodes. -159- The a p p l i c a t i o n o f a VIOWM as a d i g i t a l s w i t c h i s demonstrated by f i g u r e 4.37. In t h i s f i g u r e a 500Hz, ±50V square wave i s a p p l i e d t o the e l e c t r o d e s . The VIOWM i s a c t i n g as a f r o n t - e n d s w i t c h c o n t r o l l i n g the amount o f o p t i c a l power cou p l e d i n t o t he f i b e r . The othe r end o f the f i b e r was i n s e r t e d i n t o t he LeCroy Fibercom Analog R e c e i v e r . The output s i g n a l i n d i c a t e d t h a t t he peak-to-peak change i n o p t i c a l power was ~ 240 piW. I t was found t h a t w i t h a constant v o l t a g e a p p l i e d t o the e l e c t r o d e s over a p e r i o d o f time the waveguide would decay and, i n e f f e c t , t u r n o f f . I t took s e v e r a l seconds f o r the waveguide t o t u r n o f f although the e f f e c t s e t i n immediately. T h i s was pr o b a b l y due t o the p h o t o r e f r a c t i v e e f f e c t [22] r a t h e r than domain r e v e r s a l [70,71]. F i g u r e 4.38 shows a 0.1 Hz, ±50 V square wave a p p l i e d t o the e l e c t r o d e s . As can be seen when 50 V i s a p p l i e d t o the e l e c t r o d e s , t h e r e i s a r a p i d i n i t i a l response by the VIOWM f o l l o w e d by a slow decay o f the throughput. S i m i l a r l y , when the p o l a r i t y o f the a p p l i e d v o l t a g e i s r e v e r s e d t he throughput i s i n i t i a l l y a minimum and i n c r e a s e s w i t h time. The " m e l t i n g " away o f t h e waveguide and the dark spot can be seen q u i t e c l e a r l y when the output i s p r o j e c t e d on a scre e n . We b e l i e v e t h a t t h i s e f f e c t i s due t o the p h o t o r e f r a c t i v e e f f e c t r a t h e r than domain r e v e r s a l . I f i t were due t o domain r e v e r s a l then a f t e r a lo n g a p p l i c a t i o n o f v o l t a g e , l o n g enough f o r t he waveguide t o t u r n completely -160- Input S igna l (100V /d iv) Output S igna l Time Base (1ms/div) F i g u r e 4.37: The o u t p u t o f an o p t i c a l f i b e r w i t h a VIOWM a c t i n g as a f r o n t - e n d s w i t c h f o r a ±50 V s q u a r e wave. H e r e t h e s w i t c h e d o p t i c a l power i s ~ 240 uW. -161- Input Signal (100V/div) Output Signal (40^W/div) — I -gnd -Dark Level Time Base (1s/div) F i g u r e 4.38: The d e c a y o f t h e o u t p u t o f a VIOWM a s a f u n c t i o n o f t i m e . -162- o f f , we would have a c h i e v e d the r e v e r s a l o f a s u f f i c i e n t number o f domains so t h a t t h e r e would be v i r t u a l l y no e l e c t r o o p t i c e f f e c t . I f t h i s were the case then upon the r e v e r s a l o f the f i e l d t h e r e s h o u l d s t i l l be v e r y l i t t l e e l e c t r o o p t i c e f f e c t and the output power sh o u l d remain r e l a t i v e l y c o n s t a n t . F i g u r e 4.38 i n d i c a t e s t h a t t h i s was not the case. In f a c t we see t h a t i n i t i a l l y t he throughput i s a minimum and g r a d u a l l y i n c r e a s e s , i . e . the dark spot at f i r s t i s v e r y much t h e r e and then g r a d u a l l y d i s a p p e a r s . A g a i n i f t h i s were due t o domain r e v e r s a l then upon a second r e v e r s a l o f the a p p l i e d f i e l d t h e r e s h o u l d be l i t t l e change i n t he throughput but we see a r a p i d i n c r e a s e i n s t e a d . In f a c t f i g u r e 4.38 i n d i c a t e s t h a t d u r i n g the a p p l i c a t i o n o f a n e g a t i v e v o l t a g e the waveguide r e c o v e r s . The n e g a t i v e v o l t a g e i s t h e r e f o r e u s e f u l as a f l y - b a c k c y c l e . Furthermore domain r e v e r s a l u s u a l l y occurs i n LiNbC>3 i n which an i m p u r i t y g r a d i e n t e x i s t s [72], u s u a l l y due t o an i n - d i f f u s i o n , a l t h o u g h Pendergrass [71] has seen domain r e v e r s a l i n s u b s t r a t e s i n t o which no i m p u r i t i e s were d i f f u s e d . The p h o t o r e f r a c t i v e e f f e c t i s the name g i v e n t o a v a r i e t y o f e f f e c t s t h a t cause changes i n the r e f r a c t i v e i n d i c e s o f m a t e r i a l s when they are exposed t o l i g h t . In LiNbC>3 mobile e l e c t r o n s are produced when l i g h t i s absorbed by i m p u r i t i e s such as Fe [22]. Without any e x t e r n a l e l e c t r i c f i e l d a p p l i e d the p h o t o e x c i t e d c a r r i e r s move i n the c r y s t a l -163- and so as t o generate a v o l t a g e such t h a t the +c-end o f the c r y s t a l i s n e g a t i v e [73]. T h i s i s the b u l k p h o t o v o l t a i c e f f e c t . I f t h e c a r r i e r s are c a p t u r e d by t r a p s at the edges o f t h e i l l u m i n a t e d r e g i o n a space charge r e g i o n may be e s t a b l i s h e d t h a t a l t e r s the r e f r a c t i v e index v i a the l i n e a r e l e c t r o o p t i c e f f e c t . * K r a e t z i g and Kurz [73] found t h a t an e x t e r n a l f i e l d can be a p p l i e d t h a t w i l l stop the p h o t o c u r r e n t i n LiNb03 doped w i t h e i t h e r Fe or Cu. In our s u b s t r a t e s , i n t e n d e d f o r o p t i c a l waveguiding, the Fe d e n s i t y i s i n t e n t i o n a l l y kept low, n o m i n a l l y 5ppm, so t h a t the p h o t o r e f r a c t i v e e f f e c t w i l l be minimized. T h i s a l s o means t h a t t h e p h o t o v o l t a i c e f f e c t w i l l be s m a l l . T h e r e f o r e , as Jerominek e t a l . p o i n t out [75], when an e l e c t r i c f i e l d i s a p p l i e d the c a r r i e r s can t r a v e l i n any d i r e c t i o n . However s i n c e t h e power d e n s i t y i n the b u l k modes i s s t i l l r a t h e r h i g h , e s p e c i a l l y at the i n p u t end o f the VIOWM, the number o f p h o t o e x c i t e d e l e c t r o n s need not be s m a l l . These c a r r i e r s can be a t t r a c t e d t o the h i g h f i e l d r e g i o n s near the e l e c t r o d e edges where they w i l l c o u n t e r a c t the a p p l i e d f i e l d . When the s w i t c h i n g speed i s low enough f o r s u f f i c i e n t accumulation o f charge i n the h i g h f i e l d r e g i o n s t h e p resence o f the charge should, i n f a c t , a c t i n i t i a l l y t o enhance th e r e v e r s e d f i e l d upon s w i t c h i n g , i . e . , an extended See Guenter and Huignard [74] for a detailed discussion of the photorefractive e f f e c t and photorefractive materials. -164- a p p l i c a t i o n o f a n e g a t i v e v o l t a g e d u r i n g t he f l y - b a c k c y c l e m a y reduce t h e p o s i t i v e v o l t a g e needed t o o b t a i n a d e s i r e d degree o f o p t i c a l confinement. Such an e f f e c t may prove t o be b e n e f i c i a l i n a p p l i c a t i o n s where t h e r e i s a dead c y c l e . A t h i g h e r s w i t c h i n g r a t e s t he charges do not have time t o accumulate and the decay e f f e c t need not be a problem. F i g u r e 4.37 i l l u s t r a t e s t h a t f o r h i g h speed s w i t c h i n g t h e r e i s e s s e n t i a l l y no decay i n the output. 4.5 D i s c u s s i o n In t h i s s e c t i o n v a r i o u s r e s u l t s are h i g h l i g h t e d and o t h e r s a re compared f o r both the p r e d i c t e d and the measured r e s u l t s . The most important r e s u l t i s t h a t t he p r e d i c t e d b e h a v i o r o f t h e VIOWM, as a v o l t a g e c o n t r o l l e d waveguide, has been v e r i f i e d . I t has been demonstrated e x p e r i m e n t a l l y t h a t t h e throughput o f a VIOWM w i l l i n c r e a s e w i t h i n c r e a s i n g v o l t a g e u n t i l a p o i n t i s reached a t which the c o u p l i n g i s maximized, then, upon f u r t h e r i n c r e a s i n g the v o l t a g e t he c o u p l i n g w i l l decrease. F i g u r e s 4.1 through 4.4 show t h a t the confinement o f th e modes i s g r e a t e r f o r 442 nm r a d i a t i o n than f o r 633 nm r a d i a t i o n f o r any p a r t i c u l a r gap w i d t h / o p e r a t i n g v o l t a g e combination. T h i s i s t o be expected as the change i n the -165- r e f r a c t i v e index i s l e s s at l o n g e r wavelengths due t o the r e d u c t i o n i n t h e v a l u e o f the r e f r a c t i v e index. A l s o we see t h a t t h e confinement i s expected t o i n c r e a s e w i t h d e c r e a s i n g gap w i d t h and w i t h i n c r e a s i n g v o l t a g e f o r p l a n a r d e v i c e s . T h i s i s a re a s o n a b l e r e s u l t s i n c e the change i n the r e f r a c t i v e i ndex i s i n v e r s e l y p r o p o r t i o n a l t o the gap width and i s d i r e c t l y p r o p o r t i o n a l t o the a p p l i e d v o l t a g e . F i g u r e 4.9 i n d i c a t e s t h a t f o r as l i t t l e as 20 V a p p l i e d t o a p l a n a r VIOWM w i t h a 2 jun i n t e r e l e c t r o d e gap -3 dB c o u p l i n g can be o b t a i n e d when \ Q = 442 nm and f o r l e s s than 25 V when t h e i n t e r e l e c t r o d e gap i s 4 [un. The p o s s i b l e a p p l i c a t i o n o f the p l a n a r VIOWM as a l i n e a r modulator has been demonstrated t h e o r e t i c a l l y and l i n e a r r e g i o n s have been seen i n the outputs o f the e x p e r i m e n t a l d e v i c e s ; f i g u r e s 4.15, 4.16, and 4.19. Another r e s u l t t h a t was p r e d i c t e d by the th e o r y and co n f i r m e d by experiment was the turn-on v o l t a g e f o r r i d g e waveguides. In f a c t f i g u r e s 4.20 and 4.23 i n d i c a t e t h a t a h a l f - h e i g h t r i d g e VIOWM w i t h a 7 jun i n t e r e l e c t r o d e gap s h o u l d have a tu r n - o n v o l t a g e o f about 17 V and f i g u r e 4.33 shows t h e t u r n - o n v o l t a g e t o be between 15 and 20 V. T h i s i s where t h e knee at the onset o f the i n c r e a s e i n the throughput o c c u r s i n d i c a t i n g a r a p i d change i n the confinement o f l i g h t t o the r i d g e . While r i d g e waveguides were not p r e d i c t e d t o be very u s e f u l as l i n e a r modulators, f i g u r e s 4.31 and 4.32, we see -166- from f i g u r e 4.33 t h a t our d e v i c e i n f a c t demonstrates a very l i n e a r r e g i o n . T h i s i s perhaps due t o the w a l l s o f the ex p e r i m e n t a l d e v i c e b e i n g s l a n t e d a l l o w i n g f o r a g r e a t e r v a r i a t i o n i n the width o f the mode than i s p r e d i c t e d f o r d e v i c e s w i t h v e r t i c a l w a l l s i n f i g u r e s 4.23, 4.24, and 4.25. Another r e s u l t t h a t i s i n t e r e s t i n g i s t h a t i n c r e a s i n g the h e i g h t o f a r i d g e VIOWM r e l a t i v e t o the i n t e r e l e c t r o d e gap w i d t h can i n c r e a s e the c o u p l i n g between a d e v i c e and a f i b e r a t low v o l t a g e s but i t reduces the c o u p l i n g t h a t can be o b t a i n e d a t h i g h e r v o l t a g e s ; f i g u r e s 4.28, 4.29, and 4.30. The improved d i g i t a l a p p l i c a t i o n o f the r i d g e VIOWM over t h e p l a n a r as a d i g i t a l s w i t c h i s seen by comparing f i g u r e 4.14 w i t h 4.33. The v o l t a g e d i f f e r e n c e between the knee a t t h e onset o f t h e i n c r e a s e i n the throughput and the maximum v a l u e o f t h e throughput f o r the p l a n a r VIOWM i s a f u l l 50 V whereas f o r the r i d g e d e v i c e i t i s o n l y about 30 V. Ridge d e v i c e s are p r e d i c t e d t o be abl e t o ach i e v e -3 dB c o u p l i n g f o r as l i t t l e as 11 V. F i n a l l y f o r both d e v i c e types we were a b l e t o achieve l a r g e e x t i n c t i o n r a t i o s , > 20 dB, by a p p l y i n g a l a r g e n e g a t i v e v o l t a g e t o induce a low r e f r a c t i v e index a n t i - waveguide out o f which l i g h t i n b u l k modes was s t e e r e d . That t h e need f o r a n e g a t i v e v o l t a g e was due t o bu l k mode c o u p l i n g i n s h o r t , - 1 mm, was confirmed by t e s t s on a lo n g -167- d e v i c e , where b u l k mode c o u p l i n g i s much reduced, f o r which a 23 dB e x t i n c t i o n r a t i o was o b t a i n e d f o r a much reduced n e g a t i v e v o l t a g e . - 1 6 8 - C h a p t e r 5 Summary, C o n c l u s i o n s , a n d S u g g e s t i o n s f o r F u r t h e r Work 5.1 I n t r o d u c t i o n Here t h e contents o f t h e t h e s i s are summarized. Then the c o n c l u s i o n s t h a t were drawn from t h i s work are p r e s e n t e d . The chapter i s concluded by a s e c t i o n s u g g e s t i n g p o s s i b l e f u r t h e r r e s e a r c h i n t h i s area. 5.2 Summary In t h i s t h e s i s the VIOWM has been s t u d i e d . Both p l a n a r and r i d g e type d e v i c e s were modeled (chapter 2 ) , f a b r i c a t e d (chapter 3), and r e s u l t s p r e d i c t e d by the theo r y and t e s t s on t h e d e v i c e s were p r e s e n t e d (chapter 4 ) . The model was developed i n terms o f the th e o r y t h a t the e l e c t r i c f i e l d e s t a b l i s h e d by the a p p l i c a t i o n o f v o l t a g e t o -169- two e l e c t r o d e s , s e p a r a t e d by a s m a l l gap, can c r e a t e a waveguide i n an e l e c t r o o p t i c s u b s t r a t e . The e l e c t r i c f i e l d d i s t r i b u t i o n s e s t a b l i s h e d i n both the p l a n a r and r i d g e d e v i c e s were c a l c u l a t e d by conformal mapping methods. Using t h e r e f r a c t i v e index d i s t r i b u t i o n i n the s u b s t r a t e , caused by the a p p l i e d e l e c t r i c f i e l d v i a the e l e c t r o o p t i c e f f e c t , w idth parameters f o r Hermite-Gaussian approximations t o the o p t i c a l f i e l d d i s t r i b u t i o n s o f the guided modes o f the VIOWM were o b t a i n e d by a v a r i a t i o n a l t e c h nique. An e x p r e s s i o n f o r the c o u p l i n g c o e f f i c i e n t between an o p t i c a l f i b e r and a VIOWM was developed based upon the approximate o p t i c a l f i e l d d i s t r i b u t i o n s . I t was p o s s i b l e u s i n g a l l o f the above r e s u l t s t o study t h e p r e d i c t e d b e h a v i o r o f a VIOWM as a v o l t a g e c o n t r o l l e d c o u p l e r . One e n v i s i o n e d a p p l i c a t i o n i s as a l i n k i n g waveguide between two o p t i c a l f i b e r s another i s as a f r o n t - e n d s w i t c h between a focus e d l a s e r beam and an o p t i c a l f i b e r . The VIOWM's use as e i t h e r a l i n e a r modulator or as a d i g i t a l s w i t c h was s t u d i e d . P l a n a r d e v i c e s w i t h i n t e r e l e c t r o d e gaps o f 4 um and r i d g e d e v i c e s w i t h 7.5 um gaps and 4 um h i g h r i d g e s were f a b r i c a t e d and t e s t e d . F i n a l l y a p l a n a r d e v i c e used i n c o n j u c t i o n w i t h a s i l i c o n V-groove t h a t a c t e d as a f r o n t - e n d s w i t c h between a focuse d Gaussian l a s e r beam and an o p t i c a l f i b e r was f a b r i c a t e d and t e s t e d . 5.3 C o n c l u s i o n s The VIOWM o f f e r s c o n s i d e r a b l e p o t e n t i a l as an o p t i c a l modulator. I t s p o s s i b l e a p p l i c a t i o n as a l i n e a r modulator o r as a d i g i t a l s w i t c h has been demonstrated. A f r o n t - e n d s w i t c h has been f a b r i c a t e d . Both t h e o r e t i c a l p r e d i c t i o n s and measured r e s u l t s have shown t h a t the VIOWM a c t s as a v o l t a g e c o n t r o l l e d c o u p l e r . While t h e v o l t a g e s used are r e l a t i v e l y l a r g e when compared t o t h ose used i n d e v i c e s such as i n t e g r a t e d o p t i c a l Mach- Zehnders and c r o s s - c o u p l e r s they are o f the same magnitude as t h o s e used i n some p o l a r i z a t i o n c o n v e r t e r s . * A l s o t he dynamic r e g i o n o f t h e VIOWM i s such t h a t l a r g e changes i n the output power can be ac h i e v e d without the need t o apply t h e e n t i r e o n - o f f v o l t a g e , i . e . , b i a s v o l t a g e s c o u l d be used t o reduce t h e v o l t a g e d i f f e r e n c e s needed. Furthermore we have shown t h a t t h e VIOWM can be used near the a b s o r p t i o n edge o f LiNb03- T h i s opens the VIOWM t o a p p l i c a t i o n s a t wavelengths where o t h e r d e v i c e s would s u f f e r from t h e e f f e c t s o f t h e p h o t o r e f r a c t i v e e f f e c t . While the VIOWM s u f f e r s from t h e p h o t o r e f r a c t i v e e f f e c t as do the othe r Examples of voltage l e v e l s for these devices can be obtained from the data sheets provided for o p t i c a l guided wave devices by companies such as Crystal Technology Inc., Palo A l t o , C a l i f o r n i a , U.S.A. -171- t y p e s o f d e v i c e i t . w i l l r e c o v e r upon the a p p l i c a t i o n o f a f l y - b a c k c y c l e . The p l a n a r d e v i c e i s easy t o f a b r i c a t e . I t uses r e l a t i v e l y l e s s r e a l e s t a t e than other i n t e g r a t e d o p t i c a l d e v i c e s . I t t h e r e f o r e promises t o g i v e h i g h y i e l d s and t o be i n e x p e n s i v e . I t does not r e q u i r e an i n - d i f f u s i o n c y c l e i n t h e f a b r i c a t i o n nor do the e l e c t r o d e s r e q u i r e p r e c i s e alignment w i t h preformed waveguides. The r i d g e d e v i c e , w h i l e more d i f f i c u l t t o f a b r i c a t e , needs lower s w i t c h i n g v o l t a g e s . Furthermore the r i d g e seems t o f i x t h e w i d t h parameter i n the z d i r e c t i o n o f t h e mode r a t h e r p r e c i s e l y . T h i s c o u l d be an advantage i n d e s i g n i n g a d e v i c e t o operate w i t h a s p e c i f i c o p t i c a l f i b e r . 5.4 S u g g e s t i o n s f o r F u r t h e r W o r k . While we have attempted t o make our t h e o r e t i c a l a n a l y s i s o f t h e VIOWM as thorough as p o s s i b l e t h e r e i s s t i l l room f o r f u r t h e r study. The i s a l s o room f o r improvement i n th e f a b r i c a t i o n t e c h n i q u e s used. One a r e a which warrants f u r t h e r study i s the o p t i m i z a t i o n o f the l e n g t h o f the d e v i c e s . I t i s important t o have d e v i c e s t h a t are as s h o r t as p o s s i b l e t o minimize t h e c a p a c i t a n c e and t o reduce l o s s e s . While our d e v i c e s were s h o r t , - 1 mm, they needed a r a t h e r l a r g e r e v e r s e -172- v o l t a g e t o i n c r e a s e t h e e x t i n c t i o n r a t i o . I t i s d e s i r a b l e t o reduce t h i s v o l t a g e i f p o s s i b l e . Of course here the r e v e r s e v o l t a g e a l s o s e r v e d as a f l y - b a c k c y c l e t o cause the d e v i c e s t o r e c o v e r from the p h o t o r e f r a c t i v e e f f e c t . The e f f e c t o f the p h o t o r e f r a c t i v e e f f e c t on t h e performance o f the d e v i c e s needs t o be s t u d i e d f u r t h e r . N a t u r a l l y i t can be reduced by r e d u c i n g t he power i n t h e guid e d mode but t h i s s o l u t i o n a l s o reduces the VIOWM's u t i l i t y . As we have shown a p p l i c a t i o n o f a n e g a t i v e v o l t a g e , e q u a l i n magnitude t o the p o s i t i v e v o l t a g e , i s s u f f i c i e n t f o r t h e VIOWM t o operate as a d i g i t a l s w i t c h . However a study o f the e f f e c t o f a p p l y i n g a l a r g e v o l t a g e , e i t h e r p o s i t i v e o r n e g a t i v e , d u r i n g a dead c y c l e may be u s e f u l i n a c e r t a i n a p p l i c a t i o n s by r e d u c i n g the v o l t a g e s needed d u r i n g normal o p e r a t i o n . . . . B e t t e r p o l i s h i n g t e c h n i q u e s are needed, f o r both the VIOWMs and t h e o p t i c a l f i b e r s , than those t h a t were used. Other methods o f f a b r i c a t i n g t h e , r i d g e waveguides c o u l d a l s o be pursued. While our method was adequate f o r d e m o n s t r a t i n g t h e d e v i c e s b e h a v i o r b e t t e r throughput s h o u l d be p o s s i b l e i f t h e r i d g e w a l l roughness was reduced. F i n a l l y i t would be i n t e r e s t i n g t o i n v e s t i g a t e the e f f e c t o f r i d g e w a l l s l a n t on the performance o f r i d g e VIOWMs. The d i f f e r e n c e between our p r e d i c t e d r e s u l t s f o r r i d g e VIOWMs w i t h v e r t i c a l w a l l s and our measured r e s u l t s on a d e v i c e w i t h s l a n t e d w a l l s i n d i c a t e s t h a t d e v i c e s w i t h -173- s l a n t e d w a l l s can i n c o r p o r a t e both the turn-on v o l t a g e o f a r i d g e VIOWM w h i l e p r e s e r v i n g the h i g h l y l i n e a r r e g i o n s found i n p l a n a r d e v i c e s . A p p e n d i x A THE ELECTROOPTIC EFFECT A . l I n t r o d u c t i o n T h i s appendix covers t o p i c s r e l e v a n t t o t h i s t h e s i s r e g a r d i n g t h e l i n e a r e l e c t r o o p t i c e f f e c t . I t begins w i t h a d i s c u s s i o n o f the o p t i c a l i n d i c a t r i x and i t s r e l a t i o n s h i p t o the r e l a t i v e d i e l e c t r i c i m p e r m e a b i l i t y t e n s o r . Then 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 i s d i s c u s s e d and the reduced e l e c t r o o p t i c c o e f f i c i e n t t e n s o r f o r LiNbC>3 i s g i v e n . F i n a l l y b o t h equations 2.1 and 2.2, r e l a t i n g t he deformation o f the o p t i c a l i n d i c a t r i x t o the a p p l i e d e l e c t r i c f i e l d i n our example, are d e r i v e d . -175- A . 2 The R e l a t i v e D i e l e c t r i c I m p e r m e a b i l i t y T e n s o r The o p t i c a l i n d i c a t r i x * s p e c i f i e s the r e f r a c t i v e index o f a c r y s t a l . The e q u a t i o n d e s c r i b i n g the e l l i p s o i d i s when x, y, and z are the p r i n c i p a l axes. The c o e f f i c i e n t s o f t h e above equ a t i o n , l / n x 2 , 1/ny 2, and l / n z 2 , are the p r i n c i p a l components o f the r e l a t i v e d i e l e c t r i c i m p e r m e a b i l i t y t e n s o r . The n o t a t i o n used above i s not convenient when d i s c u s s i n g t e n s o r t r a n s f o r m a t i o n s t h e r e f o r e i n t h i s appendix t h e p r i n c i p a l axes w i l l be c a l l e d the x j , X 2 , and X3 axes f o l l o w i n g t h e convention used by Nye [23]. Us i n g t h i s n o t a t i o n each o f the components o f the r e l a t i v e d i e l e c t r i c i m p e r m e a b i l i t y t e n s o r i s g i v e n by d E i B^. — e Q ( i •» 1, 2, and 3; j •= 1, 2, and 3) 3D . 0 where e 0 i s the p e r m i t t i v i t y o f f r e e space and o p t i c a l e l e c t r i c f i e l d . The g e n e r a l r e p r e s e n t a t i o n q u a d r i c o f the r e l a t i v e d i e l e c t r i c i m p e r m e a b i l i t y can then be d e s c r i b e d by * A l s o c a l l e d the index e l l i p s o i d by some authors; see Y a r i v and Yeh [45] c h a p t e r 7 pp. 220-275. -176- B . .x . x . = 1 . where t h e E i n s t e i n summation convention i s assumed. The r e l a t i v e d i e l e c t r i c i m p e r m e a b i l i t y i s a symmetrical t e n s o r * , i . e . , B^j = Bj£. T h e r e f o r e i t i s p o s s i b l e t o reduce t h e number o f independent components. The n i n e components a r e reduced t o s i x as shown below: B l l B 1 2 B 1 3 B l B 6 B 5 B 2 1 B 2 2 B 2 3 —» B 6 B 2 B 4 B 3 1 B 3 2 B 3 3 . .  B 5 B 4 B 3 A . 3 The E l e c t r o o p t i c E f f e c t The change i n the r e f r a c t i v e index o f a c r y s t a l due t o an a p p l i e d e l e c t r i c f i e l d i s known as the e l e c t r o o p t i c e f f e c t . In g e n e r a l a change i n the r e f r a c t i v e index o f a c r y s t a l w i l l r e s u l t i n a change i n the s i z e and o r i e n t a t i o n o f t h e o p t i c a l i n d i c a t r i x . Because o f the r e l a t i o n s h i p between t h e o p t i c a l i n d i c a t r i x and the components o f the r e l a t i v e d i e l e c t r i c i m p e r m e a b i l i t y t e n s o r changes i n the o p t i c a l i n d i c a t r i x corresponds t o changes i n the c o e f f i c i e n t s B ^ j . When a c r y s t a l 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 t h e changes i n the B̂ -; are r e l a t e d t o the a p p l i e d e l e c t r i c See Nye [23] p . 2 4 6 . -177- f i e l d (the low frequency f i e l d a p p l i e d t o the c r y s t a l as opposed t o t h e h i g h frequency o p t i c a l f i e l d ) v i a the l i n e a r e l e c t r o o p t i c c o e f f i c i e n t s r ^ j j ^ by* AB . . = r . ., E, ( i 13 13k k 1, 2, 3; j - 1, 2, 3; k = 1, 2, 3) or u s i n g t h e reduced index n o t a t i o n AB. = r . . E . ( i = 1, 2, 3, 4, 5, 6; j = 1, 2, 3) .J -J L i N b 0 3 i s a member o f the t r i g o n a l system o f c r y s t a l s and i t i s o f c l a s s 3m.** When the x^ a x i s i s d e f i n e d as the a x i s p e r p e n d i c u l a r t o the m i r r o r p l a n e o f the c r y s t a l , i . e . , xi-Lm, t h e n t h e e l e c t r o o p t i c c o e f f i c i e n t t e n s o r has t h e nonzero components: r 1 2 , r 1 3 , i 2 2 r r 2 3 , r 3 3 , r 4 2 , r 5 1 , a n d rg]_. In m a t r i x form the e l e c t r o o p t i c c o e f f i c i e n t t e n s o r l o o k s l i k e r!2 r!3 r22 r23 0 0 0 0 0 r 0 0 51 33 r42 ° r61 ° ° The change i n the ( i = 1, 2, 3, 4, 5, 6) due t o the a p p l i e d e l e c t r i c f i e l d i s g i v e n b y * Here const a n t temperature and p r e s s u r e are assumed. See Y a r i v and Yeh [45] p. 232 -178- A B 1 A B 3 A B 5 A B 6 r i 2 r!3 r22 r23 0 0 0 0 0 0 0 0 33 r42 ° r 5 1 ° "61 In LiNbC«3 -r]_2 = r22 ~ ~ r61 = 3. 4xlO~ 1 2m/V, r^3 = r 2 3 = 8.6xl0~ 1 2m/V, r 4 2 = r 5 1 = 28. 0xl0~ 1 2m/V, and r 3 3 = 30.8xlO~ x^m/V. When the a p p l i e d e l e c t r i c f i e l d i s i n the X 2 X 3 p l a n e t h e components o f the r e l a t i v e d i e l e c t r i c i m p e r m e a b i l i t y t e n s o r become — r22 E2 + r23 E3 ' B 2 = + r 2 2 E 2 + r 2 3 E 3 , B 3 - + r 3 3 E 3 , B 4 " r42 E2 ' B 5 = 0 , a n d B ^ = 0 o I b i d . -179- A.4 E q u a t i o n s 2.1 a n d 2.2 I f , f o r t h i s problem, the r e l a t i v e d i e l e c t r i c i m p e r m e a b i l i t y t e n s o r were transformed t o a new s e t o f axes where X ] / = x 2 , x 2 ' = X3, and X 3 ' = x]_, then the Mohr c i r c l e c o n s t r u c t i o n as s e t out i n Nye* [23] can be a p p l i e d d i r e c t l y t o f i n d t h e d i s t o r t i o n o f the i n d i c a t r i x i n the x 2 x 3 p l a n e . The t r a n s f o r m a t i o n o f a second rank t e n s o r from one s e t o f m u t u a l l y p e r p e n d i c u l a r axes t o another i s g i v e n by** T'. . = a..a.,I, . 1 3 ik 3 I kl where t h e a's are d i r e c t i o n c o s i n e s and T and T' are t e n s o r q u a n t i t i e s . The t r a n s f o r m a t i o n o f a second rank t e n s o r from the XJ[ s e t o f axes t o the X J / s e t o f axes, d e s c r i b e d above, can be accomplished u s i n g the d i r e c t i o n c o s i n e s = 0, a ^ 2 ~ 1 r a 1 3 = °' a 2 1 ~ °r a 2 2 «= 0, a 2 3 = 1, a 3 1 = 1, a 3 2 = 0, and 833 = 0. The i m p e r m e a b i l i t y t e n s o r r e f e r r e d t o t h e new set o f axes i s 1 2 n o B l = — + r22 E2 + r23 E3 * ** Chapter I I s e c t i o n 4 pp. 43-47 Nye [23] p. 11. -180- B'2 = + r 3 3 E 3 , 2 n e 1 2 n o B 3  =  — -  r 2 2 E 2  + r 2 3 E 3 ' B'4 = 0 , B'5 = 0 , and B 6 "  r 4 2 E 2 • where the s u b s c r i p t s on 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 and the e l e c t r i c f i e l d components are unchanged ( i . e . , t hey s t i l l r e f e r t o the o r i g i n a l s e t o f axes) . With no a p p l i e d f i e l d the major and minor axes o f the e l l i p s e formed by t a k i n g a c r o s s s e c t i o n o f the i n d i c a t r i x i n the p l a n e X3' = 0 are p r i n c i p a l axes o f the i n d i c a t r i x . Upon t h e a p p l i c a t i o n o f an e l e c t r i c f i e l d the i n d i c a t r i x deforms i n b o t h s i z e and o r i e n t a t i o n . S i n c e the o n l y nonzero o f f d i a g o n a l element o f the r e l a t i v e d i e l e c t r i c i m p e r m e a b i l i t y t e n s o r under the i n f l u e n c e o f the f i e l d i s Bg' t h e r o t a t i o n o f the i n d i c a t r i x i s about the X3'-axis. T h i s b e i n g th e case the Mohr c i r c l e can be used t o f i n d the angle o f r o t a t i o n and the p r i n c i p a l components o f the new i n d i c a t r i x ( i . e . , t he i n d i c a t r i x when the f i e l d i s a p p l i e d ) . I f the new p r i n c i p a l axes o f the r e l a t i v e d i e l e c t r i c i m p e r m e a b i l i t y t e n s o r are l a b e l e d x^" and X2" and the angle -181- o f r o t a t i o n , measured from X]/' t o X ] / , i s e, then the p r i n c i p a l components o f the t e n s o r are g i v e n by B i + B2 and B2 B i + B2 + r where ( B'2 ~ ) 1/2 and t h e r e l a t i o n s h i p g i v i n g the angle e i s 2B' tan(2e) = B2 " B i By d i r e c t s u b s t i t u t i o n we o b t a i n e q u a t i o n 2.1 tan (2e) 42 y -2 -2 n - n + r - _ E - r _ „ E - r „ E e o 33 z 22 y 23 z (2.1) where Ey = E 2 , and E z = E 3 . To o b t a i n e q u a t i o n 2 . 2 we b e g i n by f i n d i n g an e x p r e s s i o n f o r B 2 " . F i r s t a simple s u b s t i t u t i o n g i v e s -182- 1 + 2 B 6 ' V B2 "  Bl 1/2 1 /2 here we can use the approximation (1 + x) ' <= 1 + x/2 f o r s m a l l x s i n c e [2Bg'/(B2' - B ] / ) ] ^ « 1. We can now w r i t e B i + B2 B2 " B i + + 1,2 B ,2 B' + B2 " B i B2 " B i or. + r _ _ E + 33 z ,2 2 _-2 _-2 { r42 Ey»" n 2 n e n - n + r _ _ E - r „ „ E - r 0 _ E e o 33 z 22 y 23 z The l a s t two terms on the r i g h t hand s i d e o f the above e q u a t i o n r e p r e s e n t t h e change i n the r e l a t i v e d i e l e c t r i c c m s a b i l i t y due t o the a p p l i c a t i o n o f an e l e c t r i c f i e l d , i . e . , -2 An •= r o r > E + e 33 z < r42V' * r 4 2 V -2 -2 • n - n + r _ , E - r _ „ E - r _ 0 E e o 33 z 22 y 23 z r , _ E + 33 z -2 -2 n — n e o The r e l a t i o n s h i p between the change i n the r e f r a c t i v e index and t h e change i n the r e l a t i v e d i e l e c t r i c i m p e r m e a b i l i t y i s g i v e n by See Spiegel [7 6] p. 110. -183- 3 A n An An o b t a i n e d from dn~ 2/dn = -2/n^ s i n c e An •* dn f o r s m a l l An. Now we can w r i t e e q u a t i o n 2.2 3 3 2 n r _ _ E (y,z) n {r.^E (y,z)} e 33 z J e 42 y •*' A n e ( y , z ) - - - . (2.2) -2 -2 2 2 ( n ^ - n e o -184- A p p e n d i x B STATIONARY FORMULAS B.1 I n t r o d u c t i o n T h i s appendix p r o v i d e s the p r o o f s t o s e v e r a l o f the a s s e r t i o n s made i n chapter 2 about the s t a t i o n a r y nature o f c e r t a i n i n t e g r a l s . B.2 E q u a t i o n 2.10 In s e c t i o n 2.4 i t i s c l a i m e d t h a t the i n t e g r a l e q u a t i o n 2 oe oo —oo —oo 2 + — , 3y t , dz , 2 2 - v ( y , z) y d y d z = 0 (2.10) i s s t a t i o n a r y and t h a t the g e n e r a l form o f the wave equ a t i o n 2 2 Vfc>l> + v (y, z ) v = 0 , i s t h e Euler-Lagrange eqaution, where -185- ,2 ,2 2 2 dy dz T h i s i s proven by t a k i n g the f i r s t v a r i a t i o n o f e q a u t i o n 2.10 which i s g i v e n by 81 OO Of ' dy > 2 ' dy s 8 + 6 — , 3y J 2 2 - 8[v (y,z)v J dydz or 81 OO oo 11 d\|f d8y dy d8y 2 2 2 +2 2v (y, z)v8y - 2y v (y, z) 8v (y, z) dy dy dz dz dydz A p p l y i n g Green's theorem t o the f i r s t two terms i n the above i n t e g r a l one o b t a i n s J J <ty d8y dy d8y 2 + 2 dy dy dz dz dydz oo oo - J J 28v ' d y d V . 2 2 V dy dz ) dy dydz + J Sy — ds C dn and t h e f i r s t v a r i a t i o n o f e q u a t i o n 2.10 becomes oo oo 61 ( i 2 i 2 d y d V - 28y + v (y,z)y 2 2 - 2y v (y, z) 8v (y, z) dydz -186- o f which t h e e x p r e s s i o n c o n t a i n e d w i t h i n the parentheses i s r e c o g n i z e d t o be the g e n e r a l form o f the s c a l a r wave e q u a t i o n and i s equal t o zero. Hence the g e n e r a l form o f the s c a l a r wave e q u a t i o n i s the Euler-Lagrange e q u a t i o n p r o v i d e d t h a t v(y,z) i s s t a t i o n a r y , i . e . , i f the f i r s t v a r i a t i o n o f i n t e g r a l i s t o be zero then the l a s t term i n the i n t e g r a l must a l s o be equal t o zero which i m p l i e s t h a t v(y,z) must be s t a t i o n a r y , i . e . 8v(y,z) = 0 . B.3 The P r o p a g a t i o n C o n s t a n t The purpose o f t h i s s e c t i o n i s t o show t h a t the p r o p a g a t i o n c o n s t a n t i s a s t a t i o n a r y v a l u e . To do t h i s one beg i n s w i t h t h e e q u a t i o n IS 2 2 a 2ln[n^ (y, z) J z dln[n z(y,z)] 9z dydz / J S dydz and t a k e s i t s f i r s t v a r i a t i o n . I f the i n t e g r a l i n the numerator on t h e r i g h t hand s i d e o f the above e q u a t i o n i s l a b e l l e d 1\ and the i n t e g r a l i n the denominator i s l a b e l l e d J-2 then the f i r s t v a r i a t i o n i s g i v e n by -187- 80 «= 2p 80 =8 V V V *1 ̂  V 1! " I 1 6 I 2 6 1 1 + P v 6 l 2 (Bl.l) The f i r s t v a r i a t i o n o f I i i s 81 dy dfiy dy d8y 2 2 +2 2p (y, z)y8y - 2y p (y, z) 8p (y, z) dy dy dz dz dydz where 2 / 2 2 d l n f n (y,z)] z d z ' z dln[n (y,z)] z dz 2 2 + n (y, z) k z o and the f i r s t v a r i a t i o n o f i n t e g r a l l£ i s oo 00 81 2 = J J 2y8y dydz so t h a t t h e terms 2 2 ~ ~ v 8 l 2 - P v J i 2y8y dydz and - / / 2p (y, z)y6y dydz c a n b e c o m b i n e d t o r e s u l t i n 00 00 2 2 2 - \ \ 2p (y,z)y8y dydz + P v J / 2y8y dydz = - J J 2v (y,z)y8y dydz -188- A p p l y i n g Green's theorem t o the f i r s t t o terms i n 611 g i v e s OO CM dy dSy dy d5y 2 + 2 dy dy dz dz dydz = oo oo j j 28y ( d 2 y d 2 y ^ -oo —oo :» 2 a 2 dy dz dy dydz + j 6y — ds dn and t h e numerator o f e q u a t i o n BI.1 becomes J J / 2 -.2 d y d y - 26y + v (y,z)y 2 2 , dy dz - 2y p(y, z)8p(y,z) dydz The e x p r e s s i o n w i t h i n the parentheses i s r e c o g n i z a b l e as the s c a l a r wave e q u a t i o n and i s equal t o zero . I t f o l l o w s t h a t SBv i s s t a t i o n a r y p r o v i d e d t h a t the f i r s t v a r i a t i o n o f p(y,z) i s zero i . e . «p(y, z) = 0 which i s o b v i o u s l y so p r o v i d e d t h a t an a p r i o r i knowledge o f w z v i s assumed. -189- 0 A p p e n d i x C THE COUPLING C O E F F I C I E N T C l I n t r o d u c t i o n In t h i s appendix e q u a t i o n 2.17 i s d e r i v e d . In s e c t i o n 2.6 we d e r i v e d the e x p r e s s i o n 2 a f P f J J 2 2 2 2 2 2 2 2 -{y /w +z /w +(y-a) /w ><z-b) /w 1 /2 yv zv yf zf , , e J J dydz T = v. w J v yv y —, 0 ^ 2 v yv .2,2 2,2, —(y /w +z /w ) yv. zv dydz by assuming t h a t the power cou p l e d t o r e f l e c t e d r a d i a t e d modes c o u l d be n e g l e c t e d . T h i s was done u s i n g the approximate f i e l d d i s t r i b u t i o n s m o t i v a t e d i n s e c t i o n 2.5 f o r the o p t i c a l f i b e r and the VIOWM. Here the v a r i a b l e s a and b l o c a t e t h e c e n t e r o f the o p t i c a l f i b e r r e l a t i v e t o the -190- c e n t e r o f t h e s u r f a c e o f the waveguiding r e g i o n o f the VIOWM. C.2 The N o r m a l i z e d A m p l i t u d e s a f a n d a v The time-averaged power p r o p a g a t i n g i n a guided mode o f where E i s the e l e c t r i c f i e l d , H i s the magnetic f i e l d , and u x i s a u n i t v e c t o r i n the d i r e c t i o n o f p r o p a g a t i o n . When the e l e c t r i c f i e l d o f the guided mode i s p o l a r i z e d i n the p l a n e normal t o the d i r e c t i o n p r o p a g a t i o n , the t r a n s v e r s e p l a n e , t h e mode i s c a l l e d a t r a n s v e r s e e l e c t r i c or TE mode. The e x p r e s s i o n f o r t h e power i n a TE mode may be reduced t o where the s u b s c r i p t t i n d i c a t e s the t r a n s v e r s e component o f th e e l e c t r i c f i e l d . In our case the t r a n s v e r s e component o f the e l e c t r i c f i e l d i s p o l a r i z e d p a r a l l e l the z - a x i s and i s an o p t i c a l waveguide may be g i v e n terms o f the f i e l d s by* l 2 dydz * See f o r example Snyder and Love [77] chapter 11 pp. 208-237. -191- g i v e n f o r t h e o p t i c a l f i b e r by eq u a t i o n 2.11 and f o r the VIOWM by e q u a t i o n 2.12. I f t h e power i n the mode i s no r m a l i z e d so t h a t the mode c a r r i e s p Watts then t h e amplitude o f the f i e l d i s c a l c u l a t e d u s i n g 2fifoop 0 0 0 0  / 2 / 2 . 2 . 2 . f , -<y /w + z / w z f ) l i e dydz f o r t h e f i b e r and u s i n g 2^v<op —00 0 yv # 2 . 2 2 . 2 - ( y /w + z /w ) e dydz f o r t h e VIOWM. The r a t i o a f / a v i s determined by t a k i n g the square r o o t o f t h e r a t i o a f / a v and by assuming t h a t n-f = = u D . I t i s g i v e n by -00 0 yv 2 . 2 2 . 2 , - ( y /w + z /w ) yv Z V ^ J e dydz . 2 , 2 . 2 , 2 . -(y / w f + z /wzf) e dydz -192- S u b s t i t u t i n g t he r a t i o a f / a v i n t o t he e x p r e s s i o n f o r the c o u p l i n g c o e f f i c i e n t g i v e s *<M«>1/2JJ 2 , 2 . 2 , 2 2 , 2 2 , 2 -{y /w +z /w +(y-a /w ,+ (z-b) /w ,}/2 l j r yv zv > J ' y f ' z f , . e 3 dydz T - v w / » 2 . 2 2 . 2 ~ ~ - ( V / W + Z / W z f ) /• r e ^ dydz J I yv 2 2 2 2 . - ( y /w +z /w ) yv zv dydz 1 2 C.3 The N u m e r a t o r o f t h e C o u p l i n g C o e f f i c i e n t The i n t e g r a l i n the numerator o f the above e x p r e s s i o n f o r t h e c o u p l i n g c o e f f i c i e n t i s ' v \ 2 2 2 2 2 2 2 2 Y -{y /w + z /w + (y-a) /w + (z-b) /w >/2 yv zv y f z f . . e dydz yv ' F i r s t t he i n t e g r a l i s t o se p a r a t e d i n t o two i n t e g r a l s one i n y and one i n z r = e 2,„ 2 .2-- 2 ~ ( y ^ - a /2w y f -b /2w z f -J 2 2 -y / n +ay/w y J ' y f ^ „ J" - z 2 / n +bz/w 2 f dy I e dz \ w i where -193- 0 2 2 2w w yv yf n - y 2 , 2 yv yf and n z r> 2 2 2w w _ zv zf 2 ^ 2 w + W _ zv zf Next t h e s u b s t i t u t i o n s o f v a r i a b l e s y y' - — " y / 2 and z n l / 2 a r e made y i e l d i n g 1/2 n n ,2,2 , .2 . 2 . / 0 ~ , 2, 0 l / 2 , . 2 ~ , 2 ,. _ l / 2 , . 2 y z - ( a /w +b / w 2 f ) / 2 f ~y + a I 1 y y / w y f f ~ z + b n 2 / w z f r •= : e J y'e dy'J e dz' w yv We s h a l l s o l v e the above i n t e g r a l s one at a time b e g i n i n g w i t h ,2 ^ _,l/2 , , 2 r -y' + an y'/w y' e y y f dy' -194- which can be w r i t t e n i n terms o f the Hermite p o l y n o m i a l rr . \ f -y' 2 + aft 1 / 2y'/w 2 1 - _ y,2 + a f l l / Z y , / 2 J y' e  y y  dy' = _ J (y') e  Y Y  dy' 0  2  0 The g e n e r a t i n g f u n c t i o n f o r Hermite p o l y n o m i a l s i s * 2 2 d n  e" x H <x) = < - l ) n e x , n = 0,1,2,... dx which can be s u b s t i t u t e d i n t o the p r e c e d i n g i n t e g r a l g i v i n g ,2 ^ „ l / 2 , . 2 - „ l / 2 , . 2 , -y' 2 • -y' • + all y' /w { an y' /w de  3 J H (y') e  y y  dy' = - J e  y y  dy' = 0 0 dy' an 1 / 2 y'/w 2 , ,2 _ J e - y J ' "yf d e - y ' t h e l a s t term o f which i s s o l v e d u s i n g i n t e g r a t i o n by . ** p a r t s ~ an 1/2 y'/w 2  ,2 - J e y  y f  de" y Lebedev [78] p. 60. A l l of t h e i n t e g r a l s i n t h i s appendix i n v o l v i n g e x p o n e n t i a l s can be s o l v e d u s i n g the formulas i n S p i e g e l [7 9] on p. 98 and u s i n g the s p e c i a l v a l u e s f o r the gamma f u n c t i o n on p. 101. -195- - y ' 2 + an 1 / 2y'/w 2 £ ~ ,~ -y' 2 + af> 1 / 2y'/w 2 y - ""y f I + f e 0 0 y y f d y ' y a fl/4w 1 + e J J e r f c ' - a n 1 / 2 N 2w y f I y f ) The second i n t e g r a l i s e a s i l y s o l v e d g i v i n g I - z ' 2 + bn 1 / 2z'/w 2 ? 4 b Cl /4w zf , , 1/2 z zf dz' = it e The numerator o f the c o u p l i n g c o e f f i c i e n t can now be w r i t t e n as r = « O 1^ 2 , 2, 2 ,.2, 2 . 2 „ 4 y z l y/ 2 - < a / w „ ^ + b /w,^)/2+b Il^/4wr K e yf zf z zf 2w yv 1/2 1/2 , an * 2_ 4 y a n /4w 1 + e y y e r f c 2w yf -a i l 1/2 "1 2w yf ) C.4 T h e D e n o m i n a t o r o f t h e C o u p l i n g C o e f f i c i e n t The i n t e g r a l s i n the denominator o f the e x p r e s s i o n f o r the c o u p l i n g c o e f f i c i e n t are -196- ~ o ° , 2 , 2 . 2 , 2 . ««.<« - J / e y + ̂  /"-,dyd, J J -oo 0 N 2 -(y 2 /w 2  + z 2 /w 2  ) e dydz yv which can be s e p a r a t e d i n t o f o u r i n t e g r a l s each o f which i s s o l v e d i n a s t r a i g h t forward manner 2, 2 e dy = w it y f 2, 2 dz = W JK zf > 2 2 . 2 w re —v /w yv yv e dy = 1/2 yv and J 2/ 2 — z /w zv e dz w TC zv 1/2 The e x p r e s s i o n f o r r' becomes W W W _W JK yv zv yf zf or f o r r ' 1 / 2 1/2 (w W W ,W -t yv zv yf zf 12 -197- C.5 E q u a t i o n 2.17 Now the c o u p l i n g c o e f f i c i e n t can be g i v e n by 2 ( P f P v ) 1 / 2 r 1/2 (P f +P v ) ' which reduces t o X K f v' y z 1/2 ,„ a , , ' «1/2 re (B, + B ) (w w w ,w c) w V H f K v ' * yv zv yf z f yv 1/2 1/2 a \ / 4 w y f f , afl rc e J e r f q y I + 2w yf (2.17) -198- R e f e r e n c e s 1. 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