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Voltage induced optical waveguide modulators in lithium niobate Jaeger, Nicolas August Fleming 1989

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VOLTAGE  INDUCED OPTICAL WAVEGUIDE MODULATORS IN LITHIUM NIOBATE by NICOLAS AUGUST FLEMING  B . S c . E . E . , The U n i v e r s i t y M.A.Sc,  The U n i v e r s i t y  A THESIS SUBMITTED  JAEGER  of the Pacific,  of British  Columbia,  IN PARTIAL FULFILLMENT  OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF E L E C T R I C A L ENGINEERING We  accept t h i s  thesis  to the required  as  conforming  standards  THE UNIVERSITY OF B R I T I S H COLUMBIA M a r c h 1989 © N i c o l a s August  1981  Fleming Jaeger,  1989  1986  In  presenting  degree freely  at  the  available  copying  of  department publication  this  of  in  partial  fulfilment  University  of  British  Columbia,  for  this or  thesis  reference  thesis by  this  for  his thesis  and  scholarly  or for  her  of  The University of British Columbia Vancouver, Canada  DE-6 (2/88)  I  1 further  purposes  gain  the  shall  requirements  agree  that  agree  may  representatives.  financial  permission.  Department  study.  of  be  It not  that  the  be  an  advanced  Library shall  permission for  granted  is  for  by  understood allowed  the that  without  make  it  extensive  head  of  copying my  my or  written  Abstract  Two t y p e s the voltage and  o f o p t i c a l m o d u l a t o r were s t u d i e d , b o t h o f  induced  o p t i c a l waveguide type  d e m o n s t r a t e d b y C h a n n i n i n 1971.  was c r e a t e d i n an e l e c t r o o p t i c  operated  planar  which r e s u l t e d i n l a r g e  a t much r e d u c e d  and r i d g e types.  e l e c t r o d e s were d e p o s i t e d i n t h e ridge type  The d e v i c e s  discussed  voltages.  They a r e o f  In the planar type the  on t o p o f a p l a n a r  s u b s t r a t e and  The t h e o r y  of operation  a n d t h e y were m o d e l e d ,  tested.  d e r i v e d r e s u l t s were o b t a i n e d  Numerically  w i t h w a v e l e n g t h s o f 442  decreasing  device  for light  increased with  increasing voltage, The t h e o r y  t o i n v e s t i g a t e the performance o f the  a s 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 w a v e g u i d e b e t w e e n two  optical function optical  f a b r i c a t e d , and  and d e c r e a s i n g wavelength.  was f u r t h e r d e v e l o p e d  types o f  a n d 633 nm w h i c h showed t h a t t h e  of the light gap w i d t h ,  separated  f o r both  ^ o T " . r o was d e v e l o p e d  confinement  and  a ridge of electrooptic material  thick electrodes.  voltage Channin  t h e s i s h a d 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  therefore the  waveguide  on t h e s u b s t r a t e .  operating voltages being necessary. in this  An o p t i c a l  proposed  s u b s t r a t e by a p p l y i n g  b e t w e e n two e l e c t r o d e s d e p o s i t e d u s e d wide e l e c t r o d e spacings  first  fibers.  The optimum c o u p l i n g e f f i c i e n c y ,  of voltage  a n d i n t e r e l e c t r o d e gap w i d t h ,  f i b e r s t o both  terms o f t h e model.  types  as a from  o f d e v i c e was c a l c u l a t e d i n  Key a s p e c t s  o f the theory  were  —iii-  confirmed  by t h e measurements made on t h e f a b r i c a t e d  devices.  A p l a n a r d e v i c e was  u s e d 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 t h e v o l t a g e i n d u c e d waveguide w i t h t h e fiber.  One  p r o b l e m was  a decay phenomenon i n w h i c h t h e  i n d u c e d waveguide d i s a p p e a r e d w h i c h a c o n s t a n t v o l t a g e was T h i s was I t was  over a p e r i o d of time d u r i n g a p p l i e d t o the electrodes.  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 .  f o u n d t h a t t h e d e v i c e w o u l d r e c o v e r upon t h e  application of a fly-back cycle.  -iv-  Table o f Contents Page Abstract Table List  i i  o f Contents  iv  of Figures  v i i  Acknowledgements  x i i  Chapter  1  INTRODUCTION  Chapter  2  THEORY  12  2.1  Introduction  12  2.2  The E l e c t r o o p t i c  2.3  The C o n f o r m a l 2.3.1  1  Effect  14  Mappings  18  The P l a n a r D e v i c e  19  2.3.1.1  The D e v i c e S t r u c t u r e  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 .  2.3.2  19 . .  The R i d g e D e v i c e  21 29  2.3.2.1  The D e v i c e S t r u c t u r e .  2.3.2.2  The E l e c t r i c  . . . . . . . .  Field Distributions.  . .  29 33  2.4  The V a r i a t i o n a l Method  41  2.5  The H e r m i t e - G a u s s i a n  49  2.5.1  The O p t i c a l  Fiber  2.5.2  The  I n d u c e d  2.6 Chapter 3.1  Approximations  V o l t a g e  51 O p t i c a l  W a v e g u i d e .  .  .  5 8  The C o u p l i n g C o e f f i c i e n t  65  3  74  FABRICATION  Introduction  .  74  —v-  3.2  T h e VIOWM  75  3.2.1  The P l a n a r  VIOWM  77  3.2.2  T h e R i d g e VIOWM  3.2.3  C u t t i n g and P o l i s h i n g .  79 .  88  3.3  The S i l i c o n V-grooves  91  3.4  Device/Optical  98  Chapter 4  F i b e r Alignment  RESULTS  105  4.1  Introduction.  105  4.2  The P l a n a r  VIOWM  106  4.2.1  Calculated Results  107  4.2.2  Measured R e s u l t s  121  4.3  T h e R i d g e VIOWM  . .  135  4.3.1  Calculated Results  135  4.3.2  Measured R e s u l t s  . . . *.  147  . . . . . . . . . . . . .  156  4.4  The F r o n t - E n d S w i t c h .  4.5  Discussion.  Chapter 5  . . . . . . .  Summary, C o n c l u s i o n s , Further  Work  5.1  Introduction.  5.2  Summary  5.3  Conclusions  5.4  Suggestions  Appendix A  164 and Suggestions f o r  . . . . . . . .. . . . . . . . ,  168 168  *  168 170 f o rFurther  Work  171  THE ELECTROOPTIC E F F E C T  A.l  Introduction  A. 2  The R e l a t i v e D i e l e c t r i c  174 174  Impermeability  Tensor  .  175  -vi-  A.3  The E l e c t r o o p t i c  A. 4  Equations  Appendix B  Effect  176  2.1 a n d 2.2  179  STATIONARY FORMULAS  184  B. l  Introduction  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 C o n s t a n t  186  Appendix C  .  THE COUPLING COEFFICIENT  189  C l  Introduction  C. 2  The N o r m a l i z e d A m p l i t u d e s  C.3  The Numerator o f t h e C o u p l i n g C o e f f i c i e n t  C.4  The Denominator o f t h e C o u p l i n g C o e f f i c i e n t  C.5  E q u a t i o n 2.17  References.  184  189 a f and a  190  v  . . . . .  192 195 197 198  -vii-  List  1.1  1.2  1.3  o f Figures  T h e p l a n a r VIOWM i n c r o s s s e c t i o n . H e r e l i g h t i s p r o p a g a t i n g o u t o f t h e page t o w a r d t h e r e a d e r . * .  5  T h e r i d g e VIOWM i n c r o s s s e c t i o n . H e r e l i g h t i s p r o p a g a t i n g o u t o f t h e page toward.the r e a d e r . . .  6  T h e VIOWM a c t i n g a s a l i n k i n g w a v e g u i d e b e t w e e n two o p t i c a l f i b e r s  10  2.1  T h e p l a n a r VIOWM s t r u c t u r e  20  2.2  A n i n t e r m e d i a t e m o d e l o f t h e p l a n a r VIOWM structure  22  T h e m o d e l u s e d t o a n a l y z e t h e p l a n a r VIOWM structure. . . . . . . . . . . . . .•* . . . . . .  23  2.4  The W a n d S - p l a n e s  25  2.5  A plot  o f E (y,z)  f o r t h e p l a n a r VIOWM  . . . . . . 30  2.6  A plot  o f E (y,z)  f o r t h e p l a n a r VIOWM  31  2.7  T h e r i d g e VIOWM s t r u c t u r e  2.8  The model used t o a n a l y z e VIOWM s t r u c t u r e  2.3  y  z  32 t h e r i d g e waveguide 34  2.9  T h e (a) £, (b) W, a n d (c) S - p l a n e s  36  2.10  A p l o t o f E ( y , z ) f o r t h e r i d g e VIOWM. Here t h e p l o t h a s been c u t a l o n g t h e l i n e z = 0 showing the f i e l d f o r z £ 0 only  42  A p l o t o f E ( y , z ) f o r t h e r i d g e VIOWM. Here t h e p l o t has been c u t a l o n g t h e l i n e z = 0 showing the f i e l d f o r z £ 0 only .  43  The r e f r a c t i v e index d i s t r i b u t i o n with a step index p r o f i l e  52  2.11  2.12 2.13  y  z  A Gaussian w  2.14  yf  field  distribution  i n which  = zf w  A Gaussian Wyf  optical  of a fiber  = w /2 zf  5  optical  field  distribution  6  i n which .  57  -viii-  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  61  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 /2  62  A p l o t o f Bv versus a 4 um i n t e r e l e c t r o d e the electrodes f o r X  66  z v  2.16  zv  2.17  2  Q  2.18  and w f o r a VIOWM w i t h g a p w i t h 50.0V a p p l i e d t o = 442 nm . . . . . . . . . . z v  T h e 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 um i n t e r e l e c t r o d e g a p f o r (a) 10.0, (b) 30.0, a n d (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 for X = 442 nm  67  The i n t e r f a c e between t h e o p t i c a l f i b e r VIOWM.  71  Q  2.19  3.1 3.2  3.3 3.4 3.5 3.6 3.7  The i n t e r e l e c t r o d e i s 4 um w i d e .  g a p o f a VIOWM.  and t h e  Here t h e gap 80  SEM p i c t u r e o f a r i d g e e t c h e d i n L i N b 0 3 - T h e s c a l e o f t h e 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  SEM p i c t u r e s h o w i n g t h a t t h e r i d g e i s a b o u t 7.5 um w i d e . -". . . . . . . . . . . . . . . . . . . . .  84  P r o f i l o l m e t e r output t h e r i d g e i s 4 urn  85  showing t h a t  The aluminum e l e c t r o d e s o f a r i d g e by t h e s e l f - a l i g n e d t e c h n i q u e  theheight o f VIOWM f o r m e d 87  The p o l i s h i n g j i g : t h e main body ( r i g h t ) a n d t h e p o l i s h i n g p l a t e ( l e f t ) . . . . . ; . '* . . . . . .  89  T h e e n d f a c e o f two r i d g e VIOWMs, e p o x i e d a f t e r a 1 um a l u m i n a p o l i s h  92  together,  3.8  A p l a n a r VIOWM w i t h t h e e n d s p o l i s h e d  93  3.9  T h e p o l i s h e d e n d o f a p l a n a r VIOWM where t h e i n t e r e l e c t r o d e gap i s seen t o r u n p e r p e n d i c u l a r to t h e endface  94  3.10  The V - g r o o v e f a b r i c a t i o n p r o c e s s  99  3.11  T h e V - g r o o v e a r r a y , f i b e r , VIOWM, p i n , a n d p r o b e s during t h e alignment procedure . . . . . . . . . .  102  -ix-  3.12  T h e V - g r o o v e a r r a y , f i b e r , VIOWM, p r o b e s , a n d 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 bonding procedure  4.1  A topographical  plot  of  4.2  A topographical  plot  of w  forK  Q  = 442 nm. . . . 109  4.3  A topographical  plot  o f Wy^. f o r X  Q  = 633 nm. . . . 110  4.4  A topographical  plot  of w  Q  = 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 a t 30 V i s m a x i m i z e d . . . . 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 t h e c o u p l i n g a t 50 V i s m a x i m i z e d . . . . 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 m a x i m i z e d . . . . 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 a t 50 V i s m a x i m i z e d . . . . 117  4.9  T h e optimum power t r a n s f e r T^ a s 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 a n d i n t e r e l e c t r o d e g a p w i d t h . . . 119  4.10  T h e p o w e r t r a n s f e r T b e t w e e n 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 m a x i m i z e d . . . . . . 122  4.11  T h e power t r a n s f e r T b e t w e e n 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 m a x i m i z e d . . . . . 123  4.12  T h e b a s i c l a b o r a t o r y a p p a r a t u s u s e d t o make m e a s u r e m e n t s o n VIOWMs . . . . . . . . . . . . . . . 125  4.13  The p o l a r i z e d o u t p u t  4.14  T h e o u t p u t 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 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 to the electrodes. . . 128  4.15  T h e o u t p u t 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 p e a k - t o - p e a k t r i a n g l e wave applied t o the electrodes  4.16  4.17  forX  103  Q  z v  z v  forX  = 442 nm. . . . 108  4  4  of the optical  fiber.  . . . . 126  T h e o u t p u t 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 p e a k - t o - p e a k t r i a n g l e wave applied t o the electrodes • • • • • A comparison o f t h e t h e o r e t i c a l  and measured  12 9  1  3  results  0  —x-  for  a planar  device  133  4.18  An a l t e r n a t e  laboratory  apparatus  setup.  4.19  T h e o u t p u t o f a l o n g VIOWM w i t h 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 f o r K = 633 nm . . .  . . . . . 134  * . . . 136  Q  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 t i m e s t h e i n t e r e l e c t r o d e gap w i d t h where X = 442 nm . . . . . . . . . 139 Q  4.21  A topographical plot of f o r a ridge height t i m e s t h e i n t e r e l e c t r o d e g a p w i d t h where X = 442 nm  1.0  0  4.22  140  A topographical plot o f w f o r a r i d g e h e i g h t 1.5 t i m e s t h e i n t e r e l e c t r o d e g a p w i d t h where X = 442 rim . . . . . . . . . . . . . . 141 v v  Q  4.23  A topographical plot of w f o r a ridge height t i m e s t h e i n t e r e l e c t r o d e g a p w i d t h where X = 442 nm . . . . . . . . . . .  0.5  A topographical plot of w f o r a ridge height t i m e s t h e i n t e r e l e c t r o d e g a p w i d t h where X = 442 nm .  1.0  z v  Q  4.24  z v  142  Q  ;  4.25  143  A topographical plot of w f o r a r i d g e h e i g h t 1.5 t i m e s t h e i n t e r e l e c t r o d e g a p w i d t h where X = 442 nm . . . . . . . . . . . . . . . . . . . . . . 144 z v  Q  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 t h e c o u p l i n g a t 30 V i s m a x i m i z e d . . . . . . . . . . . . . . . . .  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 a t 50 V i s m a x i m i z e d . . . . . . . . . . . . . . . . . 146  4.28  T h e 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 a n d 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 0.5 t i m e s t h e gap w i d t h  148  T h e 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 w i d t h f o r a r i d g e 1.0 t i m e s t h e gap w i d t h  147  T h e 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 w i d t h f o r a r i d g e 1.5 t i m e s t h e gap w i d t h  150  4.29  4.30  -xi-  4.31  T h e p o w e r t r a n s f e r T b e t w e e n two o p t i c a l f i b e r s w h e r e t h e c o u p l i n g a t 30 V i s m a x i m i z e d f o r t h e ridge device  151  4.32  T h e p o w e r t r a n s f e r T b e t w e e n two o p t i c a l f i b e r s w h e r e t h e c o u p l i n g a t 50 V i s m a x i m i z e d f o r t h e ridge device . . . . 152  4.33  T h e 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 g a p f o r a 100 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 where X = 442nm .  4  Q  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 ridge . 155  4.35  T h e 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 g a p f o r a 120 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 where X = 442nm Q  157  4.36  T h e o u t p u t o f a h a l f - h e i g h t VIOWM f o r a ±20 V s q u a r e wave a p p l i e d t o t h e e l e c t r o d e s . . . . . . . 158  4.37  T h e o u t p u t o f a n o p t i c a l f i b e r w i t h a VIOWM a c t i n g a s 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 jiW. . . . 160  4.38  The decay o f t h e output  o f a VIOWM a s a f u n c t i o n  -xii-  Acknowledgments  This part  i s easy.  h e l p t o me t h o u g h o u t guidance  I doubt  They d e s e r v e appreciation. gratitude  My p a r e n t s h a v e b e e n t h e g r e a t e s t  t h i s work.  that  Without  a n d r e c e i v e my d e e p e s t  love,  and thanks.  Without  and support I  I thoroughly enjoy.  more d i f f i c u l t .  While  There  available  discussions. as f r i e n d s ,  There  sort  made t h e h a r d e s t t i m e s  o u t t h e one g r o u p  fringes begin t o blur my m i n d s e y e o n e i t h e r  somewhat  a department  should nots. in  full  easier  When I t r y  from t h e o t h e r I f i n d t h a t t h e  a n d many o f t h e same f a c e s a p p e a r i n side.  t h o s e who s h o u l d b e m e n t i o n e d find  actions  a r e a l s o t h o s e who, b y b e i n g  a n d t h e r e b y made i t p o s s i b l e t o k e e p on g o i n g . to  as a  a r e t h o s e who h a v e  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 t h r o u g h  through  several  even e n j o y a b l e , i t i s h a r d t o  s a y who h a s c o n t r i b u t e d what. directly  This  life.  s t a n d o u t i n my m i n d a s h a v i n g made my l i f e student t o l e r a b l e ,  When I t r y t o d i f f e r e n t i a t e f r o m t h o s e who s h o u l d n o t I  o f s h o u l d s a n d an empty c o l u m n o f  I c a n t h i n k o f no one who h a s n o t t o u c h e d  some way t h a t  my  i n integrated  c o n t r i b u t i o n t o my  Now t h i n g s g e t a l i t t l e  or  thanks, and  h i s guidance  an a r e a o f r e s e a r c h t h a t  graduate  point.  N e x t t o my p a r e n t s D r . L . Young d e s e r v e s  h a s b e e n a most i m p o r t a n t  people  support and  I e v e r would have r e a c h e d t h i s  s u r e l y would n e v e r have found an i n t e r e s t optics,  their  me  h a s e f f e c t e d who I am a n d what I h a v e done.  -xiii-  Yet  i f I thank  I will  one a n d a l l a n d m e n t i o n  n o t b e b e i n g t r u e t o t h o s e who were most t r u e t o me.  I have, t h e r e f o r e , one  compiled a l i s t  reason o r another w i l l  what g r a d u a t e of  trials  linger  and t r i b u l a t i o n s  subject matter  o f t h o s e p e o p l e who, f o r be p a r t  o f my memory o f  I am s u r e t h a t  the faces  i n my memory l o n g a f t e r t h e  are forgotten,  o f my t h e s e s .  I cannot  l o n g e r even than t h e list  everything that  o f t h e f o l l o w i n g p e o p l e h a s done f o r me n o r c a n I  a t t a c h a weight them, like  always  s t u d i e s t r u l y was.  these people w i l l  each  no one i n p a r t i c u l a r  t o t h e importance  I am s u r e t h a t  o f my i n t e r a c t i o n s  with  e a c h one o f them knows, I w o u l d  just  t o g i v e them a l l my s p e c i a l t h a n k s :  I . Abdel-Motaleb,  B. A h l b o r n , N. B e a u l i e u , M. B e d d o s e , F . B e r r y , E . Bonn, D. Boshier, Dindo,  K. B r i n d a m o u r , J . C l a r k , A. C h o i , D. D a i n e s , J .  S. D i n d o ,  Fletcher, Jankowski,  H. Dommel, R. D o n a l d s o n ,  B. G u i d i c i , E. J u l l ,  D. H u i , C. J a e g e r , N. J a e g e r ,  H. K a t o ,  G. S c h m i d t ,  Townsley, the  short  P.  C. P a s s m o r e , W. P a s s m o r e , A.  C. S h e f f i e l d ,  L . S n i d e r , C. S u d h a k a r , P.  J . Weber, L . Wedepohl, a n d M. Wvong. list!).  To t h o s e n o t l i s t e d  R.  M. K h a r a d l y , A. M a c K e n z i e ,  M a t z , D. M i c h e l s o n , C. N e s b i t t , Prince,  C. Dumont, D.  above, t h a n k y o u .  (And t h a t i s  -1-  Chapter  1  INTRODUCTION  Integrated design,  study,  o p t i c s i s t h e name g i v e n and a p p l i c a t i o n o f devices  components t h a t  s i m i l a r i n scale to,  One  o f t h e m o s t commonly u s e d m a t e r i a l s  o f components t h a t  waveguides [8,9],  for thefabrication  can be f a b r i c a t e d i n LiNb03  [1,2], modulators  LiNb03-  introduced  induced  limitation  voltage  o f 300V.  i n that  (VIOWM) i n  o p t i c a l w a v e g u i d e was  b y C h a n n i n i n 1971 [ 1 2 ] .  serious  Soref  the fabrication of  o p t i c a l waveguide modulators  The v o l t a g e  include  [10,11].  This t h e s i s i s concerned with induced  The  [3-7], p o l a r i z a t i o n converters  and frequency converters  voltage  optical  i n t e g r a t e d e l e c t r o n i c components.  i n t e g r a t e d o p t i c components a n d s y s t e m s i s L i N b 0 3 .  kinds  The  made u s i n g  a r e f a b r i c a t e d b y t h e same m e t h o d s a s , a n d  are  of  to the analysis,  first  Channin's device  i t n e e d e d an minimum  C h a n n i n ' s work was f o l l o w e d  e t a l . [13] w h i c h a l s o n e e d e d s u c h l a r g e  apparent need f o r such l a r g e v o l t a g e s  had a  operating  by t h a t o f voltages.  caused t h e voltage  -2-  induced  o p t i c a l waveguide t o be d i s r e g a r d e d  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 . shown  and other  In t h i s t h e s i s i t i s  that, t h e v o l t a g e l e v e l s c a n b e r e d u c e d b y a t 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 a n d t e s t e d v o l t a g e induced  o p t i c a l w a v e g u i d e s i n KNbC-3  electrooptic 12dB  coefficient*  f o r 35V a p p l i e d .  spacings,  than  LiNbC>3) w i t h  on-off  and l a r g e o p t i c a l  field  Our d e v i c e s were a b l e t o a c h i e v e  good c h a r a c t e r i s t i c s  ratios of  These d e v i c e s had l a r g e e l e c t r o d e  large capacitances,  distributions.  (which has a l a r g e r  a t lower v o l t a g e s  equally  a n d h a d much  lower  capacitances. Savatinova electrically  e t al... [15] h a v e d e m o n s t r a t e d a n  induced  device the optical formed by m o d i f y i n g in-diffusion electrodes selectively prism  Ti:LiNb03  strip-waveguide.  characteristics  of a planar  t h e s u r f a c e l a y e r o f t h e LiNb03 by t h e  on t h e s u r f a c e o f t h e waveguide. coupled  into  intensity  can then  i n which there  device  The  o f Savatinova  using isa  be i n t e r r o g a t e d .  T h e y 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 The d e v i c e s  Light i s  a n d o u t o f a p a r t i c u l a r mode  Regions o f t h e output  change i n t h e l i g h t  the  waveguide,  o f T i , a r e changed by a p p l y i n g v o l t a g e t o  couplers.  modulation.  In t h e i r  studied i n this  a p o s s i b l e 95% thesis differ  et a l . i n that there  from  i s no  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 d i s c u s s e d i n appendix A.  -3-  diffusion  and t h a t  coupling  method o f '"butt-coupling"  i s achieved  [16] o r " e n d - f i r e - c o u p l i n g " [17]  w h i c h removes t h e need f o r p r i s m s . make f o r a n o p t i c a l alignment.  Also  distributions be  well  fibers.  coupling ability  Prisms a r e b u l k y and  s y s t e m n e e d i n g somewhat p r e c i s e  i n our devices  the optical  field  o f t h e g u i d e d modes c a n b e c o n t r o l l e d s o a s t o  matched t o t h e 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  A s s i n g l e mode f i b e r  frequently  b y t h e more d i r e c t  this  i s being  i s an i m p o r t a n t  efficiencies  advantage as regards  and p o s s i b l e  t o butt-couple  u s e d more a n d more  applications.  t o our device  will  have  advantages as f a r as t h e s i z e o f t h e device packaging o f a device  with fiber p i g t a i l s  The  further  and t h e  attached i s  concerned. Another device al.  that  h a s b e e n i n v e s t i g a t e d b y Kawabe e t  [18] c o n s i s t s o f a p l a n a r  LiNbC«3 i n w h i c h a 420 nm h i g h voltage  optical  the  g u i d e d wave i s r a d i a t e d  (a 20 v o l t  out  into the bulk of the c r y s t a l .  We h a v e b e e n a b l e  f o r the E t o achieve  z  fiber  devices  ( i nlong  t o t h e output  devices  fiber  -19dB f o r  mode o n a 3 mm  long  s i m i l a r r e s u l t s with  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  long  a r e a l t e r e d and  an e x t i n c t i o n c o e f f i c i e n t  difference)  When  on e i t h e r s i d e o f t h e r i d g e  c h a r a c t e r i s t i c s o f the substrate  Kawabe e t a l . r e p o r t  device.  r i d g e has been e t c h e d .  i s applied t o electrodes  the  ±10V  d i f f u s e d T i waveguide i n  o r a r i d g e by u s i n g  the coupling  from t h e i n p u t  due t o b u l k modes i s s m a l l ) .  Also  -4-  where t h e  throughput  continues  to  Further  increase  surface  f o r which the They d i d not  w a v e g u i d i n g and  voltages  the  saturate  d i f f e r e n c e s beyond  20V.  of  modes  r e q u i r e d were a b o u t 3 t i m e s as the  effect  t h e s i s two  t y p e s o f VIOWM a r e  of bulk  large.  mode c o u p l i n g  i s planar  and  the  the  s e c o n d has  s e c o n d has  a ridge.  other  The  devices  It  good o r b e t t e r of the  voltage  than induced  strictly  waveguide t y p e i n t h a t  surface  of the o f the  voltage substrate  predisposed In t h e  separated crystal The  by  to the  planar  has  device  LiNb03 s u b s t r a t e second device a ridge  ideally  are  been m o d i f i e d of o p t i c a l  (figure  by  are  two  isolated  a thin  (figure  1.1)  1.2)  Application  same h e i g h t  of voltage  i n the  to the  substrate.  that  electrodes,  from a s i n g l e Si02« from the  electrodes  planar  which  ridge.  electrodes The  the  i n order  metal  differs  as t h e  both  waves.  layer of  i s i n c l u d e d between t h e  of the  field  not  guiding  a n a r r o w gap,  in that  electric  induced  They a r e  is will  above.  be  in  the  first  added advantages.  shown t h a t t h e i r p e r f o r m a n c e i s as performance of the  studied;  genre w h i c h have been d i s c u s s e d  it  ours  Kawabe e t a l .  the propagation  comment on  e a s y t o make b u t be  for voltage  seems t o  device. In t h i s  first  device  p r o b l e m s t h a t were e n c o u n t e r e d by  included  their  of t h e i r  establishes  substrate  is a  an  crystal  -5-  Air  nterelectrode Gap Metal  Metal  Electrode  \  Opt i c a Buffer Layer  /-  _7\  N  Electrode  /  V///////.  W a v e g u i d i ng Region o—>  Electrooptic Substrate  z  v 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 i s p r o p a g a t i n g out o f t h e page t o w a r d t h e r e a d e r .  light  -6-  Ai r Electrode  E l ec t r o d e  \  /  Ridge  Op t  teal  Buffer Layer  Wavegu i d i n g Region  Electroopt i c Substrate  F i g u r e 1.2: propagating  0—> z  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 out o f t h e page t o w a r d t h e r e a d e r .  is  -7-  which e x h i b i t s the  electric  substrate. electrode  field  alters  The f i e l d  electrooptic  isolating  Therefore  indices  of the  i n the regions close t o the  as i n t h e immediate r e g i o n  gap hence as t h e v o l t a g e  wave i s i n c r e a s i n g l y  SiC-2 l a y e r  effect.  the refractive  i s largest  edges as w e l l  interelectrode optical  the linear  i s increased the  confined t o that  region.*  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 the optical  field  of the  from t h e metal  The  layer  electrodes.  While t h e operation o f t h e device i s conceptually straight to  f o r w a r d t o s t u d y a n d d e s i g n VIOWMs i t was n e c e s s a r y  d e v e l o p a method o f a n a l y s i s .  using  Our method c o n s i s t e d o f  c o n f o r m a l mapping t e c h n i q u e s t o c a l c u l a t e t h e  refractive  index d i s t r i b u t i o n  particular  voltage.  established  Then f o l l o w i n g t h e a p p r o a c h u s e d by  Marcuse f o r  analyzing  variational  technique t o f i n d the o p t i c a l  distributions distributions.  i n a VIOWM f o r a  step index  fibers  [19],  we u s e d a  field  f o r waveguides w i t h t h e s e r e f r a c t i v e I n t h i s method a p p r o p r i a t e t r i a l  index  functions  with variable  p a r a m e t e r s a r e assumed t o a p p r o x i m a t e t h e  optical  distributions.  field  See a p p e n d i x A f o r effect.  We h a v e u s e d  a discussion  Hermite-Gaussian  of the linear  electrooptic  As w i l l b e s e e n i n s e c t i o n 2.2 t h e v o l t a g e must h a v e t h e c o r r e c t s i g n , r e l a t i v e t o t h e c r y s t a l axes, f o r t h e r e f r a c t i v e index t o increase otherwise the r e f r a c t i v e i n d e x w i l l be d e c r e a s e d and l i g h t w i l l be r a d i a t e d o u t o f that region.  -8-  functions are  as t h e t r i a l  functions.  determined f o rthese  The v a r i a b l e  functions  parameters  by making u s e o f t h e  stationary  n a t u r e o f t h e e i g e n v a l u e s o f t h e s c a l a r wave  equation.  The w i d t h p a r a m e t e r s v a r y w i t h v o l t a g e  one  to predict  t h e development o f o p t i c a l  distributions with  increasing  allowing  field  voltage.  When d e s i g n i n g VIOWMs i n L i N b 0 3 t h e c o n t r o l l a b l e parameters include:  the substrate orientation, the  wavelength o f t h e l i g h t layer material, interelectrode  t o be guided, t h e o p t i c a l  the optical buffer gap o r i e n t a t i o n ,  width, t h e operating voltage, being used with the  layer  thickness, the  t h e i n t e r e l e c t r o d e gap  the ridge  height,  fibers the fiber location  and i f  a n d when possible  w i d t h p a r a m e t e r s o f t h e f i b e r mode. One  application  o f a VIOWM i s a s a v o l t a g e  l i n k i n g w a v e g u i d e b e t w e e n two o p t i c a l f i b e r s w h i c h o p t i c a l power i s t r a n s f e r r e d o t h e r v i a t h e VIOWM. sufficient  In t h i s application  t o know how t h e o p t i c a l f i e l d  development e f f e c t s t h e c o u p l i n g fibers.  A closed  coefficient  f o rbutt-coupling  ( f i g u r e 1.3)  i t i s not distribution i n the  t o and from t h e  f o rthe coupling  b e t w e e n an o p t i c a l f i b e r a n d a  VIOWM i s d e v e l o p e d i n t h i s t h e s i s .  Thus a c o m p l e t e  model  t h e s t u d y a n d 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 h a s  been developed.  in  know how t h e  of light  form e x p r e s s i o n  controlled  f r o m one f i b e r t o t h e  VIOWM d e v e l o p s o n l y b u t one must a l s o  for  buffer  The m o d e l a l l o w e d u s t o p r e d i c t  t h e modus  -9-  operandi  o f t h e VIOWM a s a l i n k i n g w a v e g u i d e f o r s e v e r a l  applications.  For instance  be used as a d i g i t a l being  used as a d i g i t a l  transfer  voltage  s w i t c h t h e change i n o p t i c a l  o f the device  theory  fabricating planar  described  o f operation  or the  and t e s t i n g  o f VIOWMs was c o n f i r m e d  several devices.  and r i d g e types. i n chapter  i nsilicon  fabrication technique  3.  Also  substrates  i sdescribed  by  T h e s e were o f b o t h  The m e t h o d o f f a b r i c a t i o n i s arrays  o f V-grooves  f o r t h e alignment arrangement  [20] were  of optical [21].  Their  as w e l l , as i s t h e alignment  a n d t h e method o f e u t e c t i c b o n d i n g between  e l e c t r o d e s on t h e V - g r o o v e s u b s t r a t e s  a n d on t h e e l e c t r o d e s  t h e VIOWMs. Tests  were made on t h e d e v i c e s .  was m e a s u r e d .  The c o u p l i n g  f i b e r s were m e a s u r e d .  digital  modulators.  optical  field  for a particular  distribution,  p r e d i c t e d by t h e theory.  response  were t e s t e d a s power  f a s h i o n by t h e a p p l i e d  and t h a t t h e c o u p l i n g e f f i c i e n c y  a particular voltage  input  The d e v i c e s  I t was shown t h a t t h e o u t p u t  c o u l d be modulated i n a l i n e a r voltage  Their voltage  c o e f f i c i e n t s b e t w e e n VIOWMs a n d  optical  for  level  power  t o small v a r i a t i o n s i n input  f i b e r s w i t h VIOWMs i n a f l i p - c h i p  on  When  l e v e l may b e m i n i m i z e d .  The  etched  o r as a l i n e a r modulator.  c a n be maximized f o r a g i v e n v o l t a g e  sensitivity  the  switch  i t i s shown t h a t t h e VIOWM c a n  c o u l d be  optimized  location of the  both these  phenomena  A n o t h e r p r e d i c t i o n was t h a t a  were  ^ 0 .  -11-  •"turn-on"  voltage exists  f o r r i d g e waveguides at which  r a p i d l y becomes c o n f i n e d t o t h e experimentally attached,  confirmed.  using  fabricated  and  an  tested.  a  optical  image r e c o r d e r  t o be  d.c. due  Columbia,  w  on  compensated  short devices  enhancing the  fiber also  envisioned to  optical  system of  Associates  observed  f o r by  f l y - b a c k " voltage to the  waveguide" out  optical  also  be an  as  the  of  Canada.  d e c a y phenomenon was  c o u l d be  was  s u c h as t h e F I R E 9000 p r o d u c e d by  a p p l i c a t i o n of a negative  useful  an  t o the photorefractive e f f e c t  this  negative The  f o r the  company M a c D o n a l d D e t t w i l e r and  A  that  A device with  T h i s d e v i c e was  front-end switch  Richmond, B r i t i s h  This effect  a r r a y o f s i l i c o n V - g r o o v e s , was  u s e d as  local  ridge.  light  which i s b e l i e v e d [22].  the  extinction ratio  found  application of  a  e l e c t r o d e s o f t h e VIOWM.  v o l t a g e was  a l s o found  i t created a sort  of which l i g h t  I t was  of  i n b u l k modes was of the  device.  w  to  be  anti-  refracted  -12-  2  Chapter  THEORY  2.1  Introduction  The of t h i s  theory chapter  The The The The The  the  s e c t i o n on t h e  change i n t h e  rotation  substrate.  in five  sections  entitled:  Electrooptic Effect, Conformal Mappings, V a r i a t i o n a l Method, Hermite-Gaussian Approximations, Coupling C o e f f i c i e n t .  In the for the  o f t h e VIOWM i s c o n t a i n e d  of the  electrooptic  refractive  indicatrix  Both are given  index  and  effect  expressions  a r e d e r i v e d and  about t h e X - a x i s  i n terms o f the  of  for  the  applied electric  In the c o o r d i n a t e system used here the x, y, and z-axes are c o i n c i d e n t w i t h the X, Y, and Z-axes of the LiNbC>3 s u b s t r a t e , r e s p e c t i v e l y . They correspond t o 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 t h a t the x - d i r e c t i o n becomes the d i r e c t i o n of p r o p a g a t i o n t h e r e f o r e the symbol, p, w i l l d e s i g n a t e the p r o p a g a t i o n c o n s t a n t i n the x - d i r e c t i o n .  -13-  field.  I t i s assumed t h a t Y - c u t LiNbC>3 i s t h e e l e c t r o o p t i c  substrate.  This  corresponds t o the o r i e n t a t i o n that  used i n the f a b r i c a t i o n devices  described  electric  field  Again t h i s  behavior  of the  The a p p l i e d  i s assumed t o h a v e Y a n d Z components  only.  corresponds t o the case i n our experiments. field  distribution  needs t o be a n a l y z e d o f the devices  i n s i d e the e l e c t r o o p t i c  i n order  to predict the  at various voltages.  c o n f o r m a l mappings s e c t i o n b o t h t h e p l a n a r types  experimental  i n the following chapters.  The e l e c t r i c substrate  and t e s t i n g  was  a r e modeled and t h e e l e c t r i c  established  field  In the  and r i d g e  device  distributions  i n them a r e d e r i v e d .  I n t h e v a r i a t i o n a l method s e c t i o n a t e c h n i q u e approximating the electromagnetic o p t i c a l wave p r o p a g a t i n g  field  i n a voltage  for  distribution  induced  o f an  optical  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 . I n t h e s e c t i o n on t h e H e r m i t e - G a u s s i a n the  reasons  presented. justifying optical  f o r choosing Also  for  trial  included i n this  functions are  section i s a discussion  t h e use o f a Gaussian approximation  field  distribution  In t h e c o u p l i n g analytic  these  approximation  expression  f o r the  i n a single-mode o p t i c a l  coefficient  section a closed  f o r the coupling coefficient  c o u p l i n g b e t w e e n an o p t i c a l  fiber  o p t i c a l waveguide.  This expression  approximate o p t i c a l  field  fiber.  form i s derived  and a v o l t a g e  induced  i s derived using the  distributions  assumed f o r b o t h t h e  -14-  optical is  fiber  and  assumed t h a t  polished  to  intimate  contact  the  The  ends o f t h e  and  degree t h a t  s e c t i o n we  Both the  will  i s shown t o be  as  10%  used,  and  axes c o n t a i n s  The is  r33  gap  the  [24]  are  small  The  i t .  to the The  the  applied  affected. f o r the  a quadratic  The  fields  i n appendix  of the  X-axis.  electric  substrate,  E , z  This  field was  can  the  be  as  f o r the  much  fields  for  the  A. of  LiNb03  advantage  be  chosen t o take advantage  were f a b r i c a t e d so t h a t t o the  u s e d and  t e r m t h a t may  l i n e a r term,  the  change i n  magnitude o f  desirable that  Y - c u t LiNbC"3 was  runs p a r a l l e l  Z-axis of the  one  electric  mathematical formalism  i t is clearly  devices  distribution  into  from  p r i n c i p a l a x e s and  change i n t h e  effect i s included  and  brought  r e l a t i o n s h i p between  largest linear electrooptic coefficient  taken of r33.  c h a n g e due  i s retained.  electrooptic  obtain  indicatrix  However t h e  of the  be  be  Effect  magnitudes o f the  orientation  principal  t h e y can  r e f r a c t i v e i n d e x and  of the  ignored.  waveguides can  t h a t power i s " " b u t t - c o u p l e d "  orientation  be  two  It  other.  change i n t h e  field.  i n d u c e d o p t i c a l waveguide.  Electrooptic  In t h i s the  voltage  such a high  waveguide t o the  2.2  the  the  of  interelectrode  ensured that  the  component p a r a l l e l  to  symmetric w i t h r e s p e c t  the to  -15-  the  XY-plane.  The d i s t r i b u t i o n  of the e l e c t r i c  field  component p a r a l l e l t o t h e Y - a x i s  o f t h e s u b s t r a t e , E , was  anti-symmetric  I n what f o l l o w s a n e q u a t i o n  giving An ,  t o t h e XY-plane.  t h e change i n t h e e x t r a o r d i n a r y r e f r a c t i v e  i n t e r m s o f t h e two e l e c t r i c f i e l d  e  is  general  equation  relating  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 field  components E y a n d E  i s given  i n the impermeability.  impermeability coefficients  tensor  i n a p p e n d i x A. When  crystal  the  yz-plane  i n terms o f t h e e l e c t r o o p t i c  and t h e e l e c t r i c f i e l d  diagonal plane  B4 w i l l  and t h e d i s t o r t i o n  general,  components.  orientation  is  t  only nonzero o f f -  Nye** one c a n o b t a i n  [23] p p . 43-47.  only  axes.  i n this  plane, will,  t h e magnitudes and  The c h a n g e i n  a s an a n g l e axes.  i n the yz-  The d i s t o r t i o n  o f changes i n b o t h  of the principal  When t h e  i s entirely i n  i n the indicatrix,  c a n be e x p r e s s e d  Ibid.  e  h a v e an e f f e c t  respect to the original  Nye  n  u s i n g t h e Mohr C i r c l e * . consist  orientations  **  B4 = r^^y  element.  can be found  with  then  a change i s  One c a n c o n s t r u c t t h e new  s u b s t r a t e i s LiNb03 and t h e e l e c t r i c f i e l d  in  z  t h e change i n t h e o p t i c a l  i s a p p l i e d t o an e l e c t r o o p t i c  incurred  in  index,  derived. The  a  v  of rotation,  ©,  Following t h e procedure  -16-  2r,,E 42 y  tan(2e) = - 2 - 2  n  which  - n  e  o  i s derived  coefficients denominator rotation  +  r_,E  33 z  (2.1)  -  ror>E  22 y  r»_E  23 z  A.  i n appendix  are small  -  Since  the last  c a n be i g n o r e d .  of the indicatrix  three  We  the electrooptic terms  i n the  can approximate  i n the YZ-plane  the angle  of  by  42 y n  The  -2 e  -  n  values  edges  the thick  order  Ail propagate the  f o r Ey were i n t h e r e g i o n s  f o r the thin  electrodes.  o f 10^ V/m  Therefore  the rotation the guided  two  like  i s one i n w h i c h  predominantly  like  field  parallel  analogous  modes  polarized X  of this  This  than  1°.  device  modes  A TE l i k e  mode i s o n e  and a  TM  field i s  i n t h e XY-plane. modes  into  predominantly  of the substrate  the electric  V  n o t have  can be d i v i d e d  o f t h e mode i s  and E p q  will  will  i s due t o t h e l a r g e  modes.  t o the Z-axis  to the E pq  of less  However t h e y  The g u i d e d  polarized mode  o f E y a r e on t h e  a rotation  constant.  T E a n d TM  the electric  values  the corners  i s ignored.  i n the x-direction.  basic types which  cause  optical  o f LiNbC>3.  near the  e l e c t r o d e s and near  The peak  and w i l l  same p r o p a g a t i o n  anistropy  in  o  highest  electrode of  -2  T h e s e modes a r e  of rectangular  channel  -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 t h e c o o r d i n a t e system Marcatili).  Due t o t h e d i f f e r e n c e  coefficients  r  the will  3  = 3 0 . 8 x l O " m / V a n d r 3 = 8 . 6 x l O " m / V [24] 12  2  change i n t h e r e f r a c t i v e be approximately  The l o w e s t  i nthe electrooptic  12  3  used by  3 times  index  "seen" by a TE l i k e  that  o r d e r TE l i k e mode w i l l  s e e n b y a TM l i k e b e t h e most  b e assumed t h r o u g h o u t  mode b e i n g c o u p l e d i n t o ,  used.  thesis that the  o r out o f , and propagating i n t h e  VIOWM i s a T E l i k e mode. c o u l d be used  this  mode.  important o f  t h e modes p r o p a g a t i n g i n t h e VIOWM f o r t h e v o l t a g e s Therefore i t w i l l  mode  t o ensure  Polarization-preserving that this  fiber  i s t h e case.  F o r T E l i k e modes t h e w a v e g u i d e c r e a t e d b y t h e application  o f v o l t a g e t o t h e e l e c t r o d e s o f a VIOWM i n v o l v e s  the extraordinary r e f r a c t i v e the  index, n , o f LiNb03.  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  optical causes  indicatrix  axis  due t o t h e a p p l i e d e l e c t r i c  t h e waveguide t o be c r e a t e d .  principal  axis consists  o f both  change i n o r i e n t a t i o n .  It i s  e  of the field  that  The change i n t h e  a change i n magnitude and a  However, s i n c e t h e r o t a t i o n  i s small  we h a v e i g n o r e d i t a n d h a v e t r e a t e d t h e c h a n g e a s 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  only.  The c h a n g e A n ( y , z ) i s g i v e n b y e  l  A n e ( y , z) - -  e 33Ez<y'z) r  2  n  e  { r  42Ey  ( y  '  2 ) } 2  (2.2)  -18-  w h i c h i s a l s o d e r i v e d i n a p p e n d i x A. right  hand s i d e o f equation  direction the  alone  The f i r s t  t e r m on t h e  2.2 i s t h e c h a n g e i n t h e z-  and t h e s e c o n d t e r m i s t h e added change i n  magnitude o f t h e p r i n c i p a l  axis.  We h a v e r e t a i n e d t h e  s e c o n d t e r m b e c a u s e i t may b e a s much a s 10% o f t h e f i r s t term  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 edges o r c o r n e r s  though t h e r o t a t i o n  2.3  i s small.  The C o n f o r m a l  The applied  conformal field  Mappings  m a p p i n g s t h a t were u s e d t o f i n d t h e  distributions,  E (y,z) z  and E y ( y , z ) , a r e  discussed i n the following subsections and  ridge  f o r both  the planar  devices.  First,  however, we must d e r i v e t h e L a p l a c i a n f o r t h e  orientation  o f t h e 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  variables.  This  LiNbC>3. and  ridge  i s necessary  due t o t h e a n i s o t r o p y o f  The same t r a n s f o r m a t i o n  i s good f o r b o t h  equation  with  r e s p e c t t o t h e YZ-plane o f  LiNbC>3 i s g i v e n b y  = ey  the planar  devices.  Poisson's  VD  even  + ez  = 0  which by t h e t r a n s f o r m a t i o n  of variables  -19-  ' e y'  ^  1/2  =  y  v  (2.3a)  e  y )  and z'  = z ,  where e  (2.3b)  and  v  e  diagonalized written  d  2  dy'  as  are  z  the  Y and  Z-components  p e r m i t t i v i t y tensor  the  a  and  e)  of  c  r e s p e c t i v e l y , can  the  be  Laplacian  2  v  + 2  3 v  = o  (2.4)  2  dz'  2.3.1  The Planar Device  2.3.1.1  The  The  Device  planar  Structure  VIOWM  ( f i g u r e 2.1)  substrate  s e p a r a t e d by  a n a r r o w i n t e r e l e c t r o d e gap,  i t s faces.  substrate  transmission  losses  and The  thin  the  t h i n metal  A thin optical buffer  between the  field  w i t h two  consists of  electrooptic  of  (e  and  due  the  to  electrodes, deposited  layer i s  electrodes  an  to  on  one  included  reduce  i n t e r a c t i o n s between t h e  optical  electrodes.  metal electrodes  with respect  t o the  and  the  optical buffer  i n t e r e l e c t r o d e gap  layer  w i d t h and  are their  -20-  ntereIect rode Gap Electrode Metal  Me t a I  _r<. .  Electrode  ^.„„„.™,...,..„„™,™.„ v  wv  w  Opt i ca Buffer Layer  X  0—> z  y  Substrate  Figure  2.1: The p l a n a r VIOWM s t r u c t u r e .  -21-  thickness  may  be  s t r u c t u r e was i n w h i c h two  ignored  replaced  situated directly  may  i n t u r n be  the in  the  electrodes  region the  i n t e r e l e c t r o d e gap  y  =  o  electric  This field  anisotropic  i n the  thickness  distribution i n which  give  structure  field  plane of the  in  as  exact as  i t  has  anisotropic  boundary c o n d i t i o n  applied  2.2  substrate  to those of the The  actual  structure  s u b s t i t u t i n g one  i n f i g u r e 2.2.  depicted  infinitesimal  a n i s o t r o p i c h a l f of the  boundary c o n d i t i o n s  The  shown i n f i g u r e  second s u b s t i t u t i o n w i l l  n o r m a l component o f t h e  E  the  embedded i n an  This  f o r the  and  of  device.  substrate.  c a l c u l a t e d , by  are  f i g u r e 2.3.  equivalent  structure  atop the  analyzed,  substrate  results  the  coplanar electrodes  are  in  by  i n modeling the  for  the  electrodes  is  .  y=0  for both  structures.  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  The  c o n f o r m a l mapping c o n s i s t s  u < JT i n t h e  W-plane t o t h e  with i t s f o c i a c h i e v e d by  a t ±g/2,  the  Ramo e t  region  i n the  o f mapping t h e exterior to  S-plane.  This  mapping S = - ( g / 2 ) c o s ( W ) . *  a l . [26]  section  7.6  pp.  a  strip  hyperbola,  mapping i s Figure  331-340.  0  2.4  <  -22-  X •>  V  A i r  Electrode  z  y  ntereIect Gap  rode Electrode  A n i s o t r o p i c Substrate  F i g u r e 2.2: structure.  An  intermediate  model  of the planar  VIOWM  -23-  Anisotropic Material E l e c t r o d e  E l e c t rode  n t ere Iect rode X o—> Gap Anisotropic Material  F i g u r e 2.3: structure.  The model u s e d t o a n a l y z e  y  the planar  VIOWM  z  -24-  depicts  t h e W and S-planes.  concerned with t h e l i m i t i n g  The c u r r e n t  problem i s  case i n which t h e f o c i a r e  coincident  w i t h t h e ends o f t h e e l e c t r o d e s  electrodes  being  represented  by t h e l i n e s  resulting i n the |o| 2: g/2.  I f t h e l i n e s u = 0 a n d u = JC a r e assumed t o b e electrodes potential  v V(u,v)  =  o  a t ground and V function  u  0  v o l t s r e s p e c t i v e l y then t h e  i n the strip  i s given by  .  K  The  electric  given  field  c  and E^,  are  i n terms o f t h e p o t e n t i a l f u n c t i o n by dV(u,v)  E  components i n t h e S - p l a n e , E  o  = a  Jtdo  do  and dV(u,v)  V  o  du  rukn  V cs  o  dv  jcdo  where t h e C a u c h y - R i e m a n n c o n d i t i o n used. of  H e r e du/dc a n d dv/do,  dW/dS,  dW  du  3v  dS  do  da  respectively.  du/da> = — dv/do h a s b e e n  are the real  and imaginary  parts  -25-  V  W - p I a ne  1\  ..1  ^ S - p l a n e  F i g u r e 2.4:  The  W  and  g  g  2  2  S-planes.  -26-  The  i n v e r s e mapping o f S = -(g/2)cos(W),  function  is W = n -  o f S,  derivative,  cos  dW/dS , e q u a l s  (2S/g).  - 1  {(g/2)  Therefore  - S }" /  2  giving  2  1  2  W as  a  the  or  dw 1/2  ds  2  — o  I  — ±2ooo  2 )  which a f t e r  V  2  + oo  o  rearrangement  _ tan  cos  2am  -1 {  2  gives  (g/2)  2  - o  2  2  + os  j  (2.5a)  1/4 — a  2  +OB  2  ,-22  +  ia  a  v 2 ,  and 2<T09  V sin' _ tan o  -1 (g/2)  2  2  -  a  2  +  a  -,2 O  I Obviously positive to  + oo  2  )  (2.5b)  1/4 ±  A  2  + 4o  2  OB  2 )  the p o s i t i v e z'-axis.  correspond  positive  2  2  c-axis should  Therefore  t o the p o s i t i v e  x - a x i s t o be  i f we  correspond  to  the  wish the p o s i t i v e  y'-axis this  fixes  ©-axis  the  o r i e n t e d so as t o p o i n t i n t o t h e  paper  -27-  in  figure  and  2.2.  b into  Finally  on s u b s t i t u t i o n  of equations  2.3a  2.5a a n d b one o b t a i n s 1/2  V  o  — tan  cos  2z(e /e ) z  -1  (g/2)2 -  I  2  y  y  z  + (e /e )y  2  z  y  )  2  (2.6a) r  /  \  g  —  K  2  <  -  z  2  z  —  +  y  2  1/4 c  , 2 4z  +  1« J  , 2 ,  •  •>  -  e  2  y  2  y '  v.  y  z  —  and 2z(e V sin o  — tan  -1 (g/2)2  2  -  z  z  /e  2  y  )  1/2  y  + (e /e )y z  y  J  2  (2.6b) 2  2  g  —  -  n  than  —  +  y  2  <  +  £  silicon  . 2 4z  -  e  >  z y  2  1« J  1 \  y  d i o x i d e h a s a much s m a l l e r  LiNbC>3 ( a p p r o x i m a t e l y  a t XQ  e  z  y  Sputtered  for  2  , 2 ,  -  index  z  e  1/4  = 633nm).  refractive  1.46 a s c o m p a r e d t o 2.20  F o r u s i t was a g o o d c h o i c e a s t h e  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 a s i t was a v a i l a b l e  i n the  solid  t o work  s t a t e l a b o r a t o r y here  with. the  Using  that  i t we c a n a p p r o x i m a t e t h e d e c a y c o n s t a n t f o r  evanescent  { ( 2 r c A  0  )  ( 2 . 2 0  2  the f i e l d  surface  in  a t U.B.C. a n d i s e a s y  field,  normal t o t h e s u r f a c e by  - 1 . 4 6 ) / } which i s about 2  1  2  decays t o about  2000A.  16/um t h i s  4% o f i t s v a l u e  An e v e n b e t t e r r e s u l t  means  at the  i s obtained  at X  D  =  -28-  442nm.  However, a t  low  frequencies  o f SiC-2 i s s i g n i f i c a n t l y = e /e z  and  f o r the  0  28  LiNbC-3  of the  field  distribution  field  i n the  with  the  l a y e r be  two  materials  i n the  device.  electrodes  LiNbC-3. kept t h i n  4 as  so t h a t t h e  effect  was  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 buffer  l a y e r t h i c k n e s s was  width- and in  was  t r e a t e d as  c a l c u l a t i n g the  words t h e  electric  though the  t a k e n t o be  electrodes  that  2.6  electric field  and  negative*  voltage,  o f 5%  i n t o the  Ey(y,z)  V,  and  contact  that the  small.  buffer  In  the  assumed t h a t  from the  the  gap  electrodes  distributions.  i n contact  induced  of the  In  other  substrate.  E (y,z) z  with  the  waveguide  was  gap The  w i d t h away applied  from  field  are p l o t t e d i n f i g u r e s  In b o t h o f these  applied across  so t h a t E ( y , z ) z  the  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  surface of the  respectively.  the  field  in  interelectrode  a mere o f f s e t  at a distance  distributions and  of the  e l e c t r o d e s were d i r e c t l y  substrate then the  the  5%  43  applied  compared t o  important  z  electric  I t reduces the  i t was  K  relative  are p l a c e d d i r e c t l y  Therefore  or  Q  compared t o  a f f e c t s the  r e g i o n as  constant  = ey/e  This d i f f e r e n c e i n the  high permittivity  c a s e where t h e  dielectric  l o w e r t h a n e i t h e r Ky  (approximately  respectively).  permittivities  the  the  is positive.  2.5  p l o t s i t i s assumed electrodes  is  Although the  The IRE S t a n d a r d s on P i e z o e l e c t r i c C r y s t a l s i s u s e d the s i g n convention here. See L i n e s and G l a s s [27] 147 .  y-  for p.  -2  component order  of the  9-  applied f i e l d  o f m a g n i t u d e as t h e  i s s e e n t o be  of the  z-component i n t h e  same  regions  near  o the is  electrode very  edges the  small  2.3.2  f o r the  term  field  (r42 y(y^ )} E  2  strengths  i  equation  n  used.  The Ridge Device  2.3.2.1  The D e v i c e S t r u c t u r e  While the  planar  VIOWM w o r k s t o c r e a t e  waveguide between t h e  input  device  i t i s o b v i o u s , by  figure  2.2,  field  that  i n the  the  device  can  be  and  the  output  i n s p e c t i n g the  i t does not  make t h e  i n t e r e l e c t r o d e gap  more e f f i c i e n t  the  ridge  region.  surface,  of  the  use  of the  In order  e l e c t r o o p t i c medium as  in figure  Ey,  will  be  relatively  and  t a n g e n t i a l component, E ,  will  be  essentially  uniform.  I t can  be  s e e n t h a t Ey  n o r m a l component o f t h e  boundary.  If Ey  air  b o u n d a r y and  E  ys  at the =  a  i s the  ( air/ substrate> ya e  e  E  will  field  be  s m a l l by  o  i s that  S  r  E  ys  i n the  - 0.02E . y a  the  small  comparing  on b o t h s i d e s o f  n o r m a l component o f t h e Ey  2.7.  n o r m a l component o f  field  z  large  t o make  applied the  at the  enough t h e  linking  faces  best  a ridge of the  i s high  a  s t r u c t u r e shown i n  i n c l u d e d between t h i c k e l e c t r o d e s  If  the  2.2  the  field  substrate It follows  in  then then  -30-  Figure  2.5:  A plot  of E (y,z) v  f o r the planar  VIOWM.  -31-  Figure  2.6:  A plot  of E (y,z) z  f o r the  planar  VIOWM.  -32-  X  0  >Z  Al r y  ntereIect Gap Me t o l  rode Metal  E l e c t rode ,  L  1  F i g u r e 2.7:  The  :  ^  s  R i dge|  Anisotropic^ Substrate I  ridge  VIOWM  structure.  E l e c t rode  -33-  that the  i f the actual The  ridge  enough * t h a t  i s high  structure  structure  by  that  plate  i n f i g u r e 2.8  depicted  upper e l e c t r o d e on  itself.  [28]  capacitor makes a  In the  the  edge,  and  normal t o the  this  i n t o the  section to  structure  2.3.2.2  In  of  capacitor, electrodes.  figure  order to  the  than f o l d i n g back  one  field  I t i s one  capacitor  moves away  the  from  becomes u n i f o r m  of the  t h i s i s a l s o the  f i n d the  First  strip  the  mapped t o  the  2.9b,  mapped t o  2.9c.  a  goals  case  for  of the  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  under a n a l y s i s  is  the  that  2.8.  structure  figure  and  in  difference that  parallel plate  d i s a p p e a r as  show t h a t  2.8.  fringing field  with the  case of the  approximate  i s similar to  90° a n g l e r a t h e r  fringing effects quickly  can  shown i n f i g u r e  commonly u s e d f o r c a l c u l a t i n g t h e parallel  we  applied  field  i t i s n e c e s s a r y t o do  0 < T| < n i n t h e  and  second the  the  unshaded r e g i o n to  two  intermediate  analyze the  H i g h e n o u g h b e i n g when E  v s  »  0.  S-plane, structure  is  W-plane,  upper h a l f plane o f the of the  the  mappings.  £-plane, f i g u r e 2.9a,  upper h a l f plane of the  It i s sufficient  distributions in  W-plane  figure shown i n  -34-  L  -Electrodes  F i g u r e 2.8: T h e m o d e l VIOWM s t r u c t u r e .  used  to analyze the ridge  waveguide  -35-  figure  2.9c r a t h e r  than that  symmetry o f t h e two The  shown i n f i g u r e 2.8  structures.  m a p p i n g u s e d t o map  t h e £-plane t o t h e W-plane i s W  = e^ a n d a S c h w a r t z - C h r i s t o f f e l W-plane t o t h e S - p l a n e . is  transform  i s u s e d t o map t h e  The S c h w a r t z - C h r i s t o f f e l  transform  found by i n t e g r a t i n g t h e d e r i v a t i v e  ds  (w + l )  dw  w  which  1/2  gives (w+i)  s  due t o t h e  K {  -  2(W+l)  1 / 2  +  1 / 2  -  i  In  + C (w+l)1/2  +  i  (u+l+iv)1/2 =  K |  2(u+l+iv)  i / 2  +  1  In  (2.7)  + C (u+l+iv)1/2  The  -  complex c o n s t a n t s ,  +  1  K a n d C, a r e e v a l u a t e d  by  first  l e t t i n g u -» -1 on t h e l i n e v = 0, w h i c h maps t o t h e p o i n t ig/2.  i _  =  By d o i n g t h i s  (K., +  iK,)ln(-l)  +  e q u a t i o n 2.7  (C, +  iC,)  =  becomes  iK.,* - K,»t + C,  +  iC,  2  equating  the real  and i m a g i n a r y p a r t s  o f which  gives  -36-  V  C- p I a n e  T  —  >  f  (a ) iV  W-plane :  u  -1( b ) CO  i  S-p I a ne g/2  (c ) F i g u r e 2.9: The  (a)  (b) W, and  (c)  S-plahes.  -37-  g K  l  = -  -  C  2  2K  and c K  l  2 " "  it  and  by l e t t i n g  equation  -~  on v = 0, w h i c h maps t o S =  2.7 becomes  <KX + i K 2 ) ( 2 + l n ( 0 ) )  -  the  u -> 0 +  imaginary part  0 = K 2 (2 - ~) +  C2  which implies  that  +  (C1 + i C 2 )  o f which i s  K2 must b e e q u a l t o z e r o  must C i a n d C2 s o t h a t  one  and t h e r e f o r e so  obtains  g K  = 2K  and c = 0 .  As i t t u r n s continually electric an  out t h e term  (u + 1 + i v ) ^  through out t h e s o l u t i o n s  f i e l d distributions.  expression  2  appears  f o rthe applied  Therefore  we w i l l  • • *i ft • f o r i t i n t h e f o r m Me . This  first  gives  find  -38-  M  =  {(u +  l )  2  +  v  +  1  2  }  1 / 4  and  9  =  _  -1 tan  I  2  Rewriting  u  J  equation  2.7 one o b t a i n s  g  s = _  2 M c o s (6)  + _ |  -[  ln(M  2  (  M c o s (9)  -  2 M c o s (6)  +  1)  l n(M2  -  +  2 M c o s (6)  +  D  ]  2K  +  2Msin(9)  i  + tan  (  -1  I,  M c o s (9)  -  1  tan  )  M C O S (9)  -1 M c o s (9)  k  +  1  )  or  2Mcos(9)+  _ [ l n ( M  2  -  2 M c o s (9) +  1 ) - l n (M2+  2 M c o s (9) +  (2.9a)  1)  and M c o s (9) 2Msin(9)  + tan  2K  k  Equations  M c o s (9)  -1 M C O S (9)  -  1  ,  . k  2.9a a n d b g i v e t h e c o o r d i n a t e s ,  S-plane  i n terms o f t h e coordinates,  plane.  In t h e next paragraph expressions  field  components,  E  a  W-plane c o o r d i n a t e s ,  "|  -1 tan M c o s (9)  +  1  (2.9b)  ) .  o a n d ©, i n t h e  u a n d v,' i n t h e Wfor the electric  a n d E^, i n t h e S - p l a n e i n terms o f t h e u and v w i l l  be d e r i v e d .  -39-  It is  held  i s convenient t o consider a t zero  electrode. electrode 2.8 is  in  v o l t s applied t o the other  0  c o n f i g u r a t i o n t o see that  i sused t o analyze the electrode assumed t h a t  situation  one o f t h e e l e c t r o d e s  I t i s p o s s i b l e t o u s e t h e symmetry  lower e l e c t r o d e  the  v o l t s with V  that  t h e upper e l e c t r o d e  i f t h e model o f f i g u r e configuration i s at +V  must b e a t + V / 2 v o l t s . 0  corresponds t o t h e ^-axis being  l i n e T| = w b e i n g the strip  a t +V  Q  volts.  between t h e ^ - a x i s  of the  0  and i f i t  v o l t s then the  I n t h e £-plane t h i s a t +V /2 v o l t s and Q  The p o t e n t i a l  function  a n d t h e l i n e T\ = n w i l l b e  d e n o t e d , V(^,TI) , a n d i s g i v e n b y v V($,TI)  = — -n . 2K  the  electric  field  dv($,Ti)  vQdn  do  2ndo  components a r e t h e n g i v e n b y  E  a  and  ^  vodn  vod$  2nd(o  2ndo  CD  a>a>  where t h e C a u c h y - R i e m a n n c o n d i t i o n aT\/d(o = dZ,/da h a s b e e n used. and  The p a r t i a l  imaginary parts  d e r i v a t i v e s dt,/do a n d dn/da, a r e t h e r e a l o f t h e d e r i v a t i v e d^/dS,  -40-  dS  da  do  respectively.  Thus t h e  relations  Im dS  2K  J  and  Re  L  2K  dS  are obtained. df, dW  The  d e r i v a t i v e d£/dS i s g i v e n  ln(W)  d  dS  -1  w =  dS  dW  which  dS  L  dW  W  dW  by  (W+l)  (W+l)-  1 / 2  = M-^-  1 9  1/2  gives  s i n (0)  2nM  and  E  CO  = -  COS  (9)  2nM  From f i g u r e s will  correspond  co-axis w i l l  2.8  and  2.9c  i t i s obvious  to the y'-axis of the device  correspond  t o the  z'-axis.  that the  o-axis  and  the  Therefore  that  i f we  wish  -41-  to  find the e l e c t r i c  method,  field  the transformations  i n the yz-plane, o f the equations  be a p p l i e d a f t e r t h e c o n f o r m a l Figures and of  E (y,z) z  2.10  a n d 2.11  2.3a a n d b must  mapping.  show t h e d i s t r i b u t i o n s  i n the substrate.  An o f f s e t  of  Again,  assumed t h a t t h e v o l t a g e negative the  i.e. V  applied f i e l d  z-component  Although  i s o f t h e same o r d e r  i n the regions  small  f o r the f i e l d  the  f o r ridge heights applied f i e l d  Therefore  high  The  index  o f magnitude as t h e i n the 2.2,  used.  a n d 2.11  i t i s apparent  o r g r e a t e r t h e y-component o f  i s much s m a l l e r t h a n  t h e z-component.  field  or higher.  V a r i a t i o n a l Method  In o r d e r optical  o f g/2  of  t h i s m o d e l o f t h e r i d g e VIOWM c a n b e u s e d f o r  r i d g e s g/2  2.4  t h e y-component  r4£Ey, i n e q u a t i o n  strengths  By i n s p e c t i o n o f f i g u r e s 2.10 that  f o r the  the electrodes i s  around the corner  e l e c t r o d e t h e square o f the product very  y  a s i n s e c t i o n 2.3.1.2, i t i s  applied across  i s negative.  0  E (y,z)  from t h e e l e c t r o d e s  5% o f t h e gap w i d t h h a s b e e n assumed t o a c c o u n t  optical buffer layer.  is  u s i n g t h e above  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 an o p t i c a l w a v e g u i d e w i t h  a  of the  refractive  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 d i m e n s i o n s i t i s  necessary  t o be a b l e  first  to construct the appropriate  wave  -42-  F i g u r e 2.10: A p l o t o f E ( y , z ) f o r t h e r i d g e VIOWM. Here t h e p l o t has been c u t a l o n g t h e l i n e z = 0 showing t h e f i e l d for z < 0 only. y  -43-  F i g u r e 2.11: A p l o t o f E ( y , z ) f o r t h e r i d g e VIOWM. Here the p l o t has been c u t along the l i n e z = 0 showing the f i e l d for z < 0 only. z  -44-  equation field  and second t o s o l v e i t e x a c t l y  distribution.  Indeed t h e r e e x i s t  index d i s t r i b u t i o n s [29-31].  obviously  a certain finding  f o r w h i c h an e x a c t  I t i s , however,  solution  f o r the optical certain  refractive  approach i s p o s s i b l e  f a r more common t h a t w h i l e a  exists,  as i s e v i d e n c e d by t h e f a c t  that  w a v e g u i d e g u i d e s o p t i c a l waves, t h e p r o b l e m o f  i t exactly  i s intractable.  Hence  approximate  methods a r e n e e d e d f o r o b t a i n i n g t h e o p t i c a l distributions refractive  i n o p t i c a l waveguides w i t h  index d i s t r i b u t i o n s  field  arbitrary  and over a p e r i o d o f t i m e  t h e s e have been developed. The two m a i n a p p r o x i m a t e methods t h a t h a v e e m e r g e d a r e the  effective  method.  refractive  i n d e x method and t h e f i n i t e  The d e v e l o p m e n t o f b o t h methods b e g a n c a . 1969.  The c o n c e p t  of the effective  dielectric  i n t r o d u c e d b y Knox a n d T o u l i o s  i n 1970  to  t h e method o f M a r c a t i l i  to  i t f i n d the propagation constants  waveguides.  Optical  field  [25].  c o n s t a n t * was [32] a s an  numerical techniques  H o c k e r a n d B u r n s [33] u s e d for arbitrary  distributions  equations, application  for solving  first  extension  c a n be  obtained  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 a n d any o f a number  *  element  second order  of  differential  f o r example t h e method o f R u n g e - K u t t a [ 3 4 ] . of the f i n i t e  e l e m e n t method t o  The  solving  The term e f f e c t i v e r e f r a c t i v e index was a p p l i e d somewhat l a t e r but the only d i f f e r e n c e i s t h a t the e f f e c t i v e d i e l e c t r i c constant i s d e f i n e d t o be e = p /k and the e f f e c t i v e r e f r a c t i v e index i s d e f i n e d t o be n ^ f f = p / k . 2  r e  2  0  0  -45-  e l e c t r o m a g n e t i c waveguiding problems Daly's paper shaped  [35]. Later  b e g a n w i t h Ahmed a n d  i t was a p p l i e d t o a r b i t r a r i l y  inhomogeneous o p t i c a l waveguides  b y Yeh e t a l .  More r e c e n t l y K o s h i b a e t a l . h a v e u s e d t h e f i n i t e method t o a n a l y z e a n i s o t r o p i c  functions  [38],  [40],or f i n i t e  t h e WKB  e x p a n s i o n i n terms  method*  [39], Green's  d i f f e r e n c e methods  a p p l i e d t o o p t i c a l waveguiding problems element method and t h e e f f e c t i v e h a v e r e c e i v e d b y f a r t h e most  element  o p t i c a l waveguides [ 3 7 ] .  O t h e r t e c h n i q u e s b a s e d on, f o r example, of c i r c u l a r harmonics  [36],  [41] h a v e  been  but the f i n i t e  refractive  i n d e x method  attention.  V a r i a t i o n a l methods h a v e l o n g b e e n e l e c t r o m a g n e t i c waveguiding problems  applied to  i n metal  clad  A A  waveguides. obtain  The v a r i a t i o n a l method h a s a l s o b e e n  approximate width parameters  lictributions  f o r the optical  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  used t o field core  AAA  optical itself  fibers.  In fact  a variational  the f i n i t e  t e c h n i q u e [44] .  element  method i s  Here t h e 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 t h e WKB m e t h o d c a n be u s e d 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 i n d e x and i s i n f a c t t h e same a s t h a t d e r i v e d b y H o c k e r a n d B u r n s [33] u s i n g a wave v e c t o r ( r a y ) a p p r o a c h . Therefore 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 b u t may i n f a c t be c o n s i d e r e d as p a r t o f t h e e f f e c t i v e r e f r a c t i v e i n d e x method.  Harrington  Marcuse  [42]  c h a p t e r 7 p p . 317-380.  [19] p p . 339-347 o r O k o s h i  [43] p p . 114-121.  -46-  method has been extended t o d e r i v e t h e o p t i c a l distributions  o f t h e f u n d a m e n t a l mode o f a c h a n n e l  w i t h an a r b i t r a r y In  integral the  refractive  the variational  eigenvalues  index  f o r the  o f the Euler-Lagrange equation o f a  of a functional  Using the integral eigenvalue),  i s derived.  In t h i s  i s the scalar  of the functional of a particular  waveguide  profile.  m e t h o d an e x p r e s s i o n  Euler-Lagrange equation  (the  field  wave  stationary  application equation.  an e x p r e s s i o n  for B  2  mode o f t h e w a v e g u i d e , i s  obtained.  F u r t h e r m o r e i t i s known t h a t  the eigenvalues are  stationary  values.  as a f u n c t i o n o f  certain  variable  Thus b y p l o t t i n g  as  section  variable  to  distribution stationary  field  The H e r m i t e - G a u s s i a n  parameters that  optical  chosen t o those values o f  may b e  found.  2.5 t h e c h o i c e o f H e r m i t e - G a u s s i a n  t h e approximate o p t i c a l  motivated.  the  field  p a r a m e t e r s w h i c h make p In  2  parameters o f a f u n c t i o n  approximate t h e o p t i c a l the  p  field  will  be  f u n c t i o n s u s e d h a v e two  determine the t r a n s v e r s e extent o f  distribution.  the variational  distribution  functions  Here t h e e q u a t i o n s  relevant  method a r e d e v e l o p e d f o r t h e e x a c t  solution. The expressed  general  form o f t h e s c a l a r  i n t h e form  2 2 V f c v + v (y, z) y = 0 ,  wave e q u a t i o n  c a n be  -47-  where 3  2  -.2  „2 2  adz2  3 3y  I f t h e a b o v e wave e q u a t i o n i s m u l t i p l i e d b y y a n d t h e r e s u l t is OO  f j  integrated  over t h e e n t i r e yz-plane*  i t gives  f  OO  f J  2 2 2 \|/ V^v + v (y, z) v  • dydz «= 0  a n d i f G r e e n ' s Theorem i s a p p l i e d t o t h e f i r s t integral  OO  J —oo  one g e t s  OO  J  OO  ¥  V2V  dydz - - J  2  O"  {  > ay >  of which t h e l a s t Finally  2 +  —  —OO  zero.  term o f t h i s  dydz + J  —  , az ,  t e r m on t h e r i g h t  hand s i d e  y — ds C  dn  i s equal t o  the integral 2  -I I  + ,  is  ay ,  - v  2  2 (y,z)v  dydz «= 0  (2.10)  , az ,  obtained.  The domain of i n t e g r a t i o n f o r a d i e l e c t r i c o p t i c a l waveguide i s the e n t i r e t r a n s v e r s e p l a n e . T h i s i s due t o the boundary c o n d i t i o n f o r the guided modes of these waveguides which f o r c e s the o p t i c a l f i e l d t o be zero at i n f i n i t y i f the guided power i s t o remain f i n i t e . Yariv and Yeh [45] chapter 11 pp. 405-503.  It  r e m a i n s t o be  stationary  and  shown t h a t e q u a t i o n  t h a t i t i s a minimum.  proof i s presented that equation the is  general  form  of the  the Euler-Lagrange The  f o r the exact  the  i s stationary  and  given  f o r an  optical  (the s u b s c r i p t  e  o b t a i n e d from Maxwell's e q u a t i o n s .  refractive  fact  that  above,  equation.*  <l> (x,y,z)  wave e q u a t i o n  is in  In appendix B  s c a l a r wave e q u a t i o n ,  wave e q u a t i o n  distribution,  2.10  2.10  field  e i s for exact), i s  I n t h i s work t h e  o p t i c a l waveguide i n which  index v a r i e s  i n two  dimensions,  scalar  the  i n the  yz-plane,  is 2 3  *e  2 *e  d  d ln[n (y,z) ] 2  dln^  2  ] dln[n (y,z)] 2  + dy  2  dz  2  dz  where * ( x , y , z )  =  dz  -1  distribution, From t h e  2K/\0).  t h e g e n e r a l form  and  2  v  <y,z)  i s the  free  dln[4>  [46]  space  chapter  (k  function v (y,z),  in  2  (page 4 6 ) ,  i s given  2  z  3 pp.  z  wavenumber  2 2 2 + n (y,z)k - p . dz  2  2 p  refractive  ] dln[n (y,z)]  + dz  Fox  Q  o f t h e wave e q u a t i o n  2  =  k  z  above e q u a t i o n t h e  d ln[n (y,z) ] 2  i s the  x  e  2 2 <y,z)Jc z o  dz  = e P 4 > (y, z) , n ( y , z )  e  index  + n  dz  59-79.  o  0  by  -49-  However, i n a p p e n d i x stationary  B i t i s shown t h a t  the Euler-Lagrange  In  t h e above e q u a t i o n t h i s an  optical  the  fields  i s o b v i o u s l y not (y,z),  distributions,  o f 4>  e  will  the o p t i c a l VIOWM w i l l will  be  2 .5  following  In  on t h e  invariant  In f a c t  functions  characteristics exact  solution  fiber  to the o p t i c a l  a r e e x p e c t e d t o be choice of the on  and  be u s e d .  The  are  solutions to  the  approximating  a knowledge o f  the  o f s o l u t i o n s t o s i m i l a r p r o b l e m s where i s known.  the  field  of choosing functions that  s h o u l d be b a s e d  a  functions for  Approximations  consists  The  had  approximations.  approximation  s c a l a r wave e q u a t i o n .  i f one  will  and t h e above e q u a t i o n  o f t h e waveguides employed w i l l  to those that  approximate  invariant.  distributions  similar  section  above  i n the o p t i c a l  Hermite-Gaussian  t h i s work a p p r o x i m a t i o n s  0.  and w h i c h  s e c t i o n the approximate  g i v e n a n d e q u a t i o n 2.10  In  =  introduced that i s  as b e i n g  p r e s e n t e d i n terms o f the  The  so.  e  field distributions be  6v(y,z)  f u n c t i o n <)> t h e n t h e  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 the  be  i s a c t u a l l y used.  knowledge o f t h e  In  based  and w h i c h i s t h e n  function that  priori  equation provided that  approximation to v  independent be  is  a n d t h e g e n e r a l f o r m o f t h e s c a l a r wave e q u a t i o n  is  2.5.2  e q u a t i o n 2.10  In the f o l l o w i n g  the  subsections a  -50-  discussion  o f the c h o i c e o f each o f the  functions w i l l waveguide One  g i v e n i n terms  of i t s a p p l i c a t i o n t o the  used. o f t h e m a i n g o a l s o f t h i s work i s t o show t h a t  VIOWM h a s optical  be  approximate  an a p p l i c a t i o n  fibers  as a l i n k i n g w a v e g u i d e b e t w e e n  (or o t h e r waveguides). able to predict  b e t w e e n t h e two  w a v e g u i d e t y p e s when t h e y a r e b r o u g h t  field in  terms  obtain  the e f f i c i e n c y  i n a b u t t - c o u p l i n g arrangement.  distributions  of  I f the  coupling  optical  i n b o t h w a v e g u i d e t y p e s c a n be e x p r e s s e d  of Hermite-Gaussian functions  a closed  two  It i s therefore  n e c e s s a r y t o be  together  the  form a n a l y t i c  solution  i t i s possible f o r the  to  coupling  coefficient. As w i l l coefficient the  two  be  shown i n s e c t i o n  2.6  i s o b t a i n e d i n terms  field  distributions.  the  coupling  of the overlap i n t e g r a l  The  contribution to the  overlap  i n t e g r a l made b y t h e e v a n e s c e n t f i e l d s  small.  Therefore the fact  approximate  functions w i l l  than o f t h e form exp(-r) w i l l is are  seen t h a t  *  o f t h e form e x p ( - r ) 2  have l i t t l e  i f the approximate  optical  t o be u s e d t o c a l c u l a t e t h e c o u p l i n g  b u t t - c o u p l e d waveguides  i s obviously  that the evanescent f i e l d be  that  of  effect.*  field  of the  rather Thus i t  distributions  coefficients for  i t i s most i m p o r t a n t t h a t  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 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 t h e current problem they are e n t i r e l y appropriate.  the  be  approximations  be c l o s e t o t h e a c t u a l d i s t r i b u t i o n s  r e g i o n s where t h e f i e l d s  2.5.1  i nthe  are largest.  The O p t i c a l F i b e r  It  h a s l o n g b e e n known t h a t  a high  permittivity  if dielectric  r o d c a n a c t as a w a v e g u i d e  and s o l u t i o n s t o t h e  wave e q u a t i o n h a v e b e e n o b t a i n e d f o r r o d s w i t h v a r i o u s c r o s s sections  including circular  rectangular  [25].  index p r o f i l e s elliptically  Typically  that  optical  are either  approximation  to the refractive  one i n w h i c h t h e r e f r a c t i v e  to  h a v e one v a l u e , n  to  have another v a l u e , n  2.12 i l l u s t r a t e s  these  circularly  fiber.  determined  symmetric o r  useful  b y t h e method  profile  i n t h e c l a d d i n g i s assumed c  o  > nc±.  index p r o f i l e  have a s t e p index p r o f i l e .  have r e f r a c t i v e  o f many  The s t e p i n d e x  i , such t h a t n  the refractive  i . e . a plane  i n d e x o f t h e c o r e i s assumed  , and t h a t c  A  Few, i f any, o f t h e c o m m e r c i a l l y  fibers  fibers  c o  and  refractive  index d i s t r i b u t i o n  i s the "step index" p r o f i l e .  optical  f i b e r s have  of propagation.  is  index  [30],  symmetric i n t h e t r a n s v e r s e plane,  normal t o t h e d i r e c t i o n  fibers  [29],e l l i p t i c a l  Figure  of a step available  The p r o f i l e s o f  index d i s t r i b u t i o n s  that are  of fabrication.  See O k o s h i [43] c h a p t e r development.  1 p p . 1-16 f o r an h i s t o r i c a l  -52-  Fiber Cladding R e f r o d i v e I n d e x : Pi  F i b e r Cor e Refractive  I n d e x : Pi  E x t e r n a l Medium R e f r a c t i v e I n d e x : Pi  CO  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 with step index p r o f i l e .  index d i s t r i b u t i o n  of a  fiber  -53-  ' While elliptical  exact  cross section  [29,30] t h e y functions. has  solutions exist  f o r both  circular  f i b e r s with a step index  are i n terms o f r a t h e r c o m p l i c a t e d F o r example t h e c i r c u l a r  and profile  higher  core step index  fiber  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  c o r e and  i n terms o f m o d i f i e d K e l v i n  cladding.* basis  While  such  functions useful  distributions  solutions  d e a l i n g with the  Often exact  they  light  constants  of optical  of  field  and  for  o f t h e v a r i o u s modes,  fibers  are of l i t t l e  i n a general  practical  emanating from  use  a particular  when single-  fiber.  i t i s e a s i e r t o work w i t h  solution.  useful.  study  sense,  mode o p t i c a l  a complete s e t  i n l a r g e c o r e multi-mode f i b e r s  allowing the  theoretical  form  i n d e s c r i b i n g the o p t i c a l  c a l c u l a t i n g the propagation thus  functions i n the  Gaussian**  an a p p r o x i m a t i o n  approximations  They a r e mathematical  are  s i m p l e r t o use  to  particularly having  the  O  form and  AAA  exp(-r^).  T h e y a r e known t o p r o p a g a t e  so t h e p r o p a g a t i o n  characteristics)  Marcuse  characteristics  o f a beam l a u n c h e d  [19]  chapter  8 pp.  (the  into  i n free  space  diffraction  free  space  from  G  [47]  chapter  3 pp.  a  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 t h e H e r m i t e p o l y n o m i a l H which i s equal 1. Verdeyen  the  53-69.  to  -54-  fiber, The  say  as  part  of  an  coupling efficiency  optical of  G a u s s i a n mode i s e a s i l y  a  for  is  focused laser  calculated  Gaussian p r o f i l e i s found to equation  system,  be  [48].  the  predictable. beam t o  a  guided  Furthermore  solution  to  the  fibers with a quadratic refractive  the  wave  index  profile.* It is  in  i s also  fact  demonstrable t h a t the  a good a p p r o x i m a t i o n t o  distributions  approximation  optical  field  encountered i n single-mode o p t i c a l  Comparison's o f  theoretically  have been compared t o indicated  the  Gaussian  the  predicted profiles  Gaussian  a close correlation  function  between the  for  [41]  two.  fibers  [49,50] a g a i n The  field  f i b e r may  two  degenerate,  the  directions  z-axis. two  In  Gaussian  of p o l a r i z a t i o n  modes a r e  *  Ibid.  **  Okoshi  of  a  b e i n g composed  is arbitrary  i n t h i s work i t w i l l  be  The  [43]  chapter  4 pp.  48-81.  of two  assumed t h a t  one  y-axis  longer degenerate s t i l l  choice  of  for these  and  one  case of p o l a r i z a t i o n - p r e s e r v i n g  no  emanating  circularly  s e e n as  i s c o i n c i d e n t with the the  have  correlation.  distribution  i n g e n e r a l be  and  f u n c t i o n h a v e b e e n made  orthogonal polarizations.**  modes, t h e r e f o r e polarization  the  showing a c o n v i n c i n g  optical  symmetric  and  fibers  Also  c o m p a r i s o n s b e t w e e n m e a s u r e d power d i s t r i b u t i o n s from a c t u a l  fibers.  i t will  with  fiber be  the  the  assumed  -55-  that  the fiber  polarization  i s oriented  i scoincident  so t h a t ,  a s above, o n e  with the y-axis  a n d one w i t h t h e  z-axis. Since  theelectric  field  a VIOWM i s p o l a r i z e d p a r a l l e l field  of the optical t o the z-axis  d i s t r i b u t i o n i nthe fiber that  parallel field  t o t h e z-axis  i nthe fiber  distribution i n only  the optical  i salso polarized  need be c o n s i d e r e d .  The o p t i c a l  c a n t h u s be approximated by t h e  expression -ip x <K (x,y,z) = e 4> (y,z) f  f  (2.11)  f  where t h e s u b s c r i p t  -<y /w  f stands f o r f i b e r and  + * /»* >/2 2  2  yf  f  where a f i s t h e n o r m a l i z e d a m p l i t u d e  and W y f and w f a r e t h e z  width parameters i n t h e 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  normalized amplitude  i s a constant  mode c a r r i e s u n i t power a n d w i l l  that  ensures that t h e  i sdiscussed  further i n  a p p e n d i x C. Figure in  field distribution  w h i c h W y f = w f a n d f i g u r e 2.14 shows one i n w h i c h W y f =  w /2. z f  2.13 shows a G a u s s i a n o p t i c a l z  -56-  F i g u r e 2.13: yf zfw  =  w  A  Gaussian o p t i c a l  field  distribution  in  which  -57-  2.14:  Figure w  yf  =  w  z f / 2  A  Gaussian  optical  field  distribution  in  which  -58-  2.5.2  The  Voltage Induced O p t i c a l Waveguide  B a s e d on it  can  be  rather  Since  be  the  seen t h a t complex  the  refractive  f o r both the  i t i s very unlikely  e q u a t i o n c o u l d be refractive used.  In  this  section  is  index d i s t r i b u t i o n  exact  the  and  actual  will  r i d g e VIOWM.  solution  variational  choice of  2.3  to  the  wave  a  method w i l l  be  Hermite-Gaussian optical  field  justified.  begin with,  refractive  optical  the  an  2.2  a waveguide w i t h such  approximate the  distributions To  found f o r  in sections  p l a n a r and  that  index d i s t r i b u t i o n  functions to  the  material presented  certain  o b s e r v a t i o n s can  index d i s t r i b u t i o n  f i e l d distributions  and  be  made a b o u t  a knowledge o f  in similar  the  waveguides can  be  i s symmetric  with  used: 1)  The  refractive  respect  2)  The  to  difference  substrate  3)  And  the  for  and  xy-plane,  i n the the  a certain  induced i n the larger  index d i s t r i b u t i o n  i n the  refractive  optical buffer  range of  refractive  the  layer  large,  is  a p p l i e d v o l t a g e s the index of  interelectrode  " s u r r o u n d i n g " r e g i o n and  index of  the  gap  the  change  substrate  region than i n  change i n d u c e d i n  is the  the  -59-  surrounding  region decreases  monotonically  t o zero a t  infinity.* From t h e above o b s e r v a t i o n s  certain  characteristics  of a  b o u n d mode c a n b e a n t i c i p a t e d . 1)  The o p t i c a l symmetric  field  distribution will  or anti-symmetric  with  be e i t h e r  r e s p e c t t o t h exy-  plane,  2)  The s h a p e o f t h e o p t i c a l such t h a t i t i s very substrate value  3)  field  distribution will  be  s m a l l a t t h e boundary between t h e  a n d t h e b u f f e r l a y e r as compared t o i t s peak  further into the substrate,  M o s t o f t h e power w i l l interelectrode monotonically  be c o n f i n e d t o t h e  gap r e g i o n a n d w i l l t o zero at i n f i n i t y  decrease i n the surrounding  region. Hence t r i a l  f u n c t i o n s must b e c h o s e n w h i c h w i l l  have  similar  characteristics. In order t o understand  t h e modus o p e r a n d i  o f t h e VIOWM  only t h e lowest  o r d e r mode n e e d b e c o n s i d e r e d .  The l o w e s t  o r d e r mode w i l l  b e t h e most h i g h l y c o n f i n e d mode f o r t h e  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 h e r e a s 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  c o u p l i n g t o o r from t h e f i b e r positioned.  VIOWM,  i f the fiber  The a p p r o x i m a t e  t h e VIOWM w i l l  have t h e h i g h e s t degree o f  optical  b e d e s i g n a t e d <I> (y,z), v  field  i s well distribution for  subscript  vf o r  and i s g i v e n by -iB x e  * (y,z)  ;  v  y ^ o (2.12)  ;  y < 0  where Bv i s t h e p r o p a g a t i o n c o n s t a n t o f t h e mode a n d 2  2  -<y /w  yv  2  2  + * /*  zv  )/2  yv  where a  i s t h e normalized amplitude,  v  Wy^. a n d w  2 v  are the it  width parameters T-KO  xn t h e y a n d z - d i r e c t i o n s  r!orraali.zed a m p l i t u d e  respectively.  i sdiscussed further  i n appendix  C.  F i g u r e 2.15 shows a H e r m i t e - 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 v = w z v a n d f i g u r e 2.16 shows one i n which W y  V  = w /2. z v  As c a n be seen Gaussian  approximations  of the desired respect  from  figures  2.15 a n d 2.16 t h e H e r m i t e -  t o the f i e l d  characteristics:  t o t h e xy-plane,  they  they  distributions  meet  a r e symmetric  with  each  are small, zero i n fact, a t  As c a n b e s e e n tf (y,z) has t h e form ( y ) e x p (-y ) i n t h e y - d i r e c t i o n , f o r p o s i t i v e y, a n d H g ( z ) e x p ( - z ) i n t h e zd i r e c t i o n , f o r a l l z, h e n c e t h e name H e r m i t e - G a u s s i a n . 2  v  2  -61-  F i g u r e 2.15: i n w h i c h w,.,,  A Hermite-Gaussian = w_,,.  optical  field  distribution  -62-  F i g u r e 2.16: i n w h i c h wvv  A Hermite-Gaussian = w /2. z v  optical  field  distribution  -63-  the  i n t e r f a c e between t h e s u b s t r a t e and t h e o p t i c a l b u f f e r  layer, at  and t h e evanescent  field  goes t o z e r o  monotonically  infinity. In  the  2.4  section  exact  field  presented  a l l t h e r e s u l t s were d e r i v e d i n t e r m s o f  distribution.  i n terms o f t h e approximate f u n c t i o n s .  with t h e approximation approximate o p t i c a l 2  v (y,z) v  field  2  z  d i s t r i b u t i o n by  dln[n  2  (y,z)] 2  -  + n (y,z)k z  dz  To b e g i n  t o v (y,z) i s g i v e n i n t e r m s o f t h e  3 l n [ n (y,z)] 2  Now t h e e q u a t i o n s a r e  2 o  - p  2 v  .  dz  W  zv  which g i v e s t h e s t a t i o n a r y  -/ J  2  integral  ,  approximation  2 2  +  —oo —OO  for this  ,  ay ,  2  dydz = 0  (2.13)  az ,  w h i c h becomes  /J in  (  ,  a*  ay  V  1  2  2  + ,  v  az  exactly  first  v  w2 v  dydz  function.  two t e r m s o f t h e i n t e g r a l  and a r e g i v e n by  (2.14)  t  terms o f t h e approximate The  2/  v (y,z)<(.  I  v  c a n be e v a l u a t e d  -64-  (  I I  1.5  }  v  dydz =  I ay  J  I  J  4> dydz v  J  J  6  yv  and  J  0.5  v  I  dydz dz  {  2 dydz v  )  Since  I i s a minimum I  I;  i.e. I  S I.  an  equation  v  Therefore  v  must p r o v i d e  i f 2.13  an u p p e r b o u n d  i s rewritten to  2  2  8 ln[n <y,z)] 2  z Bin [n (y, z) ]  2  2  + •  + > 8«  dz'  .  "zv  >v  v  lower  r e w r i t i n g equation bound  0.5  2 yv  2 zv  2.14  n2( z)k2  dydz  y/  3 2  dydz  in a similar  fashion provides  on 3 ln[n (y,z)] 2  1.5  provide  f o r Bv  r  then  on  J-~ J  z  2  31n[n (y, z) ] 2  + r.2(y,z))c2  v  dydz  "zv /  (  *  v  dydz  Since the propagation  constant  above e q u a t i o n  used to f i n d the width  c a n be  •  i s a stationary value  the  p a r a m e t e r s w-  a  -65-  and w .  Proof that  zv  appendix values for  B.  ^  i sa stationary value i sprovided i n  By v a r y i n g t h e w i d t h p a r a m e t e r s  a grid of  f o r fry c a n b e c r e a t e d a n d t h e v a l u e s o f Wy^. a n d w  which J ^  2  i s s t a t i o n a r y may b e f o u n d  F i g u r e 2.17 shows a p l o t  graphically.  v e r s u s Wy^. a n d w  of  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  a p p l i e d t o t h e e l e c t r o d e s a n d where XQ  = 442nm, n  n  4  0  = 2.371, r Figure  for  3  3  = 30.8xl0  - 1 2  m/V,  and r  z v  2  for  z v  50.0V e  = 28.0xl0  = 2.2884, - 1 2  m/V.  2.18 shows t h e d e v e l o p m e n t o f t h e o p t i c a l  field  t h e VIOWM w i t h t h e 4^m i n t e r e l e c t r o d e g a p 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, 50.0 v o l t s ,  2.6  as p r e d i c t e d by t h e v a r i a t i o n a l  The C o u p l i n g  Hermite-Gaussian chosen  approximate  are convenient following, expanding  Coefficient  field  t o manipulate  distributions,  mathematically.  the coupling coefficients the optical  fields  approach  as t h e  that  they  In the  a r e determined by  i n t h e two b u t t - c o u p l e d  waveguides i n terms o f both guided This  method.  f u n c t i o n s have t h e advantage,  optical  and  and r a d i a t i o n  has been used by Burns and M i l t o n  modes. [51]  to  s t u d y t h e 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 w a v e g u i d e s a n d by  H u n s p e r g e r e t a l . [52] t o s t u d y b u t t - c o u p l i n g b e t w e e n  solid  state  lasers  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 versus w „ and w 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 g a p w i t h 50.0V a p p l i e d t o t h e electrodes f o r\ = 442nm. v  0  z v  -67-  F i g u r e 2.18: T h e 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 g a p f o r (a) 10.0, (b) 30.0, a n d ( 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 ^ = 442nm. Q  -68-  The the  c o n t i n u i t y of the  i n t e r f a c e between t h e  considered.  single-mode operation  «>,. + f  R0>,  for  the  p ,«>.f f K  for  +  +  E  _ = ref  Rp£«>£ + f  H  ref  VIOWM.  transmission the  trans  the  E.  field  , =  Tp  v  r e f l e c t e d and  a  r  e  of the  *-he  «>  v  +  H.  write  (2.15b)  T are  the  from the  by  Pv^*  r e s p e c t i v e l y , and  transmitted  electric  E  r  e  f  magnetic  coupling  between t h e  the  E t  r  a  n  H  r e  f  and  field  two  waveguides  power i n t h e r e f l e c t e d that  the  major p a r t  2.15b  of  b a c k t r a v e l i n g g u i d e d mode.  i n e q u a t i o n 2.15a  the  and  field  similarly  transmitted  fiber  and  and  both are  VIOWM, w h i c h i s assumed t o be  the  and  Each  dropped,  is multiplied  integrated  i n t e r f a c e between t h e  the  terms  r e f l e c t e d r a d i a t e d mode h a v e b e e n  i n equation  boundary plane,  optical  reflection  r e m a i n i n g terms, remaining a f t e r the  m u l t i p l i e d by  designed  trans  i s made t h a t  is  VIOWM i s  r a d i a t e d modes.  f o r the  the  the  assumed t o be  for coupling  r e f l e c t e d power i s i n t h e  the  and  at  and  r a d i a t e d modes i s n e g l i g i b l e and  representing  components  (2.15a)  r e f l e c t e d and  assumption  of the  fiber  r a d i a t e d modes and  of the  solve  can  field  trans  coefficients,  distributions To  +  one  H e r e R and  distributions H  v  magnetic f i e l d  t o the  are  T<D  electric  the  optical  I f b o t h waveguides are  for  f  tangential  optical  plane x =  0.  over  the  fiber  and  The  two  g  -69-  resulting  equations are then subtracted,  one f r o m t h e o t h e r ,  leaving  P (P v  - P ){  f  v  J  »*  f  dydz - R P ( P v  f  and  solving  p, - p R = P, +  v  —OO  for R  *%>  dydz = 0  f  OO  n' - n' =  p  - T  (2.16)  n' + n'  v  optical  Fresnel  —  gives  where n f ' a n d n ' a r e t h e e f f e c t i v e the  J  + p ){  —oo — oo  refractive  f i b e r a n d t h e VIOWM, r e s p e c t i v e l y ,  reflection  coefficient  indices  of  and r i s t h e  f o r p l a n e waves a t n o r m a l  incidence. To  justify  r a d i a t e d modes  n e g l e c t i n g t h e power i n t h e one must l o o k a t t h e r e s u l t  2.16 r e p r e s e n t s .  In e f f e c t  power, r^,  reflected  t h e assumption  t h e same a s t h a t  incidence.  The p l a n e wave s o l u t i o n  the  i n the refractive  the  field  distributions  amount o f r e f l e c t e d  modes  i s , i n fact, the  i n that  I n o t h e r words as of the  o f t h e boundary p l a n e t h e  e x t e n d t o become p l a n e waves a n d  power i s e x a c t l y  o f t h e waveguides c o n s i d e r e d  guided,  i s that the  index d i s t r i b u t i o n s  w a v e g u i d e s d i s a p p e a r on b o t h s i d e s optical  equation  f o r p l a n e waves a t n o r m a l  c a s e f o r w e a k l y g u i d e d modes.  variations  that  i n t h e case o f o p t i c a l waveguides i s  essentially  limiting  reflected  equal t o  .  The  i n t h i s work a r e w e a k l y  the extents of the o p t i c a l  field  -70-  distributions amount  a r e much g r e a t e r t h a n a w a v e l e n g t h ,  o f power r e f l e c t e d  i s small,  » 5% o f t h e t o t a l  i n c i d e n t power, t h e r e f o r e t h e a p p r o x i m a t i o n reasonable  and t h e  should  lead to  results.  Using  equations  coefficient  2.15a a n d b t h e t r a n s m i s s i o n  c a n be s o l v e d  for.  First  2.15a i s m u l t i p l i e d b y  {^PfO-y* a n d 2.15b i s m u l t i p l i e d b y P v * * a n d t h e r e s u l t i n g v  equations  a r e i n t e g r a t e d over t h e boundary plane  This time,  however, t h e e q u a t i o n s  oo oo f  which a f t e r 2a  f P f  a r e added r e s u l t i n g i n  oo oo  2B B J J ©%> —oo —oo V  x = 0.  f  dydz - T P O V  F  + P ) V  J . J «>X d y d  2  —oo —oo some r e a r r a n g e m e n t  -f y  2 2 2 2 -{y /-^ + z / K 2  e  2  gives  2  2 2 - a) /w 2  v  +  (  y  +  2 f  2 2 <z - b , / w } / 2 2  2  f  ^  J - ~ /0  T (y /w 2  «Sr<Pf  +  2  yv  Pv> J / —. 0  + z /w 2  2  )  «v'  d y d z  yv  where t h e v a r i a b l e s a a n d b l o c a t e t h e c e n t e r fiber  relative  t o the center  waveguiding region af/a and  v  of the surface of the  o f t h e VIOWM, s e e f i g u r e 2.19.  i s d e t e r m i n e d by n o r m a l i z i n g  i s given  See  by*  appendix  of the optical  C.  The  ratio  t h e power i n e a c h mode  -71-  Opt i CO I Fiber  [c o r e  Subst r o t e - i Sur f o c e  VIOWM  H  Wove gu i d e  nterfoce  Edge  View  Wavegu i de 11 HVIOWM o Z V-gr  End  F i g u r e 2.19: The i n t e r f a c e VIOWM.  oove  View  between t h e o p t i c a l  fiber  and t h e  -72-  1  '  y I  2  // 2  2  —(y - o o  0  2  /w y  e  x / ^ 2  + z /w  v  ) zv  2 dydz  yv  e  The e x p r e s s i o n  .2,2 - ( y /w  , 2,2, + z /« ) dydz  f  z f  f o r thetransmission  coefficient  c a n be  r e w r i t t e n * as .  7  /  2(pj) ) n n f v y z 1/2 f r  -<a /w e 2  /  1 / 2  1  7  /  2  y  2  f  z  2  )/2 + b £l / 4 w * 2  f  2  z  f  r  1/2 v  y v zv y f z f  1/2w1/2 e V a 2  aQ  4 W  J  y  yf J  yv  ,  e r ff c |  (2.17)  1 + 2w yf  where , 2 2 2w w , yv y f 2 ^ 2 w + w _ yv yf  and  *  + b /w  2  See a p p e n d i x C.  -73-  2 ^ 2 zv zf  A  similar  identical an  t o equation  optical As  fiber  2.17 f o r l i g h t c o u p l e d  i s e v i d e n t from  from  a VIOWM t o  equation  2.17 t h e l o c a t i o n  of the  r e l a t i v e t o t h e i n p u t o f a VIOWM i s i m p o r t a n t i n  direction  the coupling.  I t i s obvious  t h a t i n t h e z-  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 a s t h i s  give the best  distributions parameters. the  an e q u a t i o n t h a t i s  fiber.  determining  will  l i n e o f reasoning yields  o v e r l a p b e t w e e n a n y two G a u s s i a n  independent  of their  respective width  On t h e o t h e r h a n d f i n d i n g  y-direction  will  the best  location i n  d e p e n d on t h e o p e r a t i n g v o l t a g e a n d  mode o f o p e r a t i o n , s e e c h a p t e r  4.  -74-  Chapter  3  FABRICATION  3.1  Introduction  In t h i s  chapter  the fabrication  r i d g e VIOWMs i n l i t h i u m n i o b a t e , polishing, included optical  o f both  i n c l u d i n g c u t t i n g and  and o f V-grooves i n s i l i c o n  i s a s e c t i o n on t h e alignment fibers  alignment  p l a n a r and  using t h eV-grooves.  information i s contained  i s described.  Also  o f t h e VIOWMs a n d  The f a b r i c a t i o n a n d i nthree  sections  entitled: The VTOWM, The S i l i c o n V - g r o o v e , a n d 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 b e g i n s  common t o t h e f a b r i c a t i o n Then t h e f a b r i c a t i o n device  type  by d i s c u s s i n g i s s u e s t h a t a r e  o f both  techniques  are contained  planar  and r i d g e  devices.  t h a t a r e unique t o each  i ntheir  own s u b s e c t i o n s .  The  -75-  cutting a  and p o l i s h i n g  separate The  o f two t y p e s  o f device  i s contained i n  subsection.  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  description  of the anisotropic etching of s i l i c o n  t o produce  V-grooves. V-grooves etched  i nsilicon  s t a b l e method o f c o u p l i n g l i g h t optic  devices  device/optical alignment  fiber  and out o f i n t e g r a t e d  technique  alignment  The combination  t h e V-groove array  switch.  into  [ 2 1 ] . The  section describes the  o f a VIOWM a n d a n o p t i c a l  V-grooves. on  by a f l i p - c h i p  can be used t o form a  f i b e r u s i n g an a r r a y o f  o f a VIOWM a n d a n o p t i c a l  i s u s e d as an o p t i c a l  M e a s u r e m e n t s on t h e s w i t c h  front-end  arepresented  fiber ^  i n chapter  4.  3.2  T h e VIOWM  The  initial  same f o r b o t h cutting  p r e p a r a t i o n o f t h e L i N b 0 3 w a f e r s was t h e  planar  and p o l i s h i n g  and r i d g e d e v i c e s stage  were f a b r i c a t e d on Y - c u t * Crystal inches  Technology  Inc.,  a s was t h e f i n a l  i nthe fabrication.  LiNb03 wafers o b t a i n e d Palo Alto,  i n d i a m e t e r a n d 0.04 i n c h e s  Ca. thick.  The d e v i c e s from  The w a f e r s were 3 In order  t o make  These s u b s t r a t e s a r e c u t and p o l i s h e d so t h a t t h e Y-axis of t h e c r y s t a l i s normal t o t h e 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 quarters  of the  by  first  face of the  (made b y  sources.  covering the  wafer with  evaporated using system  l a r g e wafers they  The  Al  during the  phosphoric The solution in  of Alconox  a 5 min.  by  The  provided been  to  avoided  water at  The  immersion i n a a 10  1%  min.  rinse  f o l l o w e d by  a  10  soak w i t h u l t r a s o n i c s a m p l e was  heated  with  agitator in LiNb03.  r e m o v a l f r o m t h e b o i l i n g m e t h a n o l t h e w a f e r s were nitrogen  immediately p r i o r to being  Sputtering  System Model  p a t t e r n i n g o f t h e w a v e g u i d e s on  substrates  was  made b y  10  Sierracin,  nm.  the  done u s i n g p h o t o l i t h o g r a p h y .  stripes varying  from 4 t o  picked-  50°C.  t h e r m a l shock s h a t t e r i n g the  a Perkin-Elmer The  a  s o l u t i o n of  f o l l o w e d by  a 5 min.  3  surface  s p e e d d i a m o n d saw.  s o a k i n b u f f e r e d HF and  and  m e t h a n o l t h e n b o t h were t r a n s f e r r e d t o t h e  blown dry w i t h  long  f o r 5 min.  of holding  t h a t had  e t c h i n g i n a 1:1  i n b o i l i n g methanol.  After  was  a high  (de-ionized)  r i n s e i n D l water,  order  into  Dl  dirt  w a f e r s were t h e n c l e a n e d  agitation the  inert  thick  optically polished  c u t t i n g process  done on  r e m o v e d by  a c i d and  D l water,  min.  any  polished  d i f f u s i o n pumped vacuum  C a r l Hermann A s s o c . ) c a p a b l e  c u t t i n g was  l a y e r was  l\xm  o f A l about  a conventional  means o f u n d e r c u t t i n g The  entire optically  A l p r o t e c t e d the  from s c r a t c h e s  up.  a coat  were d i v i d e d i n t o  S a n t a C l a r a , Ca.  i n length  On  f r o m 1 t o 23  P h o t o r e s i s t was  patterned  loaded  3140. LiNbC>3 The  mask u s e d  t h e mask were mm  and  i n width  on  the  sample  so  -77-  that  the  to the  longer  X-axis of the  parallel field,  dimension of the  to the  created  crystal.  Thus t h e  XY-plane of the by  a p p l i c a t i o n of voltage  200  (in the  nm  optical buffer  face  electric  to the  electrodes,  Z-axis.  1 hr.  99.95% p u r i t y .  preclean  between t h e  target  chamber was  18  was  done a t  100  0.5  hr.  The  of the and  the  mTorr o f Ar W  sputtered  The  deposition  The  was  target preceded  placed  atmosphere i n  5 mTorr 0 .  The  2  about 2 W  r e f r a c t i v e index of  optically  with a shutter  wafer.  f o r w a r d and  RF  LiNbC-3 w a f e r . The  target  and  SiC>2 was  system) o n t o t h e  of a s e c t i o n of the  nominally of a  layer of  Perkin-Elmer sputtering  polished  by  the  run  The Planar VIOWM  A  was  to the  parallel  waveguides would  c r y s t a l and  w o u l d h a v e a component p a r a l l e l  3.2.1  s t r i p e s would run  the  sputtering  r e f l e c t e d power  SiC"2 i s a b o u t  therefore  provides  a good o p t i c a l b u f f e r  w h i c h has  a r e f r a c t i v e i n d e x o f a b o u t 2.3.  1.5  l a y e r on See  and  for it  LiNbC>3  section  2.3.1.2. The  w a f e r was  removed f r o m t h e  patterned  with photoresist  gap  formed u s i n g  t o be  so  as  a lift-off  to  sputtering allow  the  technique  system  and  interelectrode [53].  The  photoresist  photoresist*  was  patterning  applied  was done a s f o l l o w s : t h e  i n a photoresist  spinner  a t 4000  rpm f o r 25 s e c ; a p r e b a k e was p e r f o r m e d a t 95°C f o r 25 min.;  the photoresist  w i t h a power d e n s i t y photoresist [53];  was e x p o s e d t o 320 nm r a d i a t i o n o f 25 mW/cm  2  f o r 40 s e c ; t h e  was t h e n s o a k e d i n c h l o r o b e n z e n e f o r 2.5 m i n .  a p o s t - b a k e was p e r f o r m e d f o r 25 m i n . a t 95°C; t h e  photoresist  was  d e v e l o p e d i n a 1:1  solution of  Shipley  MF-312 d e v e l o p e r a n d D l w a t e r ; t h e f i n a l  step  flowing  and narrow  D l water.  photoresist the  By t h i s method a l o n g  r i d g e was  electrodes  be f a b r i c a t e d u s i n g  of the planar  rinsing in  lift-off.  on t h e S i V - g r o o v e s b y a  T h e r e f o r e t h e y were made o u t o f AuGe w i t h a  e u t e c t i c t e m p e r a t u r e o f 363°C.  However, t h e a d h e s i o n o f  AuGe o n S i 0 2 i s p o o r a n d i t was n e c e s s a r y t o i n c l u d e l a y e r b e t w e e n t h e two.  fabricated  so t h a t  VIOWM were d e s i g n e d s o 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 e u t e c t i c bond.  was  f o r m e d w i t h some u n d e r c u t t i n g  i n t e r e l e c t r o d e gap c o u l d The  Ti  (UV)  Thus t h e e l e c t r o d e s  a thin  were  i n t h e C a r l Herrmann s y s t e m , w i t h one T i a n d one  AuGe s o u r c e , b y d e p o s i t i n g nm AuGe on t h e T i .  *  Shipley's  *  Ghandhi  S1400-30.  [ 5 4 ] p.  59.  50 nm T i on t h e S i 0 2 a n d t h e n  300  -79-  Once t h e m e t a l h a d b e e n d e p o s i t e d removed by immersion  t h e p h o t o r e s i s t was  i n a sequence o f s o l v e n t s  e a c h : h o t (95°C) M i c r o s t r i p * * , h o t a c e t o n e , Figure  wide.  The Ridge VIOWM  In  a r i d g e VIOWM t h e  r i d g e s must b e f o r m e d  Etching  LiNb03 has been achieved  several  different  plasma e t c h i n g  using  substrate After  C F 4  CHF3  CCI2F2  [57],  [58].  was s p u t t e r e t c h e d thei n i t i a l  Perkins-Elmer  between t h e t a r g e t  CF4  [56],  I n t h i s work t h e L i N b 0 3  i n an argon plasma.  a T i source.  by s p u t t e r e t c h i n g , sample u n t i l  with  being  deposited,  The  preclean  i n the  The  a shutter  t h e m e t a l was b e i n g  The plasma h a d a d i s t i n c t i v e b l u e  was  etching  c l e a n i n g t h e w a f e r was p l a c e d  and the  [55],  reactive  and r e a c t i v e i o n  s p u t t e r i n g system with  s o u r c e was p r e c l e a n e d ,  deposited.  by s e v e r a l authors by  i n Freon plasmas u s i n g  and  first.  methods i n c l u d i n g : a r g o n i o n m i l l i n g  ion-beam e t c h i n g u s i n g and  and 2-propanol.  3.1 shows t h e i n t e r e l e c t r o d e g a p o f a VIOWM.  Here t h e gap i s 4  3.2.2  f o r 5 min.  tint  p r i o r t o which i t i suniformly  when T i pink.  u s u a l l y t a k e s between 1 and 2 h r .  A product Paterson,  of Philip N. J .  A. Hunt C h e m i c a l C o r p o r a t i o n ,  West  -80-  F i g u r e 3.1: The i s 4 pint w i d e .  interelectrode  gap o f a VIOWM.  H e r e t h e gap  -81-  The  T i was d e p o s i t e d w i t h 100 W f o r w a r d a n d a b o u t  r e f l e c t e d power w h i c h g a v e a d e p o s i t i o n r a t e o f a b o u t A/min. metal  2 W 40  The d e p o s i t i o n was c o n t i n u e d f o r 1.5 h r . g i v i n g a layer  The  about  500 nm t h i c k .  T i was f o r m e d 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 t h o s e a r e a s t h a t were n o t t o b e The u n p r o t e c t e d T i was removed i n a C F 4 p l a s m a i n a  etched.  Plasma-Therm  system.  The chamber p r e s s u r e was 500 m T o r r  a n d t h e f o r w a r d power was 100 W w i t h 2 W r e f l e c t e d . e t c h t i m e v a r i e d w i t h T i t h i c k n e s s a n d was u s u a l l y 30  a n d 45 m i n .  The between  The p h o t o r e s i s t was a p p l i e d a s d e s c r i b e d  a b o v e f o r t h e p l a n a r VIOWM e x c e p t  t h a t t h e chlorobenzene  s o a k was o m i t t e d . The  remaining  T i s t r i p e s were t h e n o x i d i z e d i n a  M i n i B r u t e o v e n a t 600°C w i t h a 1 1/min. f l o w o f O2 hr.  f o r 12  T h e o x i d e p r o v i d e d t h e mask f o r t h e s p u t t e r e t c h o f t h e  LiNb03> The  s p u t t e r e t c h was done i n t h e P e r k i n - E l m e r w i t h 100  W f o r w a r d a n d about  2 W r e f l e c t e d power a t 18 m T o r r .  t o o t h e t i m e was d e t e r m i n e d Ti. about  by t h e t h i c k n e s s o f t h e o x i d i z e d  As t h e T i has a d i f f e r e n t half,  differential  Here  e t c h r a t e t h a n t h e LiNb03,  t h e t i m e was d e c i d e d upon b y m o n i t o r i n g t h e etch rate.  B e f o r e p l a c i n g t h e sample i n t h e  A M u l t i v e r s i o n Plasma, R e a c t i v e I o n E t c h and Plasma D e p o s i t i o n S y s t e m M o d e l PK-1250PE/RIE/PD.  -82-  sputterer the thickness  o f t h e o x i d i z e d T i was m e a s u r e d on a  T e n c o r A l p h a - S t e p 200 p r o f i l o m e t e r . of  sputter etching  and  t h e new h e i g h t  returned  measured.  If layer  t h e s a m p l e was removed f r o m t h e s p u t t e r e r was m e a s u r e d .  The sample was t h e n  Then i t was removed a n d t h e h e i g h t  This process i n the height  the height  was r e p e a t e d  ridge height  patterned  an i n i t i a l  o f about  l a y e r o f S i 0 2 was d e p o s i t e d  the  3.2  i s a scanning  of a sputter  etched  r i d g e c a n be o b t a i n e d  t o be about profilometer 3.3.  |im.  output  etching.  1.5  jim.  was a c h i e v e d  and o p t i c a l  as d e s c r i b e d  above f o r  e l e c t r o n m i c r o s c o p e (SEM)  ridge  i n LiNb03-  The w i d t h o f  f r o m f i g u r e 3.3 where i t i s s e e n  Figure  3.4  o f the ridge  o f the ridge  electrodes  were s p u t t e r table  7.5  The h e i g h t The  and o x i d i z e d and t h e  VIOWMs.  Figure picture  t h e n a new  T i l a y e r o f 0.5 Mm r e s u l t e d i n a n  i n t h e ridge height  planar  again  was no  o f t h e r i d g e was i n s u f f i c i e n t  Once t h e d e s i r e d r i d g e h e i g h t  the  there  was i n c r e a s e d b y f u r t h e r s p u t t e r  Typically  buffer  until  was  for  of the ridges.  o f T i was d e p o s i t e d ,  increase  4 hrs.  t o t h e s p u t t e r e r a n d t h e e t c h i n g was c o n t i n u e d  another 2 hrs.  increase  After the f i r s t  i s a photocopy o f t h e shown i n f i g u r e s 3.2 a n d  i s seen t o be 4 p .  f o r t h e r i d g e VIOWM were aluminum.  deposited  with  a bias  They  a p p l i e d t o t h e sample  i n a n a r g o n p l a s m a a t 18 m T o r r .  The b i a s  s p u t t e r - c l e a n i n g o f t h e sample's s u r f a c e which  causes removes  some  -83-  F i g u r e 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 LiNbC>3 • 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 picture. T  n  e  -84-  -85-  I fcft i  40 I.  30!:  20  !  I  /....! i  J:;:: .tit 0  Figure t h e  3.4:  r i d g e  i s  26  P r o f i l o m e t e r 4  pun  o u t p u t  4b  s h o w i n g  *  t h a t  ttr  t h e  h e i g h t  o f  -86-  a d s o r b e d g a s e s and  which i n t u r n  the  f o r w a r d power was  aluminum.  power was with  a b o u t 2 W.  a shutter  (with the few  The  placed  bias  1 and  for 2 hrs.  2 pm  thick.  c o a t e d sample  the  h a r d b a k e d f o r an Therm i n an  the  power was  2 W.  tops of the  hour at  thin photoresist  was  was  s a m p l e was sputter along  then placed  etched  the  i n an  tops of the  Figure  3.5  f a b r i c a t e d by  gap  The  the  Al  at  between selfmetal  a low  speed  ridges  and  photoresist  etched i n the  was  Plasma-  chamber p r e s s u r e  100  W and  was  320  the r e f l e c t e d  f o r a b o u t 50  min  tops of the  photoresist.  until ridges  The and  plasma u n t i l  the  aluminum  removed  hr.  longer  (1/2  Al) . aluminum  electrodes  s e l f - a l i g n e d technique described  The  c e n t r a l dark l i n e  running  the  f i g u r e i s the  of the  top  last  was  a  Perkin-Elmer system  mTorr A r  hour  the  The  i s a p i c t u r e of the  the  f o r only  spun onto t h e  ridges.  i n the  r i d g e s was  than i t took to deposit  was  of  target  m e t a l l a y e r was  continued  i n the  18  the  removed and  removed from t h e  f o r m i n g a n a r r o w , ~ 4 |im,  f o r an  were f o r m e d by  130°C and  f o r w a r d power was Etching  the r e f l e c t e d  v a l l e y s between t h e  oxygen atmosphere.  mTorr and  and  1400-30 a t 2000 rpm)  a t h i c k l a y e r i n the  a t h i n l a y e r on  the  deposited  Photoresist  adhesion  precleaned  sample t a b l e  electrodes  (Shipley's  W  the  sample and  s h u t t e r was  The  The  aligned technique.  to give  between t h e  Then t h e  100  t a r g e t was  a p p l i e d to the  minutes).  deposited  The  increases  from t h e LiNb03  top  ridge.  to the  above. bottom  of  -87  3 . 5 : The a l u m i n u m e l e c t r o d e s the s e l f - a l i g n e d technique.  F i g u r e  by  o f a r i d g e VIOWM f o r m e d  -88-  3.2.3  C u t t i n g and P o l i s h i n g  The i n d i v i d u a l VIOWMs were either  a high  c u t from t h e s u b s t r a t e  s p e e d d i a m o n d saw o r a w i r e  saw.  The c h o i c e  o f t h e c u t t i n g method depended on t h e d e s i r e d y i e l d . u s i n g t h e d i a m o n d saw t h e edge damage c a n b e s e v e r e the  w a v e g u i d e s must b e c u t i n t o a b o u t 3 mm  Using  t h e wire  sections. mm  long  using  When so t h a t  sections.  saw t h e w a v e g u i d e s may b e c u t i n t o 2 mm  In both cases  the final  device  long  i s t o be about 1  long. The l e n g t h o f t h e d e v i c e  l o s s e s due t o a b s o r p t i o n , electrode  interactions.  that  there  bulk  modes.  was k e p t  short t o minimize the  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 The drawback o f a s h o r t d e v i c e i s  i sa relatively  l a r g e amount, o f c o u p l i n g due t o  Although a device  about  f a b r i c a t e d most were b e t w e e n 1.6 mm  1 i n c h l o n g was a n d 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 m a i n body and  thepolishing plate.  screw t h a t crystal  The main body h a d a l a r g e  c o u l d be lowered t o p r o v i d e  being  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  stainless  s t e e l p l a t e about  decreased  with  slot  polishing  for the  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 .  x 10 mm  a backstop  center  excessive  1 mm t h i c k  (the t h i c k n e s s  g r i n d i n g and p o l i s h i n g ) w i t h  cut into i t s center.  Figure  a 2 mm  3.6 shows t h e  j i g : t h e m a i n body a n d t h e 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 polishing plate (left).  j i g : t h e main body  (right)  and t h e  -90-  The  samples  t h e two b i t s  of crystal  together using was  were p o l i s h e d two a t a t i m e .  The f a c e s o f  c o n t a i n i n g t h e VIOWMs were e p o x i e d  a 5 minute  epoxy.  The s a n d w i c h  thus  formed  p r e s s e d t o g e t h e r t o remove e x c e s s e p o x y f r o m b e t w e e n t h e  two.  I t i s i m p o r t a n t t o reduce t h e t h i c k n e s s o f t h e epoxy  layer  s o a s t o r e d u c e t h e edge damage t h a t  results  during  the grinding process. When t h e e p o x y b e t w e e n t h e two s a m p l e s set  t h e sandwich  plate both  was e p o x i e d i n t o t h e s l o t  of the polishing  j i g so t h a t  had had time t o i n the polishing  i t protruded equally  from  sides. The p r o t r u d i n g L i N b 0 3  g r i n d i n g paper, millimeter  until  i t extended about  o u t from e i t h e r  second g r i n d  was t h e n g r o u n d ,  i s done u s i n g  using  600  a quarter of a  side of the polishing plate. 800 g r i t  t h e e d g e damage was removed. c o n t i n u a l l y monitored, using  Figure sandwich  alumina  a m i c r o s c o p e , t o d e t e r m i n e when The f i n a l  p o l i s h was done  slurry.  3.7 shows t h e p o l i s h e d e n d f a c e o f s u c h a  f o r two r i d g e w a v e g u i d e s a m p l e s  alumina p o l i s h . of the p i c t u r e region  until  The edge o f t h e c r y s t a l was  t h e e d g e damage h a s b e e n removed. a 0.05  A  compound.  P o l i s h i n g was done u s i n g a 1 nm a l u m i n a s l u r r y  using  grit  The l a r g e  light  a r e a s a t t h e t o p and bottom  a r e t h e LiNbC>3 s u b s t r a t e s ,  i s t h e epoxy, t h e t h i n b r i g h t  aluminum e l e c t r o d e s .  f o l l o w i n g t h e 1 pm  t h e dark  stripes  central  are the  The c e n t r a l bump i s t h e L i N b 0 3  ridge.  -91-  The  ridge height  i s about  figure.  One  can  climb  the  sides of the  up  damage c a n  be  clearly  s e e n on  Once b o t h  faces  0.4 see  times the width i n that the  ridge.  the  upper  aluminum  A l s o the  the  f o r a b o u t 5 min.  they  came a p a r t ,  Figure on  3.8  end  to the  Figure  were a g a i n  about 1  i n t e r e l e c t r o d e gap  edge  end  was  soaking  had  they  them i n  been removed  hot from  hr. device  T h i s d e v i c e was  shows t h e  of  soaked i n hot M i c r o s t r i p  i n t e r e l e c t r o d e gap  other.  3.9  Once t h e y  shows a p l a n a r  b o t h ends the  results  o f b o t h VIOWMs h a v e b e e n p o l i s h e d  p o l i s h i n g p l a t e they  until  electrodes  substrate.  were removed from t h e p o l i s h i n g p l a t e by Microstrip  this  of the  t h a t had  been p o l i s h e d  i s seen t o run about  device  perpendicular  1.6  mm  where  from  one  long.  the  to the p o l i s h e d  end  face. Finally side slide  3.3  of the and  e x c e s s L i N b 0 3 c o u l d be  device  by  m o u n t i n g i t on  cutting i t using  The  The  the  Silicon  use  substrates  a low  removed from e i t h e r  a glass  speed diamond  saw.  V-grooves  of V-grooves a n i s o t r o p i c a l l y  [59,60] f o r t h e  alignment  butt-coupling.  etched  of o p t i c a l  i n - d i f f u s e d waveguides i n LiNb03 p r o v i d e s of achieving e f f i c i e n t  microscope  The  in Si fibers  a practical "flip-chip"  with method  -92-  F i g u r e 3 . 7 : The together, after  e n d f a c e o f two a l p alumina  r i d g e VIOWMs, e p o x i e d polish.  - 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 e n d s 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 e n d 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 s e e n t o r u n p e r p e n d i c u l a r t o t h e endface.  -95-  method o f c o u p l i n g between o p t i c a l w a v e g u i d e s was f i r s t  fibers  a n d Ti:LiNbC>3  r e p o r t e d b y Hsu a n d M i l t o n i n 1976  [21]. I n t h i s method a V-groove i s e t c h e d silicon that  wafer.  taking  result  i n t h e V-groove has i t s core  above t h e u n e t c h e d s u r f a c e i n efficient  35 t i m e s g r e a t e r  i s i n t h e (111) d i r e c t i o n Electrodes  coupling of light  i n t h e (100) d i r e c t i o n  so  electrical  contact  with  the electrodes  which i s p l a c e d  fields  a high  "upside  In t h i s  and i n t e g r a t e d o p t i c s device  as t o f a c i l i t a t e  optical  than  described  down" o n t h e w a f e r c o n t a i n i n g t h e V - g r o o v e s . fiber  by  a r e f o r m e d on t h e u n e t c h e d p o r t i o n s o f  wafer t o provide  optical  t o an  of the etch rate of S i  f o r the etchant  of t h e integrated optics device  the  of the  The d e p t h c o n t r o l i s o b t a i n e d  advantage o f t h e a n i s o t r o p y  below. the  lying  optic device.  which i s about it  fiber  at a height  wafer that w i l l integrated  (100) p - t y p e  The d e p t h o f t h e V - g r o o v e i s c o n t r o l l e d s o  an o p t i c a l  positioned  in a  way  are located  degree o f o v e r l a p between t h e  i n b o t h w a v e g u i d e s when t h e y  are butted  together. The optical are  variables critical fiber  the outer  concentricity to  alignment  and an i n t e g r a t e d waveguide, by t h i s diameter o f the f i b e r , and t h e V-groove depth.  c o n t r o l each o f these  (less than  to the v e r t i c a l  method,  the core-cladding I t w o u l d be  v a r i a b l e s to.sub-micron  1% i n e a c h c a s e )  o f an  necessary  tolerances  i f one were t o p r e d i c t a b l y  -96-  obtain  a certain  degree o f c o u p l i n g ,  therefore  i t is  d e s i r a b l e t o have a method o f v a r y i n g t h e h e i g h t fiber  above t h e s u r f a c e  Giallorenzi a second,  Sheem  [61] a c c o m p l i s h e d t h i s u s i n g  deeper V-groove.  f i b e r b e i n g a l i g n e d was tapered  o f the S i wafer.  fiber  I n t h i s method t h e h e i g h t  c o n t r o l l e d by  i n the deeper V-groove  of a different  depth,  and  a tapered fiber i n  sliding beneath  p r e s e n t work an a r r a y o f 13 V - g r o o v e s was is  of the  i n c r e a s i n g by  the it.  of the  second In the  used each o f  1 nm  f r o m one  which  groove  to the next. The steps:  fabrication  o x i d e growth,  sputtering, and  V-groove  o f the V-grooves electrode  consists  fabrication,  window f a b r i c a t i o n ,  of 6 basic  oxide  V-groove  etching,  oxide removal. In order t o get repeatable r e s u l t s  start  w i t h a c l e a n wafer.  p r e c e d e d by process  i t i s necessary to  Therefore the f a b r i c a t i o n  c l e a n i n g the wafer u s i n g the w e l l  known  was RCA  [62] .  A t h e r m a l o x i d e was Primarily  grown on t h e w a f e r  f o r two  i t i s n o t a t t a c k e d by t h e e t c h a n t u s e d t o  the V-grooves,  therefore  i t c a n a c t as t h e V - g r o o v e  reasons. create mask  f i x i n g t h e w i d t h o f t h e windows and t h e r e b y t h e d e p t h o f t h e V-grooves,  and  secondarily  electrodes  electrically  grown t o a t h i c k n e s s  i t serves to i s o l a t e  from t h e s u b s t r a t e .  o f 500  h e a t e d t o 1100°C w i t h an 0  2  nm.  To do t h i s  f l o w o f 1 1/min.  the  The  oxide  an o v e n The  was  slices  was  -97-  were i n t r o d u c e d  t o t h e oven and a l l o w e d  min.  A flow  "wet"  atmosphere.  o f 1.6 1/min. o f H2 was s t a r t e d g i v i n g a n O2+H2 Wet g r o w t h i s a p p r o x i m a t e l y  magnitude f a s t e r t h a t d r y growth min.  theH  2  2  There f o l l o w e d  lift-off  [63].  a f t e r t h a t t h e O2 a 20 m i n f l o w  technique  t h a t was i d e n t i c a l  ( s e c t i o n 3.2.1).  t h a t t h e e l e c t r o d e s were f o r m e d a t t h i s  lift-off A  protect  procedure  l a y e r was d e p o s i t e d was a b o u t  making  at this point t o  i n t h e Perkins-Elmer  This  s p u t t e r i n g system  700 nm t h i c k .  a p p l i e d t o cover  those  i n t h e Si02-  areas  b u f f e r e d HF f o r 20 min. HF e t c h was i n t e n d e d  e t c h t h e grown o x i d e .  The S i 0 2 was t h e n  f o l l o w e d b y a r i n s e i n DI  etched water.  t o be a b i t t o o s h o r t t o c o m p l e t e l y  T h i s was t o a v o i d o v e r e t c h i n g  i n widening t h e V-grooves.  r e m o v e d b y a 20 m i n .  P h o t o r e s i s t was  t h a t were n o t t o b e e t c h e d  ( i n c l u d i n g t h e back o f t h e wafer).  would r e s u l t  regions  from t h e V-groove e t c h a n t .  Windows were e t c h e d  The  s t a g e was  uncertain.  s e c o n d l a y e r o f S i 0 2 was d e p o s i t e d the electrodes  The  the application  o f p h o t o r e s i s t was u n e v e n i n t h e i n t e r g r o o v e  in  o f N2  t o t h a t used t o form  b e c a u s e i f t h e V - g r o o v e s were f o r m e d f i r s t  and  A f t e r 80  e l e c t r o d e s o n t h e V - g r o o v e s were f o r m e d u s i n g a  e l e c t r o d e s o f t h e p l a n a r VIOWM  reason  the  an o r d e r o f  1 1/min. The  the  ( 0 only)  f l o w was s t o p p e d a n d 5 min.  f l o w was a l s o s t o p p e d . at  t o h e a t up f o r 5  The f i n a l  which  l a y e r was  C F 4 plasma e t c h performed as d e s c r i b e d  -98-  in  s e c t i o n 3.2.2  f o r the T i etch.  Finally  the photoresist  vr?>.s r e m o v e d . The V - g r o o v e s were e t c h e d weight),  DI w a t e r  i n a s o l u t i o n o f KOH  (64%) a n d 2 - p r o p a n o l  was  h e a t e d t o 85°C i n a b e a k e r w i t h  was  immersed  blown  f o r 85  dry with The  T h i s was with  The s o l u t i o n  a condenser.  The w a f e r was  The  t h e n removed  wafer and  was  layer protecting the electrodes  was  N . 2  sputtered  t h e n removed  min.  (16%).  (20% b y  oxide  by e t c h i n g  f o r about 5 min.  i n buffered  f o l l o w e d b y a r i n s e i n DI w a t e r a n d b l o w  HF.  drying  N . 2  Figure  3.10  d e p i c t s the e n t i r e V-groove  fabrication  process. Since  the device  was  t o a c t as a f r o n t - e n d  w h i c h i t i s p l a c e d between c o n t r o l l i n g the l i g h t the  coupled  application of voltage  were d e s i g n e d  discussed  c o u p l i n g between i n s e c t i o n 2.6.  theoretical  development  in  fiber,  from t h e l a s e r t o t h e f i b e r  by  t o i t s e l e c t r o d e s , the V-grooves slightly  overhang  t h e V - g r o o v e on t h e o t h e r  Device/Optical  The  a n d an o p t i c a l  s o t h a t t h e VIOWM w o u l d  end o f t h e S i wafer w i t h  3.4  a laser  switch,  Fiber  side.  Alignment  a VIOWM a n d an o p t i c a l I t was  one  pointed  fiber  out t h a t  of the coupling c o e f f i c i e n t  the was  was  -99-  (a) Prepare Clean Si Vafer  (e) Oxide Patterning and Etch  (b) Thermal Oxide Growth  (f) V-groove Etch  (c) Metal Deposition and Patterning  (g) Protective Oxide Removal Legend: Si  (d) Sputtered Oxide Deposition  Figure  3.10: The V - g r o o v e f a b r i c a t i o n  2  Metal  process.  -100-  i n s e n s i t i v e t o whether l i g h t the  fiber  o r from  the  was c o u p l e d  from  f i b e r t o t h e VIOWM.  Therefore  possible t o align the device with the l i g h t the  reverse direction  operation  i . e . with  t h e VIOWM t o i t was  propagating i n  from t h a t i n t e n d e d f o r t h e a c t u a l  light  from t h e  fiber  coupled  into the  VIOWM. First  a length o f optical  removing t h e p l a s t i c c l e a v i n g them.  f i b e r was p r e p a r e d b y  c o a t i n g from b o t h  The o u t p u t  ends a n d t h e n  o f a l a s e r was t h e n  f o c u s e d on  one  o f t h e c l e a v e d end faces while monitoring t h e output a t  the  other.  tight,  C l a d d i n g modes were r e m o v e d b y a s e r i e s o f  about  1 i n c h i n diameter,  loops i n t h e f i b e r .  In the  c a s e t h a t t h e f i b e r was m u l t i m o d e , h a v i n g b e e n d e s i g n e d monomode o p e r a t i o n a t a l o n g e r w a v e l e n g t h , t h e s e served t o s t r i p The  t h e higher order guided  loops  for also  modes.  back, o f t h e s a m p l e VIOWM was c e m e n t e d t o a p i n  w h i c h was a t t a c h e d t o t h e boom o f a c r a n e mounted o n a two axis piezoelectric micropositioner. down w i t h The  The s a m p l e hung  upside  i t s electrodes facing the floor. S i w a f e r c o n t a i n i n g t h e V - g r o o v e s was m o u n t e d  right  s i d e up, s o 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 h e c e i l i n g , on a stage t h a t c o u l d be heated AuGe.  t o t h e e u t e c t i c temperature  of  T h e s u r f a c e s o f t h e w a f e r a n d t h e VIOWM were a l i g n e d  by p l a c i n g t h e two i n c o n t a c t a n d h e a t i n g t h e s t a g e  just  e n o u g h t o s o f t e n t h e cement h o l d i n g t h e VIOWM t o t h e p i n .  -101-  The  output  end o f the  optical  f i b e r was l a i d  g r o o v e a n d a s m a l l w e i g h t was p l a c e d  i n a V-  atop i t h o l d i ti n  place.  T h e s a m p l e VIOWM was p o s i t i o n e d s o a s t o b e i n  contact  with  the  t h e end o f the  fiber.  e l e c t r o d e s on t h e w a f e r c o n t a i n i n g t h e V - g r o o v e s and t h e  s a m p l e ' s p o s i t i o n was a d j u s t e d best  coupling  necessary  until  i t was f e l t  T h i s p r o c e d u r e was r e p e a t e d  groove u n t i l  o f AuGe, 363°C.  t h e stage  During  a permanent  bond  t o t h e e u t e c t i c temperature  theheating  c y c l e t h e cement b e t w e e n  t h e VIOWM a n d t h e p i n was o x i d i z e d a n d t h e a f t e r t h e stage  following  c o u p l i n g was f o u n d .  c o u p l i n g was o b t a i n e d  formed by h e a t i n g  I f i t was  from groove t o  t h e groove g i v i n g t h ebest  Once t h e b e s t  separated  that the  f o r t h a t groove had been obtained.  t h e s u r f a c e s were r e a l i g n e d b y h e a t i n g  rotations.  was  A s i g n a l was a p p l i e d t o  had cooled.  two were  easily  T h e b o n d 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 droplets o f cyano-acrylate glue  t o both sides o f the  crystal.  bonded i n p l a c e by a p p l y i n g fiber  The f i b e r was a l s o  a d r o p o f t h e same g l u e  a t t h e f a r e n d o f t h e V - g r o o v e away f r o m t h e  on t h e butt  couple. Figure and  3.11 shows t h e V - g r o o v e a r r a y ,  probes during t h e alignment Figure  bonding  VIOWM,  fiber,  VIOWM,  o b j e c t i v e a f t e r t h e alignment  procedure.  pin  procedure.  3.12 shows t h e V - g r o o v e a r r a y ,  probes and input  fiber,  and permanent  Figure probes  3.11: The V - g r o o v e a r r a y , f i b e r , during the alignment procedure.  VIOWM,  pin,  and  -103-  F i g u r e 3.12: The V - g r o o v e a r r a y , f i b e r , VIOWM, p r o b e s , a n d 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 b o n d i n g procedure.  -104-  The mm  on  a  c r y s t a l s shown i n f i g u r e s side.  3.11  and  3.12  are  about  1  -105-  Chapter  4  RESULTS  4.1  Introduction  The  results,  measured, and 4.3.  both  calculated  from  t h e model and  f o r t h e p l a n a r VIOWM a r e p r e s e n t e d  the results  i n s e c t i o n 4.2  f o r t h e r i d g e VIOWM a r e p r e s e n t e d  S e c t i o n 4.4 c o n t a i n s t h e r e s u l t s  i n section  o f m e a s u r e m e n t s on  t h e VIOWM p e r m a n e n t l y mounted o n t h e S i V - g r o o v e s w i t h a n optical  fiber  discussion  attached.  fibers  circularly the nominal  section  of thevarious results  To p r e d i c t optical  Finally  t h e behavior  4.5 p r o v i d e s a  obtained.  o f t h e VIOWM d e v i c e  i t was assumed t h a t t h e f i b e r s  symmetric width  optical  field  have  distributions  p a r a m e t e r s were w f = wzf v  with  and t h a t  = 1.5 nm.*  T h i s was c l o s e t o the value given t o us by t h e engineers at MacDonald D e t t w i l e r and A s s o c . f o r 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 r e c o r d e r . They measured the h a l f width at 1/e o f t h e peak power t o be - 2 nm.  -106-  Other  f i b e r s could,  o f course,  s u p p o r t modes w i t h d i f f e r e n t  mode w i d t h s . The  numerical  techniques Pascal  i n t e g r a t i o n s were done u s i n g  f o r which t h e a c c u r a c y c o u l d be s p e c i f i e d .  was u s e d on an IBM AT c o m p a t i b l e w i t h an 80287  coprocessor.  The i n t e g r a t i o n r o u t i n e s  Turbo P a s c a l  N u m e r i c a l Methods T o o l b o x .  s p e c i f i e d t o be t o e i g h t of the routines and of the  adaptive  significant  were t a k e n  The  were t e s t e d on H e r m i t e - G a u s s i a n  f o u n d t o be w i t h i n  the s p e c i f i e d tolerance.  math  from t h e  The a c c u r a c y  figures.  Turbo  was  accuracy  functions The  limits  i n t e g r a t i o n were s e t t o 5 t i m e s t h e w i d t h p a r a m e t e r o f trial  function  ensured that the  i n b o t h t h e y and z - d i r e c t i o n s .  t h e e r r o r i n c u r r e d by n e g l e c t i n g  i n t e g r a l t h a t was  outside  of the limits  This  that part  of  of integration  w o u l d be l e s s t h a n t h e t o l e r a n c e .  4 .2  The  The  Planar  VIOWM  c a l c u l a t e d r e s u l t s from t h e model f o r t h e p l a n a r  VIOWM a r e p r e s e n t e d i n s e c t i o n 4.2.1 results  are presented  i n section  and t h e measured  4.2.2.  4.2.1  Calculated Results Figures  4.1  and  4.2  w i d t h p a r a m e t e r s , Wy-^ voltage  and  are  and  width parameters  indicate that voltage,  the  Figures  index region gap  as  the  and  this  as  The optical  of the  shorter  fact  that  w i d t h and  one  applies  an  of the  the  change i n t h e  are  plots  These r e s u l t s increasing  decreasing  of  to the  high  the to  for smaller  gaps  result in  higher  wavelengths.  i s t o be  will  expected. to the  As  the  change i n  application  becomes l a r g e r t h e  turning  i n t e r e l e c t r o d e gap.  point This  i s proportional  r e f r a c t i v e index i s d i r e c t l y  increase  This  is to  and  proportional  argument i s e q u a l l y  i n confinement with  the  of  r e f r a c t i v e i n d e x d i s t r i b u t i o n [19]  applied voltage.  a p p l i e d to the  light  increasing voltage  change i n d i r e c t i o n o f a r a y  gradient  applied  field distribution  e f f e c t i s greater  electrodes  the  4.4  confined  vicinity  mode moves c l o s e r t o t h e  because the  to the  i n the  i n d e x d i s t r i b u t i o n due  to the  and  with  optical  increasing voltage  confinement  refractive voltage  at  4.3  f o r 633nm r a d i a t i o n .  gap  the  w i d t h f o r a VIOWM f o r  words t h e  interelectrode  well  of the  width parameters decrease with  In o t h e r  electrodes  p l o t s of  functions  a g u i d e d mode becomes more h i g h l y  refractive  as  as  zv  nm.  with decreasing  wavelength. of  w ,  i n t e r e l e c t r o d e gap  w i t h a w a v e l e n g t h o f 442 of the  topographical  well  decreasing  -108-  Con.tours  Figure  4.1:  of  A topographical  constant  plot  o f wyv  for X  Q  wy v  = 442  nm.  -109-  Contours  2  of  3 Gap  Figure  4.2:  constant  A topographical  4  5  Width  plot  of w  w  6  (/bin)  z v  f o r XQ  = 442  nm.  -110-  Con t our s  of  Gap  Figure  4.3:  constant  Width  A topographical plot  o f wyv  wy  (yun )  f o r Ji  Q  = 633  nm.  v  -Ill-  Contours  of  constant  wZ  50 CD  CP D  o >  40  30 —  CD  Q_ CL <  6  20  • 10•1214-  • 10 • 12 14-  10 2  3 Gap  Figure  4.4:  6 Width  A topographical plot  of w  (/un)  z  v  for X  Q  =  633 nm.  V  -112-  interelectrode substrate the  gap w i d t h f o r t h e e l e c t r i c  i s inversely  proportional  field  t o t h e gap w i d t h .  t o t h e cube o f t h e r e f r a c t i v e i n d e x o f t h e  s u b s t r a t e w h i c h i n LiNbC-3 t e n d s t o i n c r e a s e a p p r o a c h e s t h e a b s o r p t i o n edge The  location  coefficient. location  Here,  factor  application  d e p e n d on t h e  and t h e range o f a p p l i e d  voltages.  for the fiber i nthe z-direction  a l w a y s b e a t b = 0.  of the  i n determining the coupling  i n any p r a c t i c a l s w i t c h i n g  intended application  However, t h e l o c a t i o n  The  will  of the fiber i n  y - d i r e c t i o n must b e d e t e r m i n e d b y t h e a p p l i c a t i o n . In  is  [64].  of the optical fiber will  optimum p o s i t i o n  a s one  o f t h e f i b e r r e l a t i v e t o t h e input  VIOWM i s a n i m p o r t a n t  the  Also  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  proportional  the  i n the  t h e case that  as a d i g i t a l  optical  the intended application  s w i t c h s i t u a t e d b e t w e e n two s i n g l e - m o d e  f i b e r s o n e m i g h t want t o e n s u r e t h a t  amount o f p o w e r i s t r a n s f e r r e d output  from t h e i n p u t  fiber for a particular applied  want t o e n s u r e t h a t slight  variations  situations  o f t h e VIOWM  t h e maximum fiber to the  v o l t a g e o r one m i g h t  t h e o u t p u t power i s i n s e n s i t i v e t o  i nthe applied  voltage.  Both o f these  can be achieved by j u d i c i o u s l y choosing t h e f i b e r  location. In  figures  coefficient figures  4.5 a n d 4.6 a r e p l o t t e d  T as a f u n c t i o n  of applied  i t h a s b e e n assumed t h a t  the coupling voltage.  F o r these  the optical fibers are  -113-  located  s o a s t o m a x i m i z e t h e power t r a n s f e r f o r a p p l i e d  voltages  o f 30 V a n d 50 V r e s p e c t i v e l y .  In t h i s  case t h e  VIOWM i s a s s u m e d t o h a v e a 2 nm i n t e r e l e c t r o d e g a p a n d t h e wavelength o f t h e l i g h t apparent  from both  i s t a k e n t o b e 442 nm.  figures that  degree o f c o u p l i n g  will  occur a t voltages  4.5,  o r 50 V, i n f i g u r e 4.6, however t h e c o u p l i n g  at these voltages these  greater  a greater  It i s  t h a n e i t h e r 30 V, i n f i g u r e  are the greatest  achieved  t h a t may b e a c h i e v e d f o r  voltages. In  f i g u r e s 4.7 a n d 4.8 i s p l o t t e d t h e c o u p l i n g  coefficient fibers  T as a function o f voltage  positioned  obtained  so t h a t  for the optical  t h e maximum c o u p l i n g  that  c a n be  f o r 30 V a n d 50 V 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 an  i n t e r e l e c t r o d e g a p w i d t h o f 4 nm a n d 442 \xm l i g h t . In occurs the  f i g u r e 4.5 t h e a c t u a l p e a k v a l u e a t - 37 V r a t h e r t h a n 30 V.  device  would vary  by only  change i n t h e a p p l i e d v o l t a g e . f o rapplied voltages  respectively coupling for  indicates that i f  w e r e o p e r a t e d a t 37 V t h e n t h e c o u p l i n g  coefficient  show t h a t  This  of the coupling  the coupling  coefficient  a few p e r c e n t Also  f o ra larger  f i g u r e s 4.5 a n d 4.7  o f - 3 7 V a n d 43 V  coefficient  remains w i t h i n  T i s stationary.  5% o f i t s maximum  The value  c h a n g e s e x c e e d i n g ±7 v o l t s i n f i g u r e 4.5 a n d f o r e v e n  greater  v a r i a t i o n s i n f i g u r e 4.7.  The rendering  e f f e c t described  above w o u l d a l s o b e u s e f u l i n  t h e VIOWM l e s s s e n s i t i v e t o c h a n g e s 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 |im where t h e c o u p l i n g a t 30 V i s m a x i m i z e d .  for g  -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 jim where t h e c o u p l i n g a t 50 V i s m a x i m i z e d .  for g =  2  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 v s . v o l t a g e |im where t h e c o u p l i n g a t 30 V i s m a x i m i z e d .  for g  -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 um where t h e c o u p l i n g a t 50 V i s m a x i m i z e d .  for g =  4  -118-  refractive  index p r o f i l e brought  photorefractive In  about by t h e  effect.  f i g u r e 4.9 a r e p l o t t e d t h e optimum f r a c t i o n o f power  transfer that  c a n be o b t a i n e d  f o r a planar  w i t h a w a v e l e n g t h o f 442 nm a s a f u n c t i o n and  i n t e r e l e c t r o d e gap w i d t h .  transfer it  by T  i s given  .  i spossible t o obtain  optical  fiber  that  VIOWM f o r l i g h t o f both  The f r a c t i o n o f power  From t h i s  f i g u r e one c a n s e e t h a t  -3dB power t r a n s f e r b e t w e e n a n  s u p p o r t s a mode w i t h w i d t h p a r a m e t e r s W y f  = w y = 1.5 |im a n d a VIOWM w i t h a n a p p l i e d v o l t a g e Z  20 V.  Extrapolating  shows t h a t  i n t e r e l e c t r o d e gap w i d t h s i t would be  e v e n b e t t e r power t r a n s f e r .  layer thickness  o f 5% o f t h e  i n t e r e l e c t r o d e gap w i d t h would be l e s s than would n o t r e s u l t i n s u f f i c i e n t field  o f t h e guided Finally  attenuation  input  See  o f t h e evanescent  t h e u s e o f t h e VIOWM a s a s m a l l  fiber,  fiber,  chapter  P i  Pout' w i l l n  ,  fashion  linear I f the  t h e n t h e power  be' r e l a t e d t o t h e power i n  by  2 section  signal  f i b e r s i s considered.  power i s t o b e m o d u l a t e d i n a l i n e a r i n t h e output  1000 A w h i c h  modes.  m o d u l a t o r b e t w e e n two o p t i c a l  *  Calculations  i n t e r e l e c t r o d e g a p w i d t h s l e s s t h a n 2 yaa were n o t  performed because a b u f f e r  the  o f only  t h e c u r v e s p r e s e n t e d i n f i g u r e 4.9  a t smaller  possible t o obtain for  voltage  2.3.1.2.  Contours  of  constant  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 a p p l i e d v o l t a g e a n d i n t e r e l e c t r o d e gap w i d t h .  o  -120-  P  out  = TT T  .P.  xn out xn  where T i  i sthe fraction  2  the  n  input  transfer fibers  and s i m i l a r l y  T  o f power t r a n s f e r t o t h e VIOWM a t  2 o  u  t  i sthe fraction  f r o m t h e VIOWM a t t h e o u t p u t .  could,  o f course,  o f power  The i n p u t  and output  be l o c a t e d i n d e p e n d e n t l y .  For  example one c o u l d be l o c a t e d so t h a t t h e v a r i a t i o n i n t h e power t r a n s f e r was m i n 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 a n d the  other  c o u l d be l o c a t e d a t t h e p o i n t  f o r t h e same v o l t a g e .  o f steepest  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 t h e f i b e r s t o h a v e t h e same amount transfer  a t t h e same v o l t a g e  .  out  power  becomes  - T P. 4  in •  where T i s t h e c o u p l i n g the  o f power  i n which case t h e r e l a t i o n  b e t w e e n t h e o u t p u t power a n d t h e i n p u t  p  descent  output.  coefficient  I n f i g u r e s 4.10 a n d 4.11 a r e p l o t t e d T  cases i n which both o p t i c a l achieve  at e i t h e r the input or  t h e optimum c o u p l i n g  fibers  4  for the  a r e l o c a t e d so as t o  coefficient  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 g a p f o r a w a v e l e n g t h o f 442 nm when 30 V and  50 V a p p l i e d t o t h e e l e c t r o d e s  seen both f i g u r e s i n d i c a t e t h a t the  t r a n s f e r curves.  respectively.  there  i sa linear  I n f i g u r e 4.10 t h e l i n e a r  a p p e a r s t o e x t e n d from an a p p l i e d v o l t a g e V i n which t h e f r a c t i o n 0.25 t o ~ 0.55.  In t h i s  As c a n be region i n  region  o f - 22 V t o - 26  o f t h e power t r a n s f e r r e d g o e s f r o m case t h e appropriate  bias  voltage  -121-  w o u l d be figure and  4.11  the  0.6.  - 24 V w i t h a d y n a m i c r a n g e o f 4 V. the  linear  fraction  H e r e an  objective  coupled  an  (figure  optical  a p e r t u r e and into the  angles.  The  The  end  light  was  a polarizer,  The  polarized  i n two  a spool.  c l a d d i n g modes t o be was  put  f i b e r was  not  fiber with  The  percent  K  0  with  = 442  i n the  fiber  nm,  of the  polarization  state  4Ox right  loop,  about  the  This gives  lowest  order  (figure  4.13).  The  over time  f i b e r was  axis micropositioner with rotation  by  only  mounted on  i n the v e r t i c a l  of  output  a p o l a r i z e r p l a c e d between t h e o u t p u t to vary  HE^  at the output  known; however, upon m o n i t o r i n g t h e  seen  was  long  just before  l o o s e l y g u i d e d modes.  40x  lOx  at  I t was A  a  chopper,  planes  stripped.  spot c o n s i s t i n g p r i m a r i l y fiber.  light,  a  a  c l e a v e d ends, u s i n g t h e  w r a p p e d on  a d e t e c t o r t h e power was few  - 30 V  t h e VIOWM u n d e r t e s t ,  with  t o remove any  mode o f t h e  of the  to ~  consisted of a laser,  a detector.  fiber,  1 i n c h i n diameter,  an o u t p u t  4.12)  fiber,  f i b e r was  e n o u g h f o r any  the  - 0.2  V  V.  ( a c t i n g as a p r o j e c t o r ) ,  objective.  output  - 25 V t o - 35  a p p r o p r i a t e b i a s v o l t a g e w o u l d be  apparatus  objective,  an  in  Measured Results  The  and  from  o f power t r a n s f e r r e d g o e s f r o m  a d y n a m i c r a n g e o f 10  4.2.2  range extends  Similarly  and a  a three  and  -122-  Applied  V o I t age  F i g u r e 4.10: The power t r a n s f e r T b e t w e e n 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 m a x i m i z e d .  -123-  F i g u r e 4.11: The power t r a n s f e r T b e t w e e n 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 m a x i m i z e d .  -124-  horizontal planes  and t h u s c o u l d be a l i g n e d w i t h  f a c e o f t h e VIOWM u n d e r t e s t . was r i g i d .  The o u t p u t  axis micropositioner. output  aperture  The p o l a r i z e r was p l a c e d b e t w e e n t h e  a t t h e output  a s was t h e c h o p p e r .  o f a VIOWM i s f o c u s e d  t h e power t r a n s f e r r e d t o t h e d e t e c t o r w i l l  t h e shape and s i z e o f t h e a p e r t u r e  of the projected o p t i c a l If the aperture the  optical  rect(r/2c) function otherwise  h o l d i n g t h e VIOWM  o b j e c t i v e was a l s o mounted o n a t h r e e  o b j e c t i v e and t h e aperture  When l i g h t  on  The s t a g e  the input  field  distribution  function.  Since  depend  and t h e shape and s i z e  distribution  a t t h e aperture.  i s c i r c u l a r then the overlap  field  upon an  o f t h e guided  the transmission  i s 1 i n the region o f the t o t a l  i sthat  between  mode a n d t h e of the rect  transmission  and 0  i t seffect  i st o setlimits  p r o j e c t e d image t h a t  i sinterrogated.  I f the aperture i s  l a r g e enough i t w i l l  pass t h e majority  o f t h e power  projected  image o f t h e o p t i c a l  sufficiently  high voltages  m e a s u r e m e n t s may r e a s o n a b l y included  as i tblocks  which would otherwise some o f t h e power  field  on t h e r e g i o n o f t h e  distribution for  and i t s e f f e c t be i g n o r e d .  much o f t h e power  on t h e I t i s , however,  i n the bulk  be c o l l e c t e d by t h e d e t e c t o r .  i n the bulk  i n the  modes i s c o l l e c t e d  modes Still  resulting  A r e c t f u n c t i o n i s a two d i m e n s i o n a l s t e p 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 t h e r a d i u s o f t h e a p e r t u r e . See C o l l i e r , Burckhardt, a n d 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 a s t h e c i r c ( r ) f u n c t i o n b y Goodman [66] p . 14.  -125-  40x  /  Mlcroposi  Ob j e c i i v e  tloners  F i g u r e 4.12: The b a s i c l a b o r a t o r y m e a s u r e m e n t s on VIOWMs.  a p p a r a t u s u s e d t o make  -126-  light level  10  o  to u to  c dark level 7  Figure  4.13:  The  polarized  ninutes  output  of the  optical  fiber.  -127-  in  n o n z e r o c o u p l i n g when z e r o v o l t s  are applied 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 L e C r o y F i b e r c o m  Receiver  FAR-4HS.  I t s g a i n was a d j u s t e d  Analog  so t h a t  i n p u t power c o r r e s p o n d e d t o 1 V a t t h e o u t p u t .  100 nW o f Before  taking  a n y m e a s u r e m e n t s t h e f i b e r was p o s i t i o n e d s o t h a t i t s  output  spot  focused and This  passed though t h e bulk  on t h e d e t e c t o r  chopper.  after passing  The peak v a l u e  "straight  a n d was  through the p o l a r i z e r  o f t h e power was t h e n  recorded.  t h r o u g h " power c o u l d t h e n b e u s e d t o  calculate the coupling efficiency All  of the crystal  t o t h e waveguide.  o f t h e t e s t s were p e r f o r m e d on VIOWMs w i t h  a 4fim  i n t e r e l e c t r o d e gap. In with  f i g u r e s 4.14, 4.15, a n d 4.16 a 500Hz t r i a n g l e  zero  offset  VIOWM, n o m i n a l l y  voltage,  was a p p l i e d t o t h e e l e c t r o d e s  1 mm l o n g .  applied t o the electrodes  wave,  t h e peak-to-peak  of a  voltages  a r e 70, 100, a n d 130 v o l t s  respectively. Figure curves  4.14 d e m o n s t r a t e s t h e s h a p e o f t h e t r a n s f e r  a t low v o l t a g e s  a negative  voltage  as w e l l as t h e advantages o f a p p l y i n g  t o t h e VIOWM i n t h e o f f s t a t e .  In t h i s  c a s e -35V n e e d e d t o b e a p p l i e d t o r e d u c e t h e o p t i c a l t h r o u g h p u t t o a minimum. signal  The n e g a t i v e  i s u s e f u l f o r two r e a s o n s ;  first  "anti-waveguide" and second i t helps changes i n t h e r e f r a c t i v e  index  portion of the input i t a c t s t o c r e a t e an  t h e VIOWM r e c o v e r  from  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: T h e 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 g a p f o r a 70 V p e a k - t o - p e a k t r i a n g l e applied to the electrodes.  wave  -129-  Figure  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 p e a k - t o - p e a k t r i a n g l e wave applied to the electrodes.  -130-  F i g u r e 4.16: The output o f a VIOWM with a 4 \xm 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 wav a p p l i e d t o the e l e c t r o d e s .  -131-  due  to  the  photorefractive  w a v e g u i d e i s meant a low which l i g h t creating  a dark spot of the  exactly  application r a t i o was  i n the  where t h e  of the  interelectrode  output  the  term  of  steered  region.  Upon  Thus t h e  the  was  located  mechanism f o r t h e  anti-  out  dark spot  s p o t was  p o s i t i v e voltage. The  By  b u l k modes i s  negative v o l t a g e the  enhanced.  refractive  [67].  r e f r a c t i v e index region  propagating i n the  application located  effect  upon  the  extinction  change i n  the  i n d e x 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. Figure  4.15  v o l t a g e does, as  predicted  demonstrates that  in fact, i n the  demonstrates that a  reduction  coupling This the  theory,  increasing  i n the  coupling  e f f i c i e n c y h e r e was  output  power t r a n s f e r  the  voltage  efficiency.  further The  d e t e r m i n e d t o be 10  times the  applied  divided  this  way  attenuations  and  by  the  the  maximum about  -4dB.  logarithm  o u t p u t power w o u l d be  scattered  by  interactions  w o u l d be  s m a l l b a s e d on  the  However t h e s e  measurements  f o r d i f f u s e d waveguides which a l s o [68].  In  output the  this interface  of  i t s value  same w i t h  LiNb03/SiC>2 i n t e r f a c e .  at  cause  the  w i t h the  scattering  saturate,  will  r e f l e c t i o n s f o r both the  interactions  dB/cm l o s s  to  s t r a i g h t t h r o u g h power.  s t r a i g h t t h r o u g h p o w e r s w o u l d be  exception that  applied  4.16  o p t i c a l power f r o m i t s p e a k v a l u e t o  0 V  a b o u t -1  the  furthermore f i g u r e  c a l c u l a t i o n i s o b t a i n e d as  with  and  cause the  increasing  of have  -132-  f i g u r e 4.17 t h e t h e o r e t i c a l  In and is the  t h e measured data for a planar coupling  coupling coefficient T  a r e compared.  device  optimized  having  The t h e o r e t i c a l  curve  a 4 um i n t e r e l e c t r o d e g a p w i t h  f o r 50.0 V.  The measured d a t a i s  from f i g u r e 4 . 1 6 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 a n d 50.0 V.  factor  of this  to device  order  misalignment.  measured curves slight due  i s t o b e e x p e c t e d due t o a s l i g h t  t o bulk  fiber  The shapes o f t h e t h e o r e t i c a l a n d  a r e i n good agreement e x c e p t t h a t t h e r e  discrepancy mode  A  i sa  a t low v o l t a g e which i s b e l i e v e d t o be  coupling.  O t h e r e x p e r i m e n t s were p e r f o r m e d u s i n g t h e a p p a r a t u s shown i n f i g u r e 4 . 1 8 . end-fire  coupled  objective  In t h i s  a 4Ox m i c r o s c o p e  i n t o t h e VIOWM u s i n g  and t h e output  Alphametrics  c a s e p o l a r i z e d l i g h t was  was p r o j e c t e d o n t o a d e t e c t o r (an  model d c l O l O  using  head), through an aperture,  a m o d e l 1110s w i d e  as before.  The w a v e l e n g t h o f  was 633 nm a n d t h e VIOWM h a d a 4 um i n t e r e l e c t r o d e  the  light  gap  but this Figure  t i m e i t was a b o u t a n i n c h 4 . 1 9 shows t h e o u t p u t  t o - p e a k t r i a n g l e wave w i t h to the electrodes.  zero  The t o t a l  long.  when a 0.5 Hz 70 V p e a k -  offset  output  v o l t a g e was a p p l i e d  power was l o w (21 aW  peak) t o r e d u c e t h e p h o t o r e f r a c t i v e e f f e c t . t h e VIOWM e n s u r e d t h a t t h e b u l k small. occurred  spectrum  The i n p u t with  The l e n g t h o f  mode c o u p l i n g w o u l d b e  c o u p l i n g was s u c h t h a t t h e p e a k  about  30 V a p p l i e d .  This  output  f i g u r e demonstrates,  -133-  — Theory © Measured 1.0 -r 0.8 -2  T  0.6 -0.4 -0.2 + 0.0  4^0.0  10.0  20.0  Applied  30.0  40.0  50.0  Voltage  F i g u r e 4.17: A c o m p a r i s o n o f t h e t h e o r e t i c a l results f o r a planar device.  and  measured  -134-  VIOV/M 40x  Ob j e c t i v e  lOx  Objective  A p e r t ur e  Choppe r  •  Loser  \ D e i e c t or  P o Io r i z e r Microposiiioners  F i g u r e 4.18:  An  alternate  laboratory  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 coupling, VIOWM.  the voltage  and b u l k  c o n t r o l l e d coupling behavior  The e x t i n c t i o n r a t i o was 23dB.  A t even  o p t i c a l power l e v e l s l i n e a r m o d u l a t i o n o f t h i s demonstrated by a p p l y i n g  4.3  The R i d g e  The  of the  lower device  was  a 15 V b i a s t o t h e e l e c t r o d e s .  VIOWM  c a l c u l a t e d r e s u l t s from t h e model f o r t h e p l a n a r  VIOWM a r e p r e s e n t e d  i n s e c t i o n 4.3.1 a n d t h e m e a s u r e d  results  are presented  4.3.1  C a l c u l a t e d Results  The  mode  effect  i n s e c t i o n 4.3.2.  o f ridge height  on t h e o p e r a t i o n  of the  VIOWM a s r e g a r d s t h e c o n f i n e m e n t o f t h e g u i d e d mode a s a function section. height the  o f r i d g e width and v o l t a g e  I t i s important t o understand t h e e f f e c t  on t h e o p e r a t i o n  coupling  voltage,  i s studied i n this  will  o f VIOWMs.  t o an o p t i c a l  fiber,  Itwill  of ridge  b e shown t h a t  at a p a r t i c u l a r operating  d e p e n d on t h i s p a r a m e t e r a s w i l l  t h e "turn-on  voltage. Here t h e t u r n - o n v o l t a g e  refers t o the voltage  w h i c h t h e mode becomes d e t a c h e d f r o m t h e r i d g e . below t h e turn-on v o l t a g e  below  At voltage  t h e w i d t h p a r a m e t e r s o f t h e mode  -136-  Input Signal (50V/d i v )  ^- gnd  O u t p u t S i gna ( 2 1/xW p e a k ) Dark Le ve Tine  Base  ( 0 . 5 s / d i v)  F i g u r e 4.19: The o u t p u t o f a l o n g VIOWM w i t h 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 = 663 nm.  f o r XQ  -137-  increase  r a p i d l y however t h e r i d g e VIOWM w i l l  continue t o  act  as a waveguide f o r any p o s i t i v e  can  s e e t h i s b y c o n s i d e r i n g what t h e r e f r a c t i v e  distribution  applied voltage.  o f t h e VIOWM l o o k s l i k e  Using the technique  o f conformal  One  index  f a r away f r o m  t h e gap.  m a p p i n g f o r two c o p l a n a r  electrodes  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  voltage V  a p p l i e d b e t w e e n t o t h e e l e c t o r d e s , one f i n d s  Q  the e l e c t r i c  field parallel  t o t h e Z-axis  s m a l l gap, w i t h a that  i s g i v e n by  V C O S (e) Q  E (y,z) z  - - _ K  p  where p = [ ( e / e ) y + z ] 2  z  From e q u a t i o n  drop  - 1  [ (e /e ) z  1 / 2  y  y/z] .  index  o f f a s 1/r f o r any p o s i t i v e  can ignore t h e contribution  a s o n e moves away f r o m [39]  and 8 = t a n  1 / 2  2.2 one s e e s t h a t t h e r e f r a c t i v e  distribution will (one  2  y  t h e gap.  o f Ey as i t w i l l  be small)  U s i n g e i t h e r t h e WKB m e t h o d  o r t h e w a v e - v e c t o r method o f Hocker a n d Burns  finds that a raytravelling  voltage  along a r a d i a l  [33]  line will  one  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 . * Calculated ridge heights  results  arepresented  0.5, 1.0, a n d 1.5 t i m e s  together with the results analysis  of the effects  Jaeger  f o r r i d g e VIOWMs w i t h t h e gap w i d t h .  These  f o r t h e p l a n a r d e v i c e complete our  o f t h e v a r i a b l e parameters f o r  e t a l . [69] p p . 6-7.  -138-  VIOWMs f o r w h i c h t r e n d s c a n b e p r e d i c t e d operandi  explained.  Figures the  WyV  4.20, 4.21 a n d 4.22 a r e t o p o g r a p h i c a l  w i d t h parameters as f u n c t i o n s  interelectrode 0.5, 442  gap w i d t h f o r r i d g e  of applied  plots of  v o l t a g e and  VIOWMs w i t h h e i g h t s o f  1.0, a n d 1.5 t i m e s t h e g a p w i d t h r e s p e c t i v e l y nm.  plots the  a n d t h e modus  Figures  for thew  f o rX  0  =  4.23, 4.24 a n d 4.25 a r e t h e c o r r e s p o n d i n g width parameters.  z v  The s h a d e d r e g i o n s on  g r a p h s i n d i c a t e where t h e mode i s d e t a c h e d  from t h e  ridge. In  figures  coefficients with gap  4.26 a n d 4.27 a r e p l o t t e d t h e c o u p l i n g  as f u n c t i o n s  of the applied  a 7 um i n t e r e l e c t r o d e width)  that  ridge.  the optical  figures  are located  power t r a n s f e r  f o rapplied  respectively.  Again,  that  degree o f coupling  a greater  greater 4.27, the  than e i t h e r  i t has been  In fraction ridge  that  figures  assumed  v o l t a g e s o f 30 V a n d 50 V  as f o r t h e p l a n a r device, will  one s e e s  occur a t voltages  30 V, i n f i g u r e 4.26, o r 50 V, i n f i g u r e at these voltages i s  may b e a c h i e v e d .  4.28, 4.29 a n d 4.30  o f power t r a n s f e r ,  VIOWMs w i t h h e i g h t s  width respectively, voltage  (0.5 t i m e s  so as t o maximize t h e  however t h e degree o f c o u p l i n g greatest  f o r a VIOWM  gap and a h a l f - h e i g h t  F o r these fibers  voltage  T , that  a r e p l o t t e d t h e optimum c a n be o b t a i n e d f o r  o f 0.5, 1.0, a n d 1.5 t i m e s t h e g a p  f o rX  0  and i n t e r e l e c t r o d e  = 442 nm, a s f u n c t i o n s gap w i d t h .  From f i g u r e  o f both 4.28 one  -139-  Con t o u r s  of  c o n s t an t  w  y v  CD CP  O  O >  "O CD  CL <  Gap  Width  (/xn)  F i g u r e 4.20: A t o p o g r a p h i c a l p l o t o f w f o r a ridge height 0.5 t i m e s t h e i n t e r e l e c t r o d e gap w i d t h where = 442 nm. y v  -140-  Contours of constant w  4  5  Gap  6  7  8  Width  9  (/an)  F i g u r e 4.21: A t o p o g r a p h i c a l p l o t o f W y f o r a ridge height 1.0 times t h e i n t e r e l e c t r o d e gap width where X = 4 42 nm. V  Q  -141-  Gap  Width  (/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 f o r a ridge height 1.5 t i m e s t h e i n t e r e l e c t r o d e gap w i d t h where A, = 442 nm. v v  0  -142-  Con t o u r s  of  constant  wz v  F i g u r e 4.23: A t o p o g r a p h i c a l p l o t o f w f o r a ridge height 0.5 t i m e s t h e i n t e r e l e c t r o d e gap w i d t h where XQ = 442 nm. z v  Contours  of  constant  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 o f w f o r a ridge height 1.0 t i m e s t h e i n t e r e l e c t r o d e gap w i d t h where \0 = 442 nm. zv  -144-  Contours  of  wZ  constant  V  50 CD CP  40 —  00  o o > CD  CL CL  6 <£>  30-  ro O-  I  LQ  20 —  1  JU..-  OH  -  A-  /  1  'V"  < 2 ^  '  0 ^ Pi-  r 5  Gap  W  ' " . '-/V;  wmmmmmm  '  Ms  7  8  Width  (/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 f o r a ridge height 1.5 t i m e s t h e i n t e r e l e c t r o d e gap w i d t h where K = 442 nm. z v  0  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 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 a t 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 v s . 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 t h e c o u p l i n g a t 50 V i s maximized.  -147-  can  see that  i t i spossible t o obtain  between an o p t i c a l  fiber that  -3dB power t r a n s f e r  s u p p o r t s a mode w i t h  width  p a r a m e t e r s w f = w y = 1.50 nm a n d a h a l f - h e i g h t VIOWM v  2  an  applied voltage  one  can see that  V  and from  o f only  - 15 V.  t h e same c o u p l i n g  used as a small  the  fibers.  that  c a n be o b t a i n e d  f o r - 12  t h e r i d g e VIOWM c o u l d b e  s i g n a l l i n e a r m o d u l a t o r l o c a t e d b e t w e e n two I n f i g u r e s 4.31 a n d 4.32 a r e p l o t t e d T^ f o r  cases i n which both o p t i c a l  a c h i e v e t h e optimum c o u p l i n g with  f r o m f i g u r e 4.29  f i g u r e 4.30 f o r - 11.5 V.  A g a i n we h a v e c o n s i d e r e d  optical  Also  with  fibers are located  for a half-height  so as t o  ridge  VIOWM  a 7 nm i n t e r e l e c t r o d e g a p f o r a w a v e l e n g t h o f 442 nm  when 30 V a n d 50 V a p p l i e d t o t h e e l e c t r o d e s  respectively.  Again both  but neither  figure  f i g u r e s show s m a l l  linear  regions  i n d i c a t e s t h e same p o t e n t i a l a s a l i n e a r m o d u l a t o r a s  u o e s f i g u r e 4.11 f o r a p l a n a r  device  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  Figure wave, w i t h  4.33 shows a 500 Hz, 100 V p e a k - t o - p e a k t r i a n g l e zero  o f f s e t voltage,  r i d g e VIOWM w i t h a f i b e r section and  4.2.2.  the device  applied to a half-height  a t the input  The b a s e o f t h e r i d g e was n o m i n a l l y  as d e s c r i b e d i n i s a b o u t 7.5 yon w i d e  1 mm l o n g .  The knee i n t h e  -148-  Contours of constant  Gap  Width  i/ubn)  F i g u r e 4.28: The optimum power t r a n s f e r T^ a s a f u n c t i o n a p p l i e d v o l t a g e a n d 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 t i m e s t h e gap w i d t h .  of 0.5  -149-  Contours  4  of  5  Gap  constant  6  7  8  Width  T  9  (/an)  F i g u r e 4.29: The optimum power t r a n s f e r T as a f u n c t i o n 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 t i m e s t h e gap width. 2  o 1  -150-  Contours of constant T  4  5  Gap  6  7  Width  8  9  (/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 a n d 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 1.5 t i m e s t h e gap w i d t h .  -151-  1.0  -I  0.8 0.6 0.4 0.2 0.0  -J  1  .  1  .  1  0  10  20  30  40  50  Applied  Voltage  F i g u r e 4.31: The power t r a n s f e r T b e t w e e n 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 m a x i m i z e d f o r t h e r i d g e device.  -152-  A p p l i e d  VoIt age  F i g u r e 4.32: The power t r a n s f e r T b e t w e e n 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 m a x i m i z e d f o r t h e r i device. ,  ^  -153-  output  c a n 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  20 V.  This  o f about  e f f e c t i s b e l i e v e d t o b e due t o a r a p i d  change  in  t h e w i d t h p a r a m e t e r s o f t h e f u n d a m e n t a l mode o f t h e VIOWM  as  i t becomes more h i g h l y  confined  t o the region  of the  ridge. In and is  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  t h e m e a s u r e d d a t a a r e compared. f o ra rectangular  half-height  a  optimized  p  f o r 40.0 V.  f a c t o r o f a b o u t 2.2 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  due  to a slight  A factor of this f i b e r t o device  at t h e ridge walls  order  i s t o be expected  misalignment  w h i c h were n o t i d e a l l y  and s c a t t e r i n g  smooth.  m e a s u r e m e n t s show a k n e e a t a b o u t 18.0 V i n g o o d •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 agreement  The shape o f t h e  c u r v e s a b o v e 18.0 V do n o t a g r e e w e l l .  This  i sbelieved t o  due t o t h e d i f f e r e n c e i n t h e shapes o f t h e r e c t a n g u l a r  ridges walls  studied  i n t h e theory  and t h e r i d g e w i t h  slanted  t h a t was f a b r i c a t e d . Figure  4.35 shows a 500 Hz, 120 V p e a k - t o - p e a k t r i a n g l e  wave, w i t h z e r o  o f f s e t voltage,  r i d g e VIOWM a g a i n of t h e ridge mm  2  m e a s u r e d d a t a i s f r o m f i g u r e 4.33 a n d h a s b e e n s c a l e d b y  0.0 a n d 50.0 V .  be  T  The t h e o r e t i c a l c u r v e  ridge having a 7  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 The  coefficient  long.  coupling  with a fiber  applied to a half-height at the input.  i s a b o u t 7.5 nm a n d t h e d e v i c e  From t h i s  f i g u r e one c a n a g a i n  Again t h e base i s nominally  1  see that t h e  r e a c h e s a maximum a n d t h e n d e c r e a s e s .  T h e maximum  -154-  F i g u r e 4 . 3 3 : 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 i n t e r e l e c t r o d e gap f o r a 100 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 where A. = 442 nm. 0  -155-  — Theory © Measured 1.0 0.8  2  T  1 1 1  i i  _  -  0.6  L  l_  L  L  _ /  -  __  jL  L  L  /  L  _  U -w"  & ^ — I  0.0  _  iL  1  10.0  1  /_  1  1  20.0  Applied  F i g u r e 4.34: A c o m p a r i s o n results for a half-height  1  -  1 l_  1  1 1 _ 1 1 1 1 1 1  1  1  30.0  -  !  /  / /  0.2 0.0  _  —  '\  _ L  t.  0.4  rvs  40.0  50.0  Voltage  o f t h e t h e o r e t i c a l and ridge.  measured  -156-  coupling efficiency Again,  h e r e was d e t e r m i n e d t o b e a b o u t  as i n s e c t i o n  4.2.2, t h i s  calculation  -4.5dB.  i s o b t a i n e d as  10 t i m e s t h e l o g a r i t h m o f t h e o u t p u t o p t i c a l power f r o m 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 t h e straight effects  t h r o u g h power. of scattering  as c a n be seen far  T h i s was n o t c o m p e n s a t e d f o r t h e  a t t h e LiNbC>3/Si02 i n t e r f a c e  i n figure  3.2, t h e w a l l s o f t h e w a v e g u i d e a r e  rougher than t h e s u r f a c e and w i l l  probably result i n  much more s c a t t e r i n g t h a n f o r a p l a n a r d e v i c e . believed, input  therefore,  that  i sb e t t e r than t h i s Figure  although,  It i s  the coupling effieciency result  at the  indicates.  4.36 shows a 500Hz, ±20V s q u a r e wave a p p l i e d t o  the electrodes  o f a h a l f - h e i g h t VIOWM.  power i s s e e n t o b e a b o u t seen t o be about  The peak  output  50 pW a n d t h e minimum o u t p u t i s  10 jiW c o r r e s p o n d i n g t o a n e x t i n c t i o n  ratio  o f a b o u t 7dB.  4.4  The Front-End  In t h i s  section  Switch  arepresented the results o f  m e a s u r e m e n t s o n a VIOWM t h a t was p e r m a n e n t l y optical  fiber  i na flip-chip  etched V-grooves. to the electrodes  arrangement  a l i g n e d w i t h an  on an S i w a f e r  with  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 p r o b e s on t h e S i wafer.  Here K  0  = 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 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 w h e r e \ = 442 nm. Q  -158-  F i g u r e 4.36: The output of a h a l f - h e i g h t VIOWM f o r a ± 2 0 V square wave a p p l i e d t o the e l e c t r o d e s .  -159-  The  a p p l i c a t i o n o f a VIOWM a s a d i g i t a l  d e m o n s t r a t e d b y f i g u r e 4.37.  f i g u r e a 5 0 0 H z , ±50V  In t h i s  s q u a r e wave i s a p p l i e d t o t h e e l e c t r o d e s . acting  as a front-end  switch  switch i s  The VIOWM i s  c o n t r o l l i n g t h e amount o f  o p t i c a l power c o u p l e d i n t o t h e f i b e r .  The o t h e r  end o f t h e  f i b e r was i n s e r t e d i n t o t h e L e C r o y F i b e r c o m A n a l o g The  output  signal indicated that  Receiver.  t h e peak-to-peak change i n  o p t i c a l power was ~ 240 piW. It the  was f o u n d t h a t  electrodes  d e c a y and, the  waveguide t o t u r n  effect 4.38  applied to  o f time t h e waveguide would  off.  I t took s e v e r a l  seconds f o r  o f f although t h e e f f e c t s e t i n  [22] r a t h e r t h a n domain r e v e r s a l  [70,71].  Figure  shows a 0.1 Hz, ±50 V s q u a r e wave a p p l i e d t o t h e  electrodes, followed  A s c a n b e s e e n when 50 V i s a p p l i e d t o t h e there  i sa rapid i n i t i a l  r e s p o n s e b y t h e VIOWM  by a slow decay o f t h e throughput.  p o l a r i t y o f the applied voltage  throughput The  turn  voltage  T h i s was p r o b a b l y due t o t h e p h o t o r e f r a c t i v e  electrodes.  the  over a p e r i o d  i neffect,  immediately.  with a constant  i sinitially  "melting"  seen q u i t e  S i m i l a r l y , when  i s reversed the  a minimum a n d i n c r e a s e s  away o f t h e w a v e g u i d e a n d t h e d a r k s p o t  clearly  when t h e o u t p u t  We b e l i e v e t h a t photorefractive  this  i sprojected  long  on a  time. c a n be screen.  e f f e c t i s due t o t h e  e f f e c t rather  t h a n domain r e v e r s a l .  were due t o d o m a i n r e v e r s a l t h e n a f t e r a l o n g voltage,  with  enough f o r t h e waveguide t o t u r n  I fi t  application of completely  -160-  Input S i g n a l (100V/div)  Output  Signal  Time Base (1ms/div)  F i g u r e 4.37: T h e o u t p u t o f a n o p t i c a l f i b e r w i t h a VIOWM a c t i n g a s 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 w a v e . Here t h e s w i t c h e d o p t i c a l p o w e r i s ~ 240 uW.  -161-  Input Signal (100V/div)  Output Signal (40^W/div)  —  -gnd  I  -Dark Level Time Base (1s/div)  F i g u r e 4.38: T h e d e c a y function o f time.  o f t h e output  o f a VIOWM a s a  -162-  off,  we w o u l d h a v e a c h i e v e d t h e r e v e r s a l  of a  sufficient  number o f d o m a i n s s o t h a t t h e r e w o u l d b e v i r t u a l l y electrooptic reversal  effect.  of the f i e l d  electrooptic effect relatively not is  I f t h i s were t h e c a s e  upon t h e  be v e r y  power s h o u l d  little remain  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  I n f a c t we s e e t h a t i n i t i a l l y  the  was  throughput  a minimum a n d g r a d u a l l y i n c r e a s e s , i . e . t h e d a r k s p o t a t  first  i s v e r y much t h e r e a n d t h e n  Again  i ft h i s  reversal in  still  and t h e output  constant.  t h e case.  there should  then  no  of the applied f i e l d  figure  disappears.  were due t o domain r e v e r s a l t h e n  t h e throughput  fact  gradually  there  upon a s e c o n d  s h o u l d be l i t t l e  b u t we s e e a r a p i d  change  increase instead.  4.38 i n d i c a t e s t h a t d u r i n g t h e a p p l i c a t i o n  n e g a t i v e v o l t a g e t h e waveguide r e c o v e r s .  The n e g a t i v e  voltage  cycle.  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  F u r t h e r m o r e domain r e v e r s a l u s u a l l y o c c u r s w h i c h an i m p u r i t y g r a d i e n t e x i s t s in-diffusion, reversal  In  although  Pendergrass  of a  i n LiNbC>3 i n  [ 7 2 ] , u s u a l l y due t o an [71] h a s s e e n  domain  i n s u b s t r a t e s i n t o w h i c h no i m p u r i t i e s were  diffused. The  photorefractive effect  i s t h e name g i v e n t o a  variety  o f e f f e c t s t h a t cause changes i n t h e r e f r a c t i v e  indices  o f m a t e r i a l s when t h e y  LiNbC>3 m o b i l e by  e l e c t r o n s a r e p r o d u c e d when l i g h t  i m p u r i t i e s such  field  are exposed t o l i g h t .  as Fe [22]. Without  applied the photoexcited  In  i s absorbed  any e x t e r n a l  electric  c a r r i e r s move i n t h e c r y s t a l  -163-  and  so  as  crystal  i s negative  effect. of the  to generate a voltage  I f the  carriers  electrooptic external  substrates,  can  a p p l i e d the since the  be  field.  effect  and  be  When t h e  effect out  can  at the  will  [73] stop  i n the input  be  o r Cu. Fe  that  field  of the be  This  field  s p e e d i s low  charge should,  upon  VIOWM, t h e  counteract  These  near the  is  However rather number carriers  applied for  high  field  switching,  as  the  enough  i n fact,  density  Therefore,  small.  our  a l s o means  electric  regions  an  the  b u l k modes i s s t i l l end  linear  In  direction.  accumulation of charge i n the  field  be  the  small.  i n any  need not  high  switching  reversed  may  edges  found t h a t  5ppm, so  [ 7 5 ] , when an  travel  electrons  presence of the  at the  index v i a the  minimized.  e d g e s where t h e y w i l l  enhance the  traps  Kurz  nominally  will  a t t r a c t e d to the  See the  refractive  the  photovoltaic  a space charge r e g i o n  low,  power d e n s i t y  sufficient the  kept  carriers  especially  electrode  by  +c-end o f  f o r o p t i c a l waveguiding, the  a l . point  of photoexcited can  bulk  applied that w i l l  photovoltaic  Jerominek et  captured  Kraetzig  be  intended  photorefractive the  i s the  the  i n L i N b 0 3 d o p e d w i t h e i t h e r Fe  intentionally  high,  are  a l t e r s the  effect.*  field  photocurrent  that  This  illuminated region  established that  is  [73].  such t h a t  act  regions  initially  i . e . , an  to  extended  Guenter and Huignard [74] for a detailed discussion of p h o t o r e f r a c t i v e e f f e c t and p h o t o r e f r a c t i v e m a t e r i a l s .  -164-  application may  o f a negative  voltage  reduce t h e p o s i t i v e voltage  degree o f o p t i c a l be  beneficial  At  higher  confinement.  is  switching  Such an e f f e c t  essentially  a desired  may p r o v e t o  i s a dead c y c l e .  r a t e s t h e c h a r g e s do n o t h a v e t i m e t o  4.37 i l l u s t r a t e s  4.5  needed t o o b t a i n  i n a p p l i c a t i o n s where t h e r e  accumulate and t h e decay e f f e c t Figure  during the fly-back cycle  that  need n o t be a problem.  f o r high  speed s w i t c h i n g  there  no d e c a y i n t h e o u t p u t .  Discussion  In t h i s others  section various  results  a r e h i g h l i g h t e d and  a r e compared f o r b o t h t h e p r e d i c t e d and t h e measured  results. The  most i m p o r t a n t  behavior has that  until  coupling w i l l Figures  o f a VIOWM w i l l  a point  increase with  increasing  i s reached a t which t h e c o u p l i n g i s  decrease. 4.1 t h r o u g h 4.4 show t h a t t h e c o n f i n e m e n t o f f o r 442 nm r a d i a t i o n t h a n  f o r any p a r t i c u l a r  combination.  experimentally  upon f u r t h e r i n c r e a s i n g t h e v o l t a g e t h e  modes i s g r e a t e r  radiation  c o n t r o l l e d waveguide,  I t has been demonstrated  t h e throughput  maximized, then,  the  i sthat the predicted  o f t h e VIOWM, a s a v o l t a g e  been v e r i f i e d .  voltage  result  This  f o r 633 nm  gap w i d t h / o p e r a t i n g  voltage  i s t o be e x p e c t e d as t h e change i n t h e  -165-  refractive reduction that gap  index  i n t h e value  t h e confinement  i s expected t o increase  Also  with  f o rplanar  index i s i n v e r s e l y proportional  i sdirectly Figure a planar  coupling 25  o f t h e r e f r a c t i v e index.  we s e e  decreasing devices.  i s a r e a s o n a b l e r e s u l t s i n c e t h e change i n t h e  refractive  to  w a v e l e n g t h s due t o t h e  width and with i n c r e a s i n g voltage  This  and  i s l e s s at longer  proportional  t o t h e gap w i d t h  t o the applied  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  voltage. a s 20 V  applied  VIOWM w i t h a 2 jun i n t e r e l e c t r o d e g a p -3 dB  can be obtained  when \  Q  = 442 nm a n d f o r l e s s  than  V when t h e i n t e r e l e c t r o d e g a p i s 4 [un. The  possible  application of the planar  VIOWM a s a  l i n e a r m o d u l a t o r has been demonstrated t h e o r e t i c a l l y and linear  regions  have been seen i n t h e outputs o f t h e  experimental devices;  f i g u r e s 4.15, 4.16, a n d 4.19.  A n o t h e r r e s u l t t h a t was p r e d i c t e d b y t h e t h e o r y a n d c o n f i r m e d b y e x p e r i m e n t was t h e t u r n - o n v o l t a g e waveguides. half-height should  In fact  f i g u r e s 4.20 a n d 4.23 i n d i c a t e t h a t  have a t u r n - o n v o l t a g e  o f about  confinement  occurs  17 V a n d f i g u r e 4.33  t o b e b e t w e e n 15 a n d 20 V.  where t h e knee a t t h e onset  throughput  o f the increase  This  i n the  i n d i c a t i n g a r a p i d change i n t h e  of light  t o the ridge.  W h i l e r i d g e w a v e g u i d e s were n o t p r e d i c t e d useful  a  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 g a p  shows t h e t u r n - o n v o l t a g e is  f o r ridge  as l i n e a r modulators,  t o be v e r y  f i g u r e s 4.31 a n d 4.32, we s e e  -166-  from f i g u r e linear  4.33 t h a t o u r d e v i c e  region.  experimental variation devices  i n fact  demonstrates a very  T h i s i s p e r h a p s due t o t h e w a l l s o f t h e  device being  slanted allowing f o ra greater  i n t h e w i d t h o f t h e mode t h a n  with v e r t i c a l  Another r e s u l t  walls  i spredicted f o r  i n f i g u r e s 4.23, 4.24, a n d 4.25.  that i s interesting  i sthat increasing  the  height  gap  w i d t h c a n i n c r e a s e t h e c o u p l i n g between a d e v i c e  fiber be  o f a r i d g e VIOWM r e l a t i v e t o t h e i n t e r e l e c t r o d e and a  a t low v o l t a g e s b u t i t reduces t h e c o u p l i n g t h a t can  obtained  at higher voltages;  f i g u r e s 4.28, 4.29,  and  4.30. The over  improved d i g i t a l  theplanar  figure  as a d i g i t a l  4.14 w i t h  4.33.  knee a t t h e o n s e t maximum v a l u e full  a p p l i c a t i o n o f t h e r i d g e VIOWM switch  The v o l t a g e  i s seen by comparing d i f f e r e n c e between t h e  o f t h e increase i n t h e throughput  o f t h e throughput  and t h e  f o r t h e p l a n a r VIOWM i s a  50 V w h e r e a s f o r t h e r i d g e d e v i c e  i t i s o n l y a b o u t 30  V. Ridge devices coupling  a r e p r e d i c t e d t o be a b l e t o a c h i e v e  f o r as l i t t l e  Finally  f o r both  a s 11 V. device types  large extinction ratios, negative  we were a b l e t o a c h i e v e  > 20 dB, b y a p p l y i n g a l a r g e  v o l t a g e t o induce  a low r e f r a c t i v e  index  anti-  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 b u l k  coupling  -3 dB  i n s h o r t , - 1 mm,  was c o n f i r m e d  by t e s t s  mode  on a l o n g  -167-  device,  where b u l k  a 23 dB e x t i n c t i o n negative  voltage.  mode c o u p l i n g  i s much r e d u c e d ,  r a t i o was o b t a i n e d  f o r which  f o r a much r e d u c e d  -168-  Chapter  Summary, C o n c l u s i o n s ,  and Suggestions f o r  Further  5.1  possible  5.2  The c h a p t e r  i s concluded  further research  Then  i n this  by a s e c t i o n  suggesting  area.  Summary  In t h i s  t h e s i s t h e VIOWM h a s b e e n s t u d i e d .  ridge type  (chapter on  o f t h e t h e s i s a r e summarized.  c o n c l u s i o n s t h a t were drawn f r o m t h i s work a r e  presented.  and  Work  Introduction  Here t h e contents the  5  d e v i c e s were m o d e l e d  electric  m o d e l was d e v e l o p e d field  (chapter  planar  2), fabricated  3), and r e s u l t s p r e d i c t e d by t h e t h e o r y  t h e d e v i c e s were p r e s e n t e d The  (chapter  Both  and t e s t s  4).  i n terms o f t h e t h e o r y  that the  e s t a b l i s h e d by t h e 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  b y a s m a l l gap, c a n c r e a t e a  waveguide i n an e l e c t r o o p t i c distributions  substrate.  e s t a b l i s h e d i n both  theplanar  d e v i c e s were c a l c u l a t e d b y c o n f o r m a l the by  refractive  index  the applied electric  width parameters optical  field  field  mapping methods.  i n the substrate,  v i a the electrooptic  o f t h e guided  by a v a r i a t i o n a l  distributions. r e s u l t s t o study voltage  technique.  t h e predicted behavior coupler.  An e x p r e s s i o n f o r  a f r o n t - e n d s w i t c h between a f o c u s e d  ridge  fiber  and a field  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  Planar  to the  o f a VIOWM a s a  a l i n k i n g w a v e g u i d e b e t w e e n two o p t i c a l  o r as a d i g i t a l  effect,  I t was p o s s i b l e u s i n g a l l o f t h e a b o v e  controlled  fiber.  caused  b a s e d upon t h e a p p r o x i m a t e o p t i c a l  as  optical  Using  modes o f t h e VIOWM  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  VIOWM was d e v e l o p e d  field  and r i d g e  f o r Hermite-Gaussian approximations  distributions  were o b t a i n e d the  distribution  The e l e c t r i c  fibers  another i s  l a s e r beam a n d a n  The VIOWM's u s e a s e i t h e r  a linear  modulator  s w i t c h was s t u d i e d .  devices with  devices with  i n t e r e l e c t r o d e g a p s o f 4 um a n d  7.5 um g a p s a n d 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 . Finally  a planar  device  s i l i c o n V-groove that acted focused  Gaussian  fabricated  used i n c o n j u c t i o n with a as a f r o n t - e n d  l a s e r beam a n d an o p t i c a l  and t e s t e d .  s w i t c h between a f i b e r was  5.3  Conclusions  The VIOWM o f f e r s modulator. or  p o t e n t i a l as an o p t i c a l  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  as a d i g i t a l  switch  considerable  switch  has been demonstrated.  A  modulator front-end  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 t h e VIOWM a c t s a s a v o l t a g e  c o n t r o l l e d coupler.  While t h e voltages  l a r g e when c o m p a r e d  t o those used Zehnders  used a r e r e l a t i v e l y  i n devices  such as i n t e g r a t e d o p t i c a l  and cross-couplers  as t h o s e u s e d  t h e y a r e o f t h e same m a g n i t u d e  i n some p o l a r i z a t i o n  dynamic r e g i o n  converters.*  o f t h e VIOWM i s s u c h t h a t  the  o u t p u t power c a n b e a c h i e v e d  the  entire on-off voltage,  to  reduce t h e voltage  Mach-  Also the  l a r g e changes i n  w i t h o u t t h e need t o apply  i . e . , bias voltages  d i f f e r e n c e s needed.  c o u l d be used  F u r t h e r m o r e we  h a v e shown t h a t t h e VIOWM c a n b e u s e d n e a r t h e a b s o r p t i o n edge o f LiNb03wavelengths  T h i s o p e n s t h e VIOWM t o a p p l i c a t i o n s a t  where o t h e r  devices  would  s u f f e r from t h e  effects  ofthephotorefractive effect.  W h i l e t h e VIOWM  suffers  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  a s do t h e o t h e r  Examples o f v o l t a g e l e v e l s f o r these d e v i c e s can be o b t a i n e d from the data sheets p r o v i d e d f o r o p t i c a l guided wave d e v i c e s by companies such as C r y s t a l Technology I n c . , P a l o A l t o , C a l i f o r n i a , U.S.A.  -171-  types  o f device  fly-back The  recover  upon t h e a p p l i c a t i o n o f a  cycle. planar  relatively devices.  it.w i l l  device  i s easy t o f a b r i c a t e .  less real estate I t therefore  than other  integrated  promises t o give  I t does n o t r e q u i r e  I t uses  high  be  inexpensive.  in  t h e f a b r i c a t i o n n o r do t h e e l e c t r o d e s  optical  y i e l d s and t o  an i n - d i f f u s i o n require  cycle  precise  alignment w i t h preformed waveguides. The  ridge  device,  needs lower s w i t c h i n g to  w h i l e more d i f f i c u l t voltages.  to fabricate,  Furthermore t h e r i d g e  seems  f i x t h e w i d t h p a r a m e t e r i n t h e z d i r e c t i o n o f t h e mode  rather  precisely.  This  c o u l d be an advantage i n d e s i g n i n g  device  t o operate with a s p e c i f i c  5.4  Suggestions  optical  f o rFurther  a  fiber.  Work.  W h i l e we h a v e a t t e m p t e d t o make o u r t h e o r e t i c a l analysis  o f t h e VIOWM a s t h o r o u g h a s p o s s i b l e t h e r e  room f o r f u r t h e r s t u d y . the  The i s a l s o room f o r improvement i n  f a b r i c a t i o n techniques One  area  optimization  used.  which warrants f u r t h e r study o f the length  of the devices.  t o have d e v i c e s  that  the  and t o reduce l o s s e s .  capacitance  were s h o r t ,  - 1 mm,  is still  a r e as s h o r t  i s the I t i s important  as p o s s i b l e t o m i n i m i z e While our devices  t h e y needed a r a t h e r  large  reverse  -172-  voltage to  t o increase  theextinction ratio.  reduce t h i s voltage Of  effect.  also  Naturally  voltage,  equal  sufficient  needs t o be s t u d i e d  i t c a n be reduced by r e d u c i n g  s o l u t i o n a l s o reduces t h e  A s we h a v e shown a p p l i c a t i o n o f a  of the e f f e c t o f applying  e i t h e r p o s i t i v e o r negative,  during  normal o p e r a t i o n .  fibers,  t h a n t h o s e t h a t were u s e d . waveguides  could  for should  r o u g h n e s s was r e d u c e d .  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 t h e s l a n t on t h e p e r f o r m a n c e o f r i d g e  The d i f f e r e n c e b e t w e e n o u r p r e d i c t e d  r i d g e VIOWMs w i t h v e r t i c a l a device  f o r both the  behavior b e t t e r throughput  i f the ridge wall  o f ridge wall  VIOWMs.  .  W h i l e o u r method was a d e q u a t e  demonstrating t h e devices  Finally  the voltages  ..  O t h e r methods o f f a b r i c a t i n g t h e , r i d g e a l s o be pursued.  voltage,  a d e a d c y c l e may b e  p o l i s h i n g t e c h n i q u e s a r e needed,  VIOWMs a n d t h e o p t i c a l  possible  negative  switch.  a large  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  Better  effect  t h e power  i n magnitude t o t h e p o s i t i v e v o l t a g e , i s  However a s t u d y  be  from t h e  f o r t h e VIOWM t o o p e r a t e a s a d i g i t a l  needed during  as a  The e f f e c t o f t h e p h o t o r e f r a c t i v e  i n t h e g u i d e d mode b u t t h i s VIOWM's u t i l i t y .  served  t o recover  on t h e performance o f t h e d e v i c e s  further.  useful  voltage  c y c l e t o cause t h e devices  photorefractive effect  i f possible.  course here t h e reverse  fly-back  I t i s desirable  with slanted walls  walls  resultsf o r  a n d o u r m e a s u r e d r e s u l t s on  indicates that  devices  with  -173-  slanted walls  can  r i d g e VIOWM w h i l e in planar  devices.  i n c o r p o r a t e both the preserving the  turn-on  voltage  of  a  highly linear  regions  found  Appendix  A  THE E L E C T R O O P T I C  A.l  Introduction  T h i s appendix covers t o p i c s regarding the linear discussion the  Finally  our  effect  coefficient  the optical  thesis  I t begins with a  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  to  Then t h e  i s d i s c u s s e d and t h e reduced  tensor  b o t h e q u a t i o n s 2.1  example,  effect.  d i e l e c t r i c impermeability tensor.  electrooptic  electrooptic  relevant t o this  electrooptic  of the optical  relative  linear  of  EFFECT  f o r LiNbC>3 i s g i v e n .  a n d 2.2,  r e l a t i n g the deformation  indicatrix to the applied e l e c t r i c f i e l d i n  are derived.  -175-  A.2  The R e l a t i v e  Impermeability  The of  optical  a crystal.  Dielectric  Tensor  indicatrix* specifies  The e q u a t i o n  t h e above e q u a t i o n ,  principal  convenient  X2,  2 x  ,  axes.  tensor.  2  ,  are the  The n o t a t i o n u s e d a b o v e i s n o t  appendix t h e p r i n c i p a l  this  dE  2 z  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  dielectric  B^. — e  coefficients  dielectric  axes w i l l  a n d X3 a x e s f o l l o w i n g t h e c o n v e n t i o n  Using  The  1/ny , and l / n  components o f t h e r e l a t i v e  impermeability  in this  l/n  index  describing the e l l i p s o i d i s  when x, y , a n d z a r e t h e p r i n c i p a l of  the refractive  therefore  be c a l l e d t h e x j , u s e d b y Nye [ 2 3 ] .  n o t a t i o n e a c h o f t h e components o f t h e r e l a t i v e impermeability  tensor  i s g i v e n by  i  ( i •» 1, 2, and 3; j •= 1, 2, and 3)  Q  3D . 0  where e  0  i s the permittivity  electric  field.  relative  d i e l e c t r i c impermeability  *  The g e n e r a l  of free  space and  optical  representation quadric can then  of the  be d e s c r i b e d by  A l s o c a l l e d t h e i n d e x e l l i p s o i d b y some a u t h o r s ; and Yeh [45] c h a p t e r 7 p p . 220-275.  see Y a r i v  -176-  B . .x . x . = 1  .  where t h e E i n s t e i n  summation  The r e l a t i v e tensor*, reduce  convention  d i e l e c t r i c impermeability i s a  i . e . , B^j = Bj£.  B  l l  B  12  B  13  B  21  B  22  B  23  B  31  B  32  B  33  A.3  a r e reduced  The  —»  .  .  l  B  6  B  5  B  6  B  2  B  4  B  4  B  3  5  components.  Electrooptic  Effect  The change i n t h e r e f r a c t i v e an  applied electric field  effect.  will  result  the optical  i s known a s t h e  relative  i n a change i n t h e s i z e  indicatrix.  between t h e o p t i c a l  optical  index o f a c r y s t a l  Because o f t h e  indicatrix  due t o  electrooptic  I n g e n e r a l a change i n t h e r e f r a c t i v e  crystal of  The n i n e  t o s i x a s shown b e l o w :  B  B  symmetrical  Therefore i t i s possible t o  t h e number o f i n d e p e n d e n t  components  i s assumed.  index o f a  and  orientation  relationship  a n d t h e components  of the  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 t h e indicatrix  corresponds  t o changes i n t h e  coefficients B^j. When a c r y s t a l the  exhibits  the linear  c h a n g e s i n t h e B^-; a r e r e l a t e d  See Nye  [23]  p.  246.  electrooptic  to the applied  effect  electric  -177-  field  ( t h e low frequency  opposed t o t h e h i g h electrooptic AB . . =  frequency  coefficients  r . ., E, ( i 13k k  13  =  optical  field) v i a the linear  r ^ j j ^ by*  index  notation  r . . E . ( i = 1, 2, 3, 4, 5, 6; j = 1, .J -J  LiNb0 and  a p p l i e d t o t h e c r y s t a l as  1, 2, 3; j - 1, 2, 3; k = 1, 2, 3)  or u s i n g t h e reduced AB.  field  2, 3)  i s a member o f t h e t r i g o n a l  3  i t i s of class  3m.**  When t h e x^ a x i s i s d e f i n e d a s t h e  axis perpendicular t o the mirror plane xi-Lm, t h e n  the electrooptic  n o n z e r o components: rg]_.  In matrix  looks  like  0 0 0  r  !2 !3  r  22  r  r  1  2  , r  of the crystal, i . e . ,  coefficient 1  3  , i 2r 2  t e n s o r has t h e  r 3,r 3,r 2, r  form t h e e l e c t r o o p t i c  2  3  4  coefficient  5 1  ,and  tensor  r  r  23  0  0  r  system o f c r y s t a l s  33  42  °  0  0  °  °  51 r  61  The c h a n g e i n t h e  ( i = 1, 2, 3, 4, 5, 6) due t o t h e  applied electric field  *  Here constant  i s given b y  temperature  and p r e s s u r e  See Y a r i v a n d Yeh [45] p . 232  a r e assumed.  -178-  A B  0  1  0 AB3  0  A B  r  r  !3  r  22  r  23  r  51  33  42  ° 0  °  0  "61  6  i2  0  0 AB5  r  0  I n LiNbC«3 -r]_2 =  8.6xl0~ m/V,  r  22  4  30.8xlO~ ^m/V. p l  X  a  n  r  22 2 E  +  B  2  =  +  r  2  2  E  2  B  3  -  +  r  3  3  E  3  B  B  4  5  B ^ o  " 42 2 ' r  E  =  0  =  0  ,  3. 4 x l O ~ m / V , 12  and  Ibid.  tensor  r  23 3 '  +  ,  E  r  2 3  E  3  ,  r^3 = r 3 = 2  = 28. 0 x l 0 ~ m / V , a n d r 12  5  1  t h e components o f t h e r e l a t i v e  e  impermeability  —  =  When t h e a p p l i e d e l e c t r i c  x  2 3  r  r 2 = r  12  X  ~ ~ 61  become  3  field  3  = i s i n the  dielectric  -179-  A.4  Equations  If,  for this  impermeability  problem, t h e r e l a t i v e  a s s e t o u t i n Nye*  t o a new  t h e Mohr  i n the x x 2  o f a second rank t e n s o r  perpendicular  s e t o f axes  [23] c a n be a p p l i e d  of the i n d i c a t r i x  transformation  of mutually  dielectric  2  find the distortion The  2.2  x ' = X 3 , a n d X 3 ' = x]_, t h e n  2  to  and  t e n s o r were t r a n s f o r m e d  where X ] / = x , construction  2.1  axes t o another  circle  directly 3  plane.  f r o m one s e t  i s given  by**  T'. . = a..a.,I, .  ik  13  3I  kl  where t h e a ' s a r e d i r e c t i o n quantities.  cosines  The t r a n s f o r m a t i o n  a n d T a n d T' a r e t e n s o r  o f a second rank t e n s o r  the  XJ[ s e t o f a x e s t o t h e X J / s e t o f a x e s , d e s c r i b e d  can  be a c c o m p l i s h e d  ~  1  r  a  1 3 = °' 2 1  ~  a  and  8 3 3 = 0.  set  o f axes i s  using the direction °r  a  2  2  «= 0, a  The i m p e r m e a b i l i t y  2  3  =  — 2 n  +  r  22 2 E  +  r  23 3 E  o  *  Chapter  **  Nye  II section  [23] p . 11.  4 pp.  = 1, a  3  1  = 1, a  above, = 0,  3  2  a^  = 0,  t e n s o r r e f e r r e d t o t h e new  1  l  B  cosines  from  43-47  2  -180-  B'2 =  + r n  3 3  E  ,  3  2  e  1 B  3  =  — 2 n  r  22E2  +  r  23E3  '  o  B'4 = 0 ,  B'5 = 0 , a n d  B  6  "  r  42E2  •  where t h e s u b s c r i p t s the  electric field  still  refer  t o the  i n the  electrooptic  components a r e original  W i t h no a p p l i e d ellipse  on the  field  formed by t a k i n g  set the  unchanged  section  principal  deforms i n b o t h s i z e  and o r i e n t a t i o n .  d i a g o n a l element o f the  i m p e r m e a b i l i t y t e n s o r under the t h erotation  T h i s being the  o f the  case the  indicatrix  theindicatrix  new p r i n c i p a l  the  indicatrix. indicatrix  Since the relative  only  dielectric  i s about the  field i s X3'-axis.  Mohr c i r c l e c a n b e u s e d t o f i n d t h e  and the  I f the  indicatrix  i n f l u e n c e o f the  indicatrix  angle o f r o t a t i o n (i.e.,  o f the  axes o f the  o f an e l e c t r i c f i e l d  Bg'  they  major and minor axes o f t h e  Upon t h e a p p l i c a t i o n  nonzero o f f  (i.e.,  o f axes).  a cross  p l a n e X3' = 0 a r e  c o e f f i c i e n t s and  principal  axes o f the  i m p e r m e a b i l i t y t e n s o r are  components o f t h e  when t h e  field  is  relative  dielectric  new  applied).  l a b e l e d x ^ " a n d X2" a n d t h e  angle  -181-  of  m e a s u r e d f r o m X]/' t o X ] / , i s e ,  rotation,  principal  components o f t h e t e n s o r  B  i  +  B  2  B  i  +  B  2  are given  then the by  and  B  +  2  r  where  1/2 ( B'2 ~  and  )  the relationship  giving  the angle  e is  2B' tan(2e)  = B  2 "  B  i  By d i r e c t s u b s t i t u t i o n we  tan  (2e)  42 -2 n e  -  -2 n o  +  r-_E  33  where Ey = E , and E 2  To  z  z  for B 2 " .  2.1  y -  (2.1) r_„E  22  y  -  r „ E  23  z  = E3.  obtain equation  expression  obtain equation  2 . 2 we b e g i n  First  a simple  by f i n d i n g substitution  an gives  -182-  1/2 2 B  1  6  '  +  V  B  2  B  l  "  1 /2  h e r e we  can use t h e approximation  small x B  i  since +  B  2  B  2  (1 + x)  [2Bg'/(B2' - B ] / ) ] ^  " i B  +  « 1.  1,2  B  B'  + B  2  '  <= 1 + x/2 f o r  We  c a n now  write  ,2  +  " i B  B  " i B  2  or.  { 42 y»" r  + n  2 ,2  n  r__E  +  33 z  e2  n_-2  e  - n_-2 +  o  r__E  E  33 z  -  r„„E  22 y  The l a s t two t e r m s on t h e r i g h t  -  r0_E  23 z  hand s i d e  o f t h e above  e q u a t i o n r e p r e s e n t t h e change i n t h e r e l a t i v e d i e l e c t r i c cmsability  due t o 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  field,  i.e.,  An  < 42V' n  -2 -2 • - n + r_,E - r_„E e o 33 z 22 y  The r e l a t i o n s h i p and  *  r  -2 •= r o r > E + e 33 z  r,_E  33 z  r_0E  23 z  + n  r  4 2 V  -2 -2 — n e o  b e t w e e n t h e change i n t h e r e f r a c t i v e  index  t h e change i n t h e 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  given  by  See S p i e g e l  [7 6] p. 110.  -183-  n  An  3  A  An  f r o m d n ~ / d n = -2/n^  obtained Now  we  can  write  n Ane(y,z)  s i n c e An  2  -  -  3 e  r__E  33  equation  z  (J y , z )  n  3 e  f o r small  An.  2.2  {r.^E (y,z)} 42 y •*'  2  •* dn  2 .  -2 2(n ^ e  -  n  -2  o  (2.2)  -184-  Appendix  STATIONARY  B.1  B  FORMULAS  Introduction  T h i s appendix provides t h e proofs t o s e v e r a l o f t h e a s s e r t i o n s made i n c h a p t e r certain  B.2  integrals.  Equation  In s e c t i o n  oe  2.10  2.4 i t i s c l a i m e d t h a t t h e i n t e g r a l 2  oo  —oo —oo  , 3y  stationary  t  — ,  -  v  2  ( y , z) y  2  dydz  =  0  (2.10)  dz ,  and t h a t t h e g e n e r a l  form  2 2 V >l> + v (y, z ) v = 0 , fc  is  equation  2  +  is  2 about t h e s t a t i o n a r y n a t u r e o f  t h e Euler-Lagrange  eqaution,  where  o f t h e wave  equation  -185-  ,2  dy  ,2 2  This eqaution  OO  2  dz  i s p r o v e n by t a k i n g t h e f i r s t 2.10  which i s given ' dy  Of  81  2  >  8  ' dy + 6  variation  of  by  s  - 8[v  —  2  2 (y,z)v J  dydz  , 3y J or d\|f d8y  11 OO  81  oo  dy d8y  2  2 2 2v (y, z)v8y - 2y v (y, z) 8v (y, z)  +2 dy dy  dz dz  Applying  Green's theorem t o t h e f i r s t  integral  one  JJ oo  - J  J  <ty d8y dy dy  61  i n the  above  dydz dz dz  ' d y  d V  dy  28v . 2 dy  and t h e f i r s t  oo  terms  dy d8y +2  V  two  obtains  2  oo  dydz  dz  2  dydz + )  variation  oo  J  Sy — C  dn  of equation  ( di 2 y  di V  2  2  ds  2.10  becomes  2  + v (y,z)y  - 28y  - 2y v (y, z) 8v (y, z)  dydz  -186-  of  which t h e expression contained w i t h i n the parentheses i s  r e c o g n i z e d t o be t h e g e n e r a l form equation the  and i s e q u a l t o z e r o .  s c a l a r wave e q u a t i o n  p r o v i d e d t h a t v(y,z) variation the  i s stationary,  of integral  equation  i . e . , i fthe f i r s t  i s t o be z e r o t h e n  the last  term i n  i n t e g r a l must a l s o b e e q u a l t o z e r o w h i c h i m p l i e s t h a t must b e s t a t i o n a r y ,  8v(y,z)  = 0 .  B.3  The P r o p a g a t i o n  The  purpose o f t h i s  propagation begins  constant  i.e.  Constant  section  i s a stationary value.  2  IS  a l n [ n ^ (y, z) J 2  To do t h i s  one  dln[n (y,z)] z  dydz  takes  its first  1\  S  dydz  I f the integral  i nthe  hand s i d e o f t h e above e q u a t i o n i s  and t h e i n t e g r a l  the f i r s t  J  variation.  n u m e r a t o r on t h e r i g h t labelled  z  9z  /  then  i s t o show t h a t t h e  with the equation 2  J-2  wave  Hence t h e g e n e r a l f o r m o f  i s the Euler-Lagrange  v(y,z)  and  of the scalar  variation  i n t h e denominator i s g i v e n by  i s labelled  -187-  V1!  *1 ^ 80  «= 2p 80 V  1  I  =8  V  The  "  6 I  6 1  2  1  +  P  v  6 l  2 (Bl.l)  V  first  variation  of I ii s  dy dfiy  dy d8y  2 2p (y, z)y8y  +2  81 dy dy  dz dz  2 2 d l n f n (y,z)] z  z  2 - 2y p (y, z) 8p (y, z)  dydz  where  dln[n  z  (y,z)] 2 2 + n (y, z) k z o  /  2  dz  dz'  and t h e f i r s t oo  81  = J  so  that  2  2 8l  v  00  J  2y8y  of integral  l£ i s  dydz  t h e terms  2 2  variation  - P  v  ~ J  ~ i  2y8y  dydz  and  - /  /  c a n  be  -  00  \  00  \  2p (y, z)y6y  c o m b i n e d  2p  2  (y,z)y8y  dydz  t o  r e s u l t  i n  2 dydz + P J v  /  2y8y  dydz = - J  J  2 2v (y,z)y8y  dydz  -188-  Applying OO  Green's theorem t o t h e f i r s t dy dSy  CM  +2  dydz = dz dz  dy dy  j  (  oo  j  d y  d y ^  2  dydz + :»  a  2  dy  JJ  2  of equation -.2 d y  2  2  - 26y  + v , dy  dz  ds  dn  2 d y  The e x p r e s s i o n scalar  j  dz  and t h e numerator /  dy 6y —  2  28y  -oo —oo  B I . 1 becomes  (y,z)y  - 2y p(y, z)8p(y,z)  wave e q u a t i o n  p(y,z)  i s zero i . e .  «p(y, z)  = 0  and i s e q u a l  t o zero.  that the f i r s t  i s assumed.  as t h e  It follows that  variation  w h i c h i s o b v i o u s l y s o p r o v i d e d t h a t an a p r i o r i z v  dydz  within the parentheses i s recognizable  SBv i s s t a t i o n a r y p r o v i d e d  w  gives  dy d5y  2  oo  t o t e r m s i n 611  of  knowledge o f  -189-  0  Appendix  THE C O U P L I N G  C l  COEFFICIENT  Introduction  In t h i s 2.6  C  appendix equation  2.17 i s d e r i v e d .  we d e r i v e d t h e e x p r e s s i o n 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 dydz J  2 a  T  In s e c t i o n  fPf  J  J v v.  =  w  J  J yv  y  ^ —,  2  .2,2  — ( y /w  2,2,  +z /w ) yv. zv  dydz  0 v  yv  b y a s s u m i n g t h a t t h e power c o u p l e d modes c o u l d b e n e g l e c t e d .  to reflected  radiated  T h i s was done u s i n g t h e  approximate  field  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  fiber  a n d t h e VIOWM.  locate the center  of the optical  optical  Here t h e v a r i a b l e s a and b fiber  relative  t othe  -190-  center  o f t h e surface o f t h e waveguiding region o f t h e  VIOWM.  C.2  The N o r m a l i z e d A m p l i t u d e s  The t i m e - a v e r a g e d power p r o p a g a t i n g an  optical  a f and a  i n a guided  v  mode o f  w a v e g u i d e may b e g i v e n t e r m s o f t h e f i e l d s b y *  l  where E i s t h e e l e c t r i c u  x  the  H i s t h e magnetic f i e l d , and  i s a unit vector i n the direction electric  plane plane,  field  o f t h e guided  o f propagation.  t h e mode i s c a l l e d  When  mode i s p o l a r i z e d i n t h e  normal t o t h e d i r e c t i o n propagation,  The e x p r e s s i o n  the transverse  a transverse e l e c t r i c  o r TE mode.  f o r t h e power i n a TE mode may b e r e d u c e d t o 2  dydz  where t h e s u b s c r i p t the  electric  field.  the  electric  field  *  field,  t i n d i c a t e s t h e t r a n s v e r s e component o f In our case  the transverse  i spolarized parallel  See f o r e x a m p l e S n y d e r a n d L o v e 208-237.  component o f  t h e z - a x i s and i s  [77] c h a p t e r  11 p p .  -191-  given  for the optical  fiber  by equation  2.11 a n d f o r t h e  VIOWM b y e q u a t i o n 2.12. I f t h e power i n t h e mode i s n o r m a l i z e d c a r r i e s p Watts then calculated  t h e amplitude  s o t h a t t h e mode  of the f i e l d i s  using 2fi oop f  0 0 0 0 f  ,  /  2  2 . 2 . 2 .  -<y /w  +z /w  /  z f  l i e  for  the fiber  ) dydz  and u s i n g 2^ <op v  #2.2  -(y  /w  2.  2  + z /w )  e —00  dydz  0 yv  for  t h e VIOWM. The  ratio  of  the ratio  is  given by  af/a  af / a  v  v  i s determined by t a k i n g t h e square a n d b y a s s u m i n g t h a t n-f =  2.2  -(y  Z  e -00  2 . 2 ,  /w + z /w ) yv ^dydz J V  0 yv  .2,2  e  -(y  /w  .  f  2,2.  + z /w ) zf  dydz  root  = u . It D  -192-  Substituting coupling  the ratio  coefficient  af/a  the expression  f o rthe  gives  *<M«>JJ 21/  -{y  l j r  e v  into  v  2,2 . 2,2 2 , 2 2 , 2 /w +z /w + ( y - a /w ,+ (z-b) /w ,}/2 yv zv ' yf ' z f , . dydz > J  3  /  w  T -  1 »  2 . 2 2 . 2 - ( V  e  /W  +Z  / W  z  ^  f  )  ~ ~ r dydz J I  -(y  /•  2  /w  2 2 2. +z /w ) yv zv  2 dydz  yv  C.3  The Numerator o f t h e C o u p l i n g  Coefficient  The  integral  i n t h e n u m e r a t o r o f t h e above  expression  for the coupling coefficient i s '  v  Y  \  2 2 2 2 2 2 2 2 - { y /w + z /w + (y-a) /w + (z-b) /w >/2 yv zv yf zf . . e dydz  yv '  First one  the integral  i s t o separated  into  two  integrals  i n y a n d one i n z -a  r = e  J  2,„ 2 .2-- 2 ~ /2w -b /2wzf y f  (  ^  2 2 -y / n +ay/w -z /n y ' y ^dy „ J I"e 2  J  \  where  y  w  i  f  +bz/w  2 f  dz  -193-  2 2 w yv y f  0 2w  n  y  2 , 2 yv  yf  and  r> 2  2 w _ zv z f  2w  n z  w  2 ^ 2 + zv  W  _  zf  Next t h e s u b s t i t u t i o n s  of  variables  y —  y' -  "y  / 2  and z n  are  made y i e l d i n g  n  r •=  l/2  n  1/2  y z : w yv  We  e  ,2,2 - ( a /w  shall  begining  y' e  w  2  2  f  _,l/2 , , 2  + an y  J  y'e  ~y  s o l v e t h e above  with  ,2 ^  -y'  r  , .2 . 2 . / 0 ~ +b / )/ f  y'/w y  f  dy'  , 2,  0 + a I 1  l/2 , . 2 y f /  y  w  y  ~ f ~ dy'J e  z  , 2 ,. _ l / 2 , . 2 zf +  b  n  i n t e g r a l s one a t a t i m e  2  /  w  dz'  -194-  which c a n be w r i t t e n i n terms o f t h e Hermite rr .  polynomial  \  f  -y'2 + aft1/2y'/w2 y  J y' e  y  dy' = _ J 2  0  The  generating  -  1  _y,2  afll/Zy,/  +  Y  (y') e  2 Y  dy'  0  f u n c t i o n f o r Hermite polynomials i s * 2  2 H  <x) =  d n e" x  <-l)n ex  , n =  0,1,2,...  dx  which c a n be s u b s t i t u t e d i n t o t h e p r e c e d i n g  • J  ,2 ^  „l/2 , . 2 y ' /w  -  -y' • + all H  y  (y') e  y  „l/2 , . 2  an  {  dy' = - J  0  e  y  0  y' /w y  integral  giving  , -3 y ' 2 de dy' = dy'  an1/2y'/w2, ,2 - y ' "yf d e - y '  _ J  e  the  last  J  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  . ** parts ~ an 1 / 2 y'/w 2 ,2 y f - J e y de" y  Lebedev  [78] p . 60.  A l l o f 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 c a n be s o l v e d u s i n g t h e formulas i n S p i e g e l [7 9] on p . 98 a n d u s i n g t h e s p e c i a l v a l u e s f o r t h e gamma f u n c t i o n on p . 101.  -195-  -y'  + an  2  y'/w  1 / 2  y  2 £  - ""y I f  ~  ,~  a  1 + 2w  J  '  I  -an / 1  I  yf  + bn  2  1 / 2  y  2  2  f d  y'  N  )  f  i s easily  z'/w zf , , 1/2 dz' = it  b  2  The numerator  y'/w y  erfc  J  The s e c o n d i n t e g r a l -z'  1/2  0  fl/4w  e  2  f  +  0  y  - y ' + af> y e  e  ?  Cl  z  solved  /4w  giving  4  zf  of the coupling  coefficient  c a n now b e  w r i t t e n as  « O ^ y z 1  r =  2w  2  l y  K  /  < e  2  a  , 2 , 2 ,.2, 2 . 2„ 4 / „ y^ f /w,^)/2+b Il^/4w zf z r zf + b  yv  an  1/2 1/2 * y  1 +  , 2_ a n e  y  2w  C.4  w  /4w y  4  -ail  1/2  "1  yf  )  erfc 2w  yf  The Denominator o f t h e C o u p l i n g  Coefficient  The i n t e g r a l s the  coupling  i n t h e denominator o f t h e e x p r e s s i o n f o r  c o e f f i c i e n t are  -196-  -J/e ~o°  ^ "- , J J  +  y.  ,2,2  /  2,2.  ,««.<«  N2  dyd  -oo  -(y2/w2  e  + z 2 /w 2 ) dydz  0  yv  w h i c h c a n be s e p a r a t e d solved  i n a straight  2,  four integrals  f o r w a r d manner  2  e  dy = w  2,  into  it yf  2  dz =  W  JK  zf  >  w re yv  2.2  2  —v /w e  yv  1/2  dy =  yv  and  J  e  The  — z2/ /w2 zv  1/2  dz  w  expression  W W W  _W  TC  zv  f o r r ' becomes  JK  yv zv y f z f  o r f or r ' (w 1/2  1  /  2  W W ,W -t yv zv y f z f  12  each o f which i s  -197-  C.5  Equation  Now  2.17  the coupling c o e f f i c i e n t  2(PfPv)  1 / 2  c a n be g i v e n  by  r 1/2  (Pf+Pv)  '  which reduces t o  X  K  f v'  y z  ,„ + B ,),(w ' w w ,w )«1/2 w re1/2 (B, f v ' * yv zv y f z f yv a  V H  1/2 afl y  1/2 rc  c  K  a  e  \  /  4  w  J  y f  , erfq f  (2.17) I  +  2w  yf  -198-  References  1.  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