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Planar beam-steered acousto-optic light deflectors Riemann, Ernest B. 1977

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PLANAR BEAM-STEERED ACOUSTO-OPTIC LIGHT DEFLECTORS  by  E r n e s t Bruno Riemann B . E n g . ( P h y s i c s ) , M c M a s t e r , U n i v e r s i t y , 1969 M.A.Sc., U n i v e r s i t y  of B r i t i s h  C o l u m b i a , 1972  A T H E S I S SUBMITTED I N P A R T I A L FULFILMENT OF THE REQUIREMENT  FOR THE DEGREE  OF  DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES (Department o f E l e c t r i c a l  We a c c e p t t h i s the  Engineering)  t h e s i s as c o n f o r m i n g t o  required  standard  THE U N I V E R S I T Y OF B R I T I S H  COLUMBIA  J u n e , 1977 (c)  E r n e s t B r u n o R i e m a n n , 1977  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this for  thesis  scholarly purposes may be granted by the Head of my Department or  by his representatives.  It  is understood that copying or publication  of this thesis for financial gain shall not be allowed without my written permission.  Department of The University of B r i t i s h Columbia  2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date  ABSTRACT  A  t h e o r e t i c a l and  acousto-optic  light  experimental  s t u d y has  b e e n made o f  d e f l e c t o r s w i t h p a r t i c u l a r e m p h a s i s on  a means o f i m p r o v i n g  model takes  i n t o a c c o u n t t h e e l e c t r i c a l d r i v e c h a r a c t e r i s t i c s o f beam-  diffraction  o f a c o u s t i c waves and  between guided The guides  s u r f a c e a c o u s t i c w a v e (SAW)  o p t i c a l w a v e s and  experiments were c a r r i e d out  on Y - c u t L i N b O g s u b s t r a t e s .  transducer  a r r a y was  direction  centered  a t 21.8°  MHz  and  ease of f a b r i c a t i o n .  The  and  c h o s e n as  g a v e 44  excellent  d e f l e c t o r had  agreement w i t h the  theory.  advantageous technique  width  diffraction  high  a c o u s t i c waves.  propagation frequency  acousto-optic  a b a n d w i d t h o f more t h a n  an o p t i c a l w a v e 2.5  efficiency.  /  i i  I t was  of  bandwidth  mm  t h e d i f f r a c t i o n e f f i c i e n c y was  s t e e r i n g i s an and  A center  a compromise between h i g h  response of  interaction  on n i c k e l i n d i f f u s e d wave-  Z axis.  resolvable spots with  observed frequency  surface  a c o u s t i c waves w i t h  from the  The  o f the  A f o u r - s e c t i o n , three f i n g e r p a i r  used to launch  200  60 MHz  was  frequency  theoretical  transducers, anisotropic  the rigorous theory high  The  a c o u s t i c beam  s t e e r i n g as  steered i n t e r d i g i t a l  device performance.  planar  concluded  f o r devices  wide. in  t h a t beam  r e q u i r i n g l a r g e band-  TABLE OF CONTENTS Page ABSTRACT  i  TABLE OF CONTENTS  1  L I S T OF TABLES  i 1  1  v  L I S T OF I L L U S T R A T I O N S  v i  NOMENCLATURE  x i  ACKNOWLEDGEMENT  xvii  1.  INTRODUCTION  1  2.  D I E L E C T R I C O P T I C A L WAVEGUIDES  4.  .  D i e l e c t r i c S l a b Waveguides'  3  2.1  Uniform  2.2  Modes i n G r a d e d I n d e x W a v e g u i d e s  2.3  C o u p l i n g t o O p t i c a l Waveguides  11  2.4  Prism Coupler Design  15  2.5  Coupler  3  23  2.6  O p t i c a l Waveguide  2.7  Diffused  2.8  Ti/LiNb0  2.9  Ni/LiNb0  6  Fabrication Fabrication  O p t i c a l Waveguides 3  3  28  i n LiNbO^ . . . . .  30  Diffusion  31  Diffusion  35  2.10 P r o p e r t i e s  3.  . . .  of Ni/LiNb0  3  Waveguides  40  PROPAGATION AND GENERATION OF ACOUSTIC SURFACE WAVES  56  3.1  Introduction  56  3.2  S u r f a c e Waves i n P i e z o e l e c t r i c s  56  3.3  Diffraction  60  3.4  SAW G e n e r a t i o n ;  3.5  E x p e r i m e n t a l Work  o f S u r f a c e Waves t h eI n t e r d i g i t a l Transducer  63 77  BRAGG BEAM-STEERED SURFACE WAVE ACOUSTO-OPTIC LIGHT DEFLECTORS  86  4.1  Introduction  86  4.2  Theory o f t h e Surface-Wave A c o u s t o - O p t i c  i i i  Interaction  . .  87  Page 4.3  Acoustic  Beam S t e e r i n g  101  4.4  D i f f r a c t i o n E f f i c i e n c y o f Beam-Steered T r a n s d u c e r s  4.5  Acousto-Optic Overlap I n t e g r a l C a l c u l a t i o n  4.6  E x p e r i m e n t a l Work  . . .  109 I l l 121  5.  SAW TRANSDUCER FABRICATION  146  6.  CONCLUSIONS  155  . . .  APPENDIX I  157  APPENDIX I I WAVE PROPAGATION IN ANISOTROPIC MEDIA  162  REFERENCES  166  iv  LIST OF TABLES Table  Page  2.1  Ni Sputtering Calibration  36  2.2  X-Ray Fluorescence of Ni/LiNbC>  38  2.3  TE Modes a of Ni/LiNb0 OWG  46  3.1  Anisotropy Parameter b  63  3.2  Constants for LiNbO„ [40]  75  3  3  v  L I S T OF ILLUSTRATIONS Figure  Page  2.1  S l a b D i e l e c t r i c Waveguide . .  3  2.2  Z i g - Z a g Wave P r o p a g a t i o n  5  2.3  C o m p a r i s o n b e t w e e n WKB Marcuse  and E x a c t  [9])  Solutions  (after  . .  9  2.4  Z i g - Z a g Wave P r o p a g a t i o n i n G r a d e d I n d e x W a v e g u i d e s . . .  2.5  Lens Coupler  13  2.6  Prism Coupler  13  2.7  Grating Coupler  13  2.8  G e n e r a l i z e d L e a k y Wave C o u p l e r  2.9  Prism Coupler  2.10  Broadening  2.11  E f f e c t o f Beam B r o a d e n i n g o n C o u p l i n g ( a s s u m e s £/W = 1, n = 2 . 2 3 ) . . . ' m  . . . . . .  Geometry  9  13 16  o f t h e I n p u t L i g h t Beam  16 Efficiency 17  2.12  Mode S e p a r a b i l i t y v s . P r i s m A n g l e  2.13  L i m i t i n g Mode I n d i c e s v s . P r i s m A n g l e  . . . .  21  2.14  Phase V e l o c i t y Surface f o r R u t i l e w i t h c A x i s H o r i z o n t a l  22  2.15  c Axis Vertical  ,.  22  2.16  P h o t o r e s i s t Exposure f o rG r a t i n g Coupler F a b r i c a t i o n  . .  24  2.17  Underexposed G r a t i n g  26  2.18  C o r r e c t l y Exposed G r a t i n g  26  2.19 2.20  R e s u l t o f I n s u f f i c i e n t P h o t o r e s i s t Adhesion R u t i l e C o u p l i n g P r i s m (-^8X)  26 27  2.21  Ellipsometry of T i 0  2.22  Ti/LiNb0  2.23  C o u p l i n g t o a G l a s s S p u t t e r e d OWG  2.24  Gas F l o w C o n n e c t i o n s  3  2  for n  m  = 2.23 . . . . .  for Rutile  on S i l i c o n  -29  Diffusion Profile  for Ni/LiNb0  vi  20  34 34 3  OWG D i f f u s i o n  . . . .  36  Figure  Page  2.25  Absorbance o f LiNbO^ Waveguide S u b s t r a t e s  2.26  Ni/LiNb0  2.27  Stage  2.28  Two Mode N i - D i f f u s e d G u i d e S h o w i n g Mode B e a t i n g b e t w e e n t h e TE & TM Modes o o  43  2.29  C o u p l i n g i n and o u t ( T E  45  2.30  Modes o f OWG's U s e d i n A c o u s t o o p t i c E x p e r i m e n t s  45  2.31  C o o r d i n a t e R o t a t i o n i n Phase V e l o c i t y Space . .  46  2.32  Ni/LiNb0  48  2.33  The A i r y F u n c t i o n  2.34  C o m p a r i s o n o f A i r y F u n c t i o n a n d WKB S o l u t i o n s f o r TE..  3  Diffusion Profile  '  f o r C o u p l i n g t o O p t i c a l Waveguides  3  39  OWG  q  .  41 43  mode)  Index P r o f i l e  51  Mode o f 6-Mode N i / L i N b 0  OWG  3  52  2.35  TE Modes o f a N i / L i N b 0  2.36  Prism Coupler E f f i c i e n c y  55  3.1  SAW P r o p a g a t i o n  57  3.2  SAW P r o p a g a t i o n i n A n i s o t r o p i c M a t e r i a l s [ 4 0 ]  62  3.3  S e c t i o n o f an I d e a l i z e d IDT  64  3.4  SAW V e l o c i t y a n d C o u p l i n g C o n s t a n t  3.5  IDT S h u n t M o d e l E q u i v a l e n t C i r c u i t  69  3.6  Series Equivalent Circuit  70  3.7  Permittivity  3.8  S e r i e s C i r c u i t Model  75  3.9  Transducer  Conductance and Susceptance  78  3.10  Transducer  Admittance  80  3.11  R a d i a t i n g IDT E q u i v a l e n t C i r c u i t  81  3.12  Power I n s e r t i o n Loss  82  3.13  Raman-Nath D i f f r a c t i o n o f L i g h t b y S u r f a c e Waves  3  OWG  Transformation  54  f o r Y-Cut L i b N b 0  3  [40]  .'  74  n e a r Resonance  vii  68  . . . .  83  Figure  Page 2  3.14  Relative Deflected Light Intensity vs. V  4.1  D e f l e c t i o n o f a n OWG  4.2  Isotropic Acousto-Optic  4.3  I s o t r o p i c A c o u s t o - O p t i c D i s p e r s i o n Curves i n a Medium  84  Q  b y a SAW  88  D i s p e r s i o n C u r v e s f o r Ae = 0  . .  92  Modulated 93  4.4  Momentum C o n s e r v a t i o n i n A n i s o t r o p i c B r a g g D i f f r a c t i o n  4.5  Beam S t e e r i n g T r a n s d u c e r  102  4.6  P h a s e C h a n g e a c r o s s One S t e p  103  4.7  The A p e r t u r e  105  4.8  The : A p e r t u r e  4.9  Bragg-Angle Tracking  4.10  iiAcoustic Displacements  .  a n d A r r a y F u n c t i o n s f o r A = AQ a n d D ^ G . . a n d A r r a y F u n c t i o n s f o r A 4 AQ  106 107  f o r Z + 21.8° P r o p a g a t i o n  4.11  SAW  4.12  R e a l P a r t o f t h e A c o u s t i c S t r a i n s v s . Depth f o r  . . . . .  Electric Potential  113 113  f = 165 MHz  116  4.13  R e a l P a r t o f E l e c t r i c F i e l d s v s . D e p t h f o r f = 165 MHz  4.14  R e l a t i v e E l e c t r o - o p t i c and P h o t o e l a s t i c C o n t r i b u t i o n s t o . the Overlap  99  .  I n t e g r a l f o r f = 1 6 5 MHz  117  119  4.15  g as a F u n c t i o n o f Frequency  120  4.16 4.17 4.18  20X E n l a r g e m e n t o f T r a n s d u c e r P h o t o l i t h o g r a p h y M a s k . . . Acousto-Optic Deflector (Actual Size) Transducer f a i l u r e  121 122 123  4.19  Transducer  124  4.20  Raman-Nath D i f f r a c t i o n o f a G u i d e d TE wave ( t h e u p p e r arid l o w e r s p o t s on t h e l e f t a r e t h e d i f f r a c t e d b e a m s ; t h e l a r g e s p o t i s t h e u n d i f f r a c t e d TE mode, a n d t h e s m a l l s p o t on t h e r i g h t i s a TM mode)  Failure  4.21  U n d i f f r a c t e d TE ( l e f t ) o  a n d TM  4.22  Same w i t h r f D r i v e S w i t c h e d  viii  o  Modes  on (n  ^0.4)  125 126 126  Figure  Page  4.23  Frequency Response of D e f l e c t o r D i f f r a c t i o n E f f i c i e n c y  4.24  Broadband Response (experiment o n l y )  4.25  Diffraction  4.26  Beam S t e e r i n g  4.27  Acousto-Optic Deflector Drive  4.28  Transmission Line Reflections  135  4.29  TE  135  4.30  Diffraction  4.31'  Diffraction Efficiency  Modes o f  with V  f  Efficiency IDT  as  of A c o u s t i c  Power  . .  M a s k (20X)  Efficiency  130 132  Circuit  ....  Beam  vs.  128 129  a Function  the D e f l e c t e d  = 3.8  .  Frequency  a t f = 200  MHz  133  136 vs. Bragg  Frequency  V rms  138  4.32  D e f l e c t o r Bandwidth vs.  Bragg Frequency w i t h V  4.33  Peak D i f f r a c t i o n E f f i c i e n c y — Bandwidth T r a d e o f f w i t h V = 3.8 V rms C o m p a r i s o n o f R e s p o n s e o f P h a s e d - A r r a y and Conventional  139  Bragg D e f l e c t o r  139  f  = 3.8  V rms  f  4.34  4.35  Acoustic  Power v s .  f  4.36  Deviation  4.37  Diffraction Efficiency (f = 150 MHz)  138  141  from Bragg Angle vs. vs.  f  141  f for Several Drive  Voltages 142  4.38  Acoustic  Power v s . F r e q u e n c y  143  4.39  L i g h t D e f l e c t o r Beam P r o f i l e s  5.1  Artwork Ruling Apparatus  5.2  Cutter  5.3  L i f t i n g of Photoresist  148  5.4  Shorted Transducer  150  5.5  P h o t o r e s i s t P a t t e r n n e a r IDT  5.6  P o r t i o n o f Beam S t e e r i n g E x p e r i m e n t s (2000X) . .  (n^ ^  .9)  143  . .  147 . .  147  Center  Transducer used i n  150 the 151  ix  Figure  Page  5.7  Lifting  o f Aluminum  152  5.8  Photolithography  5.9  C o r r e c t M a s k A l i g n m e n t f o r Z-21.8" SAW ( t a n 21.8° = .4) .  S t a t i o n i n L a m i n a r F l o w Hood  x  153  Propagation 154  NOMENCLATURE Chapter  2.  A,B  amplitude  Ai  Airy function  c  velocity  d  OWG  D  gap b e t w e e n OWG a n d p r i s m  E  electric  k  OGW p r o p a g a t i o n v e c t o r  H  l e n g t h o f c o u p l i n g r e g i o n on p r i s m  m  OGW mode  n n n  a e m  c o e f f i c i e n t s o f OWG TE modes  of light  i n free  thickness  r e f r a c t i v e index of a i r extraordinary index , OGW mode  index  n  prism coupler  s  base  index  i n d e x a t g r a d e d - i n d e x OWG  n  coupler  field  n^ P  space  surface; also ordinary index  index  substrate index  P  p o w e r / u n i t w i d t h c a r r i e d b y a g u i d e d TE mode  V  phase v e l o c i t y o f l i g h t  i n substrate  W  width of incident light  beam  Wp  w i d t h o f l i g h t beam o n p r i s m b a s e  y  WKB t u r n i n g p o i n t  y  m  last  Q  z e r o c r o s s i n g o f WKB  solution  n o r m a l i z e d WKB t u r n i n g p o i n t a a  prism c  angle  prism coupler radiation-loss  coefficient  3  z c o m p o n e n t o f OGW p r o p a g a t i o n v e c t o r  n  prism coupler  efficiency  xi  (Chapter  2.)  9 9  a n g l e of l i g h t  i n c i d e n c e on p r i s m base  i n t e r n a l angle of i n c i d e n c e of OGW  m  X  l i g h t wavelength  y  .  a n g l e of l i g h t  i n free  space  i n c i d e n c e on couping p r i s m  y^  p e r m e a b i l i t y of f r e e  a)  a n g u l a r frequency of l i g h t waves  Chapter  on waveguide s u r f a c e  space  3.  a  IDT e l e c t r o d e m e t a l l i z a t i o n  A  IDT SAW  f i n g e r width (Section 3.4) amplitude ( S e c t i o n 3.5)  A, ,B, kq kq  SAW  p a r t i a l wave amplitudes  b  constant i n p a r a b o l i c v e l o c i t y surface approximation  B  IDT r a d i a t i o n  a  susceptance  c..,.. xjkl  elastic  C,j,  IDT  D_^  electric  e. ljk  p i e z o e l e c t r i c constants  E. x  SAW  electric  f  SAW  frequency  f  IDT c e n t e r frequency  G G  s t i f f n e s s constants  s t a t i c capacitance displacement  field  IDT r a d i a t i o n  a  conductance  IDT r a d i a t i o n conductance  o  J,  first  K  SAW  K(q),K(q)  complementary e l l i p t i c  L  s e r i e s and p a r a l l e l  1  s  ,L  p  factor  at f  o  o r d e r B e s s e l f u n c t i o n of the f i r s t  kind  wavevector i n t e g r a l s o f the f i r s t  IDT matching i n d u c t o r s  xxi  kind  3.) number o f IDT f i n g e r p a i r s a c o u s t i c power e l e c t r i c a l p a r t o f a c o u s t i c power m e c h a n i c a l p a r t o f a c o u s t i c power a c o u s t i c and e l e c t r i c a l IDT q u a l i t y f a c t o r s Fresnel  distance  IDT  radiation resistance  IDT  finger series resistance  r a d i a t i o n resistance at f o sin(x)/x acoustic IDT  strain  m e t a l l i z a t i o n thickness  acoustic  stress  p a r t i c l e displacement from e q u i l i b r i u m SAW v e l o c i t y r.m.s. v o l t a g e IDT  radiating  radiation  aperture  reactance  complex impedance Av/v, the change i n SAW v e l o c i t y when the s u b s t r a t e s u r f a c e i s covered w i t h an i d e a l , m a s s l e s s conductor. permittivity  constants  SAW wavelength SAW a n g u l a r  frequency  angle between SAW p r o p a g a t i o n v e c t o r and the power flow d i r e c t i o n SAW e l e c t r i c p o t e n t i a l f i e l d  xiii  (Chapter  3.)  p p  d e n s i t y i n e q u a t i o n 3.4 r e s i s t i v i t y i n e q u a t i o n 3.44 sheet  g  resistivity  4.  Chapter A  width of d i f f r a c t e d  B  inverse dielectric  D  w i d t h o f r a d i a t i n g a p e r t u r e o f o n e s e c t i o n o f beam -steered transducer  E-  OGW  •*»r E. l  r e a l p a r t o f SAW e l e c t r i c  field  E.(y) l  amplitude  o f SAW e l e c t r i c  field  f  acoustic  frequency  f  Af  ,f  electric  l i g h t beam a p e r t u r e permittivity  tensor  f i e l d vector  h i g h and l o w f r e q u e n c i e s a t w h i c h i s matched  the Bragg  angle  d e f l e c t o r Bragg bandwidth  m  f  IDT a r r a y  G  w i d t h o f one s e c t i o n o f b e a m - s t e e r e d IDT, i n c l u d i n g electrodes  H  b e a m - s t e e r e d IDT s t e p h e i g h t b e t w e e n a d j a c e n t s e c t i o n s  i  electric  k~Q  i n c i d e n t OGW  ic •  wavevector  K  SAW  £  '  frequency.  current wavevector  of first-order diffracted  beam  wavevector  d i f f r a c t i o n o r d e r number  L  acousto-optic interaction  length  m  TE mode  M  n u m b e r o f s e c t i o n s i n b e a m - s t e e r e d IDT  n  r e f r a c t i v e index  N  superscript denoting  index  , normalized xiv  4.) number o f r e s o l v a b l e s p o t s acoustic  power  e l e c t r i c a l power d i s s i p a t e d a c o u s t i c p o w e r f o r 100% photoelastic  in  IDT  diffraction  efficiency  constants  i n t e g e r number o f A / 2 Q  steps i n H  r e a l p a r t of a c o u s t i c s t r a i n amplitude  of r e a l part of a c o u s t i c s t r a i n  normalized  OGW  electooptic radius  electric field  as a f u n c t i o n o f  depth  constants  vector  magnitude of  impedance  acousto-optic overlap permittivity  integral  change f o r a c o u s t o - o p t i c  d i f f e r e n c e b e t w e e n i n c i d e n t and angular w i d t h of d i f f r a c t e d relative permittivity substrate relative  interaction  Bragg  angles  l i g h t beam  tensor  permittivity  acousto-optic d i f f r a c t i o n beam-steered d e f l e c t o r  efficiency  diffraction  efficiency  d i f f r a c t i o n e f f i c i e n c y when A9=0 Bragg  angle  light  angle  angle  of propagation  o f i n c i d e n c e t o SAW of f i r s t  planes  order d i f f r a c t e d  complex t r a n s m i s s i o n l i n e r e f l e c t i o n transit  of constant  coefficient  t i m e o f s o u n d wave a c r o s s l i g h t beam  Bragg angle  as a f u n c t i o n o f  xv  beam  frequency  phas  (Chapter 4 . ) <})^(f)  th coupling constant for I diffraction order  <j>(f)  angle of propagation of principal maximum of array function  CD  angular frequency of incident light  o  angular frequency of first-order diffracted light 0,  SAW angular frequency  xvi  ACKNOWLEDGEMENT  I  t h a n k my s u p e r v i s o r ,  guidance during  D r . L. Young, f o r h i s s u p p o r t and  the course of t h i s research.  Mr. A r v i d L a c i s d i d t h e  scanning e l e c t r o n microscopy and e l e c t r o n microprobe a n a l y s i s . s u g g e s t i o n s and a s s i s t a n c e were r e c e i v e d Peter  Musil  Mr.  measurements.  Professor  R. B u t t e r s  helped with  the x-ray  S p e c i a l , t h a n k s a r e due t o D r . E.V. J u l l a n d  H a n s Hogenboom f o r n u m e r o u s h e l p f u l d i s c u s s i o n s , a n d t o M i s s  Louie for typing Council is  f r o m Mr. Rodger Bennet and Mr.  on t h e c u t t i n g a n d p o l i s h i n g o f c r y s t a l s , a n d f r o m M r . J a c k  S t u b e r i n t h e machine shop. fluorescence  Helpful  t h e t h e s i s ; F i n a n c i a l s u p p o r t of. t h e N a t i o n a l  o f Canada, through a S c i e n c e S c h o l a r s h i p  g r a t e f u l l y acknowledged.  xvii  Sannifer  Research  a n d a l s o G r a n t No. A 3 3 9 2 ,  1.  1.  The  p r i n c i p a l o b j e c t i v e o f t h i s t h e s i s was t o do a c o m b i n e d  t h e o r e t i c a l and e x p e r i m e n t a l of improving  INTRODUCTION  study  o f a c o u s t i c beam s t e e r i n g a s a means  the performance of planar acousto-optic  light deflectors.  I n an i n c r e a s i n g l y d i g i t a l a g e , i n t e g r a t e d o p t i c s and s u r f a c e wave (SAW) d e v i c e s competitive. deflectors  a r e among t h e f e w a n a l o g  technologies  acoustic  likely  I n a d d i t i o n t o p o t e n t i a l . a p p l i c a t i o n s as l i g h t  t o remain  switches,  and m o d u l a t o r s , t h e d e v i c e s  studied here promise the r e a l i z a -  t i o n o f more c o m p l e x s i g n a l p r o c e s s i n g  f u n c t i o n s such as c o n v o l u t i o n and  s p e c t r a l a n a l y s i s on an e s s e n t i a l l y r e a l - t i m e b a s i s [ 7 5 ] . The and  technology  modulation of l i g h t  geometry u t i l i z i n g  acousto-optic  i s reasonably  devices  w e l l developed  i n C h a p t e r 2.  h i g h q u a l i t y o p t i c a l w a v e g u i d e s (OWG)  design  The p l a n a r  efficient  counterparts.  of selected topics of the theory  waveguides i s presented  given.  [92].  t h a t a r e p h y s i c a l l y s m a l l e r , more  t h a t have h i g h e r performance than t h e i r b u l k A review  f o r the d e f l e c t i o n  s u r f a c e o p t i c a l w a v e g u i d e s and a c o u s t i c s u r f a c e waves  promises to give devices and  of bulk  of d i e l e c t r i c  Techniques f o r the f a b r i c a t i o n of i n LiNbO^ by n i c k e l i n d i f f u s i o n a r e  Methods f o r c o u p l i n g t o o p t i c a l waveguides a r e d i s c u s s e d , and f a b r i c a t i o n  of r u t i l e  coupling prisms i s described.  m e a s u r e d p r o p e r t i e s o f a 6-mode OWG' a r e g i v e n a n d t h e e l e c t r i c t r i b u t i o n s of the three  guided  theory  i s described.  i n t e r a c t i o n i s presented field  field  The t h e o r y o f t h e s u r f a c e w a v e i n C h a p t e r 4.  Expressions  dis-  method.  transducers  a c o u s t i c s u r f a c e waves on L i N b O ^ a r e r e v i e w e d , a r i d an e x p e r i m e n t  and t h e  The  TE modes a r e c a l c u l a t e d b y t h e WKB  I n C h a p t e r 3, t h e p r o p e r t i e s o f i n t e r d i g i t a l  the  slab  and  testing  acousto-optic  describing the f a r -  a c o u s t i c r a d i a t i o n p a t t e r n of beam-steered i n t e r d i g i t a l  transducers  2.  are developed, as are equations  for predicting  t e r i s t i c s o f p l a n a r beam s t e e r e d efficiency  i s calculated  light  from f i r s t  the performance  deflectors.  The  p r i n c i p l e s , using  and e l e c t r o o p t i c p r o p e r t i e s o f L i N b O ^ and t h e d e t a i l e d OGW  a n d SAW  radiation fields.  r e s u l t s a r e compared w i t h I n C h a p t e r 5,  charac-  diffraction  the  photoelastic  description  Several experiments are described,  theoretical  techniques  and f o r f a b r i c a t i n g i n t e r d i g i t a l  of the and t h e  calculations.  f o r making h i g h r e s o l u t i o n  transducers  are  discussed.  photomasks  2.  DIELECTRIC OPTICAL WAVEGUIDES  2.1 Uniform Dielectric Slab Waveguides Consider the asymmetrical dielectric slab illustrated i n Fig. 2.1. The refractive indices in the three regions indicated are related by the inequality n where n  m  m  >n >n, s a  i s the mode index of the mth guided optical mode. If we re-  s t r i c t our consideration to guided TE waves propagating i n the z direc tion, the wave equation reduces to 8E 2  9E 23E _. x _ n x 2 ~ 2 2 * 2  x 2  D?/  9y  2  Z  c a t .  For time harmonic fields with propagation constant 3 in the z direction, the wave equation becomes  \ 22 2 — = i + (n k - 3 )E =0 y d  d  Fig. 2.1 Slab Dielectric Waveguide.  (2.  4.  Boundary c o n d i t i o n s r e q u i r e t h a t y = d, and  that E  and  v a n i s h a t y = ±°°.  .  where  x  continuous at y = 0  Solutions s a t i s f y i n g these  m e n t s h a v e b e e n shown [ 1 ] t o t a k e t h e E  be  require-  form  = A exp(Sy) ,  y < 0  = A c o s (icy) + B s i n ( i c y ) ,  0 < y < d  =  d < y < °°  .  and  (A c o s ( K d ) - B s i n ( K d ) ) e x p [ - y ( y + d ) ] ,  (2.2)  2  2 2 1/2 - n k ) a . 2,2 2.1/2 ic = ( n k - g ) m r / 2 2 2,1/2 Y = [(n - n ) k - K ] 6 =  (g  ta  fl  ,  2  X 1  m  g  and w h e r e k i s t h e w a v e v e c t o r matically  i n f r e e space.  n  N  T h i s s o l u t i o n i s mathe-  i d e n t i c a l w i t h t h a t f o r t h e quantum m e c h a n i c a l p r o b l e m  p a r t i c l e i n c i d e n t on a s q u a r e p o t e n t i a l  „ (2.3)  of  a  well.  C o n t i n u i t y r e q u i r e m e n t s on H^, g i v e t h e e i g e n v a l u e e q u a t i o n 2 t a n ( i c d ) = K(Y  + <5)/(K  - yS),  t h e r e b y l i m i t i n g wave p r o p a g a t i o n t o a d i s c r e t e s p e c t r u m o f g u i d e d modes. I n a d d i t i o n , i t can be  shown [ 1 ] t h a t t h e w a v e g u i d e a l s o s u p p o r t s a  t i n u o u s s p e c t r u m o f r a d i a t i o n modes, w h i c h  form a complete  s e t t o g e t h e r w i t h t h e d i s c r e t e modes.  propagation v e c t o r of  g u i d e d wave h a s  the magnitude  g = n  m  The  E x a m i n a t i o n o f e q u a t i o n (2.3) when g < n^k; f i n e m e n t i s no wave g u i d a n c e .  as a r e s u l t ,  M  the  (2.4) i s t h e a n g l e b e t w e e n "fe and  r e v e a l s t h a t y becomes  imaginary  t h e g u i d e d mode b e c o m e s r a d i a t i v e a n d  longer possible. The  orthonormal  ksine . m  a l o n g the d i r e c t i o n o f p r o p a g a t i o n , where 9  con-  con-  Thus, y = 0 i s the c u t o f f c o n d i t i o n f o r  equation t a n ( K d ) = 6/K c  (2.5)  y.  can then be used to determine the minimum thickness d^ that w i l l support a particular guided mode. It i s interesting to note that a symmetrical Waveguide (n = n ) always has at least one guided mode [1]. As the •cl  S  guiding layer becomes thinner, proportionately more power i s carried by .the evanescent fields. A more intuitive treatment of wave guidance i s based on a rayoptic approach to light propagation in the waveguide.  Consider a wave  in the guiding layer incident on the air-waveguide interface at an angle 6 .  m  Fig. 2.2 Zig-Zag Wave Propagation. Snell's law i s n sin9 = n sin9 . m m > a a When n sin9 > n , c r i t i c a l internal reflection occurs and the wave m m a cannot escape from the waveguide.  At the lower interface, the equivalent  condition i s n  m  sin9 > n . m s  Since n > n , satisfaction of the second inequality implies s a. satisfaction of the f i r s t . obtained:  Thus, three kinds of propagating modes are  6.  and  (1)  a i r m o d e s , when n s i n G m  (2)  s u b s t r a t e modes, when n  (3) guided  modes when n  6  Propagation  o f guided  m  < n , m a > n  s  sinB  m  siriG  > n , a  m  > n . s  m  modes i s p o s s i b l e o n l y when t h e m u l t i p l e r e f l e c t i o n s  of p l a n e waves from t h e waveguide s u r f a c e s a r e i n phase. the eigenvalue  This  imposes  condition 2kn  m  d cos6  m  - 2d) ms  - 2<J> ma  T  = 2mTr ,  (2.6)  w h e r e d> a n d d> a r e t h e phase s h i f t s on r e f l e c t i o n a t t h e a i r and s u b ms ma strate  interfaces, respectively.  The v e r t i c a l  component o f t h e g u i d e d  mode f o r m s a s t a n d i n g wave b e t w e e n t h e w a v e g u i d e b o u n d a r i e s , pagation  appears t o be i n t h e h o r i z o n t a l d i r e c t i o n For  so t h a t  pro-  only.  a l i m i t e d w a v e g u i d e t h i c k n e s s d , o n l y a l i m i t e d number o f  integer values o f m w i l l  satisfy  t h e phase m a t c h i n g c o n d i t i o n (2.6).  f a c t , when t h e a p p r o p r i a t e e x p r e s s i o n s into  (2.6),  theeigenvalue  equation  2.2  Modes i n G r a d e d I n d e x W a v e g u i d e s  and  d>  m a  are substituted  (2.3) i s o b t a i n e d .  D i f f u s i o n i s a convenient o p t i c a l surface waveguides.  f o r d>  In  technique  f o r making h i g h  I t i sp a r t i c u l a r l y  quality  advantageous f o r acousto-  o p t i c d e v i c e s , s i n c e a h i g h degree o f o v e r l a p between t h e a c o u s t i c s u r f a c e wave and o p t i c a l f i e l d s i s p o s s i b l e . guided  modes i s c o n s i d e r a b l y m o r e d i f f i c u l t ,  r e f r a c t i v e index p r o f i l e diffusion,  obtained.  s  owing t o t h e non-uniform  As a r e s u l t o f e i t h e r i n - o r o u t -  the r e f r a c t i v e index near t h e surface takes n(y)  where n  However, t h e d e s c r i p t i o n o f  = n  g  + An f ( y )  i s t h e s u b s t r a t e i n d e x , An = n  the s u r f a c e , and t h e exact  o  - n  s  t h e form (2.7)  i s t h e change i n i n d e x a t  f o r m o f f ( y ) depends on t h e d e t a i l s o f t h e  7.  diffusion process.  Waveguides with Gaussian [3], erfc [3,4] and Fermi  function refractive index profiles have been reported. In general, the solution for the electric field of a graded index waveguide takes the form  E(y,z,t) = E (y)  j ( p Z _ a ) t )  x  e  .  (2.8)  Substitution into the one-dimensional wave equation gives 2 — 5 + « (y) dy 2  = 0,  2 E  2 2 2 where K (y) = n (y)k -  (2.9)  For most index profiles of interest, solution i n terms of known functions i s not possible.  Exact solutions have been obtained for an  exponential permittivity profile by Conwell [6j and a piecewise-linear permittivity profile by Marcuse {Vj . Otherwise, i t i s expedient to solve instead the equation dE  2  2  — 2 * + [ K ( y ) - S<!yJ  -~'(—^—)]E  dy  dy  2  x  = 0,  /ic(y)  solutions of which closely approximate the exact solutions under appropriate conditions.  This i s the WKB method of Quantum Mechanics, for  which the following solutions for TE guided modes [7,9] are obtained: E  where  = A exp(yy)  y < 0  E  Y  = BA IK(y) A cos (<f)(y) - TT/4)  E  x  =  y  /K /<(y) A exp{/-|ic(y) dy} o  = g' - k , ' m 2 2 2 2  y  2  2  m  (2.10)  >y > 0  °° > y > y  (2.11) m  (2.12) (2.13)  2  <o " V " 1> 3  -  (2 14)  8.  K (y) = n ( y ) k 2  2  <Ky) = /  2  - f3  2  ,  (2.15)  k(y)dy ,  7 m  (2.16)  y A and B a r e c o n s t a n t s , k i s t h e w a v e v S c t o r i n f r e e space, n  Q  i s the sur-  f a c e i n d e x a n d 6^ i s t h e z - c o m p o n e n t o f t h e p r o p a g a t i o n v e c t o r o f t h e TE mode. m  The s o l u t i o n i s o s c i l l a t o r y o v e r J  decays t o z e r o on e i t h e r The  the i n t e r v a l  (0,y ) and m ,J  side.  t u r n i n g p o i n t o f t h e m t h mode i s y ^ , d e f i n e d b y K(y  ) = 0;  marks t h e d i v i s i o n between o s c i l l a t o r y and e x p o n e n t i a l b e h a v i o u r . The  WKB m e t h o d i s o n l y u s a b l e  s l o w l y w i t h y.  (Schiff  [ 8 ] ) i f K(y) c h a n g e s  The c o n d i t i o n f o r t h i s may b e w r i t t e n a s / 1-^-1 1  «  condition i s violated,  (2.12) have v a l i d i t y o n l y i n an a s y m p t o t i c This r e s t r i c t s  f o r l o w e r o r d e r modes. case  2  1  dy  At the t u r n i n g p o i n t (y ) , t h i s  lengths o f y .  C - ?)  k/x  1  and t h e s o l u t i o n s  sense w i t h i n s e v e r a l wave-  the accuracy o f s o l u t i o n ,  particularly  G o n w e l l , 'however ha& s t a t e d i t-frat- even i n t h i s  the solutions obtained are often of reasonable  o r d e r modes, g o o d a c c u r a c y  accuracy.  For higher  i s o b t a i n a b l e .(j.sSpec-ia.1 i t e ' c h n i q u e s m u s t be  usjed ^ %hejevic,±ntifey p€ t h e «tjii?ndng ppjtn%,xjiw.he-rerith:er e i s a p o l e i n E . m' x x s  iS  J  Hocker and Burns  >  t  [ 1 0 ] h a v e shown t h a t t h e modes o f d i f f u s e d  waveguides can be d e s c r i b e d by j u s t i o n depth which mode i n d e x ,  ,  r  v  two q u a n t i t i e s , a n e f f e c t i v e  diffus-  can c o n v e n i e n t l y be chosen t o be y , and an e f f e c t i v e n  Marcuse  = 3 /k . m m  [9] has a n a l y s e d  t h e TE modes o f g r a d e d  (2.18) index slab  w a v e g u i d e s w i t h t h e WKB m e t h o d a n d a p i e c e w i s e - l i n e a r a p p r o x i m a t i o n o f  9  Fig.  the  2.4  Z i g - Z a g Wave P r o p a g a t i o n i n G r a d e d I n d e x W a v e g u i d e s .  index p r o f i l e .  The  s o l u t i o n s w e r e f o u n d t o be  with  exact s o l u t i o n s of  the p i e c e w i s e - l i n e a r  less  cumbersome m a t h e m a t i c a l l y Comparison of eqs.  form s l a b waveguide r e v e a l s graded-index s o l u t i o n s . required.  The  to apply,  since  gradual that  the  rather  i n close  agreement  permittivity profile,  (2.10)-(2.16) w i t h the  greater  t h e TE  modes o f  mathematical complexity  N u m e r i c a l methods o f a n a l y s i s are  "reflection"  on  a of  reflection  ( F i g . 2.4). i s very  Tien  nearly  unithe  usually difficult  from the b o t t o m waveguide s u r f a c e  than instantaneous  phase s h i f t  '  ( f i g . 2.3).  r a y - o p t i c p i c t u r e o f wave g u i d a n c e i s a l s o more the  and  is  e t a l [5] h a v e shown  TT/4  at y  m  and  ir/2 a t  10.  y = 0. ing  Using  t h i s , White and H e i d r i c h developed  the turning points y  V  n  2  n^.  m  Equation  <j>(0) =  o f t h e WKB m e t h o d , g i v e n t h e mode  r / 2 2,1/2 / (n (Z) - n ) 0  4m - 1 dZ - — - g -  m  e  n  A r j  S  m  Z  7 - 7 m " m-1  Z  Z  A-i  +  >  n  f o rthe t u r n i n g points  l  16  =  J. 3 ¥  n /  (  m-l  +  2  }  3 n  m -l/2 N  2  k  1/2  A  (  > "  /  (  n  o -  l  n  »  }  V l "V  (  m  =  1  (2.19)  )  , - 1 / 2 , ,4m-!, 2 ~8~ " 3  "V  r  ]  { (  1  )  k=l  A-l,  k-l  k m = 2, 3 ... M  where Z  o  =0  of  mode:  r  +  Use o f t h e p i e c e w i s e - l i n e a r  gives the following solution  t h e mth g u i d e d  indices  Ji;Q  (2.16) then t a k e s t h e form  w h e r e m = 1 , 2 ... M, a n d w h e r e Z = y / A . approximation  a method o f d e t e r m i n -  and n  These e q u a t i o n s  o  = n(0).  (2.20)  N o t e t h a t m = 1 f o r t h e z e r o t h o r d e r mode,  r e q u i r e an e s t i m a t e o f t h e s u r f a c e i n d e x n  Q  as w e l l as  t h e mode i n d i c e s n ^ , s o t h a t s u c c e s s i v e Z's may b e c a l c u l a t e d b y iteration. In  order to determine  t h e sum o f s q u a r e s  whether the estimate of n  o f t h e second  differences,  °k+2 ~ " k + l M-2  = .1 k=0  is  calculated.  file,  Z  k+2  Z  k+2  Z  +  T h e minimum i n r  Z  n  k+1  Z  k+1  Z  k+l  +  corresponds  and the corresponding v a l u e o f n  Q  " "k  k+1  k+1  i s reasonable, o  \ Z  (2.21)  k  to the smoothest index  pro-  was somewhat a r b i t r a r i l y s e l e c t e d  by W h i t e and H e i d r i c h t o be t h e b e s t one.  11.  The  index p r o f i l e o b t a i n e d can then be used w i t h  (2.16) t o c a l c u l a t e  the e l e c t r i c  field  d i s t r i b u t i o n s o f t h e modes,  e i t h e r by assuming a p i e c e w i s e - l i n e a r p r o f i l e (i.e.,  2.3  erfc) to the points  Coupling  owing p r i n c i p a l l y  o p t i c a l modes p r e s e n t s  to the small (often less Initially,  o f l i g h t beams. i n exciting  The Schubert  lenses  some  difficulty,  t h a n 10 ym) d i m e n s i o n s  of  ( F i g . 2.5) w e r e u s e d t o r e d u c e  Low c o u p l i n g e f f i c i e n c y  g u i d e d modes made t h i s  and a l a c k  of  an u n s a t i s f a c t o r y method.  p r i s m c o u p l e r , announced by T i e n and a l s o H a r r i s and  i n 1969, overcame these l i m i t a t i o n s .  a p r i s m w i t h r e f r a c t i v e i n d e x n , where n it  a function  obtained.  t o guided  o p t i c a l waveguides.  selectivity  o r by f i t t i n g  t o O p t i c a l Waveguides  Coupling  the diameter  (2.10) t o  i s critically  internally  reflected  I n F i g . 2.6, l i g h t  > n , a t an a n g l e  a t the prism base.  y such  that  I f the spacing  b e t w e e n p r i s m a n d t h e o p t i c a l , w a v e g u i d e i s a b o u t A/2 o r l e s s , l i g h t energy i n t o the waveguide i s p o s s i b l e , through  enters  coupling of  overlap of the  e v a n e s c e n t f i e l d s o f t h e r e f l e c t e d l i g h t o u t s i d e t h e p r i s m base and g u i d e d modes o f t h e f i l m .  Mathematically,  t r a n s f e r ..resembles t h e q u a n t u m m e c h a n i c a l a potential barrier; ing.  Coupling  of l i g h t expressed  consequently,  can only occur  t h i s mechanism o f energy tunneling of a particle  i ti s frequently called o p t i c a l  i fthe horizontal e l e c t r i c  a t t h e p r i s m b a s e m a t c h e s t h a t o f a g u i d e d mode.  field  index n  m  m  b y t h e p h a s e m a t c h i n g c o n d i t i o n f o r t h e mth g u i d e d  component  mode:  i s t h e h o r i z o n t a l component o f t h e g u i d e d w a v e v e o t o r . i s  tunnel-  T h i s may b e  3 = k n sine , m p m' where $  through  (2.22) The mode  12.  n  m  = ~ k  = n  p  sin6  .  Tm O  Thus, I t i s p o s s i b l e t o s e l e c t i v e l y guided  mode b y v a r y i n g t h e a n g l e  base, provided different.  A number o f t h e o r e t i c a l  case o f a uniform a uniform able  e x c i t e any one  of incidence.0  that the indices of adjacent  have been p u b l i s h e d  (2.23)  [2,9,16,17].  o f l i g h t on t h e p r i s m  modes a r e s u f f i c i e n t l y  treatments  of coupling  Most o f these  deal with  o r G a u s s i a n l i g h t beam i n t e n s i t y  i s about 80%.  By a l t e r i n g  o r t a p e r i n g t h e gap b e t w e e n w a v e g u i d e a n d p r i s m , theoretically  i n d e x must be v e r y  For high  closely  the prism  t h e beam p r o f i l e a n d /  100% e f f i c i e n c y i s experimentally.  coupling efficiency,  to a  t h e mode a n d  matched. i s h e l d i n c l o s e p r o x i m i t y t o t h e wave-  g u i d e b y t h e use o f an a d j u s t a b l e  clamp.  the m e c h a n i c a l c o n t a c t between p r i s m o v e r 25% a r e d i f f i c u l t  into  attain-  [9] g i v e s a method o f e s t i m a t i n g t h e c o u p l i n g e f f i c i e n c y  In practice,  1 mm  the special  p r o f i l e coupling  p o s s i b l e , and o v e r 90% h a s been a c h i e v e d  graded-index s l a b waveguide. prism  efficiency  d i e l e c t r i c s l a b w a v e g u i d e , i n w h i c h c a s e t h e maximum  coupling efficiency  Marcuse  particular  to achieve,  Because o f n o n u n i f o r m i t i e s i n  and g u i d e ,  particularly  coupling e f f i c i e n c i e s of f o r l i g h t beams more  than  i n diameter. A number o f o t h e r s s u c c e s s f u l o p t i c a l  loped.  couplers have been  Dakss e t a l [18] announced t h e g r a t i n g c o u p l e r  1970.  An o p t i c a l  resist  on t h e w a v e g u i d e s u r f a c e b y e x p o s u r e i n a l a s e r  A more r e c e n t  grating of p e r i o d i c i t y  fabrication  technique  ( F i g . 2.7) i n  0.67 ym was f o r m e d i n p h o t o interferometer.  employs s p u t t e r e t c h i n g o f t h e r e s i s t  p a t t e r n , thus etching the g r a t i n g i n t o matching c o n d i t i o n f o r g r a t i n g couplers B  deve-  the waveguide i t s e l f . is  = k s i n e + 2mTr/d,  The p h a s e ^ (2.24)  13.  Fig. 2.8  Generalized Leaky Wave Coupler.  14.  where d is the grating period and 0 the angle of incidence. Selective mode coupling i s again possible.  The maximum efficiency observed by  Dakss et al was 40%; with suitable techniques, 100% i s theoretically possible. The grating coupler is mechanically more stable than the prism coupler, and coupling over larger areas is easy to achieve.  Prism  couplers, however, have the advantage that they are not attached to the waveguide and can easily be moved to different orientations. At the OSA meeting on Integrated Optics in 1972, Tamir and Bertoni presented a unified theory of coupling to optical waveguides. If the medium adjacent to the waveguide surface has suitable structure, an optical waveguide i s capable of supporting the propagation of either a guided wave or a leaky surface wave. The leaky wave propagates with a field variation exp(i3' - a)z, where a i s an attenuation constant denoting the loss of energy to regions adjacent to the guide.  If a i s  suitable, this leakage w i l l give rise to a beam of light propagating out of the guide at an angle 0, determined by a phase matching condition similar to (2.23) or (2.24), depending on details of the structure adjacent to the guide.  Because the leaky wave decays exponentially, the  emerging beam has a non-uniform intensity profile, as illustrated i n Fig. 2.8. In general, the leaky wave can be supported by either multilayered (i.e., prism) or periodic (i.e., grating) structures.  By reci-  procity, light can be coupled into the waveguide by reversing the propagation direction; however, for maximum coupling efficiency, the incident beam intensity profile must have the shape illustrated in Fig. 2.8. Thus, for optimal coupling from a Gaussian or uniform beam profile, i t i s necessary to taper a along the direction of guided mode propagation such that the required input beam profile i s symmetric.  15.  2.4  Prism Coupler Design Prisms suitable for coupling to Ni/LiNbO^ diffused waveguides  were designed and made of rutile (crystalline TiO^), one of the few materials with higher refractive indices than LiNbO^.  Rutile i s bire-  fringent (uniaxial positive), with the crystal dip tic axis-coincident with the crystallographic c axis.  The highest possible mode index anti-  cipated was 2.4, based on Kaminow and Schmidt's results [3]. of rutile are n  g  The indices  = 2.582 and n_. = 2/86 [9,19], so coupling is possible  with any relative orientation of the prism and substrate crystal axes. Rutile has a hardness of 6 to 6.5 on the Mohs scale, sufficient to resist scratching by most common materials. Threeffactors must be considered in prism coupler design: the relative crystallographic orientation of prism and substrate, the magnitude of the prism angle, and the coupling efficiency.  For the  prism illustrated i n Fig. 2.9, the phase velocity matching condition n  m  =  3 k  = n sine p :m  must be satisfied before coupling to guided modes can occur. X»rlamajicti  The angles  ±n^ctriangle.AB.G' ar,e related by the equation i  c  0 = a + y, where a  (2.23)  i s the prism angle.  (2.24)  At the upper air-prism interface, Snell's  law i s sinu = n^ sxny so that n  = n sin[a + arcsin( =•) 1 m p n P2T /2 = cosa sinu + *n - sin u sina P  (2.25)  It is more useful to write this with u as the independent variable,  16.  AN  F i g . 2.9  F i g . 2.10  Prism Coupler Geometry.  Broadening of the Input Light Beam  y = arcsin[n  /~2~ 2 cosa - vn - n s i n a l . m p m J  x  (2.26) 7  I f we know the prism index and have an estimate of the guided mode index n^, we can use t h i s equation to c a l c u l a t e the range of prism angles a 7F  over which coupling i s p o s s i b l e .  This range i s such that - y < y <  IT  >  although angles o f incidence near the l i m i t s ± ir/2 are not usable because broadening of the Input l i g h t beam reduces the coupling e f f i c i e n c y . Fig.  From  2.10, we see that the f r a c t i o n of the l i g h t incident on the a c t i v e  coupling region of length JI i s  where W is the incident beam thickness.  Usually a lens of long (>20 cm)  focal length i s used to focus the laser beam to a small spot on the prism base.  The degree of beam convergence is usually small over the  dimensions of the prism (typically ^ 0.5 cm), so i t may be neglected i n (2.27).  In Fig. 2.11,  is plotted as a function of prism angle for  the case &/W•= 1, ri = 2.582 and 2.86, and n = 2.23. If maximum coupp m ^ ling efficiency is desired, i t i s necessary that the coupling length Z be greater than the projection of the light beam diameter i n the prism.  . ry= 2.86  .5  n =2.582  .4  p  .2 30°  40°  50°  60°  70  8 0°  Fig. 2.11 Effect of Beam Broadening on Coupling Efficiency (assumes £/W = 1, n = 2.23) m  IB.  The overall coupling efficiency i s _  t or  n  2.COS0COSU  c  c  TT  Wcosy  ^  o  P  = n t  where n  _  *  W  < £,  (2.28)  p  c  i s a factor dependent on prism, gap and waveguide parameters.  For coupling across a uniform gap of width D into TE modes of a gradedindex optical waveguide (OWG), Marcuse [9] obtained the radiation-loss amplitude coefficient  |A|V a  =  where y and K  a y  2  K  — 2ioy P{[(K a-y )sinh(yD)] •+ o o :  q  r  2  [y(0+K  r , )cosh(yD)] } 2 2 2 z  o 2  are given by (2.13) and (2.14), a  - k (n  (2.29)  - n^) and  2 2 (A[ B (2up P) i s determined by the mode normalization integral, o  P =  3  CO  2uu  ;  /  o  —  E* E dy x x  *  (2.30)  00  The parameter P i s the power per unit width carried by the mode. Marcuse normalizes the electric f i e l d by setting /Piou = 1 V/m for each guided o  mode. The coupling efficiency n n  c  i s then given by  = 1 - exp(-a JQ  (2.31)  c  In the derivation of (2.29), Marcuse assumed that the electric f i e l d in the gap consisted of plane waves, and that the input beam i n tensity profile was rectangular. The f i r s t assumption i s less accurate for lower order modes, whilst the second leads to overestimation of a c for Gaussian beams. The effect of changing the prism index i n (2.29) can be seen more clearly i f we take D 2= 20. | A| B a a =  Then 2 K  c  2a)y P(a+K ) o  O  .  (2.32)  The  prism index enters this  e x p r e s s i o n through  e n t i a t i o n w i t h r e s p e c t to a r e v e a l s t h a t a  a = k/n^  has  - n  .  m  Differ-  a maximum when n  c t h e OWG  surface index of r e f r a c t i o n .  exists  (as i n the case of r u t i l e ) ,  When a c h o i c e o f p r i s m  the lower  c o u p l i n g e f f i c i e n c y , provided, of course,  = n p  index w i l l  t h a t n^ >  , o'  index  give a  higher  n . Q  In d i f f u s e d waveguides, there i s u s u a l l y l i t t l e d i f f e r e n c e b e t w e e n t h e i n d i c e s o f modes o f a d j a c e n t l i n g angles modes and The  y  a c c u r a t e d e t e r m i n a t i o n o f t h e mode i n d i c e s m o r e  s e p a r a b i l i t y of individual- guided  efficiency  The  as a f u n c t i o n o f  t o beam b r o a d e n i n g  angle  the l i m i t i n g prism angles  b y 4T°  2.15  of i n c i d e n c e y < n  < a < 86°  m  < 2.5;  for n  =  TT/2;  then  2.86.  the lower,  the l i m i t s  = 2.582 and  d e s i r a b l e to s e l e c t light  to s i m p l i f y  coupling  34°  are p l o t t e d Equation  The for y  < a < 74°  against  (2.27)  upper curves =  for n  =  was are  From [ 3 ] ,  -TT/2.  to the p r i s m angle  P  In order  rutile,  i s greatest.  s o l v e d f o r a b y N e w t o n ' s m e t h o d f o r s i n y = ±1.  expect  a for  c o i n c i d e w i t h t h e maxima o f  mode i n d e x f o r t h e p r i s m i n d i c e s 2.582 and  we  (2.33)  t h a t i s , t h e mode s e p a r a b i l i t y i s l e a s t w h e n t h e due  the  m  m i n i m a o f F i g . 2.12 °  I n F i g . 2.13,  f o r an  difficult.  modes,  T h i s e x p r e s s i o n i s p l o t t e d i n F i g . 2.12  2.11;  individual  p r o v i d e s a measure o f  m  m  Fig.  coup-  cosa + n  an  = 2.23.  the  2 2 -1/2 (n - n ) sina P . rr. —2 , /,~2 n ( P. \ 2„ . 1 L l - (n c o s a - /n - n sxna) ] m p m  dy _=  m  Consequently,  are c l o s e t o g e t h e r , making the o b s e r v a t i o n of  d e r i v a t i v e o f y, w i t h r e s p e c t t o  with n  order.  are  given  2.86.  P  the  c a l c u l a t i o n o f mode i n d i c e s , i t i s  t h e c r y s t a l l o g r a p h i c o r i e n t a t i o n i n s u c h a way  i n the p r i s m propagates w i t h e i t h e r n * or n ,  * In the r e s t of t h i s s e c t i o n , n surface index of r e f r a c t i o n .  £  Q  the extreme  that  value  r e f e r s t o the o r d i n a r y r a t h e r than  the  20.  Fig. 2.12  OC Mode Separability vs. Prism Angle for n  of the extraordinary index.  = 2.23.  When the extraordinary ray axis l i e s along  the back corner of the prism (Fig. 2.14), TE waves propagate with the index n- and TM with n . This can be shown by making the substitutions e o J  n, = n_ = 1 1  n  o  = 2.582  n„ = n = 2.86 3 e and V  ?  = 0 in the phase velocity determinant (Appendix II) to obtain V + V = n V /c x y e  2  V + V  2  2  2  2  2  2  = n  2  4  V /c 4  (TE) (2.34) (TM)  Fig. 2.13 Limiting Mode Indices vs. Prism Angle for Rutile.  The phase velocity surface i s used here (rather than the ray velocity surface) because i t i s the phase of light incident at the prism base that determines whether coupling takes place. Since both equations (2.34) represent circles, the index of propagation does not vary with the angle of incidence. This then i s the preferred orientation when both TE and TM modes are to be excited. In Fig. 2.15, the c axis i s vertical. n  l "2  n  n  z  =n  =  %  The substitutions  Fig. 2.15.  c Axis Vertical.  23.  and V  = 0  give V  and  n  2  y  + V  2  ?  3  z  Z  2  z  = n  = n  2  4  o n  o  V /c  2  2 e  (TE)  2  4 2 V/c  f o r the p r o j e c t i o n s of the phase v e l o c i t y  (2.35) (TM)  s u r f a c e on t h e y z p l a n e . T h u s ,  TE modes p r o p a g a t e w i t h t h e o r d i n a r y i n d e x .  H o w e v e r , t h e i n d e x f o r TM  waves v a r i e s w i t h t h e a n g l e . o f l i g h t p r o p a g a t i o n i n t h e p r i s m .  In the  a c o u s t o o p t i c e x p e r i m e n t s d e s c r i b e d i n C h a p t e r 4, o n l y TE modes w e r e u s e d , s o t h i s o r i e n t a t i o n was degree  2.5  o f mode s e p a r a b i l i t y a n d  Coupler  I t p r o v i d e s a somewhat h i g h e r  coupling  efficiency.  Fabrication  Initially, by  satisfactory.  an a t t e m p t was  made t o c o n s t r u c t g r a t i n g c o u p l e r s  t h e method o f s i m u l t a n e o u s e x p o s u r e and development,  positive photoresist. and deeper first  PR-102  This technique, which gives reduced exposure  grooves w i t h s h a r p e r r i d g e s than c o n v e n t i o n a l methods,  u s e d b y T s a n g a n d Wang i n 1974 [2.0J  f r o m an a r g o n l a s e r i s s p l i t i n t o filtered,  was  I n F i g . 2.16,  two beams w h i c h a r e  t h e n p a s s e d t h r o u g h an o p t i c a l  the p h o t o r e s i s t - c o a t e d s u b s t r a t e .  times  t o make:.high q u a l i t y s u b m i c r o n  g r a t i n g s w i t h S h i p l e y AZ-1350 p o s i t i v e p h o t o r e s i s t .  spatially  u s i n g Gaf  light  individually  flat  to i n t e r f e r e  I t c a n b e shown t h a t t h e  at  grating  period i s (2.36) w h e r e n ^ i s t h e d e v e l o p e r i n d e x o f r e f r a c t i o n and d e n c e on t h e o p t i c a l  8^  the angle of  inci-  flat.  B e f o r e u s e , t h e d e v e l o p e r was filter  d i l u t e d 1:3 w i t h d e i o n i z e d  a n d p a s s e d t h r o u g h a .45  um  to reduce l i g h t  scattering.  p h o t o r e s i s t , d i l u t e d 1:1  w i t h reagent grade methyl e t h y l k e t o n e ,  water  The was  24.  Fig. 2.16  Photoresist Exposure for Grating Coupler Fabrication  s p i n coated be  a t 4000 RPM  a b o u t 2000 A°  f o r 10  seconds.  c o a t i n g s were measured  t h i c k w i t h a Sloan Angstrometer.  f o r 15 m i n u t e s , t h e p h o t o r e s i s t was During  The  exposure, the  After drying i n a i r  b a k e d a t 65°C f o r h a l f an  light  to  i n t e n s i t y was  hour.  m e a s u r e d t o be  about  2 60 mW/cm .  A s e r i e s of  i n g f r o m 10  s e c o n d s t o 10 m i n u t e s .  good r e s u l t s . before  The  was  i n a scanning  obtained.  u n d e r e x p o s e d p o r t i o n , and lower  T h r e e m i n u t e s was  times  found to  g r a t i n g s w e r e r i n s e d and b l o w n d r y w i t h  examination  o f 5400 A°  t e s t exposures were undertaken w i t h  e l e c t r o n microscope.  rang-  give  nitrogen  A grating period  F i g . 2.17  shows t h e  shallow  ridges of  an  F i g . 2.18  shows a c o r r e c t l y e x p o s e d a r e a  magnification. Unfortunately,  usable  g r a t i n g s were not  obtained,  primarily  t o a l a c k o f a d h e s i o n o f t h e p h o t o r e s i s t when d e v e l o p m e n t o f t h e was  carried  n o t be was 2i  through to the s u b s t r a t e  overcome by v a r i a t i o n  ordered em  x l i  f r o m NL cm  was  ( F i g . 2.19).  of process  I n d u s t r i e s i n New  d i a m e t e r and  transmittance  quite  had  s e v e r a l 5 mm  Jersey.  The  and  c o o l i n g to avoid  m a r k e d on  the  and  i t was  the  last  diced into  5 mm  t o p , w h i c h was  for surfaces  found best  about the  not  c axis v e r t i c a l .  a f i n e d i a m o n d saw. cement.  cracking.  w e t t e d e m e r y p a p e r g r a d e s was being  crystal  high.  s l i c e s were cut w i t h  the h o l d e r  could  a rutile  c r y s t a l was  due  grooves  a very pale yellow colour, although  held i n place with a thermoplastic heating  This problem  p a r a m e t e r s , so  A n u m b e r o f p r i s m s w e r e made w i t h t h e  on  at  cubes.  The  C a u t i o n was  The  crystal  required  s l i c e s were remounted  The  ground u n t i l  First, was  during flat  c r y s t a l o r i e n t a t i o n was last.  A succession  used to shape the p r i s m s ,  requiring polishing.  with  During  t o h o l d the p r i s m s by hand ( F i g . 2.20).  this  600  of grit  process,  26.  Fig.  2.17  Underexposed  Grating.  Fig.  2.18  C o r r e c t l y Exposed Grating.  Fig.  2.19  Result of Insufficient Photoresist Adhesion.  27.  Fig. 2.20  Rutile Coupling Prism (^8X).  Frequent examination for large scratches was necessary, as these were more easily removed with coarser grades of emery paper. After considerable grinding with 600 grit paper, the crystals were washed and dried.  A variety of polishing techniques were attempted  before a successful method was found.  Five micron diamond paste lubri-  cated with a light o i l on a napless polishing cloth revolving at 250 RPM gave good results.  After prolonged polishing, the crystal faces became  clear and polishing was completed with 1 ym diamond paste.  After clean-  ing, the vertical back surface was painted black to prevent light from passing directly through to the optical waveguide.  Prisms with angles  between 55° and 82° were made; for most coupling experiments, a pair of 68° prisms were used.  28.  2.6  O p t i c a l Waveguide F a b r i c a t i o n Several  techniques  were t r i e d  waveguides, i n c l u d i n g thermal ion.  o x i d a t i o n , r e a c t i v e s p u t t e r i n g and d i f f u s -  B e c a u s e t h e u l t i m a t e o b j e c t i v e was t o s t u d y  actions  i n L i N b O ^ , i t was n e c e s s a r y  t i v e index  t h a n 2.214. Initially,  tried,  Again,  acoustooptic  the thermal  o x i d a t i o n o f 1 ym t h i c k T i f i l m s was Even  prolonged  however, d i d n o t g i v e f i l m s w i t h l o w o p t i c a l a b s o r p t i o n , so  a p p r o a c h was abandoned.  argon-oxygen mixture  at pressures  by r e a c t i v e l y s p u t t e r i n g T i i n an  o f .01 a n d .002 t o r r , r e s p e c t i v e l y .  ( S p u t t e r i n g i n p u r e oxygen gave b r o w n i s h f i l m s s i m i l a r thermally oxidized titanium.) was d e p o s i t e d  i n appearance t o  A series of films of different  on a s i l i c o n s l i c e  ellipsometry.  i n order  thicknesses  t o measure t h e o x i d e  Comparison between t h e e x p e r i m e n t a l  ip, A c u r v e  thickness (Fig.  a n d c a l c u l a t e d t a b l e s g a v e a r e f r a c t i v e i n d e x o f 2.15 a t 6 3 2 . 8 nm.  2.21) This  refrac-  T i O ^ was o n e o f t h e f e w s u i t a b l e m a t e r i a l s .  B e t t e r f i l m s were obtained  by  inter-  t o use a m a t e r i a l w i t h higher  a t 800°C w i t h an o x y g e n f l o w r a t e o f 1 A / m i n .  treatment, this  f o r fabrication of optical  s u r p r i s i n g l y l o w v a l u e was m o s t l i k e l y , c a u s e d b y t h e p r e s e n c e o f  several oxides  of titanium i n the film.  a t 500°C i n c r e a s e d higher  the index  oxygen c o n c e n t r a t i o n s .  l i k e l y have i n c r e a s e d existing  Annealing  overnight  t o about 2.25, as d i d s p u t t e r i n g a t s l i g h t l y S p u t t e r i n g onto a heated substrate would  i t f u r t h e r , b u t t h i s was n o t f e a s i b l e  s p u t t e r i n g system.  i n oxygen  Even t h e b e s t  i nthe  f i l m s a p p e a r e d t o b e somewhat  l o s s y , however, so subsequent e x p e r i m e n t s were aimed a t t h e f a b r i c a t i o n of waveguides by d i f f u s i o n . Coupling was o b s e r v e d .  to guided  modes o f a 1 ym t h i c k T i O ^ f i l m o h g l a s s  The f i l m a b s o r p t i o n  appeared t o be q u i t e h i g h , s i n c e t h e  29.  Fig. 2.21  30.  b r i g h t s t r e a k c h a r a c t e r i s t i c of guided  w a v e s was  o v e r a d i s t a n c e o f l e s s t h a n one  centimeter.  human e y e  f i l m probably  30  i s a b o u t 27  dB,  so  the  strongly  The  attenuated  dynamic range of  had  the  l o s s e s i n excess  of  dB/cm [ l ] .  2.7  D i f f u s e d O p t i c a l Waveguides i n LiNbO^ Lithium niobate  ties  (Appendix I ) .  e l e c t r i c , with  has  an u n u s u a l c o m b i n a t i o n  I t i s b i r e f r i n g e n t (uniaxial negative)  a C u r i e t e m p e r a t u r e b e t w e e n 1100  stoichiometry.  of p h y s i c a l  I t has  and  and  proper-  ferro-  1080°C, d e p e n d i n g  r e l a t i v e l y large electrooptic, acoustooptic  coefficients,  visible  T h e s e c h a r a c t e r i s t i c s make i t a v e r y d e s i r a b l e m a t e r -  ial  f o r the  transmissivity in  and  non-linear optical spectrum.  as w e l l as h i g h  on  f a b r i c a t i o n of o p t i c a l waveguides, s i n c e t h e i r  c h a r a c t e r i s t i c s may  be  a l t e r e d by  electric  fields,  waves o r n o n - l i n e a r e f f e c t s t o p r o v i d e m o d u l a t i o n , and  mode c o u p l i n g o f g u i d e d The  first  by K a m i n o w a n d  < a <  .50.  magnitude of crystals times  The  i n 1973  [4,21].  Lithium niobate  r e d u c e d s u f f i c i e n t l y by  reported  crystals  r  was  1  found to a f f e c t  a pressure  64 h o u r s , t h e i n d e x  By  the  holding  o f 6 x 10  ^torr  i n a surface layer  for was  l i t h i u m o u t d i f f u s i o n t o produce e x c e l l e n t wave-  D i s c o l o r a t i o n of the further heating  crystals  r e s u l t i n g from t h i s  i n a i r f o r two  hours before  treatment  cooling.  an e r r o r f u n c t i o n d i f f u s i o n p r o f i l e , w i t h g u i d e d  e x t e n d e d as much as 100  ex-  >  degree of non-stoichiometry  b e t w e e n 21 and  waveguides had  deflection,  f.o.rm (Li-.Q,) (( N:b„0 ). , where .2 =a .2 .5 _ l - a  a t a t e m p e r a t u r e o f 1100°C a n d  c o r r e c t e d by  surface  switching,  the e x t r a o r d i n a r y ray i n d e x o f r e f r a c t i o n .  ranging  guides.  acoustic  waves.  h i b i t ' v a r i a b l e stoichiometrvy of t h e  0.48  propagation  d i f f u s e d waveguides i n t h i s m a t e r i a l were  Carruthers  the  um  o r more i n t o t h e  substrate.  was These  modes  that  31.  I n 1974, Kaminow and S c h m i d t ' r e p o r t e d t h a t t r a n s i t i o n metals a l s o produced ion  t i m e s w e r e much s h o r t e r  (as l i t t l e  t r o l was p o s s i b l e o v e r t h e s h a p e be v a r i e d  e x c e l l e n t waveguides  from complementary  the i n d i f f u s i o n of i n LiNbO^.  as a few h o u r s ) ,  Diffus-  and g r e a t e r  of the d i f f u s i o n p r o f i l e , which  con-  could  e r r o r f u n c t i o n t o G a u s s i a n , depending on  whether  a l l t h e m e t a l was d i f f u s e d i n f r o m t h e s u b s t r a t e  surface.  Their  results  i n d i c a t e d t h a t n i c k e l d i f f u s i o n i n L i N b O ^ was p a r t i c u l a r l y  rapid,  s o t h a t w a v e g u i d i n g l a y e r s s e v e r a l m i c r o n s deep c o u l d b e made a t 800° i n a few h o u r s . Even b e t t e r o p t i c a l waveguides s o l u t i o n LiNbO^-LiTaO^ epitaxy The  [12],  the  f i l m s grown on L i T a O ^ s u b s t r a t e s  A steeper  [5] from  i n d i c e s was g r e a t e r  (An ^ 0 . 0 7 compared w i t h  An ^ 0 . 0 1 ) .  prism  or grating couplers.  however, t h a t only  600°C.  2.8  Ti/LiNb0  t h e L i T a O ^ must be r e p o l e d ,  3  than  m e t e r YZ L i N b O g  with  As a r e s u l t , excited  have t h e d i s a d v a n t a g e ,  since i t s Curie  temperature i s  Diffusion  Sputtered  View,  These waveguides  phase  obtained.  i n d i v i d u a l g u i d e d modes h a v e b e t t e r a n g u l a r s e p a r a t i o n when  with  solid-  by l i q u i d  F e r m i f u n c t i o n d i f f u s i o n p r o f i l e was  d i f f e r e n c e b e t w e e n mode a n d s u b s t r a t e  d i f f u s e d waveguides  h a v e b e e n made  T i f i l m s 500 A° t h i c k w e r e d e p o s i t e d  slices  California.  obtained  on 2 i n c h  dia-  from C r y s t a l Technology, I n c . , o f Mountain  The c r y s t a l s h a d b e e n g r o w n f r o m a c o n g r u e n t m e l t , a n d  were s p e c i f i e d t o have the f o l l o w i n g i m p u r i t y manufacturer. Cr  2.8 ppm  Fe  18  Ni  3.7 ppm  Cu  2.6 ppm .  ppm  concentration  by the  32.  The followed.  d i f f u s i o n p r o c e s s o u t l i n e d b y Kaminow a n d S c h m i d t  The s u b s t r a t e s w e r e c l e a n e d b y a m e t h o d s i m i l a r  was  to that r e -  commended b y B r a n d t e t a l [ 2 4 ] f o r t h e f a b r i c a t i o n o f l o w - l o s s o p t i c a l waveguides.  U l t r a s o n i c c l e a n i n g f o r 10 m i n u t e s  tion i n deionized one h o u r  solu-  ( D I ) w a t e r was f o l l o w e d b y t h r e e D I w a t e r r i n s e s a n d  i n a DI cascade washer.  were blown  i n a .01% A l c o n o x  B e f o r e d e p o s i t i o n , t h e LiNbO^  slices  d r y w i t h n i t r o g e n and t r e a t e d w i t h a Z e r o s t a t a n t i - s t a t i c  (LiNbO^, b e i n g p y r o e l e c t r i c , attract dust).  tends t o develop p o l a r i z a t i o n  After deposition,  charges  the q u a r t z tube o f a c o l d d i f f u s i o n  nace.  the following impurity  gas (99.995%) w i t h  which  t h e Z e r o s t a t was a g a i n u s e d b e f o r e  i n s e r t i o n of the substrate i n t o Argon  gun  0  2  < 10 ppm  H 0 <10  H  2  <  C0  N  2  < 23 ppm  2 ppm  2  2  fur-  contenta a a  ppm  < .5 ppm  CH^ < .5 ppm  was p a s s e d t h r o u g h t h e f u r n a c e t u b e a t a r a t e o f 2 5,/min.  A f t e r a 6-  h o u r d i f f u s i o n a t 960°C, t h e f u r n a c e was t u r n e d o f f a n d a l l o w e d t o c o o l f o r 12 h o u r s w h i l e o x y g e n was p a s s e d  t h r o u g h a t a r a t e o f 1 £/min.  s t e p was u s e d b y Kaminow t o r e o x i d i z e t h e L i N b O ^ , brownish  from oxygen l o s s d u r i n g the d i f f u s i o n .  which  t e n d s t o become  On r e m o v a l f r o m t h e  f u r n a c e , t h e s u b s t r a t e was c o a t e d w i t h a n o x i d e l a y e r ; by l i g h t l y  This  t h i s was  removed  p o l i s h i n g b y h a n d w i t h 1 um d i a m o n d p a s t e on a n a p l e s s n y l o n  cloth. ^ A t t e m p t s - t o c o u p l e t o g u i d e d modes w i t h a 68° r u t i l e were u n s u c c e s s f u l , even modes i n s i m i l a r l y  prism  though Kaminow and S c h m i d t h a d o b s e r v e d 6 g u i d e d  t r e a t e d samples.  The d i f f u s i o n p r o f i l e  ( F i g . 2.22)  was d e t e r m i n e d b y e x a m i n a t i o n o f a 10° t a p e r s e c t i o n o f t h e s u b s t r a t e i n an e l e c t r o n m i c r o p r o b e . estimated u s i n g [22]  The e f f e c t i v e d i a m e t e r o f t h e x - r a y s o u r c e was  33.  S =  .033(V  1 , 7  - v£  w h e r e S i s t h e s o u r c e d i a m e t e r i n ym, V,  = 5 kV. f o r t h e T i K kV,, p = 7.45  V = 10  )A/(pZ)  , 7  g / c c w i t h i t h e u n i t s i n d i c a t e d , D = 1 ym,  mately  s o r e a s o n a b l e a c c u r a c y was  fitted  atomic weight  and  Using an  A,  ym.  taper s e c t i o n i n c r e a s e d the apparent ym,  potential,  l i n e a n d D i s t h e e l e c t r o n beam d i a m e t e r .  S ^ 1.4  11.5  (2.37)  V i s the a c c e l e r a t i n g  a v e r a g e v a l u e -for " t h e - a t o m i c , number Z and  The  + D,  d i f f u s i o n depth from 2 to  possible.  The  profile  i s approxi-  by R = erfc(y/1.05),  where R i s the T i to s u r f a c e T i count r a t i o , 1.05  c l o s e l y m a t c h e s Kaminow and  ion attained  the c o r r e c t depth.  Schmidt's I t was  (2.38) and y i s i n u y m .  v a l u e o f 1.1,  concluded  The  so the  value diffus-  that e i t h e r the  l i n g p r i s m d i d n o t w o r k as e x p e c t e d , o r t h e T i c o n c e n t r a t i o n was derably less  than  the U n i v e r s i t y o f Washington,  G. M i t c h e l l  ( F i g . 2.23).  C a l c u l a t i o n s showed  t h e p r i s m s h o u l d h a v e b e e n a b l e t o c o u p l e t o mode i n d i c e s r a n g i n g t o 2.6,  was  low.  s o i t was  T h i s may  of  E l e c t r i c a l E n g i n e e r i n g D e p a r t m e n t , was  t h a t the prisms worked  1.9  consi-  required.  A s p u t t e r e d g l a s s waveguide, p r o v i d e d by Dr.  to v e r i f y  coup-  concluded that the T i c o n c e n t r a t i o n i n the  have been caused by  f o r m a t i o n o f an o x i d e , o r  of t i t a n i u m e a r l y i n the d i f f u s i o n , which  reduced  used that  from guide nitride  t h e n u m b e r o f T i atoms  available. 1  The  s u b s t r a t e s were n o t r e s t o r e d  even a f t e r p r o l o n g e d treatment i n © was  insufficient  2  to t h e i r o r i g i n a l  a t 700°C.  However, t h i s a b s o r p t i o n  t o p r e v e n t g u i d e d modes f r o m b e i n g o b s e r v e d ,  s t r a t e modes w e r e c l e a r l y v i s i b l e  transparency,  i n the c o u p l i n g attempts.  since  sub-  Fig. 2.23  Coupling to a Glass Sputtered  OWG.  35.  The m o s t l i k e l y O^j ®2 of  a n c  *  c a u s e o f d i f f i c u l t y was  presumed t o be  ^• P '- y c o n c e n t r a t i o n i n t h e a r g o n gas. m  ur:  From F i g . (6.13)  t  [ 2 6 ] , i t can be i n f e r r e d  t h a t a 500 A°  s e c o n d s o r l e s s a t 960° i n 0^ p a r t i a l p r e s s u r e o f oxygen oxidation i s d i f f i c u l t  l a y e r of T i would o x i d i z e i n  at atmospheric pressure.  At the very low  encountered during d i f f u s i o n ,  the r a t e  to e s t i m a t e , b u t samples h e a t e d b r i e f l y  (-162  2.9  compound w i t h  . . k c a l / m o l e ) , t h i s e x p l a n a t i o n seems q u i t e  Ni/LiNbQ  because  H  2°  ^  r o m  t n e  a  r  8  o  n  8  a s  i n LiNbO^.  «  used r a t h e r than  An e f f o r t was  P a s c a r d and F a b r e  that a T i - Z r m i x t u r e (50% atomic) i s e f f e c t i v e  a T i - Z r s p o n g e m i x t u r e was  held i n place with T i strips tions  a r e shown i n F i g . 2.24.  inserted into  f o r oxygen  different  and  To  nitrogen  implement  The  furnace  and  connec-  removed w i t h s i l i c a g e l .  The P e r k i n - E l m e r s p u t t e r i n g s y s t e m was (< 500 A°)  made t o r e m o v e  a 3 cm q u a r t z t u b e  cut fromusheet metal. W a t e r was  titanium  [ 2 5 ] h a v e shown  r e m o v a l f r o m a r g o n a t t e m p e r a t u r e s i n e x c e s s o f 800°C. this,  oxidation  plausible.  e x p e r i m e n t s , n i c k e l was  of i t s higher d i f f u s i v i t y  ^2  Since  Diffusion  3  In subsequent  0^,  a l a r g e f r e e energy of  of  t o 960°C  a n d c o o l e d i n a r g o n w e r e a l s o o b s e r v e d t o h a v e an o x i d e l a y e r . Ti02 i s a very s t a b l e  the  c a l i b r a t e d so t h a t  l a y e r s o f N i c o u l d b e made r e p e a t a b l y .  S i x depositions  thin of  t h i c k n e s s e s w e r e made on a c l e a n g l a s s s l i d e w i t h a n r f f o r w a r d -2  p o w e r o f 100 w a t t s a n d an a r g o n p r e s s u r e o f 1.2 nickel all  l a y e r was  then d e p o s i t e d over the e n t i r e s l i d e ,  the steps e a s i l y v i s i b l e  a r e summarized  x 10  below  i n Table  i n a Sloan Angstrometer. 2.1.  torr.  A  thin  i n o r d e r t o make The  results  SENSITIVE PRESSURE REDUCER FLOWMETER  Fig. 2.24.  Gas Flow Connections for Ni/LiNbO  Table 2.1  OWG Diffusion.  Ni Sputtering Calibration.  t (min)  d(A°)  2  196 ± 100  98  4  245 ±  41  61  +  10  4  236 ±  59  59  +  15  6  384 ±  64  54  +  10  8  433 ±  43  54  +  5  10  575 ±  72  57.5 ±  7  r = d/t(A°/min) 50  37.  Neglecting  the 2 minute r e s u l t , f = d / t = 59.1 ±  the average s p u t t e r i n g r a t e i s  3.3 A°/min,  w h e r e t h e p r o b a b l e e r r o r 3.3 was c a l c u l a t e d  P.E. = 0.67  A vital  step  ( r  - r ) / n . -J 2  ±  i n e a c h n i c k e l d e p o s i t i o n was t h e p r e l i m i n a r y  r e m o v a l o f any r e s i d u a l t a r g e t o x i d e a t maximum p o w e r  from  l a y e r by s p u t t e r i n g onto the s h u t t e r  400 W) f o r s e v e r a l m i n u t e s .  O n l y t h e n was t h e s p u t -  t e r i n g r a t e s t a b l e enough t o g i v e p r e d i c t a b l e r e s u l t s . The  T i - Z r s p o n g e f u r n a c e was t u r n e d  r a t e o f 2 l/mln) order  one h o u r b e f o r e  to reach the operating  on ( w i t h an a r g o n  flow  commencement o f t h e L i N b O ^ d i f f u s i o n i n  t e m p e r a t u r e o f 900°C.  A series of tests  w e r e made t o d e t e r m i n e t h e e f f e c t o f a r g o n p u r i f i c a t i o n o n t h e q u a n t i t y of n i c k e l d i f f u s e d i n t o the substrate tude o f t h e N i K  t o Nb K  a  line  by x-ray fluorescence.  The m a g n i -  c o u n t r a t i o was t a k e n a s a n e s t i m a t e  of  a  the n i c k e l c o n c e n t r a t i o n  i n t h e t o p (> 10 ym)  T a b l e 2.2 s u m m a r i z e s t h e r e s u l t s . r e g a r d e d as p r o p o r t i o n a l o n l y  [22] l a y e r o f the specimen.  The N i / N b r a t i o s t a b u l a t e d  t o t h e atomic Ni/Nb  Some i n t e r e s t i n g c o n c l u s i o n s  a r e t o be  ratio.  c a n b e d r a w n f r o m T a b l e 2.2.  In  e a c h c a s e , i n c l u d i n g t h e 6 h o u r d i f f u s i o n a t 950°C, m o s t o f t h e n i c k e l r e m a i n e d on t h e s u r f a c e residue.  Also,  i n t h e f o r m o f an o x i d e  nitride  t h e e f f e c t o f argon p u r i f i c a t i o n on t h e q u a n t i t y  d i f f u s e d n i c k e l i s seen t o be All  or possibly  of the substrates  of i n -  considerable. were p a l e b r o w n i s h - g r e y i n c o l o r ,  the  s a m p l e d i f f u s e d a t 950°C b e i n g  the  o p t i c a l absorption  t h e most d i s c o l o r e d .  o f LiNbO„ s u b s t r a t e s  with  Measurement  was made i n a C a r e y  dual  of  T a b l e 2.2  No.  X-Ray F l u o r e s c e n c e  of Nl/LiNbO  D e s c r i p t i o n o f Sample  Ni/Nb  Ratio  1  LiNb0  2  500  3  500 A° N i d i f f u s e d i n t o L i N b 0 3£ h r . i n t r e a t e d A r .  4  No. 3 p o l i s h e d t o remove o x i d e  5•  Same a s n o . 3 e x c e p t d i f f u s e d a t 950°C f o r 6 hr.  0.230  6  No. 5 p o l i s h e d .  0.036  7  Same a s n o . 3, b u t A r n o t t r e a t e d .  0.226  8  No. 7 p o l i s h e d .  0.01  Substrate.  3  A° N i d e p o s i t e d  0 on L i N b O ^ 3  0.278 @ 800°C f o r  0.210  residue.  0.05  beam s p e c t r o p h o t o m e t e r , a n d t h e r e s u l t s p l o t t e d i n F i g . 2 . 2 5 . s t r a t e s had l i t t l e fectly  absorption  c l e a r to the eye.  i n the v i s i b l e  New  sub-  spectrum and appeared  per-  A l l s u b s t r a t e s h e a t e d i n argon and c o o l e d i n  o x y g e n h a d a n a b s o r p t i o n p e a k a r o u n d 4 5 0 nm.  Substrates  950°C w e r e c o n s p i c u o u s l y  from C r y s t a l Technology,  Inc., suggested baking was n o t f e a s i b l e , eliminated  absorbing.  the crystals  A letter  a t 1000°C i n 0^  heated to  f o r 2-3 d a y s .  s i n c e t h e n i c k e l would have d i f f u s e d t o o deeply  the waveguide.  This and  The a b s o r p t i o n may b e t h e same a s t h a t  o b s e r v e d a t 482 nm b y B a l l m a n a n d G e r n a n d [ 2 7 ] , who a t t r i b u t e d t h e d i s 4+ c o l o r a t i o n t o Nb lattice.  i o n s f o r m e d when o x y g e n i s l o s t f r o m t h e c r y s t a l 2+ The a b s o r p t i o n c o u l d a l s o b e due t o N i (as suggested by 2+  C r y s t a l Technology, I n c . ) , o r Fe small  ( S t a e b l e r and P h i l i p s  [33]).  The  (y 1 8 ppm) i r o n i m p u r i t y l e v e l i n t h e c r y s t a l s makes t h e l a t t e r  possibility less  likely.  (1) New substrate (2) Diffused 3 hr @ 800°C id 3 hr Q 950°C annealed 4 hr in 650°C Diffused 3 times at  300 Fig. 2.25  400  500  600  700  800  Absorbance of LiNbO^ Waveguide Substrates.  There i s doubt  also regarding the exact composition of the  residue l e f t after diffusion.  Lithium atoms diffuse rapidly in LiNbO^,  so they are likely to be present.  Nickel oxide (NiO) i s a p-type semi-  conductor for which lithium is an excellent dopant [28]. The doped oxide appears black [29], although the lithium concentration may not have been great enough for this degree of discoloration, which was not observed.  A number of nitrides of nickel are stable compounds, but  most of these are formed only at higher temperatures.  ^i^N i s formed at  40.  500°C, b u t i s b l a c k a n d a c o n d u c t o r  [30].  When l i t h i u m i s p r e s e n t ,  h o w e v e r , n i c k e l a n d n i t r o g e n r e a c t a t 550°C w i t h H ^ N [31]. sis  Unfortunately, n e i t h e r x-ray  t o form ( L i , N i ) N 3  fluorescence nor microprobe  analy-  a r e s u i t e d t o t h e d e t e c t i o n o f elements o f l o w a t o m i c number, s o a  direct  test o f the residue composition  c o u l d n o t b e made b y t h e s e  methods.  2.10 P r o p e r t i e s o f N i / L i N b O ^ W a v e g u i d e s S e v e r a l N i / L i N b O ^ d i f f u s e d o p t i c a l w a v e g u i d e s w e r e made b y t h e following process.  A 350 A° l a y e r o f n i c k e l was s p u t t e r e d  onto a clean  YZ L i N b O ^ s u b s t r a t e , w h i c h w a s t h e n i n s e r t e d i n t o a d i f f u s i o n  furnace.  A r g o n , p u r i f i e d w i t h T i - Z r s p o n g e a t 900°C was p a s s e d t h r o u g h a t a r a t e o f 2 Jl/min.  The f u r n a c e  was t u r n e d  temperature had s t a b i l i z e d .  o n t o 850°C, a n d o n e h o u r l a t e r t h e  After a total  of 6 i hours, t h e temperature  was r e d u c e d t o 600°C a n d o x y g e n i n t r o d u c e d w i t h a 1 Jl/min f l o w r a t e . This  step increased  elapsed  time,  t h e f u r n a c e was s h u t  room t e m p e r a t u r e . and  t h e r e - o x i d a t i o n o f t h e LiNbO^. " A t 8 i hours o f f and a l l o w e d  to cool slowly to  T h e s u b s t r a t e was r e m o v e d f r o m t h e f u r n a c e when c o o l ,  the residue polished o f f .  were e x c e l l e n t , although  The r e s u l t i n g 6-mode o p t i c a l w a v e g u i d e s  the s u b s t r a t e remained a pale brownish c o l o r .  F i g u r e 2.26 shows t h e d i f f u s i o n p r o f i l e microprobe a n a l y s i s . reasonably  total  obtained by e l e c t r o n  The n i c k e l t o s u r f a c e n i c k e l r a t i o i s f i t t e d  w e l l b y e r f c ( y / 1 1 . 5 ) , a s was e x p e c t e d f r o m t h e n o t i c e a b l e  n i c k e l oxide  or n i t r i d e residue, which apparently  d i f f u s i o n source.  An a c c u r a t e  Ni/LiNbO^ i s d i f f i c u l t ,  acted  as a  constant  q u a n t i t a t i v e microprobe a n a l y s i s o f  s i n c e l i t h i u m was n o t d e t e c t a b l e i n t h e m i c r o -  probe used, and oxygen counts tended t o be i n a c c u r a t e .  42.  F i g u r e 2.27 acoustooptic possible, guided  experiments.  and  the  modes.  On  p o s i t i o n e d on became v i s i b l e surface  on  the bottom.  for coupling  s u b s t r a t e s were e a s i l y He-Ne S p e c t r a - P h y s i c s 25  cm l e n s .  angle  The  Coupling  was  ments u n t i l strate of  the  cracked 155  by  as  follows.  clamp a d j u s t e d  This  a r e a was  until  on  this  obtained.  used to measure the  spot  the the  L i g h t from  a  s p o t b y means o f  i n t e r n a l r e f l e c t i o n a t t h e p r i s m b o t t o m was s m a l l r o t a t i o n a l and  was  a brownish  r o t a t e d about a h o r i z o n t a l a x i s u n t i l  a b r i g h t s t r e a k was  the  found.  translational  A scale attached  coupling angles  a  adjust-  to the  w i t h i n 1/2  sub-  minute  arc. Figure  2.28  shows c o u p l i n g t o t h e TE  and  TM  o mode N i / L i N b O ^ o p t i c a l w a v e g u i d e .  The  from which i t r a d i a t e s b r i g h t l y . brightness  are  c a u s e d by  wave i s c o u p l e d TE  prism  r e q u i r e d , as  pressure.  focussed  The  to  prisms  then c l o s e enough t o  C a u t i o n was  excessive  l a s e r was  t h e n a t t a i n e d by  t a b l e was  a d j u s t a b l e arms f o r c l a m p i n g  to take p l a c e .  s t a g e was  of c r i t i c a l  positioned for coupling  c o u p l i n g p r o c e d u r e was  t h e w a v e g u i d e and  and  f r e e d o m o f movement w e r e  c o u l d e a s i l y be  t h e s t a g e w e r e two The  used i n the o p t i c a l c o u p l i n g  F i v e degrees of  substrate  to the waveguide.  OWG  shows t h e s t a g e  Q  guided  of l i g h t  mode.  into  propagation. scratches  out  guided  and  modes n o t  Most of t h i s  at a surface  I n F i g . 2.29,  scratch,  c o l l i n e a r w i t h the  the  to the r i g h t i s  through the spot  s c a t t e r i n g i s probably  i s due  initial due  a smooth s u r f a c e ;  tested with a piece c h e c k e d by  t h e minimum i n t r a n s m i t t a n c e  on  the  direction  to f i n e  of ordinary  examining l i g h t  guided  to s c a t t e r i n g of  surface  s u s t a i n e d w h i l e p o l i s h i n g o f f the d i f f u s i o n r e s i d u e .  I t s d i r e c t i o n was  two  periodic variations i n streak  the b r i g h t spot  line passing  d i r e c t i o n o f p o l a r i z a t i o n was polaroid.  beam s t o p s  s p a t i a l mode b e a t i n g .  as w e l l ,  The  The  modes o f a o  The plastic  reflected  from  r o t a t i o n of the p o l a r i o d  44.  corresponded  to the d i r e c t i o n of t r a n s m i t t a n c e of TE waves.  P r o p a g a t i o n l o s s e s were measured by p l a c i n g the output  coup-  l i n g p r i s m a t s e v e r a l p o i n t s a l o n g the path of wave p r o p a g a t i o n . output was  maximized each time by a d j u s t i n g the output p r i s m clamp,  the output power, was Use  measured w i t h an A l p h a m e t r i c s  of the r e l a t i o n P „ = P , exp[-a(z2 zz z±  1 db/cm or  waveguides used  dclOlO l i g h t m e t e r .  - z l ) ] gave l o s s c o e f f i c i e n t s o f  shows the modes coupled out of the 6-mode Ni/LiNbO^  f o r a c o u s t o o p t i c experiments.  p r o p a g a t i o n i n the c r y s t a l was q  on the l e f t  The  d i r e c t i o n of  2'1.4° from the c r y s t a l X a x i s .  spots on the r i g h t are the T E , TE.^ and T E  2  are the c o r r e s p o n d i n g TM modes.  modes, and The  the p r i s m and angle of l i g h t  one mode a t a time.  i n c i d e n c e , i t was  The p r i s m p r e s s u r e was  the c o u p l i n g angles u  m  found to be 68° 12' ± 1 '  on i t s e l f  from both  The  three  the weaker spots  By  careful  adjustment  p o s s i b l e to e x c i t e  reduced  to a minimal  were measured f o r the TE modes.  angle was  light  i n p u t l i g h t beam was  u n p o l a r i z e d , so both TE and TM modes were e x c i t e d .  and  and  less. F i g u r e 2.30  of  Light  The  just  value, prism  by r e f l e c t i n g the i n c i d e n t l i g h t back  the s u b s t r a t e and p r i s m f a c e .  The mode i n d i c e s were c a l c u l a t e d from the c o u p l i n g angles u s i n g (2.26), w i t h n  p  = 2.582.  Equations  (2.19) and  (2.20) were s o l v e d f o r the  t u r n i n g p o i n t s . o f the TE modes a c c o r d i n g to the p i e c e w i s e - l i n e a r WKB approximation.' the rms  second  The e s t i m a t e d v a l u e of the s u r f a c e index was differences  smoothest p r o f i l e .  The  s u r f a c e index used was  (2.21) were minimized,  thereby g i v i n g  r e s u l t s are summarized i n T a b l e 2.2377, which gave an rms  varied  (2-3).  until  the The  d e v i a t i o n of 6.8X10  *The e l e c t r i c OGW f i e l d component a l o n g the p r o p a g a t i o n d i r e c t i o n i g n o r e d . A simple c a l c u l a t i o n shows t h a t E = .024 E . z x  was  Fig. 2.30  Modes of OWG's Used in Acoustooptic Experiments.  Table 2.3  Mode Number TE o TE^ TE  TE Modes a of Ni/LiNbO. OWG.  ym (±1') m  y  m  (um)  21° 31'  n  l = 2.235  21° 45'  n  2 = 2.233  y  2 = 5.78  n  3 = 2.2316  y  3 = 8.41  21° 56*  2  n (±.0004) m  = 3.23  These measurements were made with an angle of light propagation of 21.4° from the X axis of LiNb0  3  in the Y-cut plane.  This  direction was chosen so the light would intersect at the Bragg angle (.4°) with a 165 MHz acoustic surface wave propagating i n the Z - 21.8° direction, as discussed i n Chapter 4. The substrate index for TE polarization along this direction can be found as follows.  l  Fig. 2.31  Coordinate Rotation i n Phase Velocity Space.  47.  Consider a coordinate rotation i n the X-Z plane of the crystal phase velocity space, V = V .. cosG - V . sine x xl zl V. = V sine + V , cose z xl zl n  For plane wave propagation along x l , we have V = Then V  X  = V . cos9 and V = V , sin6. xl z xl  and  =  = 0.  Substitution into the phase  velocity surface determinant (Appendix II) and multiplication by n  xl  =  C  / V  xl  g l v e s  t (n 2  s - n^) 2  y  r  2  2  [n n x  - n / (n 2  z  o Q M cos 6 - n • sin 9)]r, = 0,  2  x l  2  x  2  2  z  so that either n - = n xl y n for TE waves.  (TM) ,  = n n /(n s±n Q + n 2  Using D „ = n  cos 9)  2  2  (2.39)  1 / 2  = 2.214, n = n = 2.294 [4] and 6 = 21.4° , ' x o . v  e  3'/  2  or .  the substrate index of propagation for TE waves i s n  = 2.22 4.  The  waveguide index profile (Fig. 2.32) was fitted quite well by n(y) = '2.2286 + 0.0091 e r f c (y/11.5), with y in ym.  (2.40)  The agreement between the index and diffusion profiles,  and between the calculated and fitted substrate index, is reasonable. The electric f i e l d distribution of the TE modes was calculated by the WKB method, using equations (2.10)-(2.16).  To check whether this  method was in fact applicable to the problem, the index profile n(y) = n  g  + An erfc (y/b)  was differentiated to obtain = 2k-n(y)-An 2.. 2. 37" - — exp(-y lb ) /rf-b-/n(y) -n 1  r  y  2  T  , *T> U.41) 2  •2.238-h 2.237 h  2.236h  2.235h  2.224  2.223  2.222  2.221 h  2.220 h  2.229 h  2.228  Fig.  2.32  Ni/LiNb0„ OWG I n d e x  Profile.  S u b s t i t u t i o n of the v a l u e s An = .0091, b = 11.5 ym, X = .6328 ym, y = 0, n  s  = 2.2286, and n  m  = 2 . 2 3 3 (TE  j . d K / d y | = ,014k  l n  mode) gave  and  :  K/X = .23k,-  so t h a t i n e q u a l i t y (2.17) .is s a t i s f i e d .  ^  I n the r e g i o n y < 0, the e x a c t s o l u t i o n of the wave e q u a t i o n was u s e d , and the c o n s t a n t A was chosen t o match the s o l u t i o n f o r y = 0 at the waveguide s u r f a c e . (2.11) was used.  F o r y > 0 and up t o y , t h e l a s t z e r o o f E , x  The i n t e g r a l s were e v a l u a t e d n u m e r i c a l l y by t h e t r a p e -  zoidal rule. For y > y , the u s u a l approach i n the WKB l i n e a r approximation  [34] of the i n d e x p r o f i l e , s e t t i n g n  where  method i s t o make a  (y) ^ n  m  + n'(y - y ) m m  i s the i n d e x a t the t u r n i n g p o i n t and n' = |dn(y)/dy| m y=y 1  m  = ^ 2 - exp(-y2/ 2) b b  f o r the case of an e r f c p r o f i l e .  (2.43)  N e g l e c t i n g terms o f second o r d e r i n  An, K(y) becomes K (y) =.. 2 k n n ' ^ ( y - y ) 2  2  m  i  m  (2.44)  I n t h e r e g i o n of the t u r n i n g p o i n t , the wave e q u a t i o n i s t h e n a p p r o x i mated by d F 2  H. ,Lh + <y - ymj g Kx = °-  -  (2 45)  dy.  where'..g. = .2k?n n [ ^-' i  m  I  l  The change o f v a r i a b l e  z = g  gives  1/3  ( y - y) m  (2.46)  „ d  x 7 dz E  z  -z E  = 0  (2.47)  50.  The solution of this equation can be expressed in terms of fractional order Bessel or Hankel functions, or, more conveniently, in terms of the Airy function Ai Cz). of (2.47) to the WKB  The problem is to match the solution  solutions on either side.  Marcuse gives  E (y) = (AB/2j)(< Tr(y -y)/3) [exp(2uj/3)H^._(?) + exp(7rj/3)H® (?) ], x o m i/J l/j 1/2  (2.48) 312  where c, = (2/3)1z  , in the vicinity of the turning point.  identity [36] Ai(-z) = (1/2) /z/3 [exp(irj /6)H^ (C) + exp(-Trj/6)I< /3  2) /3  Use of the (?)] (2.49)  gives E (y) = (AB/g  1/6  x  )v^T Ai(-z)  = (AB/g )i47T Ai [ g 1/6  1/3  ( y - y )] m  (2.50)  The Airy function has the convenient property that z = 0 at the turning point, where Ai(0) = 3  _2/3  /r(2/3)  = 0.35502 Thus, the electric f i e l d at the turning point is E (y ) = A B v ^ / . ( g r . ( 2 / 3 ) 3 ) 1/6  x  2/3  ffi  This differs from Marcuse's expression  (2.51)  [9], which appears to under-  estimate the magnitude of the electric field, c  J  To calculate the electric field in the vicinity of the turning point, the Airy function was generated with the series representation Ai(2) = c  ±  f(2) + c  2  g(z),  where ir/^N i +. — 2? +,1.4 f(z) = 1 -gy  « z,6 +, —1.4.7 —  ~9 z +. ...  [36]  (2.52)  ~ . 2 _4 . 2.5 ~7 . 2.5.8 AO , g(z) = z + Tj- z + -yj- z + with and  1 Q  ,  z  + ...  = Ai(0) c =-dAi(0)/dz = 0.25881 2  Figure 2.33 shows this function for -8 < z < 4. Since our interest was only in the vicinity of the f i n a l maximum of the function, where |z| < 3, 8 terms of the series were adequate for five decimal place accuracy.  -.5 >Fig. 2.33 The Airy Function. In the region y > y , the WKB solution was found to agree very closely with the Airy function solution, so the latter was used exclusively for y > y . For the TE mode, most of the solution i s i n o o the vicinity of the turning point, so the Airy function was used entirely for y > 0. For higher order modes, the Airy function deviates  from the c o r r e c t  s o l u t i o n w h e n y < y , s o t h e WKB m e t h o d ( 2 . 1 1 ) was u s e d  f o r 0 < y <. y .  I n F i g . 2 . 3 4 , a c o m p a r i s o n i s made b e t w e e n  function  solutions  Fig.  a n d WKB  2.34  the A i r y  f o r t h e TE. mode.  C o m p a r i s o n o f A i r y F u n c t i o n a n d WKB o f 6-Mode N i / L i N b 0 OWG.  Solutions  f o r TE" Mode  3  To s u m m a r i z e , E  t h e b e s t a p p r o a c h seems t o b e a s f o l l o w s :  x  = A exp(-yy),  y < 0,  • m = A B / K / k ( y ) COS(TT/4 - / K(y)dy), y  (2.53)  v  f  E  x  0 < y < y , °  (2.54)  53.  and E  =  (AB/g  1 / 6  )/TT  x  Ai[g  1 / 3  (y -  y )],  o  y m  S o l u t i o n s o f these equations  < y < ».  n  (2.55)  o  f o r t h e t h r e e TE modes o f t h e 6-mode  N i / L i N b O g w a v e g u i d e s s t u d i e d a r e shown i n F i g . 2 T 3 5 f o r p r o p a g a t i o n i n t h e X + 2d.4° d i r e c t i o n so t h a t e a c h c a r r i e s tributions  o f Y c u t LINbO^.  a power o f 1 w a t t / m e t e r .  a r e used l a t e r  overlap integrals, The  The modes h a v e b e e n n o r m a l i z e d These e l e c t r i c f i e l d  ^ - s e c t i o n * 47.6,), • t o c a l c u l a t e  as d i s c u s s e d i n C h a p t e r  dis-  the acoustooptic  4.  v a l u e o f t h e n o r m a l i z a t i o n i n t e g r a l was u s e d i n ( 2 . 3 0 ) t o  estimate the c o u p l i n g e f f i c i e n c y o f the r u t i l e p r i s m used. TE modes h a d a maximum c o u p l i n g e f f i c i e n c y coupling length.  A l l three  o f a b o u t 5 5 % f o r a 1 mm  I n F i g . 2.36, t h e c o u p l i n g e f f i c i e n c y n  i s plotted  as a f u n c t i o n o f gap t h i c k n e s s f o r t h e T E ^ mode, u s i n g t h e v a l u e s W = Jl = 1 mm, n approximately efficiency light  p  = 2 . 5 8 2 , a = 68° 1 2 ' , u = 22° 45' , w h i c h  t o the experimental s i t u a t i o n .  actually  o b s e r v e d was 1 0 - 1 5 % .  l o s s on r e f l e c t i o n from  required strate,  to a t t a i n  The maximum c o u p l i n g  Since n  the prism face  between t h e o r y and experiment  t  does n o t i n c l u d e t h e  (^ 2 5 % ) , t h e a g r e e m e n t  i s quite reasonable.  were conducted  can be o b t a i n e d by  the index p r o f i l e with a constant, so that K  The  cracked t h e sub-  w i t h 5% e f f i c i e n c y o r l e s s .  A q u i c k e s t i m a t e o f t h e maximum v a l u e o f a approximating  The p r e s s u r e  t h i s degree o f c o u p l i n g e f f i c i e n c y  so f u r t h e r experiments  corresponded  (y) *  (1/2)  K  Q  v a l u e o f t h e n o r m a l i z a t i o n i n t e g r a l i s s m a l l o u t s i d e t h e range  0 <. y < y , s o we c a n w r i t e  2  X  2  2  = A B [y + -'m 2  l  The  second term i s s m a l l e r than  1  2  2  7  This expression  1 / 2  ( I / O COS(K y ) ] o'  x  crm  the f i r s t , p a r t i c u l a r l y f o r h i g h e r modes,  f o r which i t can be n e g l e c t e d . (a ) % y" (n /n -l) c max m p m  (K /2)dy - ir/4)dy °  o  X  2  (f™  cos  / E*E dyfc2 A B -» o  S e t t i n g D = 0 i n (2.30) g i v e s [l + (n /n -l) (n /n _l)p m o m 2  2  1 / 2  2  2  appears t o b e a c c u r a t e w i t h i n about 10%.  f o r the case d i s c u s s e d e a r l i e r , (2.56) g i v e s ( a )  fc  1 / 2  ]-  2  (2.56) '  F o r example,  0.88 mm ^ r a t h e r •  C UlciX than  the c o r r e c t v a l u e o f 0.79.  Use o f t h e approximate v a l u e g i v e s t h e  maximum c o u p l i n g e f f i c i e n c y as 27% r a t h e r than 25.5%.  F i g . 2.36 P r i s m  Coupler  Efficiency.  3. PROPAGATION AND GENERATION OF ACOUSTIC SURFACE WAVES 3.1 Introduction Elastic surface waves have been studied since their prediction by Lord Rayleigh in 1887. At the free surface of a solid medium, sourid .waves propagate, with ra ^reduced--phase velocity and are guided in a layer about one wavelength deep. Particle motion is e l l i p t i c a l , with components normal and parallel to the surface, and an exponential decay in amplitude away from the surface. These waves found their f i r s t application i n the study of seismic phenomena, and were later used for flaw testing in materials. They were not seriously considered for electronic signal processing applications until 1965, when White and Voltmer [37] demonstrated a simple and efficient method of direct coupling to surface acoustic waves (SAW's) by means of an interdigital electrode array on the surface of a piezoelectric solid.  Since that time, SAW devices have found applica-  tion in high performance delay lines and f i l t e r s , and promise the realization of more complex circuit functions, such as real-time convolution of two signals. In this .chapter", the-propagation and. generation characteristics of JSAW's-.on ..anisotropic piezoelectric materials' is reviewed, with particular emphasis on LiNbO^ as the acoustic medium. Experiments with an 85 MHz SAW delay line are described, and measurements are compared with predictions of an equivalent circuit model. 3.2  Surface Waves in Piezoelectrics Consider an infinite slab of piezoelectric material as i l l u s -  trated in Fig. 3.1.  57.  liiiiliiiilililiii. / Wl IWIIIimil  Fig. 3.1  SAW Propagation.  An acoustic surface wave of wavevector l£ and wavelength A propagates along the x^ coordinate axis. The slab i s assumed to be many wavelengths thick, so that the lower boundary has no effect on wave propagation.  Let u^ represent the excursion from the equilibrium point  of a particle in the solid.  The elastic strain tensor i s defined as  [38] S  where the symbols x^.  k* = I  <\,£  +  £,k> >  U  denote partial differentiation with respect to  In a piezoelectric material the relation between stress and strain  involves the piezoelectric constitutive relations, which can be written as  [39J '  T. . = c IJ  D. i  J  . S, „ - e, . . E.  ijkJl  kJl  kij  k  .= e„ „ S. „ + £., E. ik£ k£ lk k  (3.2) (3.3)  E s where T i s the stress tensor, c. ., „, e., . and e., are the elastic ij ijk£ ik£ lk stiffness tensor (at constant electric f i e l d ) , the piezoelectric tensor, and the dielectric tensor (at constant strain), respectively.  58.;  I n a s t a t i o n a r y s o l i d n o t s u b j e c t t o e x t e r n a l body f o r c e s , sideration  con-  o f an i n f i n i t e s i m a l volume element c e n t e r e d a t x ^ g i v e s t h e  p a r t i c l e equation of motion, pu.  = T  l  j  f  ,  l  ,  (3.4)  w h e r e p i s t h e d e n s i t y a n d "'•" d e n o t e s d i f f e r e n t i a t i o n w i t h r e s p e c t t o time. Use o f t h e r e l a t i o n s E  k  T> = u±  and  s u b s t i t u t i o n o f (3.2) i n t o  p  U  j  =  = -  ,  0 ,  ( 3 . 4 ) g i v e s t h e SAW e q u a t i o n s o f m o t i o n ,  hjk£  +  \ i j  *,kl  (  speaking,  than  equations  the v e l o c i t y o f l i g h t ,  and  the e l e c t r o s t a t i c  However^  8  since the  form o f  Maxwell's  t o (3.7) and (3.8) have t h e form [40]  = 3  e x p ( - a K x ) e x p j (fit - K x ) ,  <j>  = 3^ e x p ( - a K x ) e x p j ( f i t - K x )  f o r wave p r o p a g a t i o n  2  3  2  i n the x  k = 1, 2, 3  (3.9)  (3.10)  3  3  direction.  g i v e s a l i n e a r homogeneous s y s t e m For a n o n - t r i v i a l s o l u t i o n ,  m u s t b e z e r o , g i v i n g an e i g h t h - d e g r e e the f i e l d s  )  loss i n accuracy.  ^  k  7  i s some f i v e o r d e r s o f m a g n i t u d e  can be used w i t h v e r y l i t t l e Solutions  *  3  ( 3 . 5 ) ."is a p p r o x i m a t e .  p r o p a g a t i o n v e l o c i t y o f s u r f a c e waves  3  <->  e  Strictly  (3.8)  (3.6)  ° = i U \ , U - 4k *,ki •  ^  less  (3.5)  Substitution into  (3.7) and  i n t h e unknowns 3^., k = 1, 4.  the determinant  of coefficients  p o l y n o m i a l e q u a t i o n i n a.  a r e bounded, o n l y s o l u t i o n s w i t h n o n - n e g a t i v e  Since  r e a l p a r t s can  be  used.  I n general,  f a c t o r y . Each v a l u e ous  equations  Using  f o u r c o m p l e x r o o t s c a n be f o u n d w h i c h a r e s a t i s -  o f a c a n t h e n b e s u b s t i t u t e d b a c k i n t o t h e homogene-  i n 6^, g i v i n g f o u r  ( g e n e r a l l y complex) v a l u e s  the other boundary c o n d i t i o n s t h e p a r t i a l  (those  corresponding  t o each value  of  field  o f 3^.  amplitudes  a) c a n b e f o u n d b y  numerical  methods t o c o m p l e t e t h e s o l u t i o n . The  p r o b l e m c a n o n l y be s o l v e d b y i t e r a t i v e  computer  techniques  The  f i r s t s u c h s o l u t i o n s f o r SAW's o n L i N b O ^ w e r e o b t a i n e d b y C a m p b e l l  and  Jones  [41]  propagation ference  i n 1968.  T h e y c a l c u l a t e d SAW v e l o c i t i e s  d i r e c t i o n s f o r both f r e e and m e t a l l i z e d s u r f a c e s .  b e t w e e n t h e two v e l o c i t i e s  i s an i m p o r t a n t  parameter  t h e s t r e n g t h o f i n t e r a c t i o n b e t w e e n SAW's a n d e l e c t r i c by-means o f s u r f a c e e l e c t r o d e s . section  on i n t e r d i g i t a l The  on  and K o e r b e r  [42]. Fourteen  Solutions for this  further i n the  complex  SAW's coeffic-  and o t h e r major c r y s t a l  cuts a r e given  [40].  summarize, t h e d e s c r i p t i o n o f a c o u s t i c s u r f a c e waves on  have both displacement  and e l e c t r i c  to the surface.  Outside  decay e x p o n e n t i a l l y ; i n s i d e , decay.  generated  t h e e s s e n t i a l l y e x p o n e n t i a l SAW d e c a y  LiNbO^ and s i m i l a r m a t e r i a l s i s q u i t e complex.  dicular  fields  T h i s w i l l be d i s c u s s e d  t h e M i c r o w a v e A c o u s t i c s Handbook To  indicating  c o m p l e t e d e s c r i p t i o n o f Y Z (Y c u t , Z p r o p a g a t i n g )  ients are required to describe  in  The d i f -  transducers.  LiNbO^ i s given by Spaight  w i t h depth.  i n different  field  I n general,  components n o r m a l a n d p e r p e n -  thematerial, the e l e c t r i c  a l l fields  t h e waves  e x h i b i t a slower  fields  oscillatory  T h e s o l u t i o n s may b e w r i t t e n i n t h e f o r m u ^ x . t ) = 3™ exp j ( f i t - K x ) 3  (3.11)  where I \ q q=l  =  P  e X  n  ( a  q ^  K X  2  )  = - | | - = e£ exp j (fit - Kx > K  and  (3.12)  3  where 3  The [75],  k  time average  J  =  x  \ q  e  X  p  (  a  q  K X  2  '  }  power f l o w i n the SAW  one due to the m e c h a n i c a l  displacement  i s the sum o f two p a r t s  field,  00 . .  P  ma= I P  k  e  J  f l  T  i j j U  d x  2  ( 3  '  1 3 )  and the other due to t h e SAW e l e c t r i c f i e l d , P  ea  =  1  2 ZlRe J  where D. i s the complex conjugate displacement v e c t o r .  The  0  j f i § D* d x  2  ,  (3.14)  of the i t h component of the e l e c t r i c  and  coefficients  are u s u a l l y norma-  l i z e d so t h a t the t o t a l power i s one watt/m.  3.3  Diffraction The  d i f f r a c t i o n of e l a s t i c s u r f a c e waves may be t r e a t e d by  methods s i m i l a r tropic  of S u r f a c e Waves  crystals,  to those used the s i t u a t i o n  f o r e l e c t r o m a g n e t i c waves.  In aniso-  i s c o m p l i c a t e d by the f a c t t h a t the  phase and group v e l o c i t i e s o f p r o p a g a t i o n are n o n - c o l l i n e a r w i t h the e x c e p t i o n of a few symmetry (pure-mode) axes. d i f f r a c t i o n of b u l k u l t r a s o n i c materials.  Cohen [A3] s t u d i e d the  waves i n a number of a n i s o t r o p i c  He used a p a r a b o l i c f i t t o the c r y s t a l phase v e l o c i t y  surface, v(6) = V ( 1 - b e ) 2  q  (3.15)  in the vicinity of pure-mode axes.  He found excellent agreement between  calculations based on the above approximation and experiments i n a number of materials. Weglein et a l [44] applied this approximation to the spreading of 100 MHz surface waves propagating in the Z and Z ± 21.8° directions in Y-cut LiNbOy obtained.  Exact agreement between theory and experiment was not  Analytic expressions for the far-field beam divergence half-  angle ty]_j2  anc  *  t  n  e  Fresnel distance r^ have been obtained by Crabb et  al [45], 1/2  W seed) w seed) 2  r  f  4A(1 + dd>/de)  =  where W i s the source width and tand> = ^ —  ( 3  *  1 7 )  . Use of the parabolic  approximation gives *l/2 y  -^ " W (1 " 2b)  1 / 9  w f 4A(1 - 2b) '  (3.18)  2  a n d  r  =  for small angles 6 about the pure mode axis.  ( 3  *  1 9 )  Except for the factor  (1 - 2b), these equations are identical with those applicable to isotropic diffraction.  The walk-off angle  $ between the phase and group  velocities i s given by <t> = -2b e.  (3.20)  Szabo and Slobodnik [46] give an excellent review of surface wave diffraction on anisotropic substrates. They compare isotropic, parabolic and exact theoretical calculations with experimental observation (the exact calculations were done by Khar us i and Farnell [47]).  Fig. 3.2  SAW Propagation in Anisotropic Materials [40]  For propagation  i n the Z direction  does n o t g i v e a c c u r a t e r e s u l t s . in  this  o n YZ L i N b O ^ , t h e p a r a b o l i c t h e o r y  The r e a s o n  f o rthis  i s t h a t 1 + d<j>/d6 ^  c a s e , s o t h a t t h e v a l u e o f b must b e v e r y a c c u r a t e l y known.  This i s n o t p o s s i b l e a t the present time because t h e m a t e r i a l constants have n o t been determined  with sufficient  accuracy.  H o w e v e r , f o r p r o p a g a t i o n i n t h e Z ± 21.8° d i r e c t i o n 1 + dcf)/d9 ^ 1.37) g o o d a c c u r a c y  (where  i s o b t a i n a b l e , as d e m o n s t r a t e d b y  Wilkinson e t a l [48], In Table propagation  3.1, t h e a n i s o t r o p y p a r a m e t e r i s g i v e n f o r t h e s e two  directions.  F o r YZ w a v e s , beam s p r e a d i n g i s much l e s s  i n t h e case o f i s o t r o p i c d i f f r a c t i o n ; hence, t h i s delay lines. in  steered light  case.  3.1  A n i s o t r o p y P a r a m e t e r b.  Propagation Direction  b [44]  Y  Z  0.54  Y  Z ± 21.8°  -0.187  SAW G e n e r a t i o n ;  3.3.  T h i s p r o p e r t y makes t h i s c u t u s e f u l f o r beam-  Cut  Consider Fig.  the I n t e r d i g i t a l  Transducer  the i d e a l i z e d i n t e r d i g i t a l  A grid of i n f i n i t e l y  e l e c t r i c slab are alternately  t r a n s d u c e r (IDT) i n  l o n g e l e c t r o d e s on a n i n f i n i t e  connected  t o an r . f . g e n e r a t o r .  piezoThe  m a t e r i a l t h i c k n e s s i s a s s u m e d t o b e much g r e a t e r t h a n t h e e l e c t r o d e spacing.  than  deflectors.  Table  3.4  c u t i s u s e f u l f o r SAW  F o r Z ± 21.8° p r o p a g a t i o n , beam s p r e a d i n g i s g r e a t e r  the isotropic  than  64.  Fig. 3.3 Section of an Idealized IDT. The problem i s to develop an equivalent circuit model. To make the calculation at a l l reasonable, i t i s necessary to make a number of assumptions: (1)  the electrodes are massless, perfectly conducting and i n f i n i t e l y long (so that the problem becomes two-dimensional),  (2)  the quasi-static approximation i s assumed, i.e., E = -V§»  (3)  the piezoelectric i s assumed to have no non-linearity, and  (4)  the driving voltage and the SAW's have the time dependence e  jfit •  The equations of state of a linear piezoelectric, (3.1)-(3.6), were discussed in section 3.2. Since the actual problem to be solved  is essentially two-dimensional, these relations can be reduced to the following  [49] ,. using reduced matrix notation:  pfi u 2  1 +  T  pfi u + T  l j l +  5 > 3  + T  2  3  T  5 > 1  =0  (3.21)  «0  3 } 3  where T  and  l  =  ll  C  l 13 3 " 31 3 >  S  +  C  T  3 " °13 l  T  5  S  C  S  S  l • l,l  S  3 " 3,3  S  5  E  l  6  E  4> 3 " 33 3 »  +  44 5  S  S  e  15  e  E  l >  E  U  U  =  U  l,3 3,l + U  *l  fc  f  E =-$ 3  >3  Using Gauss' Law, l,l  D  + D  3,3=° >  and making the indicated substitutions gives three partial differential equations in terms of u^, u  p  Q  \  4l  +  u  l,ll  +  C  44 l,33 U  +  3  (C  and §:  13  +  C  44 3,13 )U  +  (e  15  +  e  31 ,13 )$  =  ° (3.23)  (C  13  +  c  44 l,13 )u  +  pn  S  +  c  44 3 , l l u  +  c  33 3,33 u  +  e  15 5,11 33 *,33 +  e  =  (3.24)  66.  < 15 e  +  e  31 l,13 ) u  +  e  15 3,ll u  +  e  33 3,33 " U  l l ,11 " 33 *,33  e  ?  e  =  0  (3.25) The  mechanical boundary c o n d i t i o n s a r e T  3  = 0  at  T  5  = 0  at- x  x  3  = 0  (3.26)  Most f r e q u e n t l y ,  = 0 .  t h e e l e c t r i c a l boundary c o n d i t i o n s have been  chosen t o be  n  v  = J  I  at  x  3  L-A  = 0 ,  —  i i L+A < |x I < — x  (3.27) D  The  = 0  3  at  x  0 < \x±\  = 0 ,  <  ^  s e c o n d c o n d i t i o n i s an a p p r o x i m a t i o n , s i n c e i t  the e l e c t r i c  field  lized.  i s quite accurate  This  3  outside  t h e c r y s t a l where t h e s u r f a c e i nhigh  ignores  i s not metal-  p e r m i t t i v i t y m a t e r i a l s , such as  LiNbCy Equations which i s t o f i n d  t h e c o u p l e d e l e c t r i c a l a n d m e c h a n i c a l f i e l d s when an  alternating voltage attempted. coupling e  kij  E  (3.23)-(3.27) c o n s t i t u t e a s t a t e m e n t o f t h e p r o b l e m ,  i s applied.  No e x a c t a n a l y t i c s o l u t i o n i s p o s s i b l e .  approximation  k '" ^ 1 t  A v a r i e t y o f s o l u t i o n s have been  i e  c o n s t :  '  : t u t :  Usually,  ( w h i c h means i g n o r i n g t h e p i e z o e l e c t r i c t e r m s  '-  v e  r e l a t i o n s ) i s made.  This  i sequivalent to  i g n o r i n g t h e c o u p l i n g b e t w e e n e l e c t r i c a l a n d m e c h a n i c a l SAW In  the l i t e r a t u r e ,  mechanical  t h e p r o b l e m h a s been t r e a t e d as e i t h e r  neither  fields  [51,52,53].  As p o i n t e d  fields.  entirely  [39,50], o r t h e s o l u t i o n o f t h e e l e c t r i c a l f i e l d s  L a p l a c e ' s e q u a t i o n ) has been c o n s i d e r e d strain  t h e weak  (through  independent o f t h e a c o u s t i c  o u t b y M i l s o m a n d Redwood [49],  approach i s e n t i r e l y s a t i s f a c t o r y , s i n c e  a c c u r a c y and i n f o r m a t i o n  are lost.  They solved (3.23)-(3.27) with variational techniques, which  appear to give very accurate solutions over extended frequency ranges. Unfortunately, their solution i s not convenient to apply in design problems, since i t does not lead to a simple equivalent circuit. Campbell and Jones suggested that Av, the change in SAW velocity when a conducting sheet was applied to the piezoelectric surface, could be used as a measure of coupling strength between the SAW  electri-  cal and mechanical fields, and hence between the transducer and a surface wave. Figure 3.4 shows the surface wave velocity and Av/v for Y cut LiNbOg.  They found a large Av/v value for YZ propagation, implying  that high coupling efficiency was possible. This was verified by Collins et a l [54]. A number of authors [39,51,53,55,56] have shown that the physical description can lead to an equivalent circuit model when suitable simplifying assumptions are made. A more empirical approach was taken by Smith et al [57], who used a circuit model formulated by Mason for each finger pair.  They proceeded to find an admittance matrix for the  transducer as a whole by cascading sections. Two cases were treated, which give differing equivalent circuits: (1)  the in-line model assumes surface wave generation is by compressional excitation, resulting in a series equivalent circuit, and  (2)  the crossed-field model, having a shunt equivalent circuit, which assumes SAW generation by shear excitation. Generally, transducers generate surface waves by both methods  simultaneously, but frequently one mechanism predominates.  The actual  model applicable may be determined by calculating the relative power in  68:  i  i  i  i  i  i  1  1—r  Y-CUT PLATE  3350 H  1  1  1  h  H  1  h  3.6 x I0|2  2.8 xlO  1.2 x I0  2  -2  0.4 x 10 I  18  I  I..  54  I  I  90  I  1  1  126  1—  162  PROPAGATION DIRECTION.fi (DEGREES)  Fig. 3.4  SAW Velocity and Coupling Constant for Y-Cut LiNbC» [40]. 3  the shear and compressional wave components.  For YZ LiNbO^, Smith et  al calculated that the shear component i s one order of magnitude greater, so that a shunt model is more appropriate. However, i f certain conditions are met (for example, i f the IDT has few fingers) the two models may be considered equivalent. Smith et al's model was modified slightly by Auld  and Kino  [53], who solved the physical problem by means of a perturbed normal mode expansion technique and the weak coupling approximation.  They used  Engan's [52] expression for transducer static capacitance,  c  T  = wc  s  = WN(e + e )K(q)/K'(q) o  where  (3.28)  p  W i s the transducer width, N i s the number of finger pairs, e i s the relevant dielectric constant, P q = cos[— (1 - a)], where a = A/L,  and K and K' are complementary e l l i p t i c integrals of the f i r s t kind. This expression takes into account the effect of the ratio  a of  metallized to free surface under the transducer, which Smith et al's model does not.  The shunt model has the equivalent circuit  BP) Fig.  GP)  3.5 IDT Shunt Model Equivalent Circuit.  The radiation conductance at the center frequency 9. • = •';, o L xs  n c_ - NTT A G = o T o K(q) K'(q)  (3.29)  where A = Av/v i s the change in wave velocity when the surface i s metallized.  When the frequency deviation i s small (y 20%), G (ft) = G sine x a o  and  B (fl) = G ( Q  a  sin  (3.30)  < > ~ ) , 2x 2x  2 x  2  (3.31)  where x = NTrAfi/ft = NirAf/f and sine x = sinx/x. o o It i s usually more convenient to use the series model, which has the following circuit.  C  T  FL02)  Fig. 3.6  Series Equivalent Circuit.  Here, the radiation resistance at resonance i s NIT A 2  RO = fi C D  (3.32)  K(q) IC (q)  T  For the two models to be equivalent at resonance, we require that R +-J^r-% o jn c G + jn c 1  o  T  o  o  T  This implies that R % 0  < 4 5  + a2  o  T  From (3.28) and (3.29), with a = 0.5 so K = K* =. 1.854, G ° 2 2 2  =8.24 N A . 2  2  (Smith et al's equivalent expression i s 5.1 N A ).  T h u s , when 8.24  N A 2  «  2  1,  (3.33)  G R  and  n*-r7»  t h e two m o d e l s a r e e q u i v a l e n t .  ( 3  '  3 4 )  F o r example, f o r 3 f i n g e r p a i r s and 2 2  p r o p a g a t i o n i n t h e Z ± 21.8° d i r e c t i o n , 8.24 N A equivalence  i svalid.  = .024 «  1, s o t h e  I t i s i n t e r e s t i n g t o n o t e t h a t G °<N , b u t o  R  i s  q  i n d e p e n d e n t o f N. The  f r e q u e n c y dependence o f R and X i s then g i v e n b y R  (fi) = R s i n c x a ' o  X  (fi) =  R  the  relative  radiation  the transducer  I  N  <  2  X  >  2x  -  2  X  )  .  (  3  >  3  6  )  Z  b a n d w i d t h i s t a k e n t o b e 2/N, h a l f of the central  2 1 M o r e u s e f u l i s t h e - 3 dB b a n d w i d t h , w h e r e s i n e x = - j » Then t h e a c o u s t i c b a n d w i d t h i s g i v e n b y Afi  A f _ 2.78 a, -9  o In order  o  t o reduce t h e i n s e r t i o n l o s s , a s e r i e s i n d u c t o r i s  u s u a l l y added t o t h e c i r c u i t that  S  frequency d i f f e r e n c e between t h e zeros  lobe.  o r x = 1.392.  (  °  3  Usually,  (3.35)  2  i t i sresonant with  o f F i g . 3. a t the center  The i n d u c t a n c e i s s e l e c t e d s o frequency  ( s i n c e x ( f i ) = 0) , Q  that i s ,  L  = -Tr-  1  It  i spossible  model t o i n c r e a s e  o T  t o make a n u m b e r o f e x t e n s i o n s  i t s a c c u r a c y and f r e q u e n c y range.  shown t h a t a n i n t e r d i g i t a l frequencies  (3.38)  transducer  to this  Emtage  circuit  [56] h a s  produces s u r f a c e waves a t t h e  f  n  = (2n + l ) f o  ,  n = 1, 2 ... .  (3.39)  Bahr and Lee [58] obtained expressions for the radiation conductance of the pth harmonic of the fundamental resonance, where p = 2n + 1, n = 1, 2 ... . They found 2 o  G(n ) - n CJTT/A P  P  i /9frw-n [ s ( T r a ) ] c o  '  K v  (3.40)  }  K(q) K'(q)  P  where P i s a Legendre polynomial. This expression shows reasonable agreement with experimental results by Weglein [59] and Marshall [60] for YZ LiNbO^ Bulk acoustic wave generation i s not predicted by the circuit model, although i t does occur. - Schmidt [61] estimated that as much as 10% of the input power goes into bulk wave generation in YZ LiNbO^ at 112 MHz.  However, Milsom and Redwood's calculations, which are probably  more accurate, predict only 1.6% for this configuration. They show a bulk resonance at 2 f , which was observed by Daniel and'Emtage [62]. In Q  later use of the circuit model here, bulk effects are ignored. A useful  addition to the model has been made by Lalcin [63],  who determined the effect of finger resistance on transducer impedance. The additional series resistance i s V-3TP ^e  W  ( 3  s  where p W  g  '  4 1 )  is the electrode sheet resistivity, i s the electrode ?wid'th_.inj,mete-rs,  and 1  „ =7-{ e 4 L  1  +  sinh(Wa )/Wa - sin(Wa.)/Wa. Zf m T—TT\ cosh(a W) - cos^W)  J  r  (3.42) '  Lakin inconveniently gives many of his variables in units of wavelengths or ohm-wavelengths (for example, his radiation resistance  73.  is may  given i n be  ft*A).  In  approximated  t e r m s o f MKS  units,  a = a  t = 0.1  ym  phase f a c t o r a f o r LiNbO^  by 8  For  the  r  + ja. =  A  s s  p  C  f  (  )  e x a m p l e , c o n s i d e r an  1/2 (1 +  2  X  /  j)  Z  (3.43)  aluminum t r a n s d u c e r of  e x c i t i n g s u r f a c e waves i n the  Z ± 21.8°  thickness  d i r e c t i o n on Y  cut _Q  LiNbOy and  Assuming the  u s i n g N = 3,  a = 0.5  and  e  p  bulk r e s i s t i v i t y  A = 0.018, f = 200  = 56  e  Then (3.42) g i v e s n  o  , we  obtain  = 1.0  to  This expression only applies one  finger pair.  resistors R  The  so  finger  =  e  we  e x p e c t the  the  metallization factor  definition  resistance  resistance of  g  = Wa,  may  =  i  0.269.  use  f a c t o r o f a = 0.5  and  to connect  N  W  t o d e c r e a s e by p  r  a •  \  i s increased  t o some v a l u e g r e a t e r t h a n the  f a c t o r a/.5  = p/t,  the  = 2a.  0.5,  Using  expression for  finger  t h e n becomes R  The static  ( 3 . 4 3 ) Wa  pairs,  3 N A  sheet r e s i s t i v i t y  ft-m,  v = 3427 m / s e c ,  f i n g e r p a i r s w i l l be  for N  x 10  •  W  a metallization  8 p  the  = 2.83  f i g u r e a c c u r a c y , s o we  P  •D  If  ( 3 . 2 8 ) and  8 3 T cs  e f f e c t of N  in parallel,  g  to  W = 1 cm,  from  e=  aluminum,  MHz,  five  R  of  e  =  4  P ,3NatA W  dielectric p e r m i t t i v i t y w h i c h  capacitance,  (3.28) i s g i v e n by  p f o r Z p r o p a g a t i o n i n the  =  (e  yy  e  zz  appears i n the  A u l d and - e  ( 3  *  2  yz  )  Y p l a n e of LiNbO..  1  /  Kino  [53]  f o r the  major c r y s t a l c u t s ,  equation be  2 v  They s t a t e  that  e  (3.45) '  varies  P  5  only s l i g h t l y  to  4 4 )  and  that  the  stress-free  for  74. T permittivity E i s the most appropriate.  Switching to tensor notation,  with (1,2,3) = (x,y,z), Warner et a l [64] give T T T , yy xx 11 o a  T T e = £,„ = 30 e zz 33 o T e = 0 . yz Thus, for YZ propagation,  e =50.2 e P o For Z ± 21.8° propagation, we must use the second rank tensor transformation rule, (3.46)  e. . = a., a,, E . „ ij i k j£ k£  Fig. 3.7. Thus,  Permittivity Transformation.  '  '33  =  2 a  31 11  +  £  = z  ±1  a  2  33 33 E  sin 21.8° + e 2  = 37.45 E . o  cos 21.8 2  3 3  c  Hence, e' = ( e e' ) " ^ p 11  2  i n  = 56.1 e o  In Table 3.2 below, the properties of LiNbO^ required for IDT calculations are summarized for several crystal cuts. Table 3.2  Cut  X  Constants for LiNb0 [40] 3  Propagation Direction  V(m/s)  Z  3483.092  A = Av/v 00  0.02598  E  P  50.2 e o  Y  Z  Y  3487.762  Z ± 21.8°  3427.641  0.02409  50.2  0.01727  56.1 e  E  o o  The complete circuit model, with matching series inductor, i s  O  (J&OiLr  R.  Fig. 3.8.  Series Circuit Model.  To summarize, the equations for the circuit elements are as follows: C  T  =  WN(E 4pW  o  R = e 3NatA  + e ) K(q)/K'(q) p  76.  R  =  o  R  N T r 2 A  C K(q) K'(q)  Q  T  Q  2 = R sine x o  a  =  R  (  sin(2x) - 2x 2x  3  x  }  (  3  >  4  7  )  2  = N7rAf/f o  When a v o l t a g e  V:^ e ^  f c  i s ^ a p p l i e d .to t h i s c i r c u i t , t h e  a c o u s . t i a c P A w j e C l a u n c h e d : c i n - c t h e .forward, . - d i r e c t i o n i s P where  V  a  radiation  1  characteristic  (3.48)  2  = V R / Izl, a n d t h e f a c t o r o a  The forward  = V /2R , a a *  a  of 2 arises  from the b i d i r e c t i o n a l  o f the IDT.  e f f i c i e n c y w i t h which s u r f a c e waves a r e g e n e r a t e d i n t h e  d i r e c t i o n , - r e l a t i v e t o t h e e l e c t r i c a l power P  dissipated  when  eo f=f  ,  i s  °  so  that  Pt -a eo t h e power i n s e r t i o n R.I..L.  P-'—  a- ^ V /(R + R ) o o e' v  R (R + R ) a o e 2 Z  loss i s =  -10 l o g . ( R (R,+ R 5 ) / 2 | z | )  (3.49)  2  v_3.  O  £  where  N = 2  At  (  R (a)  resonance, w i t h R  a  g  +  R ) e  2 +  ( R  L =-^ x (fi)) s  +  a  2  .  = 0, we s e e t h a t t h e m i n i m u m i n s e r t i o n  3 dB. The  Q of the e l e c t r i c a l equivalent  circuit i s  loss i s  Q  From e q u a t i o n  e  = i/n C ( R + R ) . o o e  (3.37),  the a c o u s t i c Q i s 'Xi  In order necessary  to achieve  L  s  (3.51)  a  i  %  (3.52)  '  t h e b a n d w i d t h w i l l be l i m i t e d b y Q  e <  I t i s possible to  g r e a t e r bandwidth a t the expense of i n s e r t i o n  l o s s i fQ  i s detuned, so t h a t t h e a c o u s t i c and e l e c t r i c a l  resonant  quencies  3.5  N  >\J  that  otherwise,  and  NT:  2.78 ^ .9  the greatest device bandwidth, i t i s  Q  achieve  (3.50)  >  g  Q  &  fre-  differ.  Experimental In order  some e x p e r i e n c e  Work t o check t h e v a l i d i t y  i n w o r k i n g w i t h s u r f a c e w a v e s , an 85 MHz SAW  was made on a n XZ L i N b O ^ s u b s t r a t e . transducer  o f t h e c i r c u i t models and g a i n  was c u t o n r u b y l i t h  Ottawa, Ont., f o r photographic  Artwork f o r a 10-finger  and sent  delay  line  pair  t o Shaw P h o t o g r a m m e t r i c s i n  r e d u c t i o n b y 100 X.  The r e s u l t i n g  photo-  r  mask h a d f i n g e r s 2.05 mm w i d e a n d a p e r i o d i c i t y  o f 4 1 ym.  The f i n g e r s  were c o n s i d e r a b l y t h i c k e r t h a n t h e s p a c e s between them, p r e s u m a b l y because o f a l o s s i n r e s o l u t i o n i n the photographic to  obtain satisfactory  transducers,  i t was n e c e s s a r y  process.  In order  t o reduce the  f i n g e r w i d t h b y means o f o v e r e x p o s u r e o f t h e p h o t o r e s i s t . T r a n s d u c e r s w e r e made 2 cm a p a r t  o f 0.25 ym t h i c k a l u m i n u m  u s i n g G a f PR-102 p o s i t i v e p h o t o r e s i s t a n d a n a l k a l i n e f e r r i c y a n i d e etchant.  (More d e t a i l s on p h o t o l i t h o g r a p h i c p r o c e s s i n g  C h a p t e r 5.)  Electrical  connection  was made w i t h  are given i n  f i n e g o l d w i r e and  78.  silver paint.  Tests on an aluminum film revealed that the s i l v e r paint  gave a contact resistance of no more than a few ohms. Transducer impedance measurements were made over a 10-250 MHz frequency range with a Boonton 250 A RX meter.  Correction was made for  the lcm-long leads to the bridge terminals by balancing the meter with only the wires connected.  The effect of acoustic reflections was mini-  mized by covering the ends of the substrate with vinyl e l e c t r i c a l tape. Fig. 3.9 shows the conductance and susceptance of a typical transducer.  The f i r s t maximum corresponds to SAW generation at the  50  100  150  200  f(MHz) Fig. 3.9 Transducer Conductance and Susceptance.  250  fundamental resonant [65],  = 8 4 . 5 MHz.  s h u n t c i r c u i t m o d e l was  apply to t h i s  transducer,  so t h a t ( 3 . 3 3 )  i s not  because  s e r i e s models does  2  *  of f . Q  The  t h e r e l e v a n t p a r a m e t e r s f o r XZ o f t h e IDT  LiNbO^ from Table  = 1 0 . 9 p f and  G  Q  the magnitude of R  q  and R ^  Both  By  I D T s were connected 1  cm  comparing  shown t h a t (the  series  comparison).  a p a r t was  w i t h 3 . 3 uH  0.6.  compare  transducer  i n a c c u r a t e , i s adequate f o r a rough  A s u b s t r a t e w i t h t r a n s d u c e r s 1.8 delay l i n e .  f a c t o r a was  4.173.  i n this  and  Microscopic  i n t h e s e r i e s m o d e l , i t can be  f i n g e r r e s i s t a n c e e f f e c t s were n e g l i g i b l e model, although  3.2.  = 4.20 millimhos  f a v o u r a b l y w i t h t h e c a l c u l a t e d v a l u e s , 1 0 . 7 4 and  and  (3.28)-(3.31)  r e v e a l e d t h a t the m e t a l l i z a t i o n  experimental values  admit-  c i r c l e d p o i n t s are experimental  are. c a l c u l a t e d f r o m t h e s h u n t model u s i n g  examination  not  .5 ,  s a t i s f i e d . F i g u r e 3 . 1 0 shows t h e t r a n s d u c e r  i n the v i c i n i t y  the curves  connected  slug-tuned  as  a  inductors  parallel. These c a n c e l l e d the  resonant  frequency  generation.  f , thereby Q  to minimize  i n p u t t r a n s d u c e r was  o u t p u t was  c a p a c i t i v e transducer susceptance reducing the i n s e r t i o n  A g r o u n d e d a l u m i n u m s h i e l d was  o f the d e l a y l i n e The  of  since 2  in  to the generation  used i n c a l c u l a t i o n s ,  ( 3 . 3 4 ) b e t w e e n s h u n t and  8.24 N A  The  i s due  to R e i l l y et a l  e t a l [ 5 7 ] i n d i c a t e i t i s more a p p r o p r i a t e f o r XZ s u b s t r a t e s .  simple equivalence  tance  According  waves. The  The  f  t h e s e c o n d r e s o n a n c e a t a b o u t 1 6 5 MHz  bulk shear  Smith  frequency  connected  loss  for  at SAW  across the  center  s t r a y r f c o u p l i n g b e t w e e n i n p u t and  d r i v e n w i t h a 5 0 Q r f power a m p l i f i e r , and  d e t e c t e d w i t h a low  capacitance o s c i l l o s c o p e probe.  the  Use  output. the of  80.  f(MHz) Fig. 3.10  Transducer Admittance near Resonance.  .81.  pulse excitation and a storage oscilloscope gave a propagation delay of 5 ys, i n close agreement with the predicted value of 5.17 ys. With sine-wave excitation at the center frequency, the measured power i n sertion loss at the center frequency was 26 dB.  Both transducers were  matched with parallel inductors. The equivalent drive and IDT circuit is shown i n f i g 3.11. The electrical power into the device i s given by 2 Pe = Po ( 1 - pr ), where Po i s the forward power on a matched trarismission line, and the reflection coefficient i s given by p =(Y -Y)/(Y +Y) r c c Y  c  i s the characteristic line impedance, and Y i s the IDT equivalent .  circuit  admittance.  50  OHM  COAXIAL CABLE  L P<3  G.  C.T  Fig. 3.11 Radiating IDT Equivalent Circuit The acoustic power i n the forward direction i s half of P . The ratio e of acoustic forward power to matched electrical power i s thus given by 2Y G c a (Y  + G ) + (B + QC + i/(£2L ) ) c a a T p 2  (3.53) 2  The power insertion loss i s then given by (3.54)  I.L. = -10 log(P /P ) a o At the center frequency , with Y C  =  G , the minimum insertion ci  82.  0  I  1— 80  I  60 Fig.  l o s s w o u l d b e 3 dB.  3.12  1  Power  1 100  Insertion  The c a l c u l a t e d  i n s e r t i o n loss of the input trans-  At the center frequency,  l a r g e o v e r a l l l o s s observed of the output alignment  (MHz)  Loss.  d u c e r o f t h e e x p e r i m e n t a l d e l a y l i n e i s shown i n f i g . of frequency.  1 f  between t h e i n p u t and o u t p u t  as a f u n c t i o n  t h e i n s e r t i o n l o s s i s 5,.4 dB.  e x p e r i m e n t a l l y was  t r a n s d u c e r and p r o b a b l y  3.12  The  due t o i m p e d a n c e m i s m a t c h  a l s o due t o a l a c k o f transducers.  accurate  SAW wavelength and amplitude measurements were made by diffracting light from propagating waves.  In the limit of small deflection  angles, the Raman-Nath theory [66] of light diffraction gives [67] I  d  - I J (2AK) 0  1  .  °-  2  * I (AK) o for the deflected light intensity.  (3.55)  The approximation i s valid for  small amplitudes A of the sound wave.  SCREEN  Pig. 3.13 Raman-Nath Diffraction  of Light by Surface Waves.  With reference to Fig. 3.13, Raman-Nath theory gives [67] e = — = < L for small angles. The  A cosiji  (3.56)  x  experimental values d = 3.3 ± .1 cm, x - 154.8 ± .5 cm, d, = 45° and X = .6328 ym, give A = 42 ± 1.5 ym . for the sound wavelength. spacing of 20.5 ym.  This agrees well with the measured IDT finger  84.  When the input rf power was increased to a certain point, the transducer was destroyed, apparently by arcing between the fingers. The maximum wave amplitude was found to be = 1.89 nm max by measuring the ratio of deflected to incident light intensity. 3.14 shows the I / I voltage.  0  ratio plotted against the square of the driving  As expected from (3.55), a straight line i s obtained.  30  - X l O "  20  10  200  100 V (volts*) Z  0  Fig. 3.14  Figure  Relative Deflected Light Intensity vs. V  85.  In spite of the approximations used to obtain an equivalent circuit model for interdigital transducers, the disagreement between theory and experiment is less than 5%.  This i s better than the 10%  claimed by Smith et a l , possibly because Auld and Kino's expressions for G 3.  and R , and Engan's C cl  take into account variation of the trans-  X  ducer metallization factor. In the next chapter, the series equivalent circuit modelddeveloped in section 3.4 w i l l be used for- the analysis-of-interdigital arrays in acoustooptic light deflectors.  V  86..  4. 4.1  BRAGG BEAM-STEERED SURFACE WAVE ACOUSTO-OPTIC LIGHT DEFLECTORS Introduction The phenomenon of light diffraction by ultrasonic bulk  waves was f i r s t predicted by Brillouin [68] in 1922, and experimentally confirmed ten years later by Debye and Sears [69].  Since that time, a  great deal of theoretical and experimental work has been done. More recently, advances in acoustic wave generation techniques, development of the optical laser, and the discovery of new materials have spurred the development of optical modulators, frequency shifters and deflectors., In particular, since the demonstration of efficient interaction between acoustic surface waves and guided optical surface waves by Kuhn et al in 1970 [70], the possibility of fabricating high-performance  acousto-  optic surface wave devices has become a topic of interest. The surface-wave acousto-optic interaction has a number of advantages over the corresponding bulk interaction.  In the latter case,  ultrasonic waves are generated by applying rf voltages across thinly ground piezoelectric crystals bonded to the acousto-optic medium. Stepped bulk transducer arrays are d i f f i c u l t to make, and beam diffraction limits the usable acousto-optic interaction length. In the case of surface-wave devices, interdigital transducers are more easily made, using a photolithography process.  The tight confinement of acoustic  and optical fields allows long interaction lengths to be used. Efficient deflectors can be made even in materials with unexceptional acoustooptic figures of merit.  For example, Schmidt and Kaminow [71] in 1975  reported 70% light deflection with only 50 mW of electrical drive power in a Ti/LiNbO„ diffused  OWG.  In this chapter, the improvement in deflector performance obtained when acoustic surface waves are generated by a beam-steering IDT array is investigated.  A model of the device i s developed which  takes into account the IDT equivalent circuit parameters and the nature of the acoustic and optical fields, and predictions of this model are compared with experimental observation. 4.2  Theory of the Surface-Wave Acousto-Optic Interaction The problem of diffraction of a guided optical wave by a sur-  face acoustic wave has received the attention of several workers i n recent years [72-76].  A treatment applicable to non-uniform waveguides  in anisotropic piezoelectrics w i l l be given here which combines features of the above references. Consider a light wave propagating in the mth guided mode i n c i dent on an acoustic surface wave of width L and wavelength A. Let 8 mo be the angle of incidence between the light wavevector k and the mo 6  planes of constant phase of the sound wave, which produces a phase diffraction grating i n the solid by means of a periodic perturbation of the refractive index.  When a suitable phase matching condition i s met, the  light'swill in general be deflected into a diffracted beam of order £ propagating in the nth guided mode at an angle ^ ^- The interaction n  may be regarded as a collision where conservation of energy and momentum obey the relations i and  to.  I  "  =  <o  o  ±  &fi  k =£ ± UK . nil mo  (4.1)  -> -> Here to and k refer to the incident light wave, andfiand K are the o mo e. » sound wave angular frequency and wavevector, respectively.  88.  L Fig. 4.1  Deflection of an OWG by a SAW.  In the discussion to follow, a Cartesian coordinate system (x,y,z) w i l l be used; however, when tensor properties of the acousto-optic medium are needed, this is to be considered equivalent to ( x ^ j X ^ , ^ ) . -  The diffraction problem can be solved either by integral or differential equations [77],  Solution of the wave equation in the  periodically modulated medium gives the greatest insight into the problem, so this approach w i l l be taken here. In a non-magnetic, non-conducting medium with dielectric permittivity tensor e(x, t) ,, , Maxwell-'s- Equations^are \; e  0  V X  8 =,|| " •  (4.2)  89.  VH = 0 and  V-D = 0 ->  *  ->•  ->  Use of the relation D = e e(x,t).E  and elimination of H from the f i r s t two  0  equations gives V X V X E = - - ^ r "apr- ( e & t ) .E)  (A. 3)  In the isotropic case, use of V X V X E = V(V.E) - V E and V.D = 0 = 2  -> -> e Ve.E + e eV.E gives o o ( i Ve-E) - V E = 2  ^ p - (e E)  (4.4)  It can be shown [78] that the f i r s t term on the l e f t i s of the order of Se A —j^—  times the second term.  In LiNbO^, for example, a 200 MHz SAW gives  approximately 10"^ for this factor, so the f i r s t term in (4.4) may be ignored.  The wave equation is then VE = 2  - f ^ r (e(x,t).E)  (4.5)  In the interaction region, the permittivity i s e = e + Ae (x,t) , where £ i s the unperturbed value. s  In an anisotropic, piezoelectric  solid, the SAW consists of a mechanical strain wave with up to six components and an associated electric f i e l d with up to three components. In general, perturbation of the permittivity may be treated as the sum of three parts: one due to the SAW electric f i e l d (linear electrooptic effect), another by the SAW strain field (photoelastic effect) and the third due to surface corrugation of the waveguide.  Lean [73] has  shown the latter contribution to be small with respect to the other  90.  terms f o r LiNbO^, so i t w i l l be n e g l e c t e d  e = e  + A£  + Ag  &  s  here, so that  (4.6)  P  where t h e s u p e r s c r i p t s r e f e r t o t h e e l e c t r o o p t i c and p h o t o e l a s t i c effects, use  respectively.  of the inverse  The c h a n g e i n p e r m i t t i v i t y may b e e v a l u a t e d  d i e l e c t r i c p e r m i t t i v i t y tensor  by  B, w h i c h i s d e f i n e d  by B- e = Taking d i f f e r e n t i a l s  and m u l t i p l y i n g by e g i v e s Be  In s u b s c r i p t n o t a t i o n ,  Consideration  the  the r e a l part  i* " "  £  i j  A  of equations of the e l a s t i c  B  e j  k  k£ •  (3.1),  ( 4  '  7 )  (3.9) and (3.10) e n a b l e s us  s t r a i n and e l e c t r i c  SAW f i e l d s i n  ^  ( x , t ) = S ^ ( y ) cos (fit - K z )  (4.8)  ET  ( x , t ) = E*  (4.9)  (y) cos(fit - Kz) ,  change i n t h e i n v e r s e p e r m i t t i v i t y t e n s o r  due t o t h e SAW s t r a i n  [75,79]  w h e r e P.,.,  0  the  e  E ^ y ) ,Re{E.(y)}  W H 6 R E  is  = - e AB e  form  and  The  ,  this r e l a t i o ni s  A  to w r i t e  1  are the p h o t o e l a s t i c constants  a t c o n s t a n t E.  c h a n g e due t o t h e :lin'ea<r. e l e c t r o o p t i c e f f e c t i s e * T AB. . = r . ., E, , 13 l j k k  Similarly,  field  91.  w h e r e ^^y r  a  r  e  t  n  linear electrooptic  e  i  The c h a n g e i n p e r m i t t i v i t y  coefficients  at constant  strain.  due t o t h e SAW i s t h e r e f o r e  Ae. ( x , t ) = 5e. c o s ( f i t - K z ) in ' i n  where  S e . = e. . ( p S . in i j jk-J,m £m r  0  r  (4.10)  + r . E^) e. -jk£ r k n l  (4.11)  n  S i n c e t h e a c o u s t i c a n g u l a r f r e q u e n c y fi i s much l e s s  than t h e o p t i c a l f r e -  q u e n c y co, t h e t i m e der£vat*iv.ea_of -Ag(x,-t)^ma35ebe..neglect'ed,  and the  wave e q u a t i o n b e c o m e s  dt  C  For the p e r m i t t i v i t y of Mathieu's  equation.  C  dt  function given i n (4.10), t h i s  The r i g h t - h a n d t e r m may b e r e g a r d e d  [80] f o r t h e s i m i l a r problem  o f a p l a n e e l e c t r o m a g n e t i c wave b y a c o u s t i c m i c r o w a v e s . of a graded-index  Solutions  of diffraction  mode m p r o p a g a t i n g i n t h e x z p l a n e may be w r i t t e n a s  OWG  E  form  as a source  o f t h e g u i d e d modes d e s c r i b e d b y t h e e x p r e s s i o n on t h e l e f t . a r e g i v e n b y Chu a n d T a m i r  i s a  The TE  ( x , t ) = U ( y ) e x p j (tot - k x = k z) nr m mx mz w  r J  where o n l y t h e r e a l p a r t i s assumed t o have p h y s i c a l  v  (4.13) '  significance.  F o l l o w i n g C h u a n d T a m i r , we may w r i t e s o l u t i o n s o f t h e wave e q u a t i o n ;(4.12) i n t e r m s o f an i n f i n i t e  M E(x,z,t) =  where  s e t o f c o u p l e d d i f f r a c t i o n modes,  00  I I • (x)U m=0 £=-<»  (rierpjO^t - k ^ x  to^ = co^ + SLQ ,  - k  z)  (4.14)  92.  A is the diffraction order, and ^ ( x ) i s a coupling constant dependent on Se and the width of the interaction region. Since the waveguide permittivity has a modulation of periodicity A, the transverse wavenumber in isotropic materials satisfies  k  Az  =  k  0z  +  '  £ K  A=0,  ±1,±2...  (4.16)  by the Floquet Theorem [80], where the guided-mode subscript m has been dropped.  The longitudinal wavenumber is ,2 - Ax  ,2 - kAz 0 =  k0  (k  Qz  + HQ'  (4.17)  Figure (4.2) shows the acousto-optic dispersion curves implied by this relation, for the case Se = 0.  Fig. 4.2.  Only the incident (A = 0) mode propagates;  Isotropic Acousto-Optic Dispersion Curves for A? = 0.  93.  however, the other modes may s t i l l be regarded as part of the solution even though their amplitude is zero.  The rigorous  solution of Mathieu's  equation shows that a non-zero driving term in the wave equation introduces stop bands into the dispersion curves at their intersection points, which now become the Bragg regimes where coupling between diffraction modes i s possible.  For example, when k  = K/2,  coupling is strong  UZ  between the incident and f i r s t diffracted modes. In general, where RQ  z  =  + £K/2, energy i s coupled from the incident to the SL th order d i f -  fraction mode. These modes propagate at the angles sinG^ = k / k £z  Q  » sin9 + £K/k 0  Q  (4.18)  When the acoustic frequency is well below the microwave region, the circles in the dispersion diagram are close together.  If the angle  Fig. 4.3. Isotropic Acousto-Optic Dispersion Curves in a Modulated Medium.  94.  of incidence  i s n e a r z e r o , weak c o u p l i n g o c c u r s  between  many  modes [77].  T h i s i s t h e Raman-Nath ( o r t h i n g r a t i n g ) l i m i t .  adjacent Strong  c o u p l i n g i s p o s s i b l e o n l y a t t h e B r a g g r e g i m e s , w h e r e 6Q = 6^ = the Bragg angle.  Then = W(2nA),  s i n e , = lK/(2k ) b 0 n  where A  6^,  i s t h e vacuum w a v e l e n g t h o f l i g h t  (4.19)  and n i s t h e r e f r a c t i v e  In a n i s o t r o p i c m a t e r i a l s , the s i t u a t i o n i s f u r t h e r  complicated  by t h e f a c t t h a t t h e i n c i d e n t and d i f f r a c t e d r a y s p r o p a g a t e w i t h indices of r e f r a c t i o n . Dixon  [82]  For Bragg d i f f r a c t i o n i n t o the f i r s t  l  n  8  0  2n71 a + 4  =  " and  s l n 9 i  the angles  =  _ ^  (  1  of incidence  n  _ i !  (  n  0 -  2 _  n  2  i  )  ^  )  (  I n LiNbO^  2 0  >  4  >  2  1  )  metal-diffused  d i f f e r b y 1% o r l e s s , s o t h e loss of  suppress d i f f r a c t i o n i n t o higher  orders.  diffraction  accuracy.  the s t r o n g e s t p o s s i b l e c o u p l i n g i n t o the f i r s t  i t i s e s s e n t i a l t h a t the a c o u s t i c phase  by Alphonse  l »  n  and d i f f r a c t i o n .  be t r e a t e d as i s o t r o p i c w i t h l i t t l e For  to  (  A  w a v e g u i d e s , n ^ and n ^ t y p i c a l l y  order,  order,  X  X  may  different  gives  S  for  index.  g r a t i n g be  The  diffraction  t h i c k enough  condition for this  given  [27] i s L > nA /X  (4.22)  2  2 ( f r e q u e n t l y L > 5nA  /(TTA) i s u s e d i n s t e a d ) .  it  i s assumed t h a t  (4.22) i s s a t i s f i e d ,  in  the f i r s t - o r d e r Bragg regime.  (4.14) o f t h e wave e q u a t i o n :  In the f o l l o w i n g d i s c u s s i o n ,  so t h a t d e f l e c t o r o p e r a t i o n i s  O n l y two t e r m s r e m a i n i n t h e s o l u t i o n  the i n c i d e n t wave,  E (x,t) = <j> (x)U (y)expj(u) t - k ^ x - k ^ z ) Q  0  0  0  (4.23)  and the diffracted wave, E (x,t) = (), (x)U (y)expj(a) t - k ^ x - k ^ z ) 1  1  1  1  (4.24)  We require the coefficients ^ ( x ) and <j>^(x), which describe continuous coupling between the waves i n the interaction region. The acoustic surface wave i s assumed to be propagating along the z coordinate axis.  In the following derivation, guided-mode subscripts are  omitted for brevity, and alphabetic rather than numerical coordinate axis subscripts are used for clarity.  The functions U (y) describe the  electric f i e l d variation of TE optical guided modes with depth.  They  are normalized by the integral k. °° 2 / U^(y)dy = 1 .  (4.25)  1  —00  Since E Q and E^ are also solutions of the unperturbed ( 6 e = 0) wave equation, i t follows that 2 2 2 2 0 "0 k„ + YT. + - " = 0 Ox Oz 2 c n  (4.26)  U  a n d  2  2  n  ..2 . 2 r .• i w  c In general, the two waves w i l l propagate in different directions, and may be in different guided modes, so the unperturbed permittivities 2 n' Q  2 and n^r w i l l differ. (cSubstMutiowdtc'c^^a') :°and- • (4.24) into the  wave, .equation -(4.1,2)' "such that .each ,wave:is ^regaEded?tas: the source of l  the. other gives--,  ;r.*-d as 'chc .  •  •• c..hi-.  (4.28)  c and  - —  AfeU^O e x p j ( o ) t  The p r i m e s d e n o t e d i f f e r e n t i a t i o n w i t h t i v e s may  b e n e g l e c t e d , i f we  assume  that  regime, i t i s p o s s i b l e  I f deflector  (4.29)  0 z  t o x.  The s e c o n d  varies  slowly.  deriva-  When t h e i t  operation i s w e l l i n the Bragg  to deflect l i g h t  y  i n t e g r a t i n g over a l ]  two coupled-mode equations in  <j>  0  }  i n t o o n l y one o f t h e s e a t a  and  K  Multiplying  and u s i n g  (4.11)  = - J -,<r a  n  exp(jB)  (Ak  exp(-jB)  ,  - K ) z - (Aw - fi)t,  with 01 0  2  and  {A.25)  gives  (4.30)  where B = Ak x +  ( 4 . 2 8 ) by UQ and  cj^,  0 = - Joig^ and  <Kx)  - k z.) .  as t h e sum o f e x p o n e n t i a l s ,  t i m e , s o t h e & = - 1 o r d e r may b e d r o p p e d . b y U^,  x  these equations describe d i f f r a c t i o n i n t o both the  1 = 1 and I = - 1 o r d e r s .  (4.29)  Q x  respect  that  c o s ( f i t - K z ) f a c t o r i n Ae i s w r i t t e n becomes c l e a r  - k  0  (4.31)  In f u l l subscript notation the overlap integral r ^ i s  7  rv,i = / *n mOnl mOp J  u  pq Unlqdy ,  5 e  (4.33)  1  where the subscripts of U, from left to right, refer to the TE guided mode number, the diffraction order, and the Cartesian coordinate axis. Equations (4.32) were obtained by use of the fact that <0Q fc co^ = co (since usually fi << co) , and the relation k  = k J6X  cos6 . For reasons discussed J6  As  earlier, coupling occurs only in the vicinity of Ak = K z  and Au = fi.  Differentiation of (4.30) with respect to x and substitution for <j>^ and <j)^ from (4.31) and (4.30) gives the uncoupled equation <r'o - 2jpd> + a a-f Q  where  (4.34)  Q  p = Ak /2. x  Similarly, i t can be shown that d)£ + 2jp<j) + a a^.1  Q  =• 0.  (4.35)  If we assume solutions of the form exp(j£Lx), substitution gives quadratic equations in B Q and g^, B  and  2  T 2p3 Q  3 + 2pB- 2  L  = 0  (4.36)  = 0 ,  (4.37)  from (4.34) and (4.35), respectively.  Ti Setting q = /p +  CSQCX^  , we obtain the solutions  B = P ± q  (4.38)  3 = -(p ± q) .  (4.39)  0  and  n  98..  Solutions for <$>Q and <j)^ are d>(x) = A exp (jx(p+q)) + A exp (jx(p-q))  (4 40)  ^ ( x ) = A exp(-jx(p+q)) + A exp(-jx(p-q))  (4.41)  0  ±  2  3  4  V  The boundary conditions to be satisfied are <f>(0) = 1 , 0  *!«»  = 0,  <!>Q(0) = 0 ,  d)^(0) = - j a ,  and  1  (4.42)  where the last two are obtained by applying the f i r s t two to (4.30) and (4.31).  Use of these boundary conditions with (4.40) and (4.41) gives  the constants A^ through A^. Then <j>(x) = exp(jpx) (cos(qx)- j ^; sin(qx)) 0  (4.43)  and a  l  <j>(x) =-j  exp(-jpx) sin(qx) .  1  (4.44)  In a lossless medium (a reasonable approximation for LiNbO^), the deflector diffraction efficiency is given by * * n = ty^i = 1 -<i>Q<l>0  =  CIQCI^  2 2 L sine qL ,  (4.45)  where sine x = sinx/x and L is the acousto-optic interaction length. When the angle of light incidence deviates slightly from the Bragg angles by an amount A8, ,  Ak fc KA9 , x '  (4.46)  99.  as can be inferred from the momentum conservation diagram (Fig. 4.4). (Usually, n^ ^ n^, so that  'v- k^, and the triangle in Fig. 4.4 i s  nearly isosceles.)  Fig. 4.4 Momentum Conservation in Anisotropic Bragg Diffraction. If we regard the SAW normalized in the sense discussed at the end of section 3.2, then the expressions obtained so far are for an acoustic power of 1 watt/meter.  For an acoustic beam L meters wide with  an acoustic power P , the permittivity change <5£ must be multiplied by 3.  /P_/L , the change in SAW amplitude. CL  Using this, along with (4.46) and  the definition of q, the diffraction efficiency becomes n = g s i n c [ g + (KA6L/2) ] 2  2  2  2  172  ,  (4.47)  .100.  with  4  g  2  = 4 X  For l i g h t  2  rP  i n c i d e n t a t the Bragg  a  L  2  COS9Q  .•  (4.48)  cos0^  a n g l e , A9 = 0, s o (4.47) b e c o m e s  2 n Whe g = TT/2, 100% d i f f r a c t i o n required  for this  = sing .  Q  efficiency  ioo  quency f . incident  The a c o u s t i c  Vr-'  =  4 50  COS6Q ^ c o s 0 ^ ^ 1. d e f l e c t i o n angle i s a l t e r e d by v a r y i n g the acoustic  The d i f f r a c t i o n  a t theBragg  efficiency  angle.  falls  o f f , since light  transducer.  fre-  i s no longer  The u s a b l e l i m i t s w i l l b e s e t b y e i t h e r t h e  a n g u l a r s p r e a d o f t h e s o u n d beam, o r b y t h e l i m i t e d b a n d w i d t h interdigital  power  (- >  L  n" r  The  i s obtained.  i s approximately  p  since usually  (4.49)  I neither  case,  the half-power  o f the  p o i n t s o f (4.47)  2 a r e o b t a i n e d when t h e a r g u m e n t o f t h e s i n e t h e c e n t r a l maximum. degree,  U s u a l l y t h eBragg  c h a n g e s b y 1.3916 away  angle i so f the order o f one  s o t h e change i n d e f l e c t i o n a n g l e a s a f u n c t i o n  change A f i s o b t a i n e d from  from  of frequency  (4.19) A 6  b  =  2nv  <' > 4  51  Thus, KA6L, 1 _ * 2 at  the half-power points.  limited,  so t h i s  as a f u n c t i o n  irrXLf A f 2nv2  Deflectors  equation gives half  of deflector  bandwidth,  = 1.3916  a r eu s u a l l y  transducer  bandwidth  thermaximum u s a b l e i n t e r a c t i o n l e n g t h  i(u:  Thus,  L • !; ; max Af Af o 8  v 2  A  .  (4.52)  As pointed out by Gordon [83], both spatial and temporal coherence are preserved when light undergoes acousto-optic deflection, provided that the angular spread A0j of the sound beam i s much greater., than the corresponding spread Acb of the light beam. Since the angle of deflection i s twice the Bragg angle, the number of resolvable spots to which light can be focussed i s N = 2A0,/A(f) g  (4.53)  For light emerging from an aperture of width A, the optical beam spread is ,, A Ad) = — nA for a rectangular beam intensity profile and the Rayleigh criterion of spot resolution [77]. Use of (4.51) gives N where  s  = Af A/v = Af T , m m  (4.54)  T is the transit time of the acoustic wave across the light beam and  Af i s the half-power bandwidth. m r  v  In the expressions derived so far, i t was assumed that the acoustic beam has a rectangular intensity profile in the interaction region.  In the next section, the effects of acoustic beam diffraction  and techniques for increasing deflector bandwidth w i l l be discussed. 4.3 Acoustic Beam Steering Acoustic beam steering has been in use for some time as a means of improving the performance of bulk acousto-optic deflectors. One of the earliest applications was by Korpel et a l [84], who worked  102.  with 20 MHz ultrasonic waves in water.  The underlying principle consists  of varying the direction of sound wave propagation as the acoustic frequency i s altered, so as to track the Bragg angle and thereby increase the usable deflector bandwidth. Consider the propagation of acoustic surface waves generated by a stepped interdigital transducer array (Fig. 4.5).  Each transducer  section has a radiating aperture of width D and an overall width G. Generation of surface waves i s equivalent to normal incidence of a plane wave on the array, except for a factor of 1/2 due to the bidirectional radiation from interdigital transducers. The step heights are an integer ( P ) multiple of AQ/2, where waves with A = ahead after excitation  by the array.  w i l l propagate straight  If the transducers are driven out  of phase, P must be odd; i f they are in phase, P i s even.  Fig. 4.5  Beam Steering Transducer.  103.  In the far radiation field, the sound wave amplitude i s given by the Fresnel-Kirchoff integral [85], MG  , 0  expj (Kx sincj) - y(x)) dx  (4.55)  with y(x) being a phase factor to account for the transducer steps and MG being the overall width of an M-section transducer.. Usually, the Bragg deflection of light occurs in the near radiation f i e l d , because the Fresnel distance i s  excessively long,.  However, the analysis i s done  in the far radiation f i e l d for convenience. Consider the effect of one of the steps (Fig. 4.6). The phase change across each aperture i s Kx sin<f> <v» Kijix for small angles.  H / c o s <f>  X  Fig. 4.6  Phase Change across One Step.  The phase change between corresponding points on wavefronts from adjacent apertures i s  K(P | - H)/cos4> -v £ | (A - A ) Q  = PTT(1 - K/K )  104.  for small angles <j>. Thus, the diffraction integral becomes  S(<f>) =  M-l I  nD+G / expj(K<j>x - nPTT(l - K / K _ ) ) n=0 nD  dx  U  M-l . e*p(JK»G) - 1 J  £  e x p j n f K ^ D - P.Cl - K / K . ) ]  n=0  9  . sin{y M [KD(j> = S sinc(y KG*) —  -  PTT(1 - K/K ).]}  ^  '  M s i n v y [ K D * - PTT(1 - K / K ) ] } Q  i s a constant.  where  (4.56)  The sound wave intensity i s I(4>)  where  IQ  = S (cj,)  =  2  I A (<j.)B (<(>) 2  2  0  ,  (4.56)  i s a constant, A((f>) = sincOj KG<j>) i s the aperture function, and  B(<j>) i s the array function.  The principal maximum of the array function i s at the angle <)>Q, determined by |  M  [KD*  Q  - PIT (1  - K/K^) ] =  0 ,  V=I T-f^ (  ( 4  '  5 8 )  The a r r a y f u n c t i o n i s more s i m p l y e x p r e s s e d i n t e r m s o f ty^,  BC*) =  s i n [-5—  ^  (<f> - <J) )] ~  M s i n [ ^  v  (4.59)  «, - <fr )] Q  The m a x i m a a r e a t t h e a n g l e s  *Bmax= * 0  +  q  D  »  ( 4  '  6 0 )  105. where q i s an integer.  The minima of A(<}>) are at  ^Amin  q  (4.61)  G  Figure 4.7 shows the array and aperture functions where A =  and G 'v* D. The latter condition i s desirable, since most of the  transducer width i s then utilized for SAW generation.  This gives a  longer acousto-optic interaction length and higher deflector (4.50).  efficiency  Furthermore, the secondary maxima of B(<}>) then occur near the  zeros of A(<j>) , so l i t t l e acoustic power i s carried outside the central peak.  When A 4 A Q , the maximum of B(cj>) shifts away  from the maximum  of A(<{>), and i t decreases in amplitude (Fig. 4.8).  Fig. 4.7 The Aperture and Array Functions for A = A ^ and D °~ G.  106.  I  0 Fig.  4.8  The A p e r t u r e and A r r a y F u n c t i o n s f o r A ^ A ^ .  In a n i s o t r o p i c m a t e r i a l s , general are  collinear with  t h e d i r e c t i o n o f power f l o w  the propagation vector.  I f t h e angles i n (4.56)  a l l c h a n g e d b y a f a c t o r o f a = (1 - 2 b ) , where b i s t h e p a r a b o l i c  constant defined  i n 3 . 1 5 , we o b t a i n  for anisotropic  the d i s r i b u t i o n of acoustic  power  propagation,  _ I  _  an * (  )  =  V"  . 10  2,KGa* <—>  , .1  . 2rMKDa 8  v  M  2  i  n  - l — < »  *0>J  -  . 2f"KDa,  M sxn  [--y-Cfr  maximum o f t h e a r r a y  f u n c t i o n B, w h e r e <J> = cj)^.  o f t h e SAW v a r i e s w i t h  <f> a c c o r d i n g  .  , H -  ^ ( 4  '  6 2 )  <f>)J 0  The maximum p o w e r i n t h e SAW i s i n t h e v i c i n i t y  sity  i s noti n  of the central  Thus, t h e peak  to the equation  inten-  107.  I  = I . s i n c ( i KGa<J)). 0 2  To a p p l y these r e s u l t s t o Bragg d e f l e c t o r s , we the Bragg The  angle be matched a t two  s t e e r i n g angle  the Bragg  require that  f r e q u e n c i e s , f ^ and f ^ ( F i g . 4 . 9 ) .  ( 4 . 5 8 ) i s a h y p e r b o l i c f u n c t i o n of frequency,  and  angle w i t h r e s p e c t t o an a r b i t r a r y d i r e c t i o n i s [ 8 5 ]  A  where c i s a c o n s t a n t . Bragg  (4.63)  2  an  =  c - ^ f  (The angle of SAW  ,  (4.64)  p r o p a g a t i o n r e l e v a n t to the  i n t e r a c t i o n i s <J>Q ( 4 . 5 8 ) , even f o r a n i s o t r o p i c m a t e r i a l s , s i n c e i t  i s the sound wavevector  r a t h e r than the power flow d i r e c t i o n t h a t m a t t e r s . )  S i n c e the Bragg  angle i s s a t i s f i e d at f ^ and f ^ , and i j ) ^ i s a  l i n e a r f u n c t i o n of f ,  Fig.  4.9.  Bragg-Angle  Tracking.  108.  f h Using-(4.58),  = T~-TJ  - f £  ( 4  -  6 5 )  h  we o b t a i n  Z U  r  h  O  r  Vh  SL  T  The angle of intersection of light and sound waves deviates from the Bragg angle by the steering error, A0 = 4> (f) - <|>(f) b  (4.67)  0  Let us define an, .array frequency f^ such that A e ( f p i s a maximum. The condition 4r  (A6) = 0  dt  (4.68)  gives f = f ''1 ~  f • "h  2  This d i f f e r s  from Pinnow's  (4.69)  u  0  [ 8 6 ] r e s u l t f , = -j^ (f„ + f, ) , w h i c h was i 2. SL h  o b t a i n e d b y u s e o f an a r b i t r a r y a d j u s t a b l e p a r a m e t e r t o o p t i m i z e t r a n s d u c e r p e r f o r m a n c e , r a t h e r t h a n t h e more n a t u r a l c o n d i t i o n ( 4 . 6 8 ) .  Using  ( 4 . 6 9 ) , t h e s t e e r i n g e r r o r becomes A0(f) = ~  Equating  ( y -+ j - - j h £  the slopes i nequations P  - ^ )  .  ( 4 . 6 4 ) a n d ( 4 . 6 6 ) f i x e s t h e P/D  l  Xf  nv  (4.70)  ratio,  109.  4.4  D i f f r a c t i o n E f f i c i e n c y o f Beam-Steered  Transducers  I t i s now p o s s i b l e t o w r i t e e x p r e s s i o n s f o r t h e d i f f r a c t i o n e f f i c i e n c y o f a beam-steered d e f l e c t o r as a f u n c t i o n o f SAW With reference to equation  (4.47), define  KA6L  ,  amplitude  sincOj  KGOI4>Q)  ,1 . 1 1 f . T ^ T ~ - f - - 2 ' h Jl f^  TrPLf  2T-  h = —  The  frequency.  (  )  .. ... - > (4  72  o f t h e c e n t r a l maximum o f t h e a r r a y f u n c t i o n i s from ( 4 . 6 3 ) , so t h e SAW a c o u s t i c power must i n c l u d e t h e 2 1  s p a t i a l v a r i a t i o n s i n e (-j  KGCKJIQ)  .  I f an i n t e r d i g i t a l t r a n s d u c e r t o w h i c h  the s e r i e s model a p p l i e s i s used f o r SAW g e n e r a t i o n , t h e a c o u s t i c power i s g i v e n by ( 3 . 4 8 ) , P  a  =  R  v 2 / ( 2 a  ^ '| ) Z  2  2 f o r a d r i v i n g v o l t a g e V.  g  where  <f>Q  2 b  Thus, g  ^ a 2 jr s i n e (irfGa(j> /v) , .4 X COS8Q cos8^ r  =  (4.48) becomes  2  p  L  0  i s g i v e n by ( 4 . 5 8 ) .  The a n i s o t r o p y parameter a e n t e r s i n t o  this  e x p r e s s i o n , a l t h o u g h i t i s n o t i n ( 4 . 7 2 ) , s i n c e d i f f r a c t i o n i s from t h e p l a n e s o f c o n s t a n t phase o f t h e SAW. The  d i f f r a c t i o n e f f i c i e n c y o f a beam-steered d e f l e c t o r i s  therefore n  2 . 2 , 2 , 2 l/2 = g sine ( g + h > ' L  b  b  v  b  ,  ,, .,„. (4.73)  where  h  =  -2D~  (4.74) F  I  "  f  f?  no.  and 4 2 _  g  p  2 r  a :  r-  b  32c  COSOQ  . 2 TrPGa _ ._ s x n c [-p^- ( 1 - f / f g ) ] ,  __. (4.75)  r  — cos0^  with V R 2  P  a ( 4  2[:(.R  3  a.  + R )  2  +  ($1L  6  - l/(fiC ) + X  Q  T  S  cL  X  )  2  .  7 6 )  ]  and R  X  a  a  = R  n  2 sine x ,  = R  Q  [sin(2x) - 2x]/(2  w h e r e x = N i r ( f - f )/f . u (j n  Cj, a r e o b t a i n e d individual  0  The IDT a r r a y c i r c u i t  n  from the appropriate  transducer  2 X  )  ,  elements R , R , L and a e s  s e r i e s / p a r a l l e l combination  of the  sections.  These e q u a t i o n s  f o r t h e a c o u s t i c power P  a p p l y when a  voltage  cL  Ygj^t tion  i g  a  ppii  e (  i  to the s e r i e s equivalent  of the beam-steering transducer.  coaxial transmission line take a d i f f e r e n t  form.  circuit  ( F i g . 3.8)  I f the device  i s d r i v e n by a  o f c h a r a c t e r i s t i c i m p e d a n c e Z^,  We w i l l  representa-  the  assume t h a t t h e r f g e n e r a t o r  equations  i s isolated  f r o m t h e l o a d , and t h a t t h e r e a r e no s i g n i f i c a n t m u l t i p l e r e f l e c t i o n s the  line.  T h e n t h e e l e c t r i c a l p o w e r d i s s i p a t e d i n t h e IDT w i t h  Z = R + j X i s given  o  impedance  by  p where P  e  = p a0  ig ), 2  2 = V,/Z i s t h e power t h a t w o u l d be d i s s i p a t e d i f t h e l i n e f c r  matched and V coefficient  f  on  i s the forward  p-^s i s g i v e n  by  r.m.s..voltage on.the l i n e .  The  were  reflection  After some algebra, we obtain. P  e  =  4V R 2  2" 2* (R + Z ) + X c  The current flowing i n the series equivalent circuit i s given .2 by P  e  = i R. The usable acoustic power i s half the power dissipated i n  the IDT radiation resistance, that i s , 2 2V R P =|i R = a 2 a f a [(R + R + Z ) + < f l L  (4.77)  2  2  a  4.5  e  c  X -^) ] 2  s +  a  Acousto-Optic Overlap Integral Calculation The overlap integral (4.33) was evaluated for the TE guided  modes shown i n Fig. 2.35 interacting with a Z ± 21.8° propagating SAW on Y-cut LiNbO^.  The permittivity, electro-optic and photoelastic  tensors are given i n Appendix I i n matrix form for the principal axes system (X,Y,Z).  To calculate these i n the system  ( X ^ J X ^ J X ^ ) ,  which i s  rotated by 21.8° about the Y = x^ axis, i t i s preferable to revert to tensor notation i n order to use the usual transformation laws [88], a. a. e lm jn mn i  r  (4.78)  a. a. a, r lm jn kp mnp  ijk  p'. -i a. a. a, a. p ijk£ lm jn ko £p mnop =  n  r  The transformation matrix i s  13  C03t  0  sinQ  0  1  0  -sine  0  C0s9  (4.79)  112.  with 6 = 21.8°.  The direction of OGW propagation was only about,1° off  the x| axis, so l i t t l e accuracy i s lost by assuming i t to be along x^ exactly.  In the experiments, no mode conversion was observed, so the  overlap integral reduces to 00  r = /  U  —00  (y) SE'  2  dy  (4.80)  The rotated permittivity, electro-optic and photoelastic tensors are given in Appendix I, along with the permittivity change factors Se'ij p.  E . 0  M n  3I ijkJl r  E-o  j3  and  e . r..,  E.„.  0  3i  l j k j3  The necessary SAW parameters for the complete evaluation of 6E'  33  were obtained from reference [40]. Unfortunately, this reference  does not give f u l l analytic solutions, in that the  coefficients of  equations (3.11) and (3.12) are missing. However, i t i s possible to find the acoustic strain and electric fields using  and  ,  3u.  3u.  (4.82)  s  with plots of the magnitude of the SAW mechanical displacements u^ (Fig. 4.10) and electric potential |$| (Fig. 4.11)  [40].  The quantities  shown have been normalized to remove the frequency dependence; for a mechanical power flow component P  x  U  = x^/S ,  2  il  =  l  =  $  along x^, the actual magnitudes are  Ni  ^  |$ |/v^ N  '3M  > ,  *The primes are omitted in the rest of this section. in the rotated coordinate system.  A l l quantities are  Fig.  4.11  114.  |S..|= /P,_n |S .| , ij 3M i j N  1  and  1  1  1  |E.| = ^ ~ f i |E^|  where N indicates the normalized values.  ,  (4.81)  For SAW propagation along x^,  i t i s clear from (3.11) and (3.12) that there i s no x^ variation, and that the operator identification S/^x^ -»• jfi/v can be made. Differentiation with respect to x^ must be done numerically; since only the magnitudes of the fields are plotted, complex phase factors  and C^i?  defined by alu.l  au.  9x  8x  2 1  2  2  (4.82)  and  C "4iL 7  2 are missing.  3x  3x  2  2  The magnitude and phase of the normalized fields are given  in reference [40] at y = 0, so these factors can be evaluated there. For example, , 23  8u„  8u  2 ^3x  3x '  3  =i  (  ^ f 2 U  +  C  32^T  )  '  (4  '  83)  At y = 0, Ii and  S  (0) = 2.656 x 10 (.0366 + .999j) _6  2  23  ( 0 )  =  3 , 2 2  x  1 0 _ 1 2  (-'  6 0 6  + -795j) .  Only the real parts of the strain and electric fields contribute to fie^^,  so we have  115.  I  N ReS  l  N  1 2> i(.999 - ^ |U  =  N  2 3  A +  G  2  3  »l U  A x  a t y = 0.  This  gives C  2 3  = 1.102.  I  n  l—3lf )  (4.84)  2  The e q u a t i o n s  obtained  manner f o r t h e r e a l p a r t s o f t h e s t r a i n a n d e l e c t r i c  S  l l  " ° »  Ni  nn  = 0 ,  r  t-  23  '  2  A x  S  =  "  A  (  ^  1  '  1  0  ' 3 U  I 3  |m  +  -V-  }  '  2  l  N  AT —  = "-352  ' 2>  I  — ¥ -  2  X  Sj  fields are  A|uL  Re S22 " = °22 sL = .0593 vffi"- . N Soo  i n this  ,  AI uN, sf1„2  = — .0583  N Ax  '  2  E  r  .  >  2  5  6  ^  A I ^ L ,  A x  2  U I N  E * = - . 2 9 3 & -L2-L ,. 3 v I n F i g . 4.12, t h e r e a l p a r t s o f t h e a c o u s t i c s t r a i n s  and  as  a f u n c t i o n o f d e p t h f o r an a c o u s t i c f r e q u e n c y  Fig.  4.13 t h e c o r r e s p o n d i n g  electric  fields  the  3™ c o e f f i c i e n t s ,  since the displacement  not be measured w i t h g r e a t p r e c i s i o n n e a r x c h a n g e 6~e  33  i s given by  a r e shown  o f 165 MHz, a n d i n  a r e shown.  would have been o b t a i n e d b y s o l v i n g e q u a t i o n  (4.85)  Better  accuracy  (14) o f r e f e r e n c e  [40] f o r  and p o t e n t i a l f i e l d s 2  = 0.  could  The p e r m i t t i v i t y  Fig.  4.12  Real Part of the Acoustic Strains vs. Depth for f = 165  MHz.  Fig. 4.13 Real Part of Electric Fields vs. Depth for f = 165 MHz.  118.  6e  =  3 3  =:3.96S* "^4 25S* :  2  2  -.279S  2 3  +.63S^  3  +1.29x10  9  E^  -7.15x10  8  E^' (4.86)  The p r i n c i p a l s t r a i n electric butions  field with  components a r e S  c o m p o n e n t i s E.^.  depth t o the overlap  electro-optic parts  2 3  Figure  3 >  and t h e p r i n c i p a l  4.14 shows t h e r e l a t i v e  contri-  i n t e g r a l by the p h o t o e l a s t i c and  f o r a TE g u i d e d w a v e w i t h  C o m p a r i s o n w i t h F i g . 4-10  shows t h a t  a constant  electric  field.  t h e maximum i n t h e p h o t o e l a s t i c  c o n t r i b u t i o n c o r r e s p o n d s t o t h e minimum This  and S^  in | | ,  where d | u | / d x 3  i s a t about o n e - f i f t h t h e a o o u s t i c wavelength.  2  = 0.  S i m i l a r l y , the  minimum  i n t h e e l e c t r o - o p t i c p a r t i s a t a b o u t A/2, s i n c e  minimum  t h e r e , and  |$| h a s a  | E | i s p r o p o r t i o n a l t o |$|. 3  These r e s u l t s a r e s i m i l a r  to those obtained  by T s a i e t a l  [74],  e x c e p t t h a t these authors appear t o have s c a l e d the e l e c t r i c  field  i n c o r r e c t l y ( i n f i g . 4) b y a f a c t - o n Of: .2. .a's. .a f u n c t i o n o f d e p t h .  could lead to considerable modes p r o p a g a t i n g  e r r o r i n T, p a r t i c u l a r l y  f o r guided  L = 1 meter f o r the three  the s h a l l o w  optical  very near the surface.  I n F i g . 4.15, t h e f a c t o r g i s p l o t t e d f o r P g  frequencies,  This  g u i d e d TE modes o f t h e OWG.  M  = 1 watt  and  At low a c o u s t i c  o n l y a s m a l l f r a c t i o n o f t h e a c o u s t i c power f l o w  overlaps  g u i d e d o p t i c a l waves, and t h e d i f f r a c t i o n e f f i c i e n c y  i s low.  A b o v e a b o u t 200 MHz, h o w e v e r , . p a r t i c u l a r l y  f o r the higher  order  the  factor g i s flat with  i s a desirable  characteristic  for  device  response. of  This  a p p l i c a t i o n s , since i t gives  a flatter  d e f l e c t o r frequency  In c a l c u l a t i o n s , i t i s expedient to w r i t e equations  t h e t o t a l a c o u s t i c p o w e r P , r a t h e r t h a n ^^H'  propagation by  frequency.  on L i N b 0  /l.05 i n this  3 >  case.  P^ = 1.05 P ^  M  modes,  ^  o r  y  ~ ^  ±  i n terms 21.8°  [ 4 0 ] , s o t h e f a c t o r g must be d i v i d e d  Fig. 4.14  Relative Electro-optic and Photoelastic Contributions to Overlap Integral for f = 165 MHz.  120.  Fig. 4.15.  121.  4.6  Experimental  Work  S e v e r a l beam-steered l i g h t w e r e f a b r i c a t e d on Chapter  of  first  Sections  from o p t i m a l . (Fig.  6-mode N i / L i N b O ^ d i f f u s e d OWG  4.16).  d e f l e c t o r t e s t e d was 4.3  and  details). angle  The  4.4  E a c h s e c t i o n was a center  g r a p h y of vacuum d e p o s i t e d  had  made b e f o r e  d e v i c e was  o f 21.8°  from the  1.55  frequency  mm  w i d e and  o f 155  a l u m i n u m 0.3 o r i e n t e d so  ym  MHz  had  thick.  Z a x i s of Y-cut L i N b O ^  g a v e C^ = 4 p F ,  20X  frequency  was  d r i v e n out  far  array with P =  2 1/2  (See  photolitho-  Chapter 5  With the  two  of phase, and  R  1  finger pairs.  for  t h a t s u r f a c e waves propagated  measurements a t the  center  i t s design  w e r e made b y  and  4.16  t h e beam s t e e r i n g  been c o m p l e t e d , so  ducer s e c t i o n s counected i n p a r a l l e l  Fig.  devices  characterized i n  I t c o n s i s t e d o f a 2 - s e c t i o n p h a s e d IDT  Transducers w i t h  an  The  2. The  theory  the  d e f l e c t o r s w e r e made.  trans-  impedance  + R  Enlargement of Transducer P h o t o l i t h o g r a p h y  at  = 40 fi.  Mask.  122.  Vinyl electrical tape was used to minimize acoustic reflections.  These  values are i n reasonable agreement with the parameters of the IDT series equivalent circuit, which are C = 3 . 4 6 pF, R T  Q  = 3 8 . 6 Q and R  G  = 4 . 9 ft  for a metallization factor of . 0 . 4 . Fig. 4 . 1 7 shows the assembled acousto-optic deflector. IDT i s on the right, and SAW propagation i s from right to l e f t .  The Electrical  connection was made by thin wires connected with silver conductive paint. Coupling to the optical waveguide was by means of rutile coupling prisms; in the photograph, the deflector output beam can be seen as a bright spot on the base of the lower prism.  E l e c t r i c a l tape was used to absorb the  surface waves and hold the substrate in place.  The prism clamps were  adjusted to give reasonable coupling efficiency.  Fig. 4 - 1 7 Acousto-Optic Deflector (Actual Size).  123.  Some d i f f i c u l t y was e x p e r i e n c e d b e f o r e s a t i s f a c t o r y operation able  was o b t a i n e d ,  tuning  iency  i s very low).  t o b e 0.19 with  inductor.  uH r a t h e r  owing p a r t l y t o t h e d i f f i c u l t y (Without  than the a n t i c i p a t e d  light  and p o s s i b l y  discernible and by  4.19  resulted  0.31  by a d d i t i o n a l s t r a y  from the LiNbO^ surface  d r i v e power needed  generation  uH  required  suit-  effic-  and e r r o r  to resonate  can be a c c o u n t e d f o r by t h e i n d u c t a n c e o f t h e  Attempts t o observe the surface of  t h e SAW  of finding a  T h e c o r r e c t v a l u e was o b t a i n e d b y t r i a l  C j . The d i f f e r e n c e  lead wires  this inductor,  deflector  capacitance.  waves by  were s u c c e s s f u l ;  1 watt) before  Raman-Nath d i f f r a c t i o n however, the h i g h r . f .  t h e d i f f r a c t e d beams became e a s i l y  i n a number o f b u r n t - o u t t r a n s d u c e r s .  Figs.  show two t y p i c a l f a i l u r e modes; t h e f i r s t was p r o b a b l y  4.18  caused  a r c i n g b e t w e e n f i n g e r s , a n d t h e s e c o n d b y o v e r h e a t i n g due t o f i n g e r  resistance.  Fig.  4.18  Transducer  failure  124.  Fig.  4.19  When t h e d e v i c e was f i r s t  c o n n e c t e d a s shown i n F i g . 4 . 1 7 , t h e  B r a g g a n g l e o f l i g h t p r o p a g a t i o n i n t h e s u r f a c e w a v e g u i d e was h a r d t o find,  owing t o t h e d e l i c a t e  adjustments required.  d e f l e c t o r b a n d w i d t h was a b o u t 26 MHz; a s a r e s u l t ,  T h e m e a s u r e d -3 dB t h e a n g u l a r range  o v e r w h i c h t h e i n t e r a c t i o n was v i s i b l e was o n l y a b o u t 12 m i n u t e s o f a r c , as c a l c u l a t e d w i t h e q . 4.51.  A t angles of incidence n e a r l y normal t o the  a c o u s t i c w a v e v e c t o r , t h e Raman-Nath d i f f r a c t i o n (Fig.  4.20).  The d i f f r a c t i o n  Figs.  r e g i m e was c l e a r l y  e f f i c i e n c y was v e r y l o w , a s a n t i c i p a t e d .  4.21 a n d 4.22 s h o w B r a g g d i f f r a c t i o n  g u i d e d modes w i t h  visible  of the T E  t h e r f d r i v e o f f (upper photo) and on ( l o w e r  The T E mode a p p e a r s on t h e l e f t , q  q  and T M  Q  photo).  surrounded by s c a t t e r e d l i g h t .  The  d i f f r a c t i o n e f f i c i e n c y o f t h e TE mode was a b o u t 4 0 % . The d e v i c e was d r i v e n b y a GR 1215-B o s c i l l a t o r Boonton  230 A r f power a m p l i f i e r  capable of d e l i v e r i n g  f o l l o w e d by a  5 watts into  a  125.  Fig.  50 fi l o a d . was  4.20  R a m a n - N a t h D i f f r a c t i o n o f a G u i d e d TE wave ( t h e u p p e r a n d l o w e r s p o t s on t h e l e f t a r e t h e d i f f r a c t e d beams; t h e l a r g e s p o t i s t h e u n d i f f r a c t e d TE mode, a n d t h e s m a l l s p o t on t h e r i g h t i s a TM mode)  Measurement o f t h e d i f f r a c t i o n e f f i c i e n c y  complicated by m u l t i p l e r e f l e c t i o n s on t h e l i n e  frequency  response  and t h e d i f f i c u l t y  o f o b t a i n i n g r e l i a b l e high-impedance r f v o l t a g e measurements. problems were overcome b y c o n n e c t i n g  These  a resistive voltage divider  across  the device and measuring the v o l t a g e a t t h e matching i n d u c t o r w i t h l o w c a p a c i t a n c e p r o b e on a h i g h - s p e e d Fig. response.  oscilloscope.  4.23 shows a c o m p a r i s o n o f t h e m e a s u r e d a n d c a l c u l a t e d  The i n d u c t o r was t u n e d  t o g i v e maximum d i f f r a c t i o n  a t 165 MHz w i t h t h e B r a g g a n g l e m a t c h e d a t t h a t f r e q u e n c y . was  a  calculated using equations  efficiency  The  response  ( 4 . 7 3 ) - ( 4 . 7 6 ) w i t h t h e v a l u e s P = 1, 1/2  L = 3.1 mm, R  D = 1.55 mm,  = 38.6 fi, R  G = 1.9 mm,  = 4.9 fi, C  a = 1.374, f  = 3.16 p F a n d L  = v(nP/XD)  = 1 6 3 . 4 MHz  = 0 . 3 1 uH. T h e e x p e r i m e n t a l  Fig. 4.22  Same with r f Drive Switched On (n  ** 0.4)  127.  p o i n t s a r e the average o f s e v e r a l runs, The  prism  c o u p l e r was a d j u s t e d  a n d a r e f o r a l l t h r e e TE modes.  f o r o p t i m a l c o u p l i n g o f t h e T E ^ mode, s o  t h e o v e r l a p i n t e g r a l o f t h i s mode was u s e d i n t h e c a l c u l a t i o n s .  Within  2 a b o u t 2%, g^ c a n b e a p p r o x i m a t e d b y  g  2  b  =  0.040 / f  In the c a l c u l a t i o n ,  P L a  sine  nP  2  Gn 2D  i t was f o u n d t h a t t h e m a g n i t u d e o f t h e m e a s u r e d rms  d r i v e v o l t a g e h a d t o be i n c r e a s e d b y a l m o s t 2 0 % i n o r d e r nitude o f the t h e o r e t i c a l curve  to the experimental  SAW p r o p a g a t i o n  losses or inaccuracy  been t h e source  of the disagreement.  the i n d u c t o r i s detuned t o a h i g h at a frequency  b e l o w 100 MHz.  results.  limiting 146  factor.  frequency  and t h e B r a g g a n g l e  The b a n d w i d t h i n F i g . 4.23 i s  the i n t e r d i g i t a l transducer  number o f r e s o l v a b l e s p o t s  i sN  t h e B r a g g r e g i m e b y a f a c t o r o f 1.7.  diffracted  light.  that l i m i t s the  l i g h t beam a t t h e  i n c o u p l i n g t o t h e OWG.  t o the c o n d i t i o n (4.22),  at higher  (4.52) i s  = A f x = 15, which i s r a t h e r a  g  I t c o u l d be i n c r e a s e d b y u s i n g a w i d e r  According  circuit,  F o r a l i g h t beam 2 mm w i d e , t h e a n t i c i -  expense o f speed and g r e a t e r d i f f i c u l t y  clearly visible  equivalent  , t h e IDT a c o u s t i c b a n d w i d t h o f 39 MHz i s t h e  o v e r a l l d e f l e c t o r performance.  s m a l l number.  matched  The b a n d w i d t h i s c o n s i d e r a b l y g r e a t e r , a t  The b a n d w i d t h o f t h e B r a g g i n t e r a c t i o n i t s e l f  MHz, s o i t i s c l e a r l y  pated  Either  F i g . 4.24 shows t h e r e s p o n s e when  l i m i t e d by the e l e c t r i c a l bandwidth o f the transducer I n F i g . 4.24  t o m a t c h t h e mag-  i n t h e v o l t a g e measurements c o u l d have  the expense o f d i f f r a c t i o n e f f i c i e n c y .  a b o u t 24 MHz.  (4.87)  (1 - f / f j )  d e f l e c t o r o p e r a t i o n was w i t h i n  The s e c o n d o r d e r d i f f r a c t e d beam was  d r i v e p o w e r , a n d a c c o u n t e d f o r up t o 4% o f t h e  128.  129.  .06  Ob  .04  .02  140  120  180  160 f  Fig.  (MHz)  4.24  T h e d i f f r a c t i o n e f f i c i e n c y was m e a s u r e d a s a f u n c t i o n o f d r i v e voltage  a t 165 MHz.  When t h e v o l t a g e  before,  good agreement i s f o u n d b e t w e e n e x p e r i m e n t and t h e o r y .  The s m a l l d i s c r e p a n c y order  beam.  i s c o r r e c t e d b y t h e same amount a s  t h a t e x i s t s may b e due t o d i f f r a c t i o n  The a c o u s t i c d r i v e p o w e r was circuit.  order  70% a t a c o r r e c t e d e l e c t r i c a l  which corresponds to P  The maximum d i f f r a c t i o n e f f i c i e n c y  = 600 mW. el  i n t o the second  c a l c u l a t e d on t h e b a s i s o f t h e  IDT e q u i v a l e n t beam w a s  ( F i g . 4.25)  i n t o the  d r i v e p o w e r o f 1.35  first  watts,  At higher voltage, the device burnt out.  130.  TOO  200  300  4 00  ACOUSTIC DRIVE POWER  500  600  700  (mW)  Fig. 4.25 Beam steering was not clearly observable with this device. It i s most likely that the response obtained in Fig. 4.24 i s the result of acoustic beam steering, but the experimental errors were sufficiently great that no firm conclusion was possible. On the basis of these experimental findings, the beam-steered deflector theory of Sections 4.3 and 4.4 was put in i t s f i n a l form.  An  improved transducer with a 200 MHz center frequency was designed using the theory.  A higher frequency device would have been desirable (since  131.  g r e a t e r b a n d w i d t h c o u l d h a v e b e e n o b t a i n e d ) , b u t 4 ym l i n e w i d t h s w e r e c o n sidered cess  t o be t h e r e s o l u t i o n l i m i t  used.  Since  the f i n e s t l i n e s  m a t e r i a l w e r e f o u n d t o b e 0.5 mm, 125:1.  The maximum w i d t h  a b o u t 1cm. cluded  Consequently,  On t h e b a s i s o f f r e q u e n c y  This  diffraction  o f P, t h e w i d t h  c u l a t e d , and v a r i o u s v a l u e s were found.  of the transducer  accurate.  array  (4.71).  The f i n a l  requirement i n  i m p e d a n c e n e a r 50 ft. F o r a s e r i e s o f i n t e g e r s e c t i o n were  s e r i e s / p a r a l l e l combinations were t r i e d The f i n a l  (4 s e c t i o n s ) and R  + R  = 50 ft. The t o t a l  cal-  untillsuitable  d e s i g n p a r a m e t e r s w e r e P = 2, D = 2.54  mm,  r e s i s t a n c e was  e  f o r c e d t o 5 0 ft b y s e l e c t i n g s u i t a b l e v a l u e s t h i c k n e s s a n d m e t a l l i z a t i o n f a c t o r a. process  o f t h e aluminum e l e c t r o d e  C o n t r o l o f t h e f o r m e r was  used, but the l a t t e r  difficult  c o u l d be v a r i e d  will. Three f u n c t i o n a l devices were f a b r i c a t e d u s i n g  described The  frequency  f ^ , s o t h e s e l e c t i o n f ^ = 205  and impedance o f each t r a n s d u c e r  the photolithography  was  s e c t i o n s were r e q u i r e d so t h a t t h e approximate  o  at  o f the transducer  t h e r a t i o D/G was c h o s e n t o b e 0.9.  theory would be reasonably  t h e d e s i g n was a t r a n s d u c e r  L = 10.2 mm  frequency  i n S e c t i o n 4.3.,  A minimum o f f o u r t r a n s d u c e r  with  illuminating  r e s p o n s e c a l c u l a t i o n s , i t was c o n -  f i x e d t h e P/D r a t i o  reasons discussed  artwork  r a t i o was  t h a t t h e g r e a t e s t b a n d w i d t h was o b t a i n a b l e when t h e a r r a y  MHz was made.  i n the next  chapter.  A l l had s i m i l a r  techniques  characteristics.  r e m a i n d e r o f t h i s s e c t i o n d e s c r i b e s e x p e r i m e n t s c a r r i e d o u t o n one  device.  pro-  t h a t c o u l d b e h a n d l e d w a s a b o u t 1.25  t h e maximum w i d t h  t h a n t h e IDT c e n t e r  values  t h e maximum p h o t o r e d u c t i o n  o f c u t t i n g and u n i f o r m l y  f ^ was g r e a t e r  For  mask m a k i n g  t h a t c o u l d be r u l e d on r u b y l i t h  of artwork  meters, owing t o t h e d i f f i c u l t y greater widths.  of the photolithography  132.  Figure 4.26 shows an enlargement of the photolithography mask used.  The transducer had three finger pairs and four sections, which were  Fig. 4.26  Beam Steering IDT Mask (10X).  connected series-parallel and driven in phase. High-frequency impedance measurements with a Boonton 250A RX meter gave C  T  = 5.7 pf and R = 59 fi.  Examination of the transducer in a scanning electron microscope showed an average metallization factor of about 0.45 and an aluminum thickness of 0.4 ym.  Four-point probe measurements indicated an aluminum resistivity —8  of 4.5 x 10 fi-m, about 1.6X the bulk value. dimensions D = 2.54 mm and G = 2.83 mm.  The transducer had the  Comparison with the mask artwork  gave a reduction ratio of 1:118.2, which implied that A f  = 17.19 ym and  = 199.4 MHz for propagation along the Z-21.8° direction on Y-cut  LiNbO . Q  The calculated parameters of the equivalent series circuit model  133.  are C  T  = 3.55 pF, R  Q  = 33.2 ti and R  = 16.5 Q.  £  The discrepancy between  these and the measured values can be accounted for approximately when the aluminum conductors and contact pads of the device are taken into account (Fig. 4.26).  These had an estimated resistance of 4.5 ohms as well as  a capacitance of about 2 pf to the ground plane of the impedance bridge, through the LiNbO^, which has a low-frequency permittivity of about 55. The extra capacitance  appears in parallel with the transducer; because the  Q i s reasonably high, the total equivalent series capacitance would be expected to be comparable i n magnitude to the sum of the parallel capacitances.  In addition, the silver paint gave a measured contact resistance  of 3 ti at 200 MHz,  so the differences are accounted for.  For the acousto-optic experiments, a plexiglass substrate stage was used in order to minimize stray capacitance. circuit used to drive the device.  BNC I ADAPTOR-  GR874-LBA  HP  shows the  A 10 dB power attenuator was used to  reduce reflections on the transmission line.  500 MHz LOW -PASS FILTER  Figure 4.27  230  SLOTTED  SWR measurements were made  10 dB ATT EN.  AMP  LINE  GR1216A IF  Fig. 4.27  AMP  Acousto-Optic Deflector Drive Circuit.  134.  a t 200 MHz w i t h a v a r i e t y o f s m a l l h a n d - w o u n d i n d u c t o r s c o n n e c t e d i n s e r i e s w i t h t h e IDT u n t i l  an o p t i m a l match was f o u n d .  and was a p p r o x i m a t e l y  4 mm i n d i a m e t e r ,  This  i n d u c t o r h a d 3i t u r n s  w i t h a measured i n d u c t a n c e o f  0.057 uH. A d d i t i o n a l i n d u c t a n c e w a s p r o v i d e d w i t h t h e c o n n e c t i n g This  c o u l d not be measured d i r e c t l y , b u t a p i e c e  same t o t a l  l e n g t h had a measured inductance The  sequently  minimum SWR o b t a i n e d  discovered  a n SWR o f 2, s o t h e t r a n s d u c e r  was  necessary  to consider  p2  o f .035 uH.  a t 200 MHz was a b o u t 2.2;  was a p p a r e n t l y  thee f f e c t  l i n e between t h e connector  i t was s u b -  matched.  connector However, i t  o f s t a n d i n g waves o n t h e t r a n s m i s s i o n  a n d t h e IDT.  c o e f f i c i e n t o f magnitude  d e f l e c t o r half-power  o f about t h e  t h a t t h e GR t o BNC a d a p t o r a n d t h e BNC  had  reflection  of wire  leads.  A n SWR o f 2 c o r r e s p o n d s t o a  = (2-1)/(2+1) = 0.33. A t t h e l i g h t  |pj  p o i n t s , the magnitude  of thereflection  coefficient  a t t h e l o a d c a l c u l a t e d f r o m t h e m a t c h e d IDT e q u i v a l e n t c i r c u i t was l e s s  2 t h a n 0.5. (Fig.  Using  4.28)  P = P ( 1 - |p| ) t o f i n d Q  that neglect  the forward  p o w e r , i t c a n b e shown  o f the m u l t i p l e r e f l e c t i o n s leads  t o a maximum  e r r o r o f o n l y a few percent. F i g u r e 4.29 the  shows a s c a n a c r o s s . t h e  d e f l e c t e d l i g h t beam,  Auto-Photometer.  u s i n g a Gamma S c i e n t i f i c M o d e l  A t an a c o u s t i c frequency  t h e u n d i f f r a c t e d beam i n t e n s i t i e s parable  This  theoverlap  TE modes o f  2900 S c a n n i n g  o f 200 MHz, c o m p a r i s o n w i t h  i n d i c a t e d t h a t a l l t h r e e modes h a d com-  d i f f r a c t i o n e f f i c i e n c y , w i t h t h e TE.^.. mode b e i n g  efficient. of  three guided  i s i n agreement w i t h  somewhat more  the calculated relative  i n t e g r a l s f o r r thethree  magnitudes  modes.  When t h e b e a m - s t e e r i n g I D T was d e s i g n e d , m e t e r a was e r r o n e o u s l y  incorporated  into  c u l a t e d array frequency  w a s a c t u a l l y 180.4  (4.71).  theanisotropy As a r e s u l t ,  para-  the cal-  MHz r a t h e r t h a n t h e 205 MHz  135.  136.  expected.  Figure 4.30 shows the frequency response of the diffraction  THEORY  Q 270  E X P E R I M E N T  f  h  150  2  Fig. 4.30  0  = 32 ft, V  L = 10.16 mm,  f  250  f(MHz)  *i  The theoretical curves were calculated with (4.73)-  (4.75.) and (4.78), using the values C e  250  or f , the frequency at which the Bragg  JO  R  te  Diffraction Efficiency vs. Frequency.  efficiency for three values of f angle i s matched.  0  or  160  = 3.8 Vvrms, f =177 1 n  P = 2,  T  = 3.55pF, L  g  = 1.8uH, R  MHz, D = 2.54 mm,  G = 2.83  q  = 33.2 fi, mm,  = 53 fi, n = 2.23 and X = .6328 urn. The forward  voltage on the line was measured with the meter in the rf power amplifier, which was calibrated with an HP 430-C Microwave Power Meter using the 477B thermistor head.  The equivalent circuit parameters are within 10% of  the best estimates available for the total equivalent impedance of the  137.  t r a n s d u c e r , e l e c t r o d e s and matching c i r c u i t . is  Additional series  resistance  due t o t h e s k i n e f f e c t i n t h e c o n n e c t i n g w i r e s , a n d a d d i t i o n a l  induc-  t a n c e d e r i v e s f r o m t h e w i r e s a n d f e e d t h r o u g h s ( e s t i m a t e d t o b e . 1 uH) . The  lower v a l u e o f f ^ used  fit  t o the data.  in  (2% b e l o w  The d i f f e r e n c e c o u l d be due t o s m a l l n o n - u n i f o r m i t i e s  t h e t r a n s d u c e r geometry.  best f i t  t h e c a l c u l a t e d v a l u e ) gave a b e t t e r  The e x a c t v a l u e o f i n d u c t a n c e u s e d g a v e t h e  t o the observed frequency response.  This approach permits a  b e t t e r comparison between t h e o r y and e x p e r i m e n t .  V a r i a t i o n s o f ± 10% i n  the c i r c u i t parameters would have g i v e n s i m i l a r r e s u l t s , w i t h differences  i n d i f f r a c t i o n e f f i c i e n c y , bandwidth  and o v e r a l l  minor response  shape. The  e f f e c t s o f beam s t e e r i n g a r e c l e a r l y e v i d e n t i n F i g .  When t h e d i f f e r e n c e b e t w e e n f ^ a n d t h e B r a g g f r e q u e n c y i s l a r g e , s t e e r i n g b e c o m e s more p r o n o u n c e d ;  a t t h e expense  beam  the d e f l e c t o r bandwidth i n c r e a s e s  51 MHz when f„ = 160 MHz t o 68 MHz when f, = 270 MHz. I h is  This  F i g u r e 4.31  shows t h e d i f f r a c t i o n  which i l l u s t r a t e s and b a n d w i d t h .  These a r e combined  t h e i n h e r e n t t r a d e o f f between d i f f r a c t i o n  These c h a r a c t e r i s t i c s  are a l l predictable  Increasing f ^ o r decreasing f ^ has the e f f e c t of r a i s i n g  to  .04  efficiency  a t f = 200 MHz a s a f u n c t i o n o f t h e B r a g g f r e q u e n c y , a n d F i g . 4.32 d e f l e c t o r bandwidth v s . Bragg frequency.  from  increase  o f d i f f r a c t i o n e f f i c i e n c y , w h i c h d r o p s f r o m 0.34  a t t h e IDT c e n t e r f r e q u e n c y .  4.30.  shows  i n Fig.  4.33,  efficiency  from F i g . the  4.9.  curve,  w h i c h i n c r e a s e s t h e f r e q u e n c y r a n g e o v e r w h i c h a c o u s t i c beam s t e e r i n g i s effective. F i g u r e 4.34  g i v e s a comparison between t h e c a l c u l a t e d  of a c o n v e n t i o n a l and a beam-steered  deflector.  response  The l a t t e r h a s t h e B r a g g  f r e q u e n c y m a t c h e d a t 160 MHz a n d t h e f o r m e r a t 200 MHz.  The d r i v e  voltage  138.  -3  -e—-e—er  -6  -12  j  L  TOO  200  300  f (MHz)  Fig. 4.31 Diffraction Efficiency at f = 200 MHz vs. Bragg Frequency with V = 3.8 V rms.  70  60  ' N  §50  40  0  100  -I  —I  1  o o  -J  I  200  '  I  I  J  BRAGG FREQUENCY (MHz)  Fig. 4.32 Deflector Bandwidth vs. Bragg Frequency with V = 3.8 V rms.  I  300  140.  is  8 V r m s , g i v i n g a maximum d i f f r a c t i o n e f f i c i e n c y o f 0.93.  d e f l e c t o r d i f f r a c t i o n e f f i c i e n c y was f o u n d w i t h modified  versions of  (4775) a n d (4.76).  The u n s t e e r e d  (4.74) a n d (4.77), u s i n g  The a n g u l a r  d e v i a t i o n from t h e  Bragg angle i s A 9  - 2nV  b  KA9  so  that  h  A  f  L  =  = -JLA_  2nv  2  In  >  F  A  F  L  (4. )  .  88  2  (4.76), t h e f a c t o r s i n e [-ZQ- (- ~ 1  f  /  f 0  )l  must b e r e p l a c e d b y t h e d i f f r a c t i o n p a t t e r n o f t h e s t e p l e s s a r r a y , w h i c h c a n b e shown t o b e sinc (KA9,L/4) b  (4.89)  = s i n c ( h /c) , c  2  2  when f, = f . b o The great.  b a n d w i d t h o f t h e beam-steered d e f l e c t o r i s almost t w i c e as  T h e a c o u s t i c p o w e r ( F i g . 4.35)  i s t h e same f o r b o t h d e v i c e s .  p r i n c i p a l reason f o r t h e d i f f e r e n t bandwidths i s evident w h i c h shows t h e B r a g g - a n g l e d e v i a t i o n v s . f r e q u e n c y . bandwidth o f t h e unsteered  d e f l e c t o r through h i n  i nFig.  The  4.36,  This l i m i t s the  (4.74).  Of l e s s e r  2 i m p o r t a n c e i s t h e f a c t o r s i n e (h^/2), w h i c h f a l l s r a p i d l y on e i t h e r s i d e o f t h e IDT c e n t e r Calculations achieved  frequency.  i n d i c a t e t h a t g r e a t e r bandwidths c o u l d have been  i f f ^ had been c l o s e t o f  d u c e r w i t h D = 2.18  o f f somewhat m o r e  Q  .  F o r example, a s i x - s e c t i o n t r a n s -  mm, P = 2 a n d f ^ = 195 MHz i s c a p a b l e  of giving a  78-MHz b a n d w i d t h a maximum d i f f r a c t i o n e f f i c i e n c y o f 0.5 w i t h a d r i v e v o l t a g e o f 8 V rms when t  Q  = 150 MHz.  The b a n d w i d t h i s a l s o g r e a t e r a t  141.  Fig. 4.36  Deviation from Bragg Angle vs. f.  142.  higher drive voltages, as shown i n Fig. 4.37 for the device mode. With an r f forward voltage of 10 V rms on the line (P = 465 mW) and f = 150 MHz,  Fig. 4.37  Diffraction Efficiency vs. f for Several Drive Voltages (f = 150 MHz).  100% diffraction into the f i r s t order beam was observed.  The deflector  was somewhat unstable at such high input power, due to thermal expansion of the substrate which altered the coupling efficiency into the optical waveguide.  The I = -1 and £ = +2 diffracted beams were observable,  although less than 1% of the total light was diffracted into these orders. Figure 4.38 shows the dependence of diffraction efficiency on drive voltage when f = 150 MHz and f = 200 MHz.  J_  8  12  16 V (VOLTS rms)  Fig. 4.38.  t  I  5  10  L  15  DEFLECTION ANGLE fmr) Fig. 4.39 Light Deflector Beam Profiles ( r ^ ^ .9).  144.  The deflector light beam profiles are shown i n Fig. 4.39 for an rms forward voltage of 8 V at 200 MHz.  The range of angular deflection -3  over a 61 MHz deflector bandwidth i s 2A9^ =5.05 x 10  radians. Using  the Rayleigh criterion, the number of resolvable spots i s N  g  = 2A6^/A<J) = 43.  The maximum number obtainable at either high drive voltage or reduced diffraction efficiency (due to beam steering) i s over 50. Optical coupling was effected without a lens over the entire ^2.5 mm width of the input coupling prism.  The theoretical number of resolvable spots i s N = Afx = 44 s  for a bandwidth of 61 MHz, i n good agreement with the observed number. The access time i s limited by T , which i s .7 usee.  With a light beam  diameter of li.mm, the device is usable as a modulator with a bandwidth of 3.4 MHz. The theory of beam^-steered deflectors developed i n Sections 4.2-4.4 appears to agree within about 10% with the observed characteristics of the device tested. In further experiments, i t would be desirable to compare the device impedance obtained from reflection coefficient measurements on the transmission line to the anticipated values based on the IDT equivalent circuit model, matching inductor and connecting wires. Experiments at higher frequency would be a better test of the acoustic diffraction theory, since beam steering would be over a greater angular range. Better performance could be obtained at higher frequencies, since the IDT bandwidth would be greater. For example, a beam-steered transducer with a center frequency of 500 MHz would give in excess of 500 resolvable spots with a light beam 1 cmawide. Wide bandwidth i s likely  145.  to be more easily obtained by using several beam-steered transducers with different center frequencies in parallel, or possibly one transducer with different center frequencies in adjacent sections.  146.  5.  SAW TRANSDUCER FABRICATION  In this chapter, a brief summary of the procedures used for interdigital SAW transducer fabrication are given. Photolithography masks were made on Kodak 649-F 35 mm holography film by the following process.  First, the artwork pattern was generated  on cut and strip Stabilene film, using the improvised ruling apparatus shown i n Fig. 5.1. Linewidths as small as 0.5 mm could be accurately drawn by controlling the straightedge position with verniers, which were adjustable within 0.001 inch.  The cutter used i s shown i n Fig. 5.2. The  completed artwork was taped to a sheet of translucent white plexiglass and illuminated from the rear with five 600 W quartz-halogen floodlamps. The level of illumination had to be very uniform across the artwork because both the film and photoresist used were very high contrast materials.  The maximum variation tolerable was about 10%. The photo-  reduction was done with a Canon FTb, using a 28 mm f/2.8 Canon lens. The reduction ratio of a lens of focal length F with a distance x between subject and lens is given by R = x/F— 1. The use of a wideangle lens permitted large photoreductions to be made with reasonably small subject to camera separation. Kodak 649-F film has a panchromatic emulsion, whereas the orange Stabilene film of the artwork was designed for uses with orthochromatic materials.  To obtain better contrast, a green f i l t e r with strong absorp-  tion at-wavelengths--g-reater than .55 um was used during exposure. A series of time exposures was made and the developed images were examined microscopically for uniformity, contrast and an absence of fog in the clear areas. pf 6 minutes at f/4.5.  Best results were obtained with an exposure time Vibration due to the building air conditioning ~  Fig. 5.2.  148.  proved to be a problem, so i t was necessary to work at night with a l l machinery shut off. The film was developed i n Kodak D-19 for 7 minutes at 20°C in a spiral tank with constant agitation.  This was followed by a 30 second  deionized water rinse and immersion for 45 seconds in a fixing bath. After a five minute wash i n flowing water, the film was rinsed with a wetting agent and hung up to dry. No special problems were encountered with the masks made for the 2-section IDT. Gaf PR-102 positive photoresist was used according to the manufacturers recommendations.  Lifting of the photoresist was  encountered i n acid etchants (Fig. 5.3) or alkaline etchants with strong  Fig.  5.3  Lifting  of Photoresist.  149.  gas evolution. Good results were obtained in an alkaline ferricyanide etchant made with 7ig.K  3  Ee(CN)g,2i g NaOH and 200 ml deionized water.  The photoresist was exposed under a high-pressure mercury vapour lamp. The optimal exposure and development times were found by t r i a l and error. For the second beam-steered transducer, the artwork was about 1.3 meters wide and had 0.5 mm linewidths. These dimensions were reduced 118X to about 11 mm and 4.3 ym respectively.  Because of the large over-  a l l width and the narrow linewidth required, usable masks proved difficult to make. Although the resolving power of the lens was very high, there was a small loss in image contrast 5 mm from the center of the negative. This was sufficient to prevent the photolithography from working. negative contrast was increased by intensification.  The  Chromium intensifier  was tried f i r s t , but this actually reduced the image density.  The most  likely reason was a loss of silver during redevelopment in D-19 by the solvent action of sodium sulfite on the thin (6 ym) film emulsion. Excellent results were obtained with Ansco 331. intensifier [90].  Re-  development was for 15 seconds; i f this step was carried too far, the image density could easily be reduced in a 5% hypo solution.  Caution in  handling was required with this intensifier, since i t contained mercuric chloride and potassium cyanide.  Although the intensified masks appeared  to have very high contrast, problems were s t i l l encountered near the ends of the transducer, probably on account of a faint residual optical density there, between fingers in the photomask. result in an etched transducer.  Figure 5.4 shows the typical  Increasing the etching time resulted in  open fingers before a l l the aluminum islands dissolved. Figure 5.5 shows the clean photoresist pattern obtained near the center of the mask, where no problems were encountered.  The difficulty was finally solved by  150.  Fig. 5.4  Fig. 5.5  Shorted Transducer.  Photoresist Pattern near IDT Center.  151.  adding a 10 minute bake at 100°C between the exposure and development of the photoresist [91].  This had the effect of smoothing out the inhibitor  concentration in the exposed areas of the photoresist, thereby increasing the rate and uniformity of development i n these areas.  This is particu-  larly true near the highly reflective aluminum surface, where standing waves during exposure result in a maximum in inhibitor concentration. Figure 5.6 shows a photograph (taken in a scanning electron microscope) of the excellent results obtained.  Fig. 5.6 Portion of Beam Steering Transducer used in the Experiments (2000X). To make this device, Hunt Chemicals  L S I 395 Waycoat positive  photoresist was used because of i t s superior adhesion during etching in Transene aluminum etchant type A. Use of this etchant rather than the alkaline ferricyanide resulted in less undercutting of the fingers. The complete procedure was as follows.  152.  (1)  The substrate was cleaned ultrasonically in chromic acid, boiled  in reagent grade methyl ethyl ketone and blown dry with nitrogen. Figure 5.7 shows the aluminum film l i f t i n g during etching on account of inadequate cleaning of the substrate.  Fig. 5.7 (2)  Lifting of Aluminum Film.  A layer of aluminum a few tenths of a micron thick was deposited  in a Veeco vacuum system using a tungsten c o i l as the evaporation source. The thickness was monitored with an Ificon 321 quartz crystal film thickness monitor. (3)  The freshly deposited aluminum film was coated with a 1 ym  coating of photresist by spinning at 5000 rpm for 20 seconds. (4)  The photoresist was baked for 90°C for 30 minutes in a convec-  tion oven. (5)  The substrate was placed in a vacuum holder designed to assure  153.  close contact between mask and substrate during exposure of the photoresist.  The alignment of the transducer with respect to the substrate  crystal axes was done by mounting the holder on a microscope stage (Fig. 5.8) and aligning a square grid in the eyepiece with the straight  Fig. 5.8  Photolithography Station in Laminar Flow Hood.  edge of the substrate, which was normal to the crystal Z axis.  The mask  was moved by hand until i t was aligned in the appropriate direction (Fig. 5.9), at which time the vacuum was turned on. (6) An exposure of 41 seconds was given at a distance of 10 cm from a high pressure mercury vapor lamp.  The lamp required 10 minutes  of operating time to reach a stable output level. (7)  The sample was baked for 10 minutes at 100°C.  (8)  The sample was developed in Waycoat Positive LSI developer  diluted 1:1 with deionized water at 24°C for 60 seconds.  154.  Fig. 5.9  (9) (10)  Correct Mask Alignment for Z-21.8° SAW (tan 21.8° = .4).  Propagation  A postbake of 30 minutes at 100°C was given. The aluminum was etched in Transene aluminum etchant (type A)  at f u l l strength with constant agitation to remove small bubbles. (11)  The device was rinsed in deionized water, blown dry and examined  by microscope. (12)  The photoresist was stripped in methyl ethyl ketone and the  finished transducer was blown dry with nitrogen gas.  155.  6. CONCLUSIONS Acoustic beam steering has been found effective i n improving the bandwidth and diffraction efficiency of planar acousto-optic light deflectors.  A viable device model has been proposed.  The effects of  anisotropic acoustic diffraction were taken into account through the parabolic velocity surface approximation.  Auld and Kino's equations [53]  for the series IDT equivalent circuit model were used so that variations in the electrode metallization factor could be taken into account. The IDT loss resistance was calculated using a modified form of Lakin's theory [63].  Equations for the acoustic power as a function of trans-  mission line and equivalent circuit parameters were developed.  A rigor-  ous treatment of the acousto-optic interaction was given, and the d i f fraction efficiency was calculated using the known and measured properties of LiNb0 and the SAW and OGW fields. 3  Addition of a modified theory  of acoustic beam steering suitable for the analysis of IDT arrays completed the device model. experiments.  Good agreement was found between theory and  As pointed out i n Chapter 4, better performance could have  been obtained at higher acoustic frequencies.  The fine linewidths re-  quired above 500 MHz could be obtained with electron beam lithography. The indiffusion of nickel was found to be an excellent method for making high quality optical surface waveguides with losses below 1 dB/cm. Longer diffusion times at higher temperatures were found to increase optical absorption, an effect that was not entirely reversible by subsequent baking in oxygen.  Small quantities of 0^ or ^  i n the  argon gas needed to provide an inert diffusion atmosphere were found to reduce the impurity concentration considerably, probably due to the formation of oxides or nitrides of nickel on the LiNbO„ surface.  156.  The primary limitation i n device performance stems from the limited bandwidth of interdigital transducers.  It may be possible to  improve this by mechanically loading the LiNbO^ surface. Alternatively, i f adjacent sections in the beam-steering IDT had different center frequencies, improved performance could probably be obtained.  At widely  separated frequencies, different parts of the array would radiate surface waves. Before these devices become commercially viable stable, efficient coupling to the OWG must be solved.  the problem of Use of grating  couplers should improve the performance i n this regard.  If a laser diode  could be used as the light source, i t could be permanently fixed to have the correct orientation with respect to the coupler and a small, rugged device would be obtained.  157.  APPENDIX I In this appendix, a brief summary is given of the properties of LiNbO^ used i n the calculations. There i s some variation i n the published values of the refractive indices. n  g  At A = 0.6328 ym, Kaminow and Carruthers [4] give  = 2.214 and n = 2.294. These values give the best agreement between Q  experiment and theory.  The dielectric permittivity tensor used i s  therefore r 15 J;2;624 OD  0  0  S. 2-623}  0  0  0  4\ 9018,1  For the electrooptic tensor, Turner's values [93] were used. In matrix form, these are 0  -3.4  8.6  0  -3.4  8^ 6  0  0  30.8  0  28  0  28  0  0  -3.4  0  0  in the principal axes system,  x 10  1 2  m/V  The matrix form of the elastooptic tensor  used i s 2036  .072  .092  .055  0  0  .072  .036  .092  -.055  0  0  .178  .178  .088  0  0  .155  -.155  0  0  0  0 .019  0  0  0  0  0  0  0  0  .019 .31  .11 .048  158.  Most of these a r e Dixon do n o t g i v e [9 3]. and  P44 ^ Pg6» an  On t h e n e x t  elastooptic  crystal , axis.  and Cohen's v a l u e s s  o  [93].  However, t h e s e  K l u d z i n ' s numbers w e r e u s e d f o r t h e s e  three pages, the d i e l e c t r i c p e r m i t t i v i t y ,  The r o t a t i o n b y 21.8°  the o v e r l a p i n t e g r a l c a l c u l a t i o n .  corresponds  given.  about the  t o the v a l u e s used i n  The p e r m i t t i v i t y / c h a n g e f a c t o r s f o r  » a n d f o r t h e SAW  electric  field,  33*3klm k n are also  electrooptic  tensors a r e g i v e n i n c o o r d i n a t e systems r o t a t e d  a e o u s t i G c s f e r a i - n v e,.P., , e,  authors  T h e y a r e c a l l e d DEPSS(L,M) a n d D E P S E ( L ) ,  e„.r., ,e, „,  3j j k l k3 respectively.  159  to  ac c . c- ci © i IT C C if > c  1  i c e  • ^  1  c  ' \ '-I •  t  j r- (\) c l c  rvj •  rv n.! f — —  i -  fx,  I  , f. fl . n|a.  F  i i  i  o c o • If* c C 1 f>. © © I c|  o  fl C ll 11 .'I; tl II  - c o -  i II III i! I  cr  c  c.! f*. «~ <-z ; i ll <i ill  I  _ C  Z  11  rv rvjrv A j  i:l  - i I V ru ; rvj f • r  al a  ru; f\- rv rv)  a  a  j — ry -j ru  I o  c  a  a  rv  H -  i i © ©'c j©c i© c o c; iri c ir- ©J u'ir c o © r- o| r- o o 7 i d e r .  o  !  j o © rvi  ©  o  Ci f! o ©{ I I III II -J A.- A J fj IM • fi rd  ^  o (\\ f\j•• o r- (V c II II M II M »- — — r\j rvj r v A J fi fi i — rv A ; f. — -j rv f» f.: — rv; rvj f A J — | A , — fl| A J — f J rv RF  < _ < © o cl  c c © c c. ci  o © c c]  : t o t n i u  iu  u  «-J C  iii  — <  II II  it]  *-' " -~- n !CC C © ©  — . A J f| i ^ i (V . f l r", r**. f  l  (TJ : J O rv X ©  © tr> c,  It  : ii ll i<, *- r— j rvrv' r*iA J  I  t  r  Q  ci >~\ c-i  t  i  t  c. c "  CS  r  !  ft  °I  lit it  11  M  .-•  ru r\  f  Aj  f  a  a  a . a. a  ci  c © c e r- c-i r- c c c ll it It: ii n li ti  c c © c < C I/'. C. c  c © © 5 LT> LTi U" ' i— r~~i T— r- r^j c o r- AJ r-  i t  tl n.  C! —  I  1  a I Ui  v. VI  a n. LU Ul  f, f] f. fi  -a  f  a  "  c. c  o c. c  c  c  c  © © IP d <; c o X c O C' M  C C © I  © c; c c c f\,  — rvj —  £J £i ; C C c, c © c © o © ©. c \r C  (- Aj AJ O  fi  i  II  rv *J  ni a u Q "4tr a a 1  c — o ^ c c O — =3' =J < II II ll' II II IIII II If. II II tl] r\ A . rvj f f fj I A , A j fi — — rvi r • — A i f. fl — f» A i f Aj • : • f A J — fi r\J, n ( / , a or a-, rr a  ©• <  | f . © e -a r- © ©- © • H ll 11' ll ll ll It ll »- w-i rvj fA J f) rvj r" rvi Aj f! f - j r e rv, f< — | — f i A ; —• f - A ) — fi rv; — — AJ f . f i A) -  II  m— *~  n ft iv; IT o nj ci f rfl'o | a a n. a o, oJ a a CL a.  11  II M ll  • r\j ru rv• f  f  a; cr.' ; n c/') f/: w, t/; to *r. w. <o u a a u a u j a a (U U.I VlJ U UJ U ' Iki  I  o  c  o  L>  a  o  c.  (/I  160  c  at  e  c  y- r»- -j", rrv ir> - *7 AJ / ; a rv. y i IT. 3- | — c !  •i  II II II 11 tl li I M II • rv r , rv rv •: rv rvif i" . i i — rv , •, m K r r 1  ;  a  I rvi rv fj-r  K.  i , a  a.  n i  a.  i^ r", 1  a <  P^I  r  i  i t i a a CJ c. C c C" c - c - c  rri -r -a —1  L/i  -  at  Ki  • C if' a ' C -f * 1/1 r»: «. I *- u! cr. — i/ — rv -C — rv i rv i/v J>rv a a. I II !l| 11 n it II n ll n ll it rv rvj rvrv rv- r^i (V A ! f — -| r\i rv rvi '1 »- K f\" , — rv — H r\.f^. K. :— rv j rv r^t rv rvj tvr-~vi t* <v r\jrv (V ru • i a; cr : rx rxja ;  —  r-j  1  —• f\' fV f l A.!  t*\ a j  rv tv  I  rvt  r. ! — oi o c c  <--!  K>  c;t oCV  C,a O •J"' c. ir Ki o =3 o tr AJ °_ cr o  It  H  - rvj —< ^  rv cr  I  t!  !I  -c a'^  »J  a) a  a.1Or  u.  li  II llj II : tv r v r-A i , — rv. •  : rv rv  Or  r • r~ o a- r • a rv - i ; .C, rv ; • o o e  i  tl  oj ru" o II li tl it Zl rvi K I f\j  rv -  j rv .-o  KI  —  II  rv y. a Cr rt cr. n ..CLa U-' L J U J —  i . l  . c c r • — r— a- ' -c C rv  —i  m  -  ll ll ll it li II  oo° Ci O c o c .c ~ rv °" irn ii  II  II  a a am  cu a a: a:  cr  C1*o1 O r. •r- urv K X a rv• o• fVU.' ft IT•--  at-. (_ LL'  _;  tt— v C  C  c el tl i U W • - f l r*i K rv 1  1  II  ... -~  H  A  O  f  0 r\| A . =Jt ^  ii (VI  r-n  i\. f-  'J"'  W N I  II  rr Ci" tr. I T> a. a. UJ u.  X  t»; a a• a l C —« — rv tr. ^: — — -i — K'I  ; 11  rvt C, ^  I 1  It ll|  (J-I f^l  a. a\  K C  a  O  i  if' c-|  1  rv  >  C M  **  rv rv rvi rv rv rv| r\, r  I  (. K |  >  f  Kl Al -  -c rvj r-— rj tr rv i r- ^7 —• n r •-! r\ N AJ ^^ rv - rv M | li ti u it it . rvj rv -tl U | 11 II  C I,  ru r\ A.< C c: C*IoT; a G ii a.' Lil  I A rv iv K "i --t rv rv < ^ «^ rv — i  I  as?.  7-  Ct c  tl  II  I  z z z' z  ,  . (\  N  •  K  (  •  i\. •  iv rv' r. rv rv r\ rv n rv r4 I a a a. a. a a a a a rv r\ rvi iv rv r\j | c c c. c c  rv u' o' v rK\ t cj J". O ' OL. i rv v «— a -o -L> Ln ^ i I ^ -i — • X rv- *- — rvj I ' I I j l i tt it ti it n ti ii ii ii n 1 rv rv rvi K ' N j rv rv — . A,.* - r >  ^ci r;t ^"-cr r<x--r ^ /:c-xco -C Z" IT CI i j A.  t  A'  C; Aj  A.  —  a .  I.'' -f C - I 0 -f >c =t m A . r, rv rv c o c c o <z cr  i  rv: - ^  -  d it ii it H It tr ll tl it n it • — • > rv • Ul u .: 11. <r| v.i i-j t/; (rjtr. tO tT; 'JT. y a. a aj a. o a cJa a U, LiJ UJ ILI L c c o c; c q i  ANGLE  OF R O T A T I O N  IN XZ O L A N E  -  R0.00  DEGREES  O '  DJELECTRTC FDS»(11): _FP_SR('2)= EPS*(13)=  PERMITTIVITY  TEtiSnR  a, R n i s o no 0. O O O O n - O ! o", O O O O D - O I  FPso(21) = n,ooooo - o i EoSR(?2)= _5.?J>2£0 »«_ ~£BSP(23) = 0." 110 Of: 0-61  ELFCTSnOoTIC RRf1 1 I ) = PRf'!!)= R »{1221s R9IUJ):  RPfl33j=  Z  P P ( 1 322 ) »R(1232) P p f 1 1 3 3) PR f1513) PBfJ223) z PO(it33) z  -  P R f i 3 3 3 )s  _0EPSEI3)s__ DERoS ( 1 i 1 = DEPS?(12)= DFRSS(13)= 0FPSSC21)= OFPSS(22 1 = PFPS?(23)= 0FPSS<31)= 0F.P S(32)^ O E P S S C33) =  R R ( 3 1 1 l :O . O O O D - 0 1 RRM21 ) = R R f 3^ 1 ): B . a 0 01) - 1 2 P R ( I 1 2 ) = JLRJ322JJ 3 . U O O D - 1 2 J? C132„)^. RRf313): RRf123)= 1.uoon-l1 R»(333): RRf o. o o » o - n t  in  CHANGE  0 . 0 0 0 0-01 P R ( P 31 1 1 0 . 0 0 0 0-01 RR ( 2 2 2 1 ) - 2 . 7 5 ( , 0 -02 0 PP0131) 0 . 0 0 0 -01 P R >' 2 33 1 ) l ' . 3 7 5 0 - 0 2 -0? P R f ? 2 l 2 1 - 2 . 7 5 0" P R ( 2 1 2 2 ) - 7 .750" - 0 2 0 . 0 0 0 0 -01 PRi?322) 0. oooo - n i PP f 2 2 3 2 ) 0. 0 0 0 0 -01 PR(21131  --  0 , 0 0 0 0 - 01 o . 0 0 0 0 - 01 0 . 0 0 0 0 -01 a. 7500_ 0 .00 0 0 1 , 7 B 0 0 - 01 0, 0 0 0 0 - 0 1  0EP3EI 1 )= DE°SF f2) =  S  P R C 21 t i )  a  PERMITTIVITY  O.ooon-0! o. ooon-o i ijioon-1 2 0.0000-01 O.OOOD-01  H  O.OOOD-01 O.OOOD-oi n.oonp-ni o.oobn-oi  PRf221)= RR(212)= Rgj_?32)= RR(223)=  e.60 OC-12 j.fiOOD-11 3."QOD-I2 O.OOOD-01  RR.(321) = O.OOOD-01 R R ( 3 1 2 ) = O.ncoD-01 RR ( 3 3 2 )--i. a o o o - I 2 R « f 3 2 3 ) = O.OOOD-01  TE'JSP.R  fl.aoon. 02 PRfiiin P 3 f 1 5 1 1 ) 0 . 0 0 0 n-01 PP f 1 2 2 1 ) . 7 5 0 0 - 03 P R f 1 1 31 )= 0 . 0 0 0 0 - 01 « ,75on- "3 PR<1331) ,7500- 0 3 P R I i 2 I 2) P R f1 ( 2 ? ) - I ,79no- 01  -  9  TENSOR  i.oeon-i i R R ( 2 i 1 1= 0. o o 0 ti - n 1 » P ( 2 3 1 1 = RR(2?2i= 1 . aoon-1 I RR C 2 ! 3 1 = O.OOOD-01 RR(233l= l.aOOD-li  I t ASTOnPT'C  FPSRt31)= 0.OOOOD-Ol E P S ( 3 2 ) =_ _ 0 ' . O O O O D - 0 1 _ E P s R ( 3 3 ) = " 5'. 2 h 2 ' l 0 0 0  z z z PRf3131) z P R f 3 3 3 1 )z RR f 3 2 ! 2 ) Z  0 .fl0 0 D -01 Q . 2 0 0 0 - 02 0 . 0 0 0 0 -0 1 11 . 7 5 0 0 - 03 0 . 0. AO •01 0,(11)00- 0 1 P R ( 3 1 2z 2 ) 0 . 0 0 0 0 - 0! 7 . 2 0 0 0 - 02 PR(3322) 0 . 0 0 0 0 - 01 PR f 3 2 3 2 ) 11 , 7 5 0 D - 0 3 PR(3l13)  PR(3111) RRf3311) RRC3221)  0  z z z z s z .3750 P R (?3l_3l z .1 -!!2_ „ P R f 3 3 ! 3 ) Z_fl , 0 0 o 00_L_ 1 . 0 0 0 0 P R C 3 2 2 3 ) 0 1 P R ( 2 2 2 31z 0 Z 0 ,. 000n(>-0 0 0 0 .750O 7 0 01 -02 = P R (2 i 33 ) PR(3133) z 0 P R C 3 3 3 3 ) z 3 ,<>0flD-02 P R (P.333) .!) . 0 0 n -01  FACTORS  F O R T E MODE  WITH  ELECTRIC  FIELD  PR(12111 = 0 . 0 0 0 0 - 0 1 0 , 0 0 0 0 - 01 P R C 1 1 ,! 1 ) PR(1321) = 0 . 0 0 0 0 - 0 1 PBd?:?] ) 0 .0000- 0 1 P P ( 1 1 1 2 ) _0 .0000- 0 1 P R f 1 3 ) 2 ) " 0 .0000- 0 1 P R C 1222) = - 7 .750n- 0 2 0 . 0 0 0 0 - 1! I PR(1132) PR(1332) = 7 . 5 0 0 0 - 0 2 ' 0.0000- 0 1 PR(12!3) PRO 12?) _0 . 0 0 0 0 - 0 1 P R ( 1 3 2 3 ) = 7 , 5 0 0 0 - 02 7 . 7 5 0 0 - 02 PR(1233)  01  0 .0 0 0 D P R ( 2 2 I 1 ) = 9 , 2 0 0 [.) 0- 2 P R (321 1 ) 0 ,0 0 0 0 - P I . PR(2121 ) " ,75 0!i-0 3 P R ( 3 1 2 I ) 0 . 0 0 0 D ~ 0 1 P R ( 3 3 2 1 ) 2 • 7 5 0 0 - 02 PR(2321 ) P P ( 3 2 3 1 )= 1 .3 7 5 0 - 0 2 PR(2231 ) = 0 , 0 0 0 0 - 0 1 0 .0 0 0 0 -01 P R ( 2 ! | 2 ) .31.7.5 n.o_- 0 3' P R r31 12) P R C 2 3 1 2 ) = 0 .0 0 0 0 -0 ) R R ( 3 3 1 2 ) = 2. 7 5 0 0 - 0 2 PI- ( 2 2 2 2 ) — 3 , 6 0 0 0 - »?. P R ( 3 2 2 2 ) 0 , 0 0 0 001 0 , 0 0 0 D 0-1 R»(2132) P R ( 3 1 3 2 ) 7 ,5 0 0 0 - 02 R R ( 3 3 3 2 ) 0 , 0 000- 01 P R ( 2 3 ? 2 ) = 0,0000- 0] P R ( 2 2 ! 3 ) = 0 , 0 0 0 001P R ( 3 2 1 3 ) 1 .3 7 5 0 - 0 2 0 . f n . 0 0 -0 1 _ 7 . SJlO.Or.02 e?(?!23) . P R C 3 12 3.) =. 0 . OOOO- 0 1 PR(2323) = 01 PR(3323) PR(3233) PR(2233) = 7.2000- 02 O.OOOO- 0 1  -  0,0006-  z z  R ( R  ALONG  ROTATED  Z  AVIS  .0.2361epD-OP 0.o«157ftO-10 - o . o o o o o o O on , ? 5 ' i 77fio 01 . 7hl5h"0 0 0 -o.noonnco 00 - n , 7hi 5>,«n 00 . 0 , 1 0 0 3 0 1 0 01 . 0.0 0 0 0 0 0 0 00 -o.onodoOO 0 0 - 0 , 000 0000 00 - 0 Q9oR5nD 0 0  ON  162.  APPENDIX II WAVE PROPAGATION IN ANISOTROPIC MEDIA An anisotropic, non-absorbing medium is characterized by the dielectric tensor E  e =  0  l l  0  E  0  0  0 0  22  e  33  in the principal axes system. The wave equation has the form 3S 2  V X (V X E)= C  (II.l)  at  For plane harmonic waves of the type, e^^'  r  the following opera-  tor identification can be made: rf ——  3t  -*•  -10) J  The wave equation then becomes 2 kX  (k X E) + ^  D= 0,  (II .2)  which can only be satisfied i f the wavevector k" is perpendicular to the electric displacement vector 15, and i f J5, It and k are i n the same plane. Expansion of the triple cross product and use of £•$ = 0 gives 2 2 E. [)= ED cosG = -y-2 D = < ^- D , k c c >f  (II .3)  where v = ^ is the phase velocity of the wave. Since- the power flow direction is given by the Poynting vector  163.  -y  1  ->  -> j ,  S=| E XH,  we see that planes of constant phase propagate at an angle 6 to the  ~y -y ->  -y -y ->  direction of power flow, and that (D, H, k) as well as (E, H, S) constitute a mutually orthogonal triad of vectors. Fig. II.1.  This i s illustrated i n  The ray velocity (defined as the velocity of power flow) i s  then given by u = v/cos9 .  (11.4)  The principal indices of refraction are defined by (II.5)  n.  Fig. II.1 When equation  (II.2) i s written i n Cartesian components, three  homogeneous linear equations i n E  x >  E^ and E^ result.  A nontrivial  solution exists only i f the determinant of coefficients[85]  164.  [(n^/c^-ky-k ]  k  2  k k  k  y  2  2  k  2  2  k  x  where the s u b s t i t u t i o n a  k  y  [(n o)/c) -k -k ]  x  k  z  k  x  k  z  t h e n be  mutually  b e e n made.  i s made i n ( I I . 6 ) ,  the determinant  2 4. 2 2 2 Ln,v / c - v - v ] 1 y z n  v  represents those  corresponding  to  It two  polarization.  substitution k. = v . i i  v  This determinant  k v e c t o r s can e x i s t ,  i  r  (11,-6)  2  In a l l d i r e c t i o n s but  o r t h o g o n a l d i r e c t i o n s o f wave the  2  t h e medium, t h e s u r f a c e i s d o u b l e - v a l u e d .  shown t h a t two  If  = 0,  z  2  t h r e e - d i m e n s i o n a l f i g u r e i n It s p a c e .  can  k  y  z  [(n,w/c) -k -k ] 3 x y  y  ( I I . 5 ) has  a l o n g the o p t i c axes o f  k  x  V  u  —7T  v  2  assumes t h e  vxv  V  x  form  X  y  2 4. 2 2 2, [ n v Ic -v -v ]  z = 0,  (II.7)  Two  values  o f p h a s e v e l o c i t y a r e p o s s i b l e f o r o r t h o g o n a l l y p o l a r i z e d waves  travel-  V  y Z  V  r  x  2  x  V  X  z  i n the  same  V  y  z  2 4. 2 2 2 [ n v / c -V - v ] 3 x y r  V  n  0  y  which represents the double-sheeted  ling  V  z  phase v e l o c i t y  surface.  direction.  When I) i s e x p r e s s e d  i n t e r m s o f i t s p r o j e c t i o n s a l o n g E~ a n d  u, -4.  3  Writing  n = E . f cose +  -v  .T\  u ^  t h i s i n t e r m s o f C a r t e s i a n c o m p o n e n t s and  of c o e f f i c i e n t s equal to zero  gives  (II.8)  s e t t i n g the  determinant  165.  r 2. 2 2 -2, Ic /n -u -u / 1 y z  u u y z  u u xy , 2. 2 {c /n -u2-u2, }  u u xz  u u zy  / //n„-u \ {c -u } 3 x y  2  u u Z  X  x  u u y z  z  2  2  2  = 0,  (H-8)  2  the equation for the ray velocity surface. These equations may be utilized as follows. for example, in the xy plane, setting k , v  z  and u  z  If propagation i s ,  equal to zero gives three  sets of equations: k  z  =0:  k + k = (n.u/c) x y 3 • 2  2  k /(n /c) 2  l W  v  z  = 0:  + ky/(n u/c) = 1 2  2  2 2 2 v + v = c /n x y 3 0  V  n u=0: z  2  2  2. 2 . 2. 2 x 2 y l / n  +  V  / n  =  V  4.2 / C  2 , 2 2.2 u + u = c /n, x y 3 2 2,2 2 2 n.u + n.u = c , 2 x 1y  which describe the propagation characteristics of waves i n the xy plane. Waves with E = E z^ propagate with refractive index n , whilst waves with E = E x + E^y propagate with a refractive index with magnitude between x  n^ and n 2  16.6.  REFERENCES  1]  D. M a r c u s e , T h e o r y o f D i e l e c t r i c O p t i c a l W a v e g u i d e s , N.Y., C h a p t e r one ( 1 9 7 4 ) .  2]  P.K. T i e n , A p p l .  3]  I . P . Kaminow a n d R.V. S c h m i d t , A p p l . P h y s . L e t t .  4]  I . P . Kaminow a n d J . R . C a r r u t h e r s , A p p l . P h y s . . 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