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UBC Theses and Dissertations

Planar beam-steered acousto-optic light deflectors 1977

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PLANAR BEAM-STEERED ACOUSTO-OPTIC LIGHT DEFLECTORS by E r n e s t Bruno Riemann B.Eng. ( P h y s i c s ) , McMaster, U n i v e r s i t y , 1969 M.A.Sc., U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department o f E l e c t r i c a l E n g i n e e r i n g ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA Ju n e , 1977 (c) E r n e s t Bruno Riemann, 1977 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ABSTRACT A t h e o r e t i c a l and e x p e r i m e n t a l s t u d y has been made o f p l a n a r a c o u s t o - o p t i c l i g h t d e f l e c t o r s w i t h p a r t i c u l a r emphasis on a c o u s t i c beam s t e e r i n g as a means o f i m p r o v i n g d e v i c e p e r f o r m a n c e . The t h e o r e t i c a l model t a k e s i n t o a c c o u n t the 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- s t e e r e d i n t e r d i g i t a l s u r f a c e a c o u s t i c wave (SAW) t r a n s d u c e r s , a n i s o t r o p i c d i f f r a c t i o n o f a c o u s t i c waves and t h e r i g o r o u s t h e o r y o f the i n t e r a c t i o n between g u i d e d o p t i c a l waves and h i g h f r e q u e n c y s u r f a c e a c o u s t i c waves. The e x p e r i m e n t s were c a r r i e d out on n i c k e l i n d i f f u s e d wave- g u i d e s on Y - c u t LiNbOg s u b s t r a t e s . A f o u r - s e c t i o n , t h r e e f i n g e r p a i r t r a n s d u c e r a r r a y was used t o l a u n c h a c o u s t i c waves w i t h p r o p a g a t i o n d i r e c t i o n c e n t e r e d a t 21.8° f r o m t h e Z a x i s . A c e n t e r f r e q u e n c y o f 200 MHz was chosen as a compromise between h i g h a c o u s t o - o p t i c b a n d w i d t h and ease o f f a b r i c a t i o n . The d e f l e c t o r had a b a n d w i d t h o f more t h a n 60 MHz and gave 44 r e s o l v a b l e s p o t s w i t h an o p t i c a l wave 2.5 mm w i d e . The o b s e r v e d f r e q u e n c y r e s p o n s e 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 was i n e x c e l l e n t agreement w i t h t h e t h e o r y . I t was c o n c l u d e d t h a t beam s t e e r i n g i s an advantageous t e c h n i q u e f o r d e v i c e s r e q u i r i n g l a r g e band- w i d t h and h i g h d i f f r a c t i o n e f f i c i e n c y . / i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS 1 1 1 LIST OF TABLES v LIST OF ILLUSTRATIONS v i NOMENCLATURE x i ACKNOWLEDGEMENT x v i i 1. INTRODUCTION 1 2. DIELECTRIC OPTICAL WAVEGUIDES . . . . 3 2.1 U n i f o r m D i e l e c t r i c S l a b Waveguides' 3 2.2 Modes i n Graded Index Waveguides 6 2.3 C o u p l i n g t o O p t i c a l Waveguides 11 2.4 P r i s m C o u p l e r D e s i g n 15 2.5 C o u p l e r F a b r i c a t i o n 23 2.6 O p t i c a l Waveguide F a b r i c a t i o n 28 2.7 D i f f u s e d O p t i c a l Waveguides i n LiNbO^ . . . . . 30 2.8 T i / L i N b 0 3 D i f f u s i o n 31 2.9 N i / L i N b 0 3 D i f f u s i o n 35 2.10 P r o p e r t i e s o f N i / L i N b 0 3 Waveguides 40 3. PROPAGATION AND GENERATION OF ACOUSTIC SURFACE WAVES 56 3.1 I n t r o d u c t i o n 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 D i f f r a c t i o n o f S u r f a c e Waves 60 3.4 SAW G e n e r a t i o n ; t h e I n t e r d i g i t a l T r a n s d u c e r 63 3.5 E x p e r i m e n t a l Work 77 4. BRAGG BEAM-STEERED SURFACE WAVE ACOUSTO-OPTIC LIGHT DEFLECTORS 86 4.1 I n t r o d u c t i o n 86 4.2 Theory o f t h e Surface-Wave A c o u s t o - O p t i c I n t e r a c t i o n . . 87 i i i Page 4.3 A c o u s t i c 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 of Beam-Steered Transducers . . . 109 4.5 Acousto-Optic Overlap I n t e g r a l C a l c u l a t i o n I l l 4.6 Experimental Work 121 5. SAW TRANSDUCER FABRICATION 146 6. CONCLUSIONS . . . 155 APPENDIX I 157 APPENDIX I I WAVE PROPAGATION IN ANISOTROPIC MEDIA 162 REFERENCES 166 i v LIST OF TABLES Table Page 2.1 Ni Sputtering Calibration 36 2.2 X-Ray Fluorescence of Ni/LiNbC>3 38 2.3 TE Modes a of Ni/LiNb03 OWG 46 3.1 Anisotropy Parameter b 63 3.2 Constants for LiNbO„ [40] 75 v LIST OF ILLUSTRATIONS F i g u r e 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 Comparison between WKB and E x a c t S o l u t i o n s ( a f t e r Marcuse [ 9 ] ) . . 9 2.4 Z i g - Z a g Wave P r o p a g a t i o n i n Graded I n d e x Waveguides . . . 9 2.5 Lens C o u p l e r 13 2.6 P r i s m C o u p l e r 13 2.7 G r a t i n g C o u p l e r 13 2.8 G e n e r a l i z e d L eaky Wave C o u p l e r . . . . . . 13 2.9 P r i s m C o u p l e r Geometry 16 2.10 B r o a d e n i n g o f t h e I n p u t L i g h t Beam 16 2.11 E f f e c t o f Beam B r o a d e n i n g on C o u p l i n g E f f i c i e n c y (assumes £/W = 1, n = 2 . 2 3 ) . . . 17 ' m 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 f o r n m = 2.23 . . . . . 20 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 f o r R u t i l e . . . . 21 2.14 Phase V e l o c i t y S u r f a c e 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 A x i s V e r t i c a l , . 22 2.16 P h o t o r e s i s t E x p o s u r e f o r G r a t i n g C o u p l e r 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 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 A d h e s i o n 26 2.20 R u t i l e C o u p l i n g P r i s m (-^8X) 27 2.21 E l l i p s o m e t r y o f T i 0 2 on S i l i c o n -29 2.22 T i / L i N b 0 3 D i f f u s i o n P r o f i l e 34 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 34 2.24 Gas F l o w C o n n e c t i o n s f o r N i / L i N b 0 3 OWG D i f f u s i o n . . . . 36 v i F i g u r e Page 2.25 Absorbance o f LiNbO^ Waveguide S u b s t r a t e s 39 2.26 N i / L i N b 0 3 D i f f u s i o n P r o f i l e ' 41 2.27 S t a g e f o r C o u p l i n g t o O p t i c a l Waveguides . 43 2.28 Two Mode N i - D i f f u s e d G u i d e Showing Mode B e a t i n g between the TE & TM Modes 43 o o 2.29 C o u p l i n g i n and o u t ( T E q mode) 45 2.30 Modes o f OWG's Used 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 N i / L i N b 0 3 OWG I n d e x P r o f i l e 48 2.33 The A i r y F u n c t i o n 51 2.34 Comparison o f A i r y F u n c t i o n and WKB S o l u t i o n s f o r TE.. Mode o f 6-Mode N i / L i N b 0 3 OWG 52 2.35 TE Modes o f a N i / L i N b 0 3 OWG 54 2.36 P r i s m C o u p l e r 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 [40] 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 and C o u p l i n g C o n s t a n t f o r Y-Cut L i b N b 0 3 [40] 68 3.5 IDT Shunt Model E q u i v a l e n t C i r c u i t 69 3.6 S e r i e s E q u i v a l e n t C i r c u i t 70 3.7 P e r m i t t i v i t y T r a n s f o r m a t i o n .' 74 3.8 S e r i e s C i r c u i t M o d e l 75 3.9 T r a n s d u c e r Conductance and S u s c e p t a n c e 78 3.10 T r a n s d u c e r A d m i t t a n c e n e a r Resonance 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 L o s s 82 3.13 Raman-Nath D i f f r a c t i o n o f L i g h t by S u r f a c e Waves . . . . 83 v i i F i g u r e Page 2 3.14 R e l a t i v e D e f l e c t e d L i g h t I n t e n s i t y v s . V Q 84 4.1 D e f l e c t i o n o f an OWG by a SAW 88 4.2 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 f o r Ae = 0 . . 92 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 M o d u l a t e d Medium 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 . 99 4.5 Beam S t e e r i n g T r a n s d u c e r 102 4.6 Phase Change a c r o s s One Step 103 4.7 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 = AQ and D ^ G . . 105 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 4 AQ 106 4.9 B r a g g - A n g l e T r a c k i n g 107 4.10 i i A c o u s t i c D i s p l a c e m e n t s f o r Z + 21.8° P r o p a g a t i o n . . . . . 113 4.11 SAW E l e c t r i c P o t e n t i a l 113 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 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 . Depth f o r f = 165 MHz . 117 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 . t h e O v e r l a p I n t e g r a l f o r f = 165 MHz 119 4.15 g as a F u n c t i o n o f Fr e q u e n c y 120 4.16 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 Mask . . . 121 4.17 A c o u s t o - O p t i c D e f l e c t o r ( A c t u a l S i z e ) 122 4.18 T r a n s d u c e r f a i l u r e 123 4.19 T r a n s d u c e r F a i l u r e 124 4.20 Raman-Nath D i f f r a c t i o n o f a Guided TE wave ( t h e upper 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 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, and t h e s m a l l s p o t on the r i g h t i s a TM mode) 125 4.21 U n d i f f r a c t e d TE ( l e f t ) and TM Modes 126 o o 4.22 Same w i t h r f D r i v e S w i t c h e d on (n ^0 .4 ) 126 v i i i F i g u r e Page 4.23 F requency Response o f 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 . 128 4.24 Broadband Response ( e x p e r i m e n t o n l y ) 129 4.25 D i f f r a c t i o n E f f i c i e n c y as a F u n c t i o n of A c o u s t i c Power . . 130 4.26 Beam S t e e r i n g IDT Mask (20X) 132 4.27 A c o u s t o - O p t i c D e f l e c t o r D r i v e C i r c u i t . . . . 133 4.28 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 s 135 4.29 TE Modes o f t h e D e f l e c t e d Beam 135 4.30 D i f f r a c t i o n E f f i c i e n c y v s . F r e q u e n c y 136 4.31' D i f f r a c t i o n E f f i c i e n c y a t f = 200 MHz v s . B r a g g Frequency w i t h V f = 3.8 V rms 138 4.32 D e f l e c t o r Bandwidth v s . B r a g g Frequency w i t h V f = 3.8 V rms 138 4.33 Peak D i f f r a c t i o n E f f i c i e n c y — B a n d w i d t h T r a d e o f f w i t h V f = 3.8 V rms 139 4.34 Comparison o f Response o f P h a s e d - A r r a y and C o n v e n t i o n a l B r a g g D e f l e c t o r 139 4.35 A c o u s t i c Power v s . f 141 4.36 D e v i a t i o n f r o m B r a g g A n g l e v s . f 141 4.37 D i f f r a c t i o n E f f i c i e n c y v s . f f o r S e v e r a l D r i v e V o l t a g e s ( f = 150 MHz) 142 4.38 A c o u s t i c 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 ( n ^ ^ .9) 143 5.1 A r t w o r k R u l i n g A p p a r a t u s . . 147 5.2 C u t t e r . . 147 5.3 L i f t i n g o f P h o t o r e s i s t 148 5.4 S h o r t e d T r a n s d u c e r 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 C e n t e r 150 5.6 P o r t i o n of Beam S t e e r i n g T r a n s d u c e r used i n t h e E x p e r i m e n t s (2000X) . . 151 i x F i g u r e Page 5.7 L i f t i n g o f Aluminum 152 5.8 P h o t o l i t h o g r a p h y S t a t i o n i n L a m i n a r Flow Hood 153 5.9 C o r r e c t Mask A l i g n m e n t f o r Z-21.8" SAW P r o p a g a t i o n ( t a n 21.8° = .4) . 154 x NOMENCLATURE Ch a p t e r 2. A,B a m p l i t u d e c o e f f i c i e n t s o f OWG TE modes A i A i r y f u n c t i o n c v e l o c i t y o f l i g h t i n f r e e space d OWG t h i c k n e s s D gap between OWG and p r i s m c o u p l e r E e l e c t r i c f i e l d 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 base m OGW mode i n d e x n r e f r a c t i v e i n d e x o f a i r a n e x t r a o r d i n a r y i n d e x e , n OGW mode i n d e x m n^ i n d e x a t g r a d e d - i n d e x OWG s u r f a c e ; a l s o o r d i n a r y i n d e x n p r i s m c o u p l e r i n d e x P n s u b s t r a t e i n d e x s P p o w e r / u n i t w i d t h c a r r i e d by 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 s u b s t r a t e W w i d t h o f i n c i d e n t l i g h t beam Wp w i d t h o f l i g h t beam on p r i s m base y WKB t u r n i n g p o i n t m y Q l a s t z e r o c r o s s i n g o f WKB s o l u t i o n n o r m a l i z e d WKB t u r n i n g p o i n t a p r i s m a n g l e a c p r i s m c o u p l e r r a d i a t i o n - l o s s c o e f f i c i e n t 3 z component o f OGW p r o p a g a t i o n v e c t o r n p r i s m c o u p l e r e f f i c i e n c y x i (Chapter 2.) 9 angle of l i g h t incidence on prism base 9 i n t e r n a l angle of incidence of OGW on waveguide surface m X l i g h t wavelength i n free space y . angle of l i g h t incidence on couping prism y^ permeability of free space a) angular frequency of l i g h t waves Chapter 3. a IDT electrode m e t a l l i z a t i o n factor A IDT finger width (Section 3.4) SAW amplitude (Section 3.5) A, ,B, SAW p a r t i a l wave amplitudes kq kq b constant i n parabolic v e l o c i t y surface approximation B IDT r a d i a t i o n susceptance a c..,.. e l a s t i c s t i f f n e s s constants x j k l C,j, IDT s t a t i c capacitance D_̂  e l e c t r i c displacement e. p i e z o e l e c t r i c constants l j k E. SAW e l e c t r i c f i e l d x f SAW frequency f IDT center frequency G IDT r a d i a t i o n conductance a G IDT r a d i a t i o n conductance at f o o J, f i r s t order Bessel function of the f i r s t kind 1 K SAW wavevector K(q),K(q) complementary e l l i p t i c i n t e g r a l s of the f i r s t kind L ,L series and p a r a l l e l IDT matching inductors s p x x i 3.) number of IDT finger p a i r s acoustic power e l e c t r i c a l part of acoustic power mechanical part of acoustic power acoustic and e l e c t r i c a l IDT q u a l i t y factors Fresnel distance IDT r a d i a t i o n resistance IDT finger s e r i e s resistance r a d i a t i o n resistance at f o sin(x)/x acoustic s t r a i n IDT 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 equilibrium SAW v e l o c i t y r.m.s. voltage IDT r a d i a t i n g aperture r a d i a t i o n reactance complex impedance Av/v, the change i n SAW v e l o c i t y when the substrate surface i s covered with an ideal,massless conductor. p e r m i t t i v i t y constants SAW wavelength SAW angular frequency angle between SAW propagation vector 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 x i i i ( C h a p t e r 3.) 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 p g s h e e t r e s i s t i v i t y C h a p t e r 4. A w i d t h o f d i f f r a c t e d l i g h t beam a p e r t u r e B i n v e r s e d i e l e c t r i c p e r m i t t i v i t y t e n s o r 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 one s e c t i o n o f beam - s t e e r e d t r a n s d u c e r E- OGW e l e c t r i c f i e l d v e c t o r •*»r E. r e a l p a r t o f SAW e l e c t r i c f i e l d l E . ( y ) a m p l i t u d e o f SAW e l e c t r i c f i e l d l f a c o u s t i c f r e q u e n c y f ,f h i g h and low f r e q u e n c i e s a t w h i c h t h e Bragg a n g l e i s matched Af d e f l e c t o r Bragg b a n d w i d t h m f IDT a r r a y f r e q u e n c y . G w i d t h o f one s e c t i o n o f beam-steered IDT, i n c l u d i n g e l e c t r o d e s H beam-steered IDT s t e p h e i g h t between a d j a c e n t s e c t i o n s i e l e c t r i c c u r r e n t k~Q i n c i d e n t OGW w a v e v e c t o r ic • w a v e v e c t o r o f f i r s t - o r d e r d i f f r a c t e d beam K SAW w a v e v e c t o r £ ' d i f f r a c t i o n o r d e r number L a c o u s t o - o p t i c i n t e r a c t i o n l e n g t h m TE mode i n d e x M number o f s e c t i o n s i n beam-steered IDT n r e f r a c t i v e i n d e x , N s u p e r s c r i p t d e n o t i n g n o r m a l i z e d x i v 4.) number of r e s o l v a b l e s p o t s a c o u s t i c power e l e c t r i c a l power d i s s i p a t e d i n IDT a c o u s t i c power f o r 100% d i f f r a c t i o n e f f i c i e n c y p h o t o e l a s t i c c o n s t a n t s i n t e g e r number o f A Q / 2 s t e p s i n H r e a l p a r t o f a c o u s t i c s t r a i n a m p l i t u d e o f r e a l p a r t o f a c o u s t i c s t r a i n n o r m a l i z e d OGW e l e c t r i c f i e l d as a f u n c t i o n o f d e p t h e l e c t o o p t i c c o n s t a n t s r a d i u s v e c t o r magnitude of impedance a c o u s t o - o p t i c o v e r l a p i n t e g r a l p e r m i t t i v i t y change f o r a c o u s t o - o p t i c i n t e r a c t i o n d i f f e r e n c e between i n c i d e n t and Bragg a n g l e s a n g u l a r w i d t h o f d i f f r a c t e d l i g h t beam r e l a t i v e p e r m i t t i v i t y t e n s o r s u b s t r a t e r e l a t i v e p e r m i t t i v i t y a c o u s t o - o p t i c d i f f r a c t i o n e f f i c i e n c y b eam-steered 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 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 a n g l e l i g h t a n g l e o f i n c i d e n c e t o SAW p l a n e s of c o n s t a n t phas a n g l e o f p r o p a g a t i o n o f f i r s t o r d e r d i f f r a c t e d beam 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 c o e f f i c i e n t t r a n s i t t i m e o f sound wave a c r o s s l i g h t beam Bragg a n g l e as a f u n c t i o n o f f r e q u e n c y xv (Chapter 4 . ) th <})̂ (f) coupling constant for I diffraction order <j>o(f) angle of propagation of principal maximum of array function CD angular frequency of incident light angular frequency of first-order diffracted light 0, SAW angular frequency xvi ACKNOWLEDGEMENT I tha n k my s u p e r v i s o r , Dr. L. Young, f o r h i s s u p p o r t and g u i d a n c e d u r i n g t h e c o u r s e o f t h i s r e s e a r c h . Mr. A r v i d L a c i s d i d t h e s c a n n i n g e l e c t r o n m i c r o s c o p y and e l e c t r o n m i c r o p r o b e a n a l y s i s . H e l p f u l 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 f r o m Mr. Rodger Bennet and Mr. P e t e r M u s i l on t h e c u t t i n g and p o l i s h i n g o f c r y s t a l s , and f r o m Mr. J a c k S t u b e r i n t h e machine shop. P r o f e s s o r R. B u t t e r s h e l p e d w i t h t h e x - r a y f l u o r e s c e n c e measurements. S p e c i a l , t hanks a r e due t o Dr. E.V. J u l l and Mr. Hans Hogenboom f o r numerous h e l p f u l d i s c u s s i o n s , and t o M i s s S a n n i f e r L o u i e f o r t y p i n g t he 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 R e s e a r c h C o u n c i l o f Canada, t h r o u g h a S c i e n c e S c h o l a r s h i p and a l s o G r a n t No. A3392, i s g r a t e f u l l y acknowledged. x v i i 1. 1. INTRODUCTION 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 combined t h e o r e t i c a l and e x p e r i m e n t a l s t u d y o f a c o u s t i c beam s t e e r i n g as a means o f i m p r o v i n g t h e p e r f o r m a n c e o f p l a n a r a c o u s t o - o p t i c l i g h t d e f l e c t o r s . I n an i n c r e a s i n g l y d i g i t a l age, i n t e g r a t e d o p t i c s and s u r f a c e a c o u s t i c wave (SAW) d e v i c e s a r e among t h e few a n a l o g t e c h n o l o g i e s l i k e l y t o re m a i n c o m p e t i t i v e . 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 s w i t c h e s , d e f l e c t o r s and m o d u l a t o r s , t h e d e v i c e s s t u d i e d h e r e p r o m i s e t h e r e a l i z a - t i o n o f more complex 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 s u c h 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 t e c h n o l o g y o f b u l k a c o u s t o - o p t i c d e v i c e s f o r t h e d e f l e c t i o n and m o d u l a t i o n o f l i g h t i s r e a s o n a b l y w e l l d e v e l o p e d [ 9 2 ] . The p l a n a r geometry u t i l i z i n g s u r f a c e o p t i c a l w aveguides and a c o u s t i c s u r f a c e waves p r o m i s e s t o g i v e d e v i c e s 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 e f f i c i e n t and t h a t have h i g h e r p e r f o r m a n c e t h a n t h e i r b u l k c o u n t e r p a r t s . A r e v i e w o f s e l e c t e d t o p i c s o f t h e t h e o r y o f d i e l e c t r i c s l a b w aveguides i s p r e s e n t e d i n C h a p t e r 2. T e c h n i q u e s f o r t h e f a b r i c a t i o n o f h i g h q u a l i t y o p t i c a l waveguides (OWG) i n LiNbO^ by n i c k e l i n d i f f u s i o n a r e g i v e n . 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 t h e d e s i g n and f a b r i c a t i o n o f r u t i l e c o u p l i n g p r i s m s i s d e s c r i b e d . The measured p r o p e r t i e s o f a 6-mode OWG' a r e g i v e n and t h e e l e c t r i c f i e l d d i s - t r i b u t i o n s o f t h e t h r e e g u i d e d TE modes a r e c a l c u l a t e d by t h e WKB method. 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 t r a n s d u c e r s and a c o u s t i c s u r f a c e waves on LiN 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 t e s t i n g t h e t h e o r y i s d e s c r i b e d . The t h e o r y o f t h e s u r f a c e wave a c o u s t o - o p t i c i n t e r a c t i o n i s p r e s e n t e d i n C h a p t e r 4. E x p r e s s i o n s d e s c r i b i n g t h e f a r - f i e l d a c o u s t i c r a d i a t i o n p a t t e r n o f bea m - s t e e r e d i n t e r d i g i t a l t r a n s d u c e r s 2. a r e d e v e l o p e d , as a r e e q u a t i o n s f o r p r e d i c t i n g t h e p e r f o r m a n c e c h a r a c - 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 l i g h t d e f l e c t o r s . The d i f f r a c t i o n e f f i c i e n c y i s c a l c u l a t e d f r o m f i r s t p r i n c i p l e s , u s i n g t h e p h o t o e l a s t i c 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 LiNbO^ and t h e d e t a i l e d d e s c r i p t i o n o f t h e OGW and SAW r a d i a t i o n f i e l d s . S e v e r a l e x p e r i m e n t s a r e d e s c r i b e d , and t h e r e s u l t s a r e compared w i t h t h e o r e t i c a l c a l c u l a t i o n s . I n C h a p t e r 5, t e c h n i q u e s f o r making h i g h r e s o l u t i o n photomasks 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 t r a n s d u c e r s a r e d i s c u s s e d . 2. DIELECTRIC OPTICAL WAVEGUIDES 2.1 Uniform Dielectric Slab Waveguides Consider the asymmetrical dielectric slab illustrated in Fig. 2.1. The refractive indices in the three regions indicated are related by the inequality n > n > n , m s a where n m is the mode index of the mth guided optical mode. If we re- strict our consideration to guided TE waves propagating in the z direc tion, the wave equation reduces to 82E 92E 2 32E x _. x _ n x 2 2 ~ 2 2 * D?/ 9yZ c a t . For time harmonic fields with propagation constant 3 in the z direction, the wave equation becomes d \ 2 2 2 — = i + (n k - 3 )E =0 (2. dy Fig. 2.1 Slab Dielectric Waveguide. 4. Boundary c o n d i t i o n s r e q u i r e t h a t and be c o n t i n u o u s a t y = 0 and y = d, and t h a t E v a n i s h a t y = ±°°. S o l u t i o n s s a t i s f y i n g t h e s e r e q u i r e - ments have been shown [1] t o t a k e t h e f o r m E x = A exp(Sy) , y < 0 = A cos (icy) + B s i n ( i c y ) , 0 < y < d = (A cos(Kd) - B s i n ( K d ) ) e x p [ - y ( y + d ) ] , d < y < °° (2.2) . . ta2 2 2 1/2 where 6 = (g - n k ) a . 2,2 fl2.1/2 ic = (n k - g ) m r /  2 2 X 1 2 2,1/2 , n „ N Y = [(n m - n g ) k - K ] (2.3) and where k i s t h e w a v e v e c t o r i n f r e e s p a c e . T h i s s o l u t i o n i s mathe- m a t i c a l l y 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 o f a 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 w e l l . C o n t i n u i t y r e q u i r e m e n t s on Ĥ , 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 tan(icd) = K(Y + <5)/(K - y S ) , 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 waveguide a l s o s u p p o r t s a con- 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 o r t h o n o r m a l 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. The p r o p a g a t i o n v e c t o r o f t h e g u i d e d wave has t h e magnitude g = n k s i n e (2.4) m . m a l o n g t h e d i r e c t i o n o f p r o p a g a t i o n , where 9 M i s t h e a n g l e between "fe and y. E x a m i n a t i o n o f e q u a t i o n (2.3) r e v e a l s t h a t y becomes i m a g i n a r y when g < n^k; as a r e s u l t , t h e g u i d e d mode becomes r a d i a t i v e and con- f i n e m e n t i s no l o n g e r p o s s i b l e . Thus, y = 0 i s t h e c u t o f f c o n d i t i o n f o r wave g u i d a n c e . The e q u a t i o n tan(Kd ) = 6/K (2.5) c can then be used to determine the minimum thickness d^ that wi l l support a particular guided mode. It is 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 is carried by .the evanescent fields. A more intuitive treatment of wave guidance is based on a ray- optic 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 is n sin9 = n sin9 . m m > a a When n sin9 > n , critical internal reflection occurs and the wave m m a cannot escape from the waveguide. At the lower interface, the equivalent condition is n sin9 > n . m m s Since n > n , satisfaction of the second inequality implies s a. satisfaction of the fir s t . Thus, three kinds of propagating modes are obtained: 6. (1) a i r modes, when n s i n G < n , m m a (2) s u b s t r a t e modes, when n > n sir i G > n , s m m a and (3) g u i d e d modes when n s i n B > n . 6 m m s P r o p a g a t i o n o f g u i d e d 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 o f p l a n e waves fr o m t h e waveguide s u r f a c e s a r e i n phase. T h i s imposes t h e e i g e n v a l u e c o n d i t i o n 2kn d co s 6 - 2d) - 2<J> = 2mTr , (2.6) m m Tms ma where d> and 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 sub-ms ma s t r a t e i n t e r f a c e s , r e s p e c t i v e l y . The v e r t i c a l component o f t h e g u i d e d mode forms a s t a n d i n g wave between t h e waveguide b o u n d a r i e s , so t h a t p r o - p a g a t i o n a p p ears t o be i n the h o r i z o n t a l d i r e c t i o n o n l y . F o r a l i m i t e d waveguide 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 i n t e g e r v a l u e s o f m w i l l s a t i s f y t h e phase m a t c h i n g c o n d i t i o n (2.6). I n 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 f o r d> and d> m a a r e s u b s t i t u t e d i n t o (2.6), the e i g e n v a l u e e q u a t i o n (2.3) i s o b t a i n e d . 2.2 Modes i n Graded I n d e x Waveguides D i f f u s i o n i s a c o n v e n i e n t t e c h n i q u e f o r making h i g h q u a l i t y o p t i c a l s u r f a c e w a v e g u i d e s . I t i s p a r t i c u l a r l y advantageous f o r a c o u s t o - 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 . However, t h e d e s c r i p t i o n o f g u i d e d modes i s c o n s i d e r a b l y more d i f f i c u l t , owing t o t h e n o n - u n i f o r m r e f r a c t i v e i n d e x p r o f i l e o b t a i n e d . As a r e s u l t o f e i t h e r i n - o r o u t - d i f f u s i o n , t h e r e f r a c t i v e i n d e x n e a r t h e s u r f a c e t a k e s t h e fo r m n ( y ) = n g + An f ( y ) (2.7) where n i s t h e s u b s t r a t e i n d e x , An = n - n i s t h e change i n i n d e x a t s o s the s u r f a c e , and the e x a c t f o r m o f f ( y ) depends on the 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 x(y) e j ( p Z _ a ) t ) . (2.8) Substitution into the one-dimensional wave equation gives 2 —5 2 E+ « (y) = 0, (2.9) dy 2 2 2 2 where K (y) = n (y)k - For most index profiles of interest, solution in terms of known functions is 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 is expedient to solve instead the equation d2E 2 —2* + [ K 2 ( y ) - S<!yJ - ~ ' ( — ^ — ) ] E x = 0, dy dy /ic(y) solutions of which closely approximate the exact solutions under appro- priate conditions. This is the WKB method of Quantum Mechanics, for which the following solutions for TE guided modes [7,9] are obtained: E = A exp(yy) y < 0 (2.10) E Y = BA IK(y) A cos (<f)(y) - TT/4) y m > y > 0 (2.11) E x = /Ko/<(y) A exp{/-|ic(y) dy} °° > y > y m (2.12) where y 2 = g'2 - k2, ' (2.13) m 2 2 2 2 <o " V " 31> (2-14) 8. K 2(y) = n 2 ( y ) k 2 - f3 2 , (2.15) <Ky) = / 7 m k( y ) d y , (2.16) y A and B a r e c o n s t a n t s , k i s the w a v e v S c t o r i n f r e e s p a c e , n Q i s t h e s u r - f a c e i n d e x and 6^ i s t h e z-component 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. 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 t h e i n t e r v a l (0,y ) and m J ,Jm decays t o z e r o on e i t h e r s i d e . The t u r n i n g p o i n t o f t h e mth 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 method i s o n l y u s a b l e ( S c h i f f [ 8 ] ) i f K(y) changes s l o w l y w i t h y. The c o n d i t i o n f o r t h i s may be w r i t t e n as / 1-̂-1 « k/x C 2 - 1 ? ) 1 dy 1 A t the t u r n i n g p o i n t (y ) , t h i s c o n d i t i o n i s v i o l a t e d , and t h e s o l u t i o n s (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 sense w i t h i n s e v e r a l wave- l e n g t h s o f y . T h i s r e s t r i c t s t h e a c c u r a c y o f s o l u t i o n , p a r t i c u l a r l y f o r l o w e r o r d e r modes. 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 c a s e t h e s o l u t i o n s o b t a i n e d a r e o f t e n o f r e a s o n a b l e a c c u r a c y . F o r h i g h e r o r d e r modes, good a c c u r a c y i s o b t a i n a b l e .(j.sSpec-ia.1 ite'chniques must be usjed ^s%hejevic,±ntifeyiSp€ v t h e «tjii?ndng rppjtn%,xjiw.he-re,rith:er>e i s a p o l e i n E . Jm' t x x H o c k e r and Burns [10] have shown t h a t t h e modes o f d i f f u s e d w aveguides can be d e s c r i b e d by j u s t two q u a n t i t i e s , an e f f e c t i v e d i f f u s - i o n d e p t h w h i c h 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 mode i n d e x , n = 3 /k . (2.18) m m Marcuse [9] has a n a l y s e d t h e TE modes o f graded i n d e x s l a b waveguides w i t h t h e WKB method and 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 F i g . 2.4 Z i g - Z a g Wave P r o p a g a t i o n i n Graded I n d e x Waveguides. t h e i n d e x p r o f i l e . The s o l u t i o n s were f o u n d t o be i n c l o s e agreement w i t h e x a c t s o l u t i o n s o f t h e p i e c e w i s e - l i n e a r p e r m i t t i v i t y p r o f i l e , and ' l e s s cumbersome m a t h e m a t i c a l l y ( f i g . 2.3). C o m p a r i s o n o f eqs. ( 2 . 1 0 ) - ( 2 . 1 6 ) w i t h t h e TE modes o f a u n i - f o r m s l a b waveguide r e v e a l s t h e g r e a t e r m a t h e m a t i c a l c o m p l e x i t y o f t h e g r a d e d - i n d e x s o l u t i o n s . N u m e r i c a l methods o f a n a l y s i s a r e u s u a l l y r e q u i r e d . The 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 d i f f i c u l t t o a p p l y , s i n c e t h e " r e f l e c t i o n " f r o m the b o t t o m waveguide s u r f a c e i s g r a d u a l r a t h e r t h a n i n s t a n t a n e o u s ( F i g . 2 . 4 ) . T i e n e t a l [ 5 ] have shown t h a t t h e phase s h i f t on r e f l e c t i o n i s v e r y n e a r l y TT/4 a t y m and ir/2 a t 10. y = 0. U s i n g t h i s , W h i t e and H e i d r i c h d e v e l o p e d a method o f d e t e r m i n - i n g t h e t u r n i n g p o i n t s y o f t h e WKB method, g i v e n the mode i n d i c e s m V n 2 n^. E q u a t i o n (2.16) t h e n t a k e s t h e fo r m Ji;Q <j>(0) r m / 2 e n S 2,1/2 A r j 4m - 1 = / (n (Z) - n m ) dZ - — - g - 0 where m = 1, 2 ... M, and where Z = y / A . Use o f the 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 g i v e s t h e f o l l o w i n g s o l u t i o n f o r t h e t u r n i n g p o i n t s o f the mth g u i d e d mode: Z l = 1 6 ( 2 } ( n o - n l } » ( m = 1 ) (2.19) 7 - 7 J. r 3 / n m - l + 3 n m N - l / 2 / ,-1/2, r,4m-!, 2 "V1 Zm " Zm-1 + ¥ 2 > ( V l " V ] { ( ~ 8 ~ ) " 3 k = l A - i + > n k 1/2 A " A - l , k - l k m = 2, 3 ... M (2.20) where Z = 0 and n = n ( 0 ) . Note t h a t m = 1 f o r t h e z e r o t h o r d e r mode, o o These e q u a t i o n s 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 the mode i n d i c e s n^, so t h a t s u c c e s s i v e Z's may be c a l c u l a t e d by i t e r a t i o n . I n o r d e r t o d e t e r m i n e whether t h e e s t i m a t e o f n i s r e a s o n a b l e , o the sum o f s q u a r e s o f t h e second d i f f e r e n c e s , °k+2 ~ " k + l n k + l " "k M-2 = .1 k=0 Zk+2 Zk+1 Zk+1 \ Zk+2 + Z k + 1 Zk+1 + Z k (2.21) i s c a l c u l a t e d . The minimum i n r c o r r e s p o n d s t o t h e smoothest i n d e x p r o - f i l e , and t h e c o r r e s p o n d i n g v a l u e o f n Q 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 i n d e x p r o f i l e o b t a i n e d can t h e n be used w i t h (2.10) t o (2.16) t o c a l c u l a t e t h e e l e c t r i c f i e l d d i s t r i b u t i o n s 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 o r by f i t t i n g a f u n c t i o n ( i . e . , e r f c ) t o t h e p o i n t s o b t a i n e d . 2.3 C o u p l i n g t o O p t i c a l Waveguides C o u p l i n g t o g u i d e d o p t i c a l modes p r e s e n t s some d i f f i c u l t y , owing p r i n c i p a l l y t o t h e s m a l l ( o f t e n l e s s t h a n 10 ym) d i m e n s i o n s o f o p t i c a l w a v e g u i d e s . I n i t i a l l y , l e n s e s ( F i g . 2.5) were used t o reduc e t h e d i a m e t e r o f l i g h t beams. Low c o u p l i n g e f f i c i e n c y and a l a c k o f s e l e c t i v i t y i n e x c i t i n g g u i d e d modes made t h i s an u n s a t i s f a c t o r y method. The 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 S c h u b e r t i n 1969, overcame t h e s e l i m i t a t i o n s . I n F i g . 2.6, l i g h t e n t e r 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 > n , a t an a n g l e y such t h a t i t i s c r i t i c a l l y i n t e r n a l l y r e f l e c t e d a t the p r i s m b a s e . I f t h e s p a c i n g between p r i s m and t h e o p t i c a l , waveguide i s about A/2 o r l e s s , c o u p l i n g o f l i g h t e n e r g y i n t o t h e waveguide i s p o s s i b l e , t h r o u g h o v e r l a p o f t h e 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 . M a t h e m a t i c a l l y , t h i s mechanism o f energy t r a n s f e r ..resembles t h e quantum m e c h a n i c a l t u n n e l i n g o f a p a r t i c l e t h r o u g h a p o t e n t i a l b a r r i e r ; c o n s e q u e n t l y , i t i s f r e q u e n t l y c a l l e d o p t i c a l t u n n e l - i n g . C o u p l i n g can o n l y o c c u r i f t h e h o r i z o n t a l e l e c t r i c f i e l d component o f l i g h t a t the p r i s m base matches t h a t o f a g u i d e d mode. T h i s may be e x p r e s s e d by t h e phase 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 mode: 3 = k n s i n e , (2.22) m p m' where $ m 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 . The mode i n d e x n i s m 12. n = ~ = n s i n 6 T O . (2.23) m k p m 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 e x c i t e any one p a r t i c u l a r g u i d e d mode by v a r y i n g the a n g l e o f i n c i d e n c e . 0 o f l i g h t on t h e p r i s m b a s e , p r o v i d e d t h a t t h e i n d i c e s o f a d j a c e n t modes a r e s u f f i c i e n t l y d i f f e r e n t . A number o f t h e o r e t i c a l t r e a t m e n t s o f c o u p l i n g e f f i c i e n c y have been p u b l i s h e d [2,9,16,17]. Most o f t h e s e d e a l w i t h t h e s p e c i a l c a s e o f a u n i f o r m 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 p r o f i l e c o u p l i n g i n t o a u n i f o r m d i e l e c t r i c s l a b w aveguide, i n w h i c h c a s e t h e maximum a t t a i n - a b l e c o u p l i n g e f f i c i e n c y i s about 80%. By a l t e r i n g t h e beam p r o f i l e and/ o r t a p e r i n g t h e gap between waveguide and p r i s m , 100% e f f i c i e n c y i s t h e o r e t i c a l l y p o s s i b l e , and o v e r 90% has been a c h i e v e d e x p e r i m e n t a l l y . Marcuse [9] g i v e s a method o f e s t i m a t i n g the c o u p l i n g e f f i c i e n c y t o a g r a d e d - i n d e x s l a b waveguide. F o r h i g h c o u p l i n g e f f i c i e n c y , t h e mode and p r i s m i n d e x must be v e r y c l o s e l y matched. I n p r a c t i c e , t h e p r i s m 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 by t h e use o f an a d j u s t a b l e clamp. Because o f n o n u n i f o r m i t i e s i n the m e c h a n i c a l c o n t a c t between p r i s m and g u i d e , c o u p l i n g e f f i c i e n c i e s o f o v e r 25% a r e d i f f i c u l t t o a c h i e v e , p a r t i c u l a r l y f o r l i g h t beams more t h a n 1 mm i n d i a m e t e r . 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 c o u p l e r s have been deve- l o p e d . Dakss e t a l [18] announced t h e g r a t i n g c o u p l e r ( F i g . 2.7) i n 1970. An o p t i c a l g r a t i n g o f p e r i o d i c i t y 0.67 ym was formed i n p h o t o - r e s i s t on t h e waveguide s u r f a c e by e x p o s u r e i n a l a s e r i n t e r f e r o m e t e r . A more r e c e n t f a b r i c a t i o n t e c h n i q u e 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 , t h u s e t c h i n g t h e g r a t i n g i n t o t h e waveguide i t s e l f . The phase m a t c h i n g c o n d i t i o n f o r g r a t i n g c o u p l e r s i s ^ B = k s i n e + 2mTr/d, (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 is again possible. The maximum efficiency observed by Dakss et al was 40%; with suitable techniques, 100% is 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 is 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 is an attenuation constant denoting the loss of energy to regions adjacent to the guide. If a is suitable, this leakage will 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 in Fig. 2.8. In general, the leaky wave can be supported by either multi- layered (i.e., prism) or periodic (i.e., grating) structures. By reci- procity, light can be coupled into the waveguide by reversing the propa- gation 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 is necessary to taper a along the direction of guided mode propagation such that the required input beam profile is 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 is 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]. The indices of rutile are n g = 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 in Fig. 2.9, the phase velocity matching condition 3 n = = n sine (2.23) m k p :m must be satisfied before coupling to guided modes can occur. The angles X»rlamajicti ±n̂ ctri angle.cAB.G' ar,e related by the equation 0 = a + y, (2.24) where a is the prism angle. At the upper air-prism interface, Snell's law is so that sinu = n^ sxny n = n sin[a + arcsin( =•) 1 m p n P / 2 2T = cosa sinu + *n - sin u sina (2.25) P It is more useful to write this with u as the independent variable, 16. A N Fig. 2.9 Prism Coupler Geometry. Fig. 2.10 Broadening of the Input Light Beam /~2~ 2 y = arcsin[n cosa - vn - n s i n a l . (2.26) m p m J x 7 If we know the prism index and have an estimate of the guided mode index n^, we can use this equation to calculate the range of prism angles a 7F IT over which coupling i s possible. This range i s such that - y < y < > although angles of incidence near the limits ± ir/2 are not usable because broadening of the Input light beam reduces the coupling efficiency. From Fig. 2.10, we see that the fraction of the l i g h t incident on the active coupling region of length JI i s where W is the incident beam thickness. Usually a lens of long (>20 cm) focal length is 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 in (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 coup- p m ^ ling efficiency is desired, i t is necessary that the coupling length Z be greater than the projection of the light beam diameter in the prism. .5 .4 .2 30° 40° 50° . ry= 2.86 60° np=2.582 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 is _ _ 2.COS0COSU TT ^ o t * c c Wcosy P or n = n W < £, (2.28) t c p where n is a factor dependent on prism, gap and waveguide parameters. For coupling across a uniform gap of width D into TE modes of a graded- index optical waveguide (OWG), Marcuse [9] obtained the radiation-loss amplitude coefficient |A|V a y 2 K 2 a = : — r - r , (2.29) 2ioy P{[(K a-y )sinh(yD)] •+ [y ( 0+K )cosh(yD)]z} o o o 2 2 2 2 where y and K q are given by (2.13) and (2.14), a - k (n - n^) and 2 2 (A[ B (2upoP) is determined by the mode normalization integral, 3 CO P = / E* E dy * (2.30) 2uu ; x x o —00 The parameter P is the power per unit width carried by the mode. Marcuse normalizes the electric field by setting /Piouo = 1 V/m for each guided mode. The coupling efficiency n is then given by nc = 1 - exp(-acJQ (2.31) In the derivation of (2.29), Marcuse assumed that the electric field in the gap consisted of plane waves, and that the input beam in- tensity profile was rectangular. The f i r s t assumption is 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 in (2.29) can be seen more clearly i f we take D = 0. Then | A| B 2 2 2 a K a c = . (2.32) 2a)yoP(a+KO) The p r i s m i n d e x e n t e r s t h i s e x p r e s s i o n t h r o u g h a = k/n^ - n m . 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 a r e v e a l s t h a t a has a maximum when n = n , c p o' the OWG s u r f a c e i n d e x o f r e f r a c t i o n . When a c h o i c e o f p r i s m i n d e x e x i s t s (as i n the c a s e o f r u t i l e ) , t h e l o w e r i n d e x w i l l g i v e a h i g h e r c o u p l i n g e f f i c i e n c y , p r o v i d e d , o f c o u r s e , t h a t n^ > n Q . I n d i f f u s e d w a v e g u i d e s , t h e r e 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 between t h e i n d i c e s o f modes o f a d j a c e n t o r d e r . C o n s e q u e n t l y , t h e coup- l i n g a n g l e s y a r e c l o s e t o g e t h e r , making t h e o b s e r v a t i o n o f i n d i v i d u a l modes and 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 more d i f f i c u l t . The 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 p r o v i d e s a measure o f t h e s e p a r a b i l i t y o f i n d i v i d u a l - g u i d e d modes, 2 2 -1/2 dy c o s a + n (n - n ) s i n a _ = m P m . (2.33) an rr. — „ , ,~ m n ( P- 2 . \2.1/2 L l - (n c o s a - /n - n sxna) ] m p m 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 as a f u n c t i o n o f a f o r r u t i l e , w i t h n = 2.23. The minima o f F i g . 2.12 c o i n c i d e w i t h t h e maxima o f m ° F i g . 2.11; 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 when the c o u p l i n g e f f i c i e n c y due t o beam b r o a d e n i n g i s g r e a t e s t . I n F i g . 2.13, t h e l i m i t i n g p r i s m a n g l e s a r e p l o t t e d a g a i n s t 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 2.86. E q u a t i o n (2.27) was s o l v e d f o r a by Newton's method f o r s i n y = ±1. The upper c u r v e s a r e f o r an a n g l e o f i n c i d e n c e y = T T / 2 ; t h e l o w e r , f o r y = - T T / 2 . From [ 3 ] , we e x p e c t 2.15 < n m < 2.5; t h e n t h e l i m i t s t o the p r i s m a n g l e a r e g i v e n by 4T° < a < 86° f o r n = 2.582 and 34° < a < 74° f o r n = 2.86. P P I n o r d e r t o s i m p l i f y t h e 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 d e s i r a b l e t o s e l e c t the 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 t h a t l i g h t i n t h e p r i s m p r o p a g a t e s w i t h e i t h e r n * o r n £ , t h e extreme v a l u e * I n t h e r e s t o f t h i s s e c t i o n , n Q r e f e r s t o the o r d i n a r y r a t h e r t h a n t h e s u r f a c e i n d e x o f r e f r a c t i o n . 20. OC Fig. 2.12 Mode Separability vs. Prism Angle for n = 2.23. of the extraordinary index. When the extraordinary ray axis lies 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_ = n = 2.582 1 1 o n„ = n = 2.86 3 e and V ? = 0 in the phase velocity determinant (Appendix II) to obtain V 2 + V 2 = n 2 V 4/c 2 (TE) x y e (2.34) V 2 + V 2 = n 2 V 4/c 2 (TM) Fig. 2.13 Limiting Mode Indices vs. Prism Angle for Rutile. The phase velocity surface is used here (rather than the ray velocity surface) because i t is 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 is the preferred orientation when both TE and TM modes are to be excited. In Fig. 2.15, the c axis is vertical. The substitutions n l " n2 = % n = n z Fig. 2.15. c Axis Vertical. 23. and V = 0 g i v e and V 2 + V 2 = n 2 V 4 / c 2 (TE) y z o 2 ? 2 2 4 2 n Z = n n V / c (TM) 3 z o e (2.35) f o r t h e p r o j e c t i o n s o f t h e phase v e l o c i t y s u r f a c e on t h e y z p l a n e . Thus, 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 . However, 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 . I n t h e 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 were used, so t h i s o r i e n t a t i o n was s a t i s f a c t o r y . I t p r o v i d e s a somewhat h i g h e r degree o f mode s e p a r a b i l i t y and c o u p l i n g e f f i c i e n c y . 2.5 C o u p l e r F a b r i c a t i o n I n i t i a l l y , 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 by 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, u s i n g Gaf PR-102 p o s i t i v e p h o t o r e s i s t . T h i s t e c h n i q u e , w h i c h g i v e s r e d u c e d e x p o s u r e t i m e s and deeper grooves w i t h s h a r p e r r i d g e s t h a n c o n v e n t i o n a l methods, was f i r s t used by Tsang and Wang i n 1974 [2.0J 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 . I n F i g . 2.16, l i g h t f r om an argon l a s e r i s s p l i t i n t o two beams w h i c h a r e i n d i v i d u a l l y s p a t i a l l y f i l t e r e d , 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 f l a t t o i n t e r f e r e a t t h e 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 . I t can be shown t h a t t h e g r a t i n g p e r i o d i s where 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 8^ t h e a n g l e o f i n c i - dence on t h e o p t i c a l f l a t . and p a s s e d t h r o u g h a .45 um f i l t e r t o r e d u c e l i g h t s c a t t e r i n g . The p h o t o r e s i s t , d i l u t e d 1:1 w i t h r e a g e n t grade m e t h y l e t h y l k e t o n e , was (2.36) B e f o r e u s e , t h e d e v e l o p e r was d i l u t e d 1:3 w i t h d e i o n i z e d w a t e r 24. Fig. 2.16 Photoresist Exposure for Grating Coupler Fabrication s p i n c o a t e d a t 4000 RPM f o r 10 seconds. The c o a t i n g s were measured t o be about 2000 A° t h i c k w i t h a S l o a n A n g s t r o m e t e r . A f t e r d r y i n g i n a i r 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 baked a t 65°C f o r h a l f an h o u r . D u r i n g e x p o s u r e , the l i g h t i n t e n s i t y was measured t o be about 2 60 mW/cm . A s e r i e s o f t e s t e x p o s u r e s were u n d e r t a k e n w i t h t i m e s r a n g - i n g f r o m 10 seconds t o 10 m i n u t e s . Three m i n u t e s was f o u n d t o g i v e good r e s u l t s . The g r a t i n g s were r i n s e d and blown d r y w i t h n i t r o g e n b e f o r e e x a m i n a t i o n i n a s c a n n i n g e l e c t r o n m i c r o s c o p e . A g r a t i n g p e r i o d o f 5400 A° was o b t a i n e d . F i g . 2.17 shows the s h a l l o w r i d g e s o f an u nderexposed p o r t i o n , and F i g . 2.18 shows a c o r r e c t l y exposed a r e a a t l o w e r m a g n i f i c a t i o n . U n f o r t u n a t e l y , u s a b l e g r a t i n g s were n o t o b t a i n e d , p r i m a r i l y due 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 development o f the grooves was c a r r i e d t h r o u g h t o t h e s u b s t r a t e ( F i g . 2.19). T h i s p r o b l e m c o u l d n o t be overcome by v a r i a t i o n o f p r o c e s s p a r a m e t e r s , so a r u t i l e c r y s t a l was o r d e r e d f r o m NL I n d u s t r i e s i n New J e r s e y . The c r y s t a l was about 2i em x l i cm d i a m e t e r and had a v e r y p a l e y e l l o w c o l o u r , a l t h o u g h t h e t r a n s m i t t a n c e was q u i t e h i g h . A number o f p r i s m s were made w i t h t h e c a x i s v e r t i c a l . F i r s t , s e v e r a l 5 mm s l i c e s were c u t w i t h a f i n e diamond saw. The c r y s t a l was h e l d i n p l a c e w i t h a t h e r m o p l a s t i c cement. C a u t i o n was r e q u i r e d d u r i n g h e a t i n g and c o o l i n g t o a v o i d c r a c k i n g . The s l i c e s were remounted f l a t on the h o l d e r and d i c e d i n t o 5 mm cubes. The c r y s t a l o r i e n t a t i o n was marked on t h e t o p , w h i c h was n o t ground u n t i l l a s t . A s u c c e s s i o n o f w e t t e d emery p a p e r grades was used to shape the p r i s m s , w i t h 600 g r i t b e i n g t h e l a s t f o r s u r f a c e s r e q u i r i n g p o l i s h i n g . D u r i n g t h i s p r o c e s s , i t was f o u n d b e s t t o h o l d t h e p r i s m s by hand ( F i g . 2.20). 26. F i g . 2.17 Underexposed G r a t i n g . F i g . 2.18 C o r r e c t l y E x posed G r a t i n g . F i g . 2.19 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 A d h e s i o n . 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 S e v e r a l t e c h n i q u e s were t r i e d f o r f a b r i c a t i o n o f o p t i c a l w a v e g u i d e s , i n c l u d i n g t h e r m a l 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 - i o n . Because 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 a c o u s t o o p t i c i n t e r - a c t i o n s i n LiNbO^, i t was n e c e s s a r y t o use a m a t e r i a l w i t h h i g h e r r e f r a c - t i v e i n d e x t h a n 2.214. A g a i n , T i O ^ was one of t h e few s u i t a b l e m a t e r i a l s . I n i t i a l l y , t h e t h e r m a l 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 t r i e d , a t 800°C w i t h an oxygen f l o w r a t e o f 1 A/min. Even p r o l o n g e d t r e a t m e n t , however, d i d n o t g i v e f i l m s w i t h l ow o p t i c a l a b s o r p t i o n , so t h i s a p p r o a c h was abandoned. B e t t e r f i l m s were o b t a i n e d 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 argon-oxygen m i x t u r e a t p r e s s u r e s o f .01 and .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 pur e oxygen gave b r o w n i s h f i l m s s i m i l a r i n appearance t o t h e r m a l l y o x i d i z e d t i t a n i u m . ) A s e r i e s o f f i l m s o f d i f f e r e n t t h i c k n e s s e s was d e p o s i t e d on a s i l i c o n s l i c e i n o r d e r t o measure t h e o x i d e t h i c k n e s s by e l l i p s o m e t r y . Comparison between t h e e x p e r i m e n t a l ip, A c u r v e ( F i g . 2.21) and c a l c u l a t e d t a b l e s gave a r e f r a c t i v e i n d e x o f 2.15 a t 632.8 nm. T h i s s u r p r i s i n g l y l o w v a l u e was most l i k e l y , caused by t h e p r e s e n c e o f s e v e r a l o x i d e s o f t i t a n i u m i n t h e f i l m . A n n e a l i n g o v e r n i g h t i n oxygen a t 500°C i n c r e a s e d t h e i n d e x 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 h i g h e r oxygen c o n c e n t r a t i o n s . S p u t t e r i n g o n t o a h e a t e d s u b s t r a t e w o u l d l i k e l y have i n c r e a s e d 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 i n t h e e x i s t i n g s p u t t e r i n g system. Even t h e b e s t f i l m s appeared t o be 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 o f waveguides by d i f f u s i o n . C o u p l i n g t o g u i d e d modes o f a 1 ym t h i c k T i O ^ f i l m oh g l a s s was o b s e r v e d . 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 o f g u i d e d waves was s t r o n g l y a t t e n u a t e d o v e r a d i s t a n c e o f l e s s t h a n one c e n t i m e t e r . The dynamic range o f t h e human eye i s about 27 dB, so t h e f i l m p r o b a b l y had l o s s e s i n e x c e s s o f 30 dB/cm [l] . 2.7 D i f f u s e d O p t i c a l Waveguides i n LiNbO^ L i t h i u m n i o b a t e has an u n u s u a l c o m b i n a t i o n o f p h y s i c a l p r o p e r - t i e s ( A p p e n d i x I ) . I t i s b i r e f r i n g e n t ( u n i a x i a l n e g a t i v e ) and f e r r o - e l e c t r i c , w i t h a C u r i e t e m p e r a t u r e between 1100 and 1080°C, d e p e n d i n g on s t o i c h i o m e t r y . I t has r e l a t i v e l y l a r g e e l e c t r o o p t i c , a c o u s t o o p t i c and n o n - l i n e a r o p t i c a l c o e f f i c i e n t s , as w e l l as h i g h t r a n s m i s s i v i t y i n t h e v i s i b l e s p e c t r u m . These 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 - i a l f o r the f a b r i c a t i o n o f o p t i c a l w a v e g u i d e s , s i n c e t h e i r p r o p a g a t i o n c h a r a c t e r i s t i c s may be a l t e r e d by e l e c t r i c f i e l d s , a c o u s t i c s u r f a c e 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 , s w i t c h i n g , d e f l e c t i o n , and mode c o u p l i n g o f g u i d e d waves. The f i r s t d i f f u s e d waveguides i n t h i s m a t e r i a l were r e p o r t e d by Kaminow and C a r r u t h e r s i n 1973 [ 4 , 2 1 ] . L i t h i u m n i o b a t e c r y s t a l s ex- h i b i t ' v a r i a b l e s t o i c h i o m e t r v y o f t h e f.o.rm (Li-.Q,) ((>N:b„0r).1 , where .2 =a .2 .5 _ l - a 0.48 < a < .50. The degree o f n o n - s t o i c h i o m e t r y was f ound t o a f f e c t the magnitude o f the e x t r a o r d i n a r y r a y i n d e x o f r e f r a c t i o n . By h o l d i n g c r y s t a l s a t a t e m p e r a t u r e o f 1100°C and a p r e s s u r e of 6 x 10 ^ t o r r f o r t i m e s r a n g i n g between 21 and 64 h o u r s , t h e i n d e x i n a s u r f a c e l a y e r was r e d u c e d s u f f i c i e n t l y by 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- g u i d e s . D i s c o l o r a t i o n o f t h e c r y s t a l s r e s u l t i n g f r o m t h i s t r e a t m e n t was c o r r e c t e d by f u r t h e r h e a t i n g i n a i r f o r two h o u r s b e f o r e c o o l i n g . These waveguides had 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 modes t h a t e x t e n d e d as much as 100 um o r more i n t o t h e s u b s t r a t e . 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 the i n d i f f u s i o n o f t r a n s i t i o n m e t a l s a l s o p r o d u c e d e x c e l l e n t waveguides i n LiNbO^. D i f f u s - i o n t i m e s were much s h o r t e r ( a s l i t t l e as a few h o u r s ) , and g r e a t e r con- t r o l was p o s s i b l e o v e r t h e shape o f t h e d i f f u s i o n p r o f i l e , w h i c h c o u l d be v a r i e d f r o m complementary e r r o r f u n c t i o n t o G a u s s i a n , d e p e n d i n g 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 the s u b s t r a t e s u r f a c e . T h e i r r e s u l t s 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 LiNbO^ was p a r t i c u l a r l y r a p i d , so 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 be 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 have been made [5] fr o m s o l i d - s o l u t i o n L i N b O ^ - L i T a O ^ f i l m s grown on L i T a O ^ s u b s t r a t e s by l i q u i d phase e p i t a x y [ 1 2 ] , A s t e e p e r 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 o b t a i n e d . The d i f f e r e n c e between mode and s u b s t r a t e i n d i c e s was g r e a t e r t h a n w i t h d i f f u s e d waveguides (An ̂ 0 . 0 7 compared w i t h An ^ 0.01). As a r e s u l t , the i n d i v i d u a l g u i d e d modes have b e t t e r a n g u l a r s e p a r a t i o n when e x c i t e d w i t h p r i s m o r g r a t i n g c o u p l e r s . These waveguides have t h e d i s a d v a n t a g e , however, t h a t t h e L i T a O ^ must be r e p o l e d , s i n c e i t s C u r i e t e m p e r a t u r e i s o n l y 600°C. 2.8 T i / L i N b 0 3 D i f f u s i o n S p u t t e r e d T i f i l m s 500 A° t h i c k were d e p o s i t e d on 2 i n c h d i a - meter YZ LiNbOg s l i c e s o b t a i n e d from C r y s t a l T e c h n o l o g y , I n c . , o f M o u n t a i n View, C a l i f o r n i a . The c r y s t a l s had been grown f r o m a c o n g r u e n t m e l t , and 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 c o n c e n t r a t i o n b y the m a n u f a c t u r e r . Cr 2.8 ppm Fe 18 ppm N i 3.7 ppm Cu 2.6 ppm . 32. The d i f f u s i o n p r o c e s s o u t l i n e d by Kaminow and Schmidt was f o l l o w e d . The s u b s t r a t e s were c l e a n e d by a method s i m i l a r t o t h a t r e - commended by B r a n d t e t a l [24] 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 w a v e g u i d e s . 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 i n a .01% A l c o n o x s o l u - t i o n i n d e i o n i z e d (DI) w a t e r was f o l l o w e d by t h r e e DI w a t e r r i n s e s and one h o u r i n a DI cascade washer. B e f o r e d e p o s i t i o n , t h e LiNbO^ s l i c e s were b l o w n 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 gun (LiNbO^, b e i n g p y r o e l e c t r i c , tends t o d e v e l o p p o l a r i z a t i o n c h a r g e s w h i c h a t t r a c t d u s t ) . A f t e r d e p o s i t i o n , t h e Z e r o s t a t was a g a i n used b e f o r e i n s e r t i o n o f t h e s u b s t r a t e i n t o t h e q u a r t z tube o f a c o l d d i f f u s i o n f u r - nace. Argon gas (99.995%) w i t h t h e f o l l o w i n g i m p u r i t y c o n t e n t a a a 0 2 < 10 ppm H 20 <10 ppm H 2 < 2 ppm C 0 2 < .5 ppm N 2 < 23 ppm CH^ < .5 ppm was p a s s e d t h r o u g h the f u r n a c e tube a t a r a t e o f 2 5,/min. A f t e r a 6- hou 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 and 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 oxygen 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. T h i s s t e p was used by Kaminow to r e o x i d i z e t h e LiNbO^, w h i c h tends t o become b r o w n i s h from oxygen l o s s d u r i n g t h e d i f f u s i o n . On r e m o v a l from t h e f u r n a c e , the s u b s t r a t e was c o a t e d w i t h an o x i d e l a y e r ; t h i s was removed by l i g h t l y p o l i s h i n g by hand w i t h 1 um diamond p a s t e on a n a p l e s s n y l o n c l o t h . ^ 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 p r i s m were u n s u c c e s s f u l , even though Kaminow and Schmidt had o b s e r v e d 6 g u i d e d modes i n s i m i l a r l y 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 by 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 the 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 . 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 e s t i m a t e d u s i n g [22] 33. S = . 0 3 3 ( V 1 , 7 - v £ , 7 ) A / ( p Z ) + D, (2.37) where S i s t h e s o u r c e d i a m e t e r i n ym, V i s t h e a c c e l e r a t i n g p o t e n t i a l , V, = 5 kV. f o r t h e T i K l i n e and D i s t h e e l e c t r o n beam d i a m e t e r . U s i n g V = 10 kV,, p = 7.45 g/cc 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, and an average v a l u e -for " t h e - a t o m i c , number Z and a t o m i c w e i g h t A, S ^ 1.4 ym. The t a p e r s e c t i o n i n c r e a s e d t h e a p p a r e n t d i f f u s i o n d e p t h f r o m 2 t o 11.5 ym, so r e a s o n a b l e a c c u r a c y was p o s s i b l e . The p r o f i l e i s a p p r o x i - m a t e l y f i t t e d by R = e r f c ( y / 1 . 0 5 ) , (2.38) where R i s t h e T i t o s u r f a c e T i count r a t i o , and y i s inuym. The v a l u e 1.05 c l o s e l y matches Kaminow and S c h m i d t ' s v a l u e o f 1.1, so t h e d i f f u s - i o n a t t a i n e d t h e c o r r e c t d e p t h . I t was c o n c l u d e d t h a t e i t h e r t h e coup- l i n g p r i s m d i d n o t work 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 c o n s i - d e r a b l y l e s s t h a n r e q u i r e d . 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. G. M i t c h e l l o f t h e U n i v e r s i t y o f W a s h i n g t o n , E l e c t r i c a l E n g i n e e r i n g Department, was used t o v e r i f y t h a t t h e p r i s m s worked ( F i g . 2.23). C a l c u l a t i o n s showed t h a t t h e p r i s m s h o u l d have been 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 f r o m 1.9 t o 2.6, so i t was c o n c l u d e d t h a t t h e T i c o n c e n t r a t i o n i n t h e g u i d e was low. T h i s may have been caused by f o r m a t i o n o f an o x i d e , o r n i t r i d e o f 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 , w h i c h r e d u c e d t h e number o f T i atoms a v a i l a b l e . 1 The s u b s t r a t e s were n o t r e s t o r e d t o t h e i r o r i g i n a l t r a n s p a r e n c y , even a f t e r p r o l o n g e d t r e a t m e n t i n © 2 a t 700°C. However, t h i s a b s o r p t i o n was i n s u f f i c i e n t 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 i n c e sub- s t r a t e modes were c l e a r l y v i s i b l e i n the c o u p l i n g a t t e m p t s . Fig. 2.23 Coupling to a Glass Sputtered OWG. 35. The most l i k e l y cause o f d i f f i c u l t y was presumed t o be t h e O^j ®2 a n c * ^• mP u r :'- ty c o n c e n t r a t i o n i n the argon gas. From F i g . (6.13) o f [ 2 6 ] , i t can be i n f e r r e d t h a t a 500 A° l a y e r o f T i w o u l d o x i d i z e i n seconds o r l e s s a t 960° i n 0^ a t a t m o s p h e r i c p r e s s u r e . A t t h e v e r y l o w p a r t i a l p r e s s u r e o f oxygen e n c o u n t e r e d d u r i n g d i f f u s i o n , t h e r a t e o f o x i d a t i o n i s d i f f i c u l t 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 t o 960°C and c o o l e d i n argon were a l s o o b s e r v e d t o have an o x i d e l a y e r . S i n c e T i 0 2 i s a v e r y s t a b l e compound w i t h a l a r g e f r e e e nergy o f o x i d a t i o n (-162 . . 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 p l a u s i b l e . 2.9 N i / L i N b Q 3 D i f f u s i o n I n subsequent e x p e r i m e n t s , n i c k e l was used r a t h e r t h a n t i t a n i u m because o f i t s h i g h e r d i f f u s i v i t y i n LiNbO^. An e f f o r t was made t o remove 0^, ^2 H 2 ° ^ r o m t n e a r 8 o n 8 a s« P a s c a r d and F a b r e [25] have shown t h a t a T i - Z r m i x t u r e (50% a t o m i c ) i s e f f e c t i v e f o r oxygen and n i t r o g e n 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. To implement t h i s , a T i - Z r sponge m i x t u r e was i n s e r t e d i n t o a 3 cm q u a r t z tube and h e l d i n p l a c e w i t h T i s t r i p s c u t f r o m u s h e e t m e t a l . The f u r n a c e connec- t i o n s a r e shown i n F i g . 2.24. Water was 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 c a l i b r a t e d so t h a t t h i n (< 500 A°) l a y e r s o f N i c o u l d be made r e p e a t a b l y . S i x d e p o s i t i o n s o f d i f f e r e n t t h i c k n e s s e s were made on a c l e a n g l a s s s l i d e w i t h an r f f o r w a r d -2 power o f 100 w a t t s and an argon p r e s s u r e o f 1.2 x 10 t o r r . A t h i n n i c k e l l a y e r was t h e n d e p o s i t e d o v e r the e n t i r e s l i d e , i n o r d e r t o make a l l t h e s t e p s e a s i l y v i s i b l e i n a S l o a n A n g s t r o m e t e r . The r e s u l t s a r e summarized below i n T a b l e 2.1. SENSITIVE PRESSURE REDUCER FLOWMETER Fig. 2.24. Gas Flow Connections for Ni/LiNbO OWG Diffusion. Table 2.1 Ni Sputtering Calibration. t (min) d(A°) r = d/t(A°/min) 2 196 ± 100 98 50 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 3 7 . N e g l e c t i n g t h e 2 minu t e r e s u l t , t h e ave r a g e s p u t t e r i n g r a t e i s f = d/t = 59.1 ± 3.3 A°/min, where 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 f r o m P.E. = 0.67 ( r ± - r ) 2 / n . -J A v i t a l s t e p i n each 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 l a y e r by s p u t t e r i n g o n t o t h e s h u t t e r a t maximum power 400 W) f o r s e v e r a l m i n u t e s . Only 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 sponge f u r n a c e was t u r n e d on ( w i t h an a r g o n f l o w r a t e o f 2 l/mln) one h o u r b e f o r e commencement o f t h e LiNbO^ d i f f u s i o n i n o r d e r t o r e a c h t h e o p e r a t i n g t e m p e r a t u r e o f 900°C. A s e r i e s o f t e s t s were made t o d e t e r m i n e 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 the q u a n t i t y o f n i c k e l d i f f u s e d i n t o t h e s u b s t r a t e b y x - r a y f l u o r e s c e n c e . The magni- tude o f the N i K t o Nb K l i n e count r a t i o was t a k e n as an e s t i m a t e o f a a t h e n i c k e l c o n c e n t r a t i o n i n the top (> 10 ym) [22] l a y e r o f the specimen. T a b l e 2.2 summarizes t h e r e s u l t s . The Ni/Nb r a t i o s t a b u l a t e d a r e t o be 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 t o t h e a t o m i c Ni/Nb r a t i o . Some i n t e r e s t i n g c o n c l u s i o n s can be drawn f r o m T a b l e 2.2. I n each 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, most 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 i n t h e f o r m o f an o x i d e o r p o s s i b l y n i t r i d e r e s i d u e . A l s o , 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 the q u a n t i t y o f i n - d i f f u s e d n i c k e l i s seen t o be c o n s i d e r a b l e . A l l o f t h e s u b s t r a t e s were p a l e b r o w n i s h - g r e y i n c o l o r , w i t h the sample d i f f u s e d a t 950°C b e i n g t h e most d i s c o l o r e d . Measurement o f the o p t i c a l a b s o r p t i o n o f LiNbO„ s u b s t r a t e s was made i n a Carey d u a l T a b l e 2.2 X-Ray F l u o r e s c e n c e o f N l / L i N b O No. D e s c r i p t i o n o f Sample Ni/Nb R a t i o 1 L i N b 0 3 S u b s t r a t e . 0 2 500 A° N i d e p o s i t e d on L i N b O ^ 0.278 3 500 A° N i d i f f u s e d i n t o L i N b 0 3 @ 800°C f o r 3£ h r . i n t r e a t e d A r . 0.210 4 No. 3 p o l i s h e d t o remove o x i d e r e s i d u e . 0.05 5 • Same as no. 3 e x c e p t d i f f u s e d a t 950°C f o r 6 h r . 0.230 6 No. 5 p o l i s h e d . 0.036 7 Same as no. 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 beam s p e c t r o p h o t o m e t e r , and the r e s u l t s p l o t t e d i n F i g . 2.25. New sub- s t r a t e s had l i t t l e a b s o r p t i o n i n t h e v i s i b l e s p e c t r u m and appeared p e r - f e c t l y c l e a r t o t h e e y e . 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 oxygen had an a b s o r p t i o n peak around 450 nm. S u b s t r a t e s h e a t e d t o 950°C were c o n s p i c u o u s l y a b s o r b i n g . A l e t t e r f r o m C r y s t a l T e c h n o l o g y , I n c . , s u g g e s t e d b a k i n g t h e c r y s t a l s a t 1000°C i n 0^ f o r 2-3 days. T h i s was n o t f e a s i b l e , s i n c e t h e n i c k e l w o u l d have d i f f u s e d t o o d e e p l y and e l i m i n a t e d t h e waveguide. The a b s o r p t i o n may be t h e same as t h a t o b s e r v e d a t 482 nm by B a l l m a n and Gernand [ 2 7 ] , who a t t r i b u t e d the d i s - 4+ c o l o r a t i o n t o Nb i o n s formed when oxygen i s l o s t f r o m t h e c r y s t a l 2+ l a t t i c e . The a b s o r p t i o n c o u l d a l s o be due t o N i (as s u g g e s t e d by 2+ C r y s t a l T e c h n o l o g y , I n c . ) , o r Fe ( S t a e b l e r and P h i l i p s [ 3 3 ] ) . The s m a l l (y 18 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 p o s s i b i l i t y l e s s l i k e l y . (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 300 400 500 600 700 800 Fig. 2.25 Absorbance of LiNbO^ Waveguide Substrates. There is doubt also regarding the exact composition of the residue left after diffusion. Lithium atoms diffuse rapidly in LiNbO^, so they are likely to be present. Nickel oxide (NiO) is 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 is formed at 40. 500°C, b u t i s b l a c k and a c o n d u c t o r [ 3 0 ] . When l i t h i u m i s p r e s e n t , however, n i c k e l and n i t r o g e n r e a c t a t 550°C w i t h H ^ N t o form ( L i , N i ) 3 N [ 3 1 ] . U n f o r t u n a t e l y , n e i t h e r x - r a y f l u o r e s c e n c e n o r m i c r o p r o b e a n a l y - s i s 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, so a d i r e c t t e s t o f t h e r e s i d u e c o m p o s i t i o n c o u l d n o t be made by 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 ^ Waveguides 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 waveguides were made by t h e f o l l o w i n g p r o c e s s . 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 o n t o a c l e a n YZ LiNbO^ s u b s t r a t e , w h i c h was 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 f u r n a c e . A r g o n , p u r i f i e d w i t h T i - Z r sponge 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 on t o 850°C, and one h o u r l a t e r t h e t e m p e r a t u r e had s t a b i l i z e d . A f t e r a t o t a l o f 6 i h o u r s , t h e t e m p e r a t u r e was r e d u c e d t o 600°C and oxygen 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 . T h i s s t e p i n c r e a s e d t h e r e - o x i d a t i o n o f t h e LiNbO^. " A t 8 i h o u r s t o t a l e l a p s e d t i m e , the f u r n a c e was s h u t o f f and a l l o w e d t o c o o l s l o w l y t o room t e m p e r a t u r e . The s u b s t r a t e was removed f r o m t h e f u r n a c e when c o o l , and t h e r e s i d u e p o l i s h e d o f f . The r e s u l t i n g 6-mode o p t i c a l w aveguides were e x c e l l e n t , a l t h o u g h t h e s u b s t r a t e remained a p a l e b r o w n i s h c o l o r . F i g u r e 2.26 shows the d i f f u s i o n p r o f i l e o b t a i n e d by e l e c t r o n m i c r o p r o b e a n a l y s i s . 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 r e a s o n a b l y w e l l b y e r f c ( y / 1 1 . 5 ) , as 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 o x i d e o r n i t r i d e r e s i d u e , w h i c h a p p a r e n t l y a c t e d as a c o n s t a n t d i f f u s i o n s o u r c e . An a c c u r a t e q u a n t i t a t i v e m i c r o p r o b e a n a l y s i s o f N i / L i N b O ^ i s d i f f i c u l t , 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 u s e d , and oxygen c o u n t s t e n d e d t o be i n a c c u r a t e .  42. F i g u r e 2.27 shows t h e s t a g e used i n t h e o p t i c a l c o u p l i n g and a c o u s t o o p t i c e x p e r i m e n t s . F i v e degrees o f freedom o f movement were p o s s i b l e , and the s u b s t r a t e c o u l d e a s i l y be p o s i t i o n e d f o r c o u p l i n g t o g u i d e d modes. On the s t a g e were two a d j u s t a b l e arms f o r c l a m p i n g p r i s m s to t h e w a v e g u i d e . The c o u p l i n g p r o c e d u r e was as f o l l o w s . The p r i s m was p o s i t i o n e d on t h e waveguide and t h e clamp a d j u s t e d u n t i l a b r o w n i s h s p o t became v i s i b l e on t h e bottom. T h i s a r e a was t h e n c l o s e enough t o t h e OWG s u r f a c e f o r c o u p l i n g t o t a k e p l a c e . C a u t i o n was r e q u i r e d , as t h e s u b s t r a t e s were e a s i l y c r a c k e d by e x c e s s i v e p r e s s u r e . L i g h t f r o m a He-Ne S p e c t r a - P h y s i c s 155 l a s e r was f o c u s s e d on t h i s s p o t by means o f a 25 cm l e n s . The s t a g e was 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 t h e a n g l e of c r i t i c a l 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 f o u n d . C o u p l i n g was t h e n a t t a i n e d by s m a l l r o t a t i o n a l and t r a n s l a t i o n a l a d j u s t - ments u n t i l a b r i g h t s t r e a k was o b t a i n e d . A s c a l e a t t a c h e d t o t h e sub- s t r a t e t a b l e was used t o measure the c o u p l i n g a n g l e s w i t h i n 1/2 m i n u t e o f a r c . F i g u r e 2.28 shows c o u p l i n g t o t h e TE and TM modes o f a two o o mode N i / L i N b O ^ o p t i c a l waveguide. The beam s t o p s a t a s u r f a c e s c r a t c h , f r o m w h i c h i t r a d i a t e s b r i g h t l y . The p e r i o d i c v a r i a t i o n s i n s t r e a k b r i g h t n e s s a r e caused by s p a t i a l mode b e a t i n g . I n F i g . 2.29, t h e g u i d e d wave i s c o u p l e d o u t as w e l l , and t h e b r i g h t s p o t t o t h e r i g h t i s t h e T E Q g u i d e d mode. The l i n e p a s s i n g t h r o u g h t h e s p o t i s due t o s c a t t e r i n g o f l i g h t i n t o g u i d e d modes n o t c o l l i n e a r w i t h t h e i n i t i a l d i r e c t i o n of p r o p a g a t i o n . Most o f t h i s s c a t t e r i n g i s p r o b a b l y due t o f i n e s u r f a c e s c r a t c h e s 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 t h e d i f f u s i o n r e s i d u e . The d i r e c t i o n o f p o l a r i z a t i o n was t e s t e d w i t h a p i e c e o f o r d i n a r y p l a s t i c p o l a r o i d . I t s d i r e c t i o n was checked by e x a m i n i n g l i g h t r e f l e c t e d f r o m a smooth s u r f a c e ; t h e minimum i n t r a n s m i t t a n c e on r o t a t i o n o f t h e p o l a r i o d  44. corresponded to the d i r e c t i o n of transmittance of TE waves. Propagation losses were measured by placing the output coup- l i n g prism at several points along the path of wave propagation. Light output was maximized each time by adjusting the output prism clamp, and the output power, was measured with an Alphametrics dclOlO lightmeter. Use of the r e l a t i o n P „ = P , exp[-a(z2 - z l ) ] gave loss c o e f f i c i e n t s of zz z± 1 db/cm or l e s s . Figure 2.30 shows the modes coupled out of the 6-mode Ni/LiNbO^ waveguides used f o r acoustooptic experiments. The d i r e c t i o n of l i g h t propagation i n the c r y s t a l was 2'1.4° from the c r y s t a l X axis. The three spots on the r i g h t are the T E q , TE.̂  and TE 2 modes, and the weaker spots on the l e f t are the corresponding TM modes. The input l i g h t beam was unpolarized, so both TE and TM modes were excited. By c a r e f u l adjustment of the prism and angle of l i g h t incidence, i t was possible to excite j u s t one mode at a time. The prism pressure was reduced to a minimal value, and the coupling angles u m were measured for the TE modes. The prism angle was found to be 68° 12' ± 1 ' 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 on i t s e l f from both the substrate and prism face. The mode indices were calculated from the coupling angles using (2.26), with n p = 2.582. Equations (2.19) and (2.20) were solved for the turning points.of the TE modes according to the piecewise-linear WKB approximation.' The estimated value of the surface index was varied u n t i l the rms second differences (2.21) were minimized, thereby giving the smoothest p r o f i l e . The r e s u l t s are summarized i n Table (2-3). The surface index used was 2.2377, which gave an rms deviation of 6.8X10 *The e l e c t r i c OGW f i e l d component along the propagation d i r e c t i o n was ignored. A simple c a l c u l a t i o n shows that E = .024 E . z x Fig. 2.30 Modes of OWG's Used in Acoustooptic Experiments. Table 2.3 TE Modes a of Ni/LiNbO. OWG. Mode Number y m (±1')  n m (±.0004) y m (um) TE o 21° 31' n l = 2.235 = 3.23 TE^ 21° 45' n2 = 2.233 y2 = 5.78 TE 2 21° 56* n 3 = 2.2316 y 3 = 8.41 These measurements were made with an angle of light propaga- tion of 21.4° from the X axis of LiNb03 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 in the Z - 21.8° direction, as discussed in Chapter 4. The substrate index for TE polarization along this direction can be found as follows. l Fig. 2.31 Coordinate Rotation in Phase Velocity Space. 47. Consider a coordinate rotation in the X-Z plane of the crystal phase velocity space, V = V .. cosG - V . sine x xl zl V. = V n sine + V , cose z xl z l For plane wave propagation along xl, we have V = and = = 0. Then V = V . cos9 and V = V , sin6. Substitution into the phase X x l z xl velocity surface determinant (Appendix II) and multiplication by n x l = C / V x l g l v e s t 2 2 s r 2 2 2 / 2 2 o 2 • 2 Q M r, (n y - n^) [n xn z - n x l(n x cos 6 - n z sin 9)] = 0, so that either n - = n (TM) , or xl y . n = n n /(n 2 s±n2Q + n 2 cos 29) 1 / 2 (2.39) for TE waves. Using D „ = n = 2.214, n v = n = 2.294 [4] and 6 = 21.4° , 3'/ e ' x o . the substrate index of propagation for TE waves is n = 2.22 4. The waveguide index profile (Fig. 2.32) was fitted quite well by n(y) = '2.2286 + 0.0091 erfc (y/11.5), (2.40) with y in ym. The agreement between the index and diffusion profiles, and between the calculated and fitted substrate index, is reasonable. The electric field 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. , 2 *T> 37" - — 1 r- exp(-y lb ) U.41) y /rf-b-/n(y)2-n T •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 F i g . 2.32 Ni/LiNb0„ OWG I n d e x P r o f i l e . S u b s t i t u t i o n of the values An = .0091, b = 11.5 ym, X = .6328 ym, y = 0, n = 2.2286, and n =2.233 (TE n mode) gave s m l j . dK/dy | = :,014k and K/X = .23k,- so that i n e q u a l i t y (2.17) .is s a t i s f i e d . ^ In the region y < 0, the exact s o l u t i o n of the wave equation was used, and the constant A was chosen to 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 . For y > 0 and up to y , the l a s t zero of E x , (2.11) was used. The i n t e g r a l s were evaluated n u m e r i c a l l y by the trape- z o i d a l r u l e . For y > y , the usual approach i n the WKB method i s to make a l i n e a r approximation [34] of the index p r o f i l e , s e t t i n g n ( y ) ^ n + n'(y - y ) m m m where i s the index at the t u r n i n g p o i n t and n' = |dn(y)/dy| m 1 y=y m = ^ 2 - exp(-y2/ b2) (2.43) b f o r the case of an e r f c p r o f i l e . N e g l e c t i n g terms of second order i n An, K(y) becomes K 2 (y) =.. 2k 2n mn' i^(y - y m) (2.44) In the region of the t u r n i n g p o i n t , the wave equation i s then approxi- mated by d 2F H h + <y - y j g K = °- (2-45) . ,L m x dy. where'..g. = i.2k?n mn I[ l^-' 1/3 The change of v a r i a b l e z = g ( y m - y) (2.46) gives „ d E x 7 - z E = 0 (2.47) d z z 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). The problem is to match the solution of (2.47) to the WKB solutions on either side. Marcuse gives E (y) = (AB/2j)(< Tr(y -y)/3)1/2[exp(2uj/3)H^._(?) + exp(7rj/3)H® (?) ], x o m i / J l / j (2.48) 312 where c, = (2/3)1z , in the vicinity of the turning point. Use of the identity [36] Ai(-z) = (1/2) /z/3 [exp(irj /6)H^/3(C) + exp(-Trj/6)I< 2 ) / 3(?)] (2.49) gives E x(y) = (AB/g 1 / 6)v^T Ai(-z) = (AB/g1/6)i47T Ai [g 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 field at the turning point is Ex(yffi) = ABv^/.(g 1 / 6r.(2/3)3 2 / 3) (2.51) This differs from Marcuse's expression [9], which appears to under- estimate the magnitude of the electric field, cJ To calculate the electric field in the vicinity of the turning point, the Airy function was generated with the series representation [36] Ai(2) = c± f(2) + c 2 g(z), (2.52) where ir/̂ N i . 2? ,1.4 «,6 , 1.4.7 ~9 . f(z) = 1 + — + - g y z + — — z + ... ~ . 2 _4 . 2.5 ~7 . 2.5.8 AO , g(z) = z + Tj- z + -yj- z + 1 Q , z + ... with = Ai(0) and c 2 =-dAi(0)/dz = 0.25881 Figure 2.33 shows this function for -8 < z < 4. Since our interest was only in the vicinity of the final 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 is in 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 f r o m t h e c o r r e c t s o l u t i o n when y < y , so t h e WKB method (2.11) was u s e d f o r 0 < y <. y . I n F i g . 2.34, a c o m p a r i s o n i s made between t h e A i r y f u n c t i o n and WKB s o l u t i o n s f o r t h e TE. mode. F i g . 2.34 Comparison o f A i r y F u n c t i o n and WKB S o l u t i o n s f o r TE" Mode o f 6-Mode N i / L i N b 0 3 OWG. To summarize, t h e b e s t a p p r o a c h seems t o be as f o l l o w s : E x = A e x p ( - y y ) , y < 0, (2.53) • f vm E x = AB/K / k ( y ) COS(TT/4 - / K ( y ) d y ) , 0 < y < y , (2.54) y ° 53. and E = (AB/g 1 / 6 ) / T T A i [ g 1 / 3 ( y - y )], y n < y < ». (2.55) x o m o S o l u t i o n s o f t h e s e e q u a t i o n s f o r the t h r e e TE modes o f the 6-mode N i / L i N b O g waveguides s t u d i e d a r e shown i n F i g . 2T35 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 o f Y c u t LINbO^. The modes have been n o r m a l i z e d so t h a t each c a r r i e s a power o f 1 w a t t / m e t e r . These e l e c t r i c f i e l d d i s - t r i b u t i o n s a r e used l a t e r ^ - s e c t i o n * 47.6,), • t o c a l c u l a t e t h e a c o u s t o o p t i c o v e r l a p i n t e g r a l s , as d i s c u s s e d i n C h a p t e r 4. The 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 used i n (2.30) t o e s t i m a t e t h e 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 u s e d . A l l t h r e e TE modes had a maximum c o u p l i n g e f f i c i e n c y o f about 55% f o r a 1 mm c o u p l i n g l e n g t h . 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 p l o t t e d 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 TE^ mode, u s i n g t h e v a l u e s W = Jl = 1 mm, n p = 2.582, a = 68° 12', u = 22° 45' , w h i c h c o r r e s p o n d e d a p p r o x i m a t e l y t o the e x p e r i m e n t a l s i t u a t i o n . The maximum c o u p l i n g e f f i c i e n c y a c t u a l l y o b s e r v e d was 10-15%. S i n c e n t does n o t i n c l u d e the l i g h t l o s s on r e f l e c t i o n f r o m t h e p r i s m f a c e (^ 2 5 % ) , t h e agreement between t h e o r y and e x p e r i m e n t i s q u i t e r e a s o n a b l e . The p r e s s u r e r e q u i r e d t o a t t a i n 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 c r a c k e d t h e sub- s t r a t e , so f u r t h e r e x p e r i m e n t s were c o n d u c t e d 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 can be o b t a i n e d by a p p r o x i m a t i n g t h e i n d e x p r o f i l e w i t h a c o n s t a n t , so t h a t K(y) * (1/2) KQ The 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 , so we can w r i t e  / E*E dy fc 2A 2B 2 cos2 (f™ (K /2)dy - ir/4)dy -» X X o o ° = A 2 B 2 [ y + ( I / O COS(K y )] l-'m o' x crm The second term i s smaller than the f i r s t , p a r t i c u l a r l y f o r higher modes, for which i t can be neglected. S e t t i n g D = 0 i n (2.30) gives (a ) % y " 1 ( n 2 / n 2 - l ) 1 / 2 [ l + ( n 2 / n 2 - l ) 1 / 2 ( n 2 / n 2 _ l ) - 1 / 2 ] - 2 (2.56) c max 7 m p m p m o m ' This expression appears to be accurate w i t h i n about 10%. For example, f o r the case discussed e a r l i e r , (2.56) gives (a ) fc 0.88 mm ^ rather • C UlciX than the corre c t value of 0.79. Use of the approximate value gives the maximum coupling e f f i c i e n c y as 27% rather than 25.5%. F i g . 2.36 Prism Coupler E f f i c i e n c y . 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 elliptical, with components normal and parallel to the surface, and an exponential decay in amplitude away from the surface. These waves found their first application in 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 filters, and promise the reali- zation of more complex circuit functions, such as real-time convolution of two signals. In this .chapter", the-propagation and. generation char- acteristics 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 com- pared with predictions of an equivalent circuit model. 3.2 Surface Waves in Piezoelectrics Consider an infinite slab of piezoelectric material as ill 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 is 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 is defined as [38] Sk* = I <\,£ + U£,k> > where the symbols denote partial differentiation with respect to x^. In a piezoelectric material the relation between stress and strain involves the piezoelectric constitutive relations, which can be written as [39J ' T. . = cJ . S, „ - e, . . E. I J i j k J l kJl k i j k (3.2) D. .= e„ „ S. „ + £., E. i ik£ k£ lk k (3.3) E s where T is the stress tensor, c. ., „, e., . and e., are the elastic i j ijk£ ik£ lk stiffness tensor (at constant electric field), 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 , c o n - s i d e r a t i o n 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 e q u a t i o n o f m o t i o n , pu . = T l j f l , , (3.4) where p i s the d e n s i t y and "'•" denotes 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 t i m e . Use o f t h e r e l a t i o n s E k = - , (3.5) T>u±= 0 , (3.6) and s u b s t i t u t i o n o f (3.2) i n t o (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 , p U j = h j k £ + \ i j * , k l ( 3 * 7 ) ^ ° = e i U \ , U - 4 k * , k i • <3-8> S t r i c t l y s p e a k i n g , (3.5) ."is a p p r o x i m a t e . However^ s i n c e t h 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 i s some f i v e o r d e r s o f magnitude l e s s t h a n the v e l o c i t y o f l i g h t , t h e e l e c t r o s t a t i c f o r m o f M a x w e l l ' s e q u a t i o n s can be u s e d w i t h v e r y l i t t l e l o s s i n a c c u r a c y . S o l u t i o n s t o (3.7) and (3.8) have t h e f o r m [40] ^ = 3 k e x p ( - a K x 2 ) e x p j ( f i t - K x 3 ) , k = 1, 2, 3 (3.9) and <j> = 3^ e x p ( - a K x 2 ) e x p j ( f i t - K x 3 ) (3.10) f o r wave p r o p a g a t i o n i n t h e x 3 d i r e c t i o n . S u b s t i t u t i o n i n t o (3.7) and (3.8) g i v e s a l i n e a r homogeneous s y s t e m i n the unknowns 3̂., k = 1, 4. F o r a n o n - t r i v i a l s o l u t i o n , t h e d e t e r m i n a n t o f c o e f f i c i e n t s must be z e r o , g i v i n g an e i g h t h - d e g r e e p o l y n o m i a l e q u a t i o n i n a. S i n c e the f i e l d s 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 r e a l p a r t s can be u s e d . I n g e n e r a l , f o u r complex r o o t s can be fo u n d w h i c h a r e s a t i s - f a c t o r y . Each v a l u e o f a can t h e n be s u b s t i t u t e d back i n t o t h e homogene- ous e q u a t i o n s 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 o f 3̂ . U s i n g t he o t h e r boundary c o n d i t i o n s t h e p a r t i a l f i e l d a m p l i t u d e s ( t h o s e c o r r e s p o n d i n g t o each v a l u e o f a) can be found by n u m e r i c a l methods t o complete t h e s o l u t i o n . The p r o b l e m can o n l y be s o l v e d by i t e r a t i v e computer t e c h n i q u e s The f i r s t s u c h s o l u t i o n s f o r SAW's on LiNbO^ were o b t a i n e d b y Ca m p b e l l and Jones [41] i n 1968. They c a l c u l a t e d SAW v e l o c i t i e s i n d i f f e r e n t p r o p a g a t i o n d i r e c t i o n s f o r b o t h f r e e and m e t a l l i z e d s u r f a c e s . The d i f - f e r e n c e between 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 p a r a m e t e r i n d i c a t i n g t h e s t r e n g t h o f i n t e r a c t i o n between SAW's and e l e c t r i c f i e l d s g e n e r a t e d by-means o f s u r f a c e e l e c t r o d e s . T h i s w i l l be d i s c u s s e d f u r t h e r i n the s e c t i o n on i n t e r d i g i t a l t r a n s d u c e r s . The complete d e s c r i p t i o n o f YZ (Y c u t , Z p r o p a g a t i n g ) SAW's on LiNbO^ i s g i v e n by S p a i g h t and K o e r b e r [42]. F o u r t e e n complex c o e f f i c - i e n t s a r e r e q u i r e d t o d e s c r i b e t he e s s e n t i a l l y e x p o n e n t i a l SAW decay w i t h d e p t h . S o l u t i o n s f o r t h i s and o t h e r major c r y s t a l c u t s a r e g i v e n i n t he Microwave A c o u s t i c s Handbook [ 4 0 ] . To summarize, the 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 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. I n g e n e r a l , t h e waves have b o t h d i s p l a c e m e n t and e l e c t r i c f i e l d components n o r m a l and p e r p e n - d i c u l a r t o t h e s u r f a c e . O u t s i d e t h e m a t e r i a l , t h e e l e c t r i c f i e l d s decay e x p o n e n t i a l l y ; i n s i d e , a l l f i e l d s e x h i b i t a s l o w e r o s c i l l a t o r y decay. The s o l u t i o n s may be w r i t t e n i n t h e fo r m u ^ x . t ) = 3™ exp j ( f i t - K x 3 ) (3.11) where = I \ q e X P ( a q K X 2 ) q=l n ^ and = - | | - = e£ exp j (fit - Kx3> (3.12) K where 3k = J x \ q e X p ( a q K X 2 } ' The time average power flow i n the SAW i s the sum of two parts [75], one due to the mechanical displacement f i e l d , 00 . . Pma= I Pe k J f l T i j U j d x 2 ( 3 ' 1 3 ) and the other due to the SAW e l e c t r i c f i e l d , 1 P = ea 2 ZlRe J 0 jfi § D* dx 2 , (3.14) where D. i s the complex conjugate of the i t h component of the e l e c t r i c displacement vector. The and c o e f f i c i e n t s are usually norma- l i z e d so that the t o t a l power i s one watt/m. 3.3 D i f f r a c t i o n of Surface Waves The d i f f r a c t i o n of e l a s t i c surface waves may be treated by methods s i m i l a r to those used f o r electromagnetic waves. In aniso- t r o p i c c r y s t a l s , the s i t u a t i o n i s complicated by the fac t that the phase and group v e l o c i t i e s of propagation are non-collinear with the exception of a few symmetry (pure-mode) axes. Cohen [A3] studied the d i f f r a c t i o n of bulk u l t r a s o n i c waves i n a number of anisotropic materials. He used a parabolic f i t to the c r y s t a l phase v e l o c i t y surface, v(6) = V q ( 1 - b e 2 ) (3.15) in the vicinity of pure-mode axes. He found excellent agreement between calculations based on the above approximation and experiments in a num- ber of materials. Weglein et al [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 Exact agreement between theory and experiment was not obtained. Analytic expressions for the far-field beam divergence half- angle ty]_j2 a n c* t n e Fresnel distance r^ have been obtained by Crabb et al [45], 1/2 W seed) w2 seed) r f = 4A(1 + dd>/de) ( 3 * 1 7 ) where W is the source width and tand> = ̂  — . Use of the parabolic approximation gives yl/2 " W * 1 / 9 - ̂  (1 " 2b) (3.18) w2 a n d r f = 4A(1 - 2b) ' ( 3 * 1 9 ) for small angles 6 about the pure mode axis. Except for the factor (1 - 2b), these equations are identical with those applicable to iso- tropic diffraction. The walk-off angle $ between the phase and group velocities is 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 obser- vation (the exact calculations were done by Khar us i and Farnell [47]). Fig. 3.2 SAW Propagation in Anisotropic Materials [40] F o r p r o p a g a t i o n i n t h e Z d i r e c t i o n on YZ LiNbO^, t h e p a r a b o l i c t h e o r y does n o t g i v e a c c u r a t e r e s u l t s . The r e a s o n f o r t h i s i s t h a t 1 + d<j>/d6 ̂ i n t h i s c a s e , so t h a t t h e v a l u e o f b must be v e r y a c c u r a t e l y known. T h i s i s n o t p o s s i b l e a t the p r e s e n t t i m e because t h e m a t e r i a l c o n s t a n t s have n o t been d e t e r m i n e d w i t h s u f f i c i e n t a c c u r a c y . However, 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 (where 1 + dcf)/d9 ^ 1.37) good a c c u r a c y 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 by W i l k i n s o n e t a l [ 4 8 ] , I n T a b l e 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 p r o p a g a t i o n d i r e c t i o n s . F o r YZ waves, beam s p r e a d i n g i s much l e s s t h a n 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 c u t i s u s e f u l f o r SAW d e l a y l i n e s . 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 t h a n i n t h e i s o t r o p i c c a s e . 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- s t e e r e d l i g h t d e f l e c t o r s . T a b l e 3.1 A n i s o t r o p y P a r a m e t e r b. Cut P r o p a g a t i o n D i r e c t i o n b [44] Y Z 0.54 Y Z ± 21.8° -0.187 3.4 SAW G e n e r a t i o n ; t h e I n t e r d i g i t a l T r a n s d u c e r C o n s i d e r the i d e a l i z e d i n t e r d i g i t a l t r a n s d u c e r (IDT) i n F i g . 3.3. A g r i d o f i n f i n i t e l y l o n g e l e c t r o d e s on an i n f i n i t e p i e z o - e l e c t r i c s l a b a r e a l t e r n a t e l y c o n n e c t e d t o an r . f . g e n e r a t o r . The m a t e r i a l t h i c k n e s s i s assumed t o be much g r e a t e r t h a n the e l e c t r o d e s p a c i n g . 64. Fig. 3.3 Section of an Idealized IDT. The problem is to develop an equivalent circuit model. To make the calculation at a l l reasonable, i t is necessary to make a number of assumptions: (1) the electrodes are massless, perfectly conducting and infinitely long (so that the problem becomes two-dimensional), (2) the quasi-static approximation is assumed, i.e., E = -V§» (3) the piezoelectric is assumed to have no non-linearity, and (4) the driving voltage and the SAW's have the time dependence jfit e • 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: where Using Gauss' Law, p f i 2 u 1 + T l j l + T 5 > 3 = 0 (3.21) pfi 2u 3 + T 5 > 1 + T 3 } 3 « 0 T l = C l l S l + C13 S3 " 631 E3 > T3 " °13 S l + 4> S3 " e33 E3 » T5 C44 S5 e15 E l > S l • U l , l S3 " U3,3 S5 = U l , 3 + U 3 , l E l fc*fl and E 3 =-$>3 D l , l + D 3 , 3 = ° > and making the indicated substitutions gives three partial differential equations in terms of u^, u 3 and §: p Q \ + 4l u l , l l + C44 Ul,33 + (C13 + C44)U3,13 + (e15 + e31)$,13 = ° (3.23) ( C13 + c44 ) ul,13 + pnS + c44 u 3 , l l + c33 u3,33 + e15 5,11 + e33 *,33 = (3.24) 66. <e15 + e31 ) ul,13 + e 1 5 u 3 , l l + e33 U3,33 " e l l ?,11 " e33 *,33 = 0 (3.25) The m e c h a n i c a l boundary c o n d i t i o n s a r e T 3 = 0 a t x 3 = 0 T 5 = 0 a t - x = 0 . (3.26) Most f r e q u e n t l y , t he 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 v n L-A i i L+A I = J a t x 3 = 0 , — < | x x I < — (3.27) D 3 = 0 a t x 3 = 0 , 0 < \x±\ < ^ The 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 i g n o r e s the e l e c t r i c f i e l d o u t s i d e the c r y s t a l where t h e s u r f a c e i s n o t m e t a l - l i z e d . T h i s i s q u i t e a c c u r a t e i n h i g h p e r m i t t i v i t y m a t e r i a l s , s u c h as L i N b C y E q u a t i o n s (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 pro b l e m , w h i c h i s t o f i n d t he c o u p l e d e l e c t r i c a l and m e c h a n i c a l f i e l d s when an a l t e r n a t i n g v o l t a g e i s a p p l i e d . A v a r i e t y o f s o l u t i o n s have been a t t e m p t e d . 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 . U s u a l l y , t h e weak c o u p l i n g a p p r o x i m a t i o n ( 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 terms e k i j E k '"1 t^ i e c o n s t : ' : t u t : ' - v e r e l a t i o n s ) i s made. T h i s i s e q u i v a l e n t t o i g n o r i n g t h e c o u p l i n g between e l e c t r i c a l and m e c h a n i c a l SAW f i e l d s . I n t he l i t e r a t u r e , t he p r o b l e m has been t r e a t e d as e i t h e r e n t i r e l y m e c h a n i c a l [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 ( t h r o u g h 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 i n d e p e n d e n t o f t h e a c o u s t i c s t r a i n f i e l d s [51,52,53]. As p o i n t e d o u t by M i l s o m and Redwood [49], n e i t h e r 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 is not convenient to apply in design pro- blems, since i t does not lead to a simple equivalent circuit. Campbell and Jones suggested that Av, the change in SAW velo- city 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 sur- face 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 al [54]. A number of authors [39,51,53,55,56] have shown that the phy- sical 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: 3350 i i i i i i 1 1 — r Y -CUT PLATE H 1 1 1 h H 1 h 3.6 x I02|- 2.8 xlO 1.2 x I02 -2 0.4 x 10 I I I.. I I I 1 1 1— 18 54 90 126 162 PROPAGATION DIRECTION.fi (DEGREES) Fig. 3.4 SAW Velocity and Coupling Constant for Y-Cut LiNbC»3 [40]. the shear and compressional wave components. For YZ LiNbO^, Smith et al calculated that the shear component is one order of magnitude greater, so that a shunt model is more appropriate. However, i f certain condi- tions 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 = wcs = WN(eo + ep)K(q)/K'(q) (3.28) where W is the transducer width, N is the number of finger pairs, e is the relevant dielectric constant, P q = cos[— (1 - a)], where a = A/L, and K and K' are complementary elliptic integrals of the first 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) GP) Fig. 3.5 IDT Shunt Model Equivalent Circuit. xs The radiation conductance at the center frequency 9. • = •';, o L G = n c_ - NTT A o T o K(q) K'(q) where A = Av/v is the change in wave velocity when the surface is metallized. When the frequency deviation is small (y 20%), (3.29) G (ft) = G sine x a o (3.30) and B (fl) = G Q( s i n< 2 x> ~ 2 x) , a 2x2 (3.31) where x = NTrAfi/ft = NirAf/f and sine x = sinx/x. o o It is usually more convenient to use the series model, which has the following circuit. CT FL02) Fig. 3.6 Series Equivalent Circuit. Here, the radiation resistance at resonance is R = NIT 2 A O fiDCT K(q) IC (q) (3.32) For the two models to be equivalent at resonance, we require that R +-J^r-% 1 o j n o c T Go + j n o c T This implies that R % 5 0 < + a24 o T From (3.28) and (3.29), with a = 0.5 so K = K* =. 1.854, G2 ° =8.24 N2A2 . 2 2 (Smith et al's equivalent expression is 5.1 N A ). Thus, when 8.24 N 2 A 2 « 1, (3.33) G R n * - r 7 » ( 3' 3 4 ) and t h e two models a r e e q u i v a l e n t . 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 the Z ± 21.8° d i r e c t i o n , 8.24 N A = .024 « 1, so t h e e q u i v a l e n c e i s v a l i d . 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 o°<N , b u t R q i s 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 t h e n g i v e n b y R (fi) = R s i n c 2 x (3.35) a ' o X (fi) = R ( S I N < 2 X > - 2 X ) . ( 3 > 3 6 ) 3 ° 2 x Z U s u a l l y , the t r a n s d u c e r b a n d w i d t h i s t a k e n t o be 2/N, h a l f t h e r e l a t i v e f r e q u e n c y d i f f e r e n c e between the z e r o s o f t h e c e n t r a l 2 1 r a d i a t i o n l o b e . More u s e f u l i s t h e -3 dB b a n d w i d t h , where s i n e x = - j » o r x = 1.392. 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 by Afi Af _ 2.78 a, -9 o o I n o r d e r 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 the c i r c u i t o f F i g . 3. The i n d u c t a n c e i s s e l e c t e d so t h a t i t i s r e s o n a n t w i t h a t t h e c e n t e r f r e q u e n c y ( s i n c e x ( f i Q ) = 0) , t h a t i s , L = -Tr- (3.38) 1 o T I t i s p o s s i b l e t o make a number o f e x t e n s i o n s t o t h i s c i r c u i t model t o i n c r e a s e i t s a c c u r a c y and f r e q u e n c y r a n g e . Emtage [56] has shown t h a t an i n t e r d i g i t a l t r a n s d u c e r p r o d u ces s u r f a c e waves a t the f r e q u e n c i e s f = (2n + l)f , n = 1, 2 ... . (3.39) n o 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 P i /9frw-n [ c o s ( T r a ) ] G(n ) - n CJTT/A ' K v } (3.40) P P K(q) K'(q) where P is a Legendre polynomial. This expression shows reasonable agreement with experimental results by Weglein [59] and Marshall [60] for YZ LiNbO^ Bulk acoustic wave generation is 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 2f Q, which was observed by Daniel and'Emtage [62]. In 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 is V-3TP s^e W ( 3 ' 4 1 ) where p g is the electrode sheet resistivity, W is the electrode ?wid'th_.inj,mete-rs, and „ sinh(Wa )/Wa - sin(Wa.)/Wa. 1 =7-{ 1 + Zf m T—TT\ (3.42) e 4 L cosh(arW) - cos^W) J ' Lakin inconveniently gives many of his variables in units of wavelengths or ohm-wavelengths (for example, his radiation resistance 73. i s g i v e n i n ft*A). I n terms o f MKS u n i t s , the phase f a c t o r a f o r LiNbO^ may be a p p r o x i m a t e d by 8 A p s C s f 2 1/2 a = a r + j a . = ( ) X / Z ( 1 + j ) (3.43) F o r example, c o n s i d e r an aluminum t r a n s d u c e r o f t h i c k n e s s t = 0.1 ym e x c i t i n g s u r f a c e waves i n t h e Z ± 21.8° d i r e c t i o n on Y c u t _Q L i N b O y Assuming th e b u l k r e s i s t i v i t y o f aluminum, = 2.83 x 10 ft-m, and u s i n g N = 3, A = 0.018, f = 200 MHz, W = 1 cm, v = 3427 m/sec, a = 0.5 and e = 56 e , we o b t a i n f r o m (3.28) and (3.43) Wa = Wa, = 0.269. p o r i Then (3.42) g i v e s n = 1.0 t o f i v e f i g u r e a c c u r a c y , so we may use 8 c s R e = 3 T P - W • T h i s e x p r e s s i o n o n l y a p p l i e s t o a m e t a l l i z a t i o n f a c t o r o f a = 0.5 and one f i n g e r p a i r . The e f f e c t o f N f i n g e r p a i r s w i l l be t o c o n n e c t N r e s i s t o r s R g i n p a r a l l e l , so f o r N f i n g e r p a i r s , 8 p W •D = a • e 3 N A \ I f the m e t a l l i z a t i o n f a c t o r i s i n c r e a s e d t o some v a l u e g r e a t e r t h a n 0.5, we e x p e c t the r e s i s t a n c e t o d e c r e a s e by t h e f a c t o r a/.5 = 2a. U s i n g th e d e f i n i t i o n o f s h e e t r e s i s t i v i t y p g = p / t , t h e e x p r e s s i o n f o r f i n g e r r e s i s t a n c e then becomes R = 4 P W ,- ( 3 4 4 ) e 3NatA * The d i e l e c t r i c p e r m i t t i v i t y w h i c h a p p e ars i n t h e e q u a t i o n f o r s t a t i c c a p a c i t a n c e , (3.28) i s g i v e n by A u l d and K i n o [53] t o be = (e e - e 2 ) 1 / 2 (3.45) p yy zz yz v ' f o r Z p r o p a g a t i o n i n the Y p l a n e of LiNbO.. They s t a t e t h a t e v a r i e s 5 P o n l y s l i g h t l y f o r the major c r y s t a l c u t s , and t h a t t h e s t r e s s - f r e e 74. T permittivity E is the most appropriate. Switching to tensor notation, with (1,2,3) = (x,y,z), Warner et al [64] give T T T a, yy xx 11 o 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, e. . = a., a,, E . „ i j i k j£ k£ (3.46) Thus, Fig. 3.7. Permittivity Transformation. ' = 2 + 2 '33 a31 £11 a33 E33 = z±1 sin221.8° + e 3 3 cos221.8c = 37.45 E . o Hence, e' = (e i n e' )"^ 2 = 56.1 e p 11 o In Table 3.2 below, the properties of LiNbO^ required for IDT calculations are summarized for several crystal cuts. Table 3.2 Constants for LiNb03 [40] Cut Propagation Direction V(m/s) A = Av/v 00 E P X Z 3483.092 0.02598 50.2 e o Y Z 3487.762 0.02409 50.2 E o Y Z ± 21.8° 3427.641 0.01727 56.1 e o The complete circuit model, with matching series inductor, is O (J&OiLr R. Fig. 3.8. Series Circuit Model. To summarize, the equations for the circuit elements are as follows: CT = W N ( E o + ep) K(q)/K'(q) R = 4 p W e 3NatA 76. R = N T r 2 A o QQ CT K(q) K'(q) 2 R = R s i n e x a o = R ( s i n ( 2 x ) - 2 x } ( 3 > 4 7 ) 3 2 x 2 x = 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 acous.tiacPAwjeC launched:cin - c the .forward, . - d i r e c t i o n i s P = V 2/2R , (3.48) a a a * where V = V R / Izl, and the f a c t o r o f 2 a r i s e s f r o m t h e b i d i r e c t i o n a l a o a 1 r a d i a t i o n c h a r a c t e r i s t i c o f t h e IDT. The e f f i c i e n c y w i t h w h i c h 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 f o r w a r d 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 d i s s i p a t e d when eo f=f , i s ° P t P-'- R (R + R ) -a — a- - ^ a o e eo V /(R + R ) 2 Z o v o e' so t h a t the power i n s e r t i o n l o s s i s R.I..L. = -10 l o g . ( R (R,+ R5)/2|z| 2 ) (3.49) v_3. O £ where N 2 = (R a(a) + R e) 2 + ( R L s = - ^ + x a ( f i ) ) 2 . A t r e s o n a n c e , w i t h Rg = 0, we see t h a t the minimum i n s e r t i o n l o s s i s 3 dB. The Q o f t h e e l e c t r i c a l e q u i v a l e n t c i r c u i t i s Q = i/n C(R + R ) . e o o e (3.50) From e q u a t i o n ( 3 . 3 7 ) , the a c o u s t i c Q i s 'Xi NT: >\J N (3.51) 2.78 ^ .9 I n o r d e r t o a c h i e v e the g r e a t e s t d e v i c e b a n d w i d t h , i t i s n e c e s s a r y t h a t o t h e r w i s e , t h e b a n d w i d t h w i l l be l i m i t e d by Q e < I t i s p o s s i b l e t o a c h i e v e g r e a t e r b a n d w i d t h a t the expense of i n s e r t i o n l o s s i f Q g > Q & and L i s detuned, so t h a t the a c o u s t i c and e l e c t r i c a l r e s o n a n t f r e -s q u e n c i e s d i f f e r . 3.5 E x p e r i m e n t a l Work I n o r d e r t o check t h e v a l i d i t y o f t h e c i r c u i t models and g a i n some e x p e r i e n c e i n w o r k i n g w i t h s u r f a c e waves, an 85 MHz SAW d e l a y l i n e was made on an XZ LiNbO^ s u b s t r a t e . A r t w o r k f o r a 1 0 - f i n g e r p a i r t r a n s d u c e r was c u t on r u b y l i t h and s e n t t o Shaw P h o t o g r a m m e t r i c s i n Ottawa, Ont., f o r p h o t o g r a p h i c r e d u c t i o n by 100 X. The r e s u l t i n g p h o t o - r mask had f i n g e r s 2.05 mm w i d e and a p e r i o d i c i t y o f 41 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 the s p a c e s between them, p r e s u m a b l y b e c a u s e o f a l o s s i n r e s o l u t i o n i n the p h o t o g r a p h i c p r o c e s s . I n o r d e r t o o b t a i n s a t i s f a c t o r y t r a n s d u c e r s , i t was n e c e s s a r y t o reduc e t h e f i n g e r w i d t h by means o f o v e r e x p o s u r e o f the p h o t o r e s i s t . u s i n g Gaf PR-102 p o s i t i v e p h o t o r e s i s t and an a l k a l i n e f e r r i c y a n i d e e t c h a n t . (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 a r e g i v e n i n C h a p t e r 5.) E l e c t r i c a l c o n n e c t i o n was made w i t h f i n e g o l d w i r e and Q a i % ' (3.52) T r a n s d u c e r s were made 2 cm a p a r t o f 0.25 ym t h i c k aluminum 78. silver paint. Tests on an aluminum film revealed that the silver 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 electrical tape. Fig. 3.9 shows the conductance and susceptance of a typical transducer. The first maximum corresponds to SAW generation at the 50 100 150 200 250 f ( M H z ) Fig. 3.9 Transducer Conductance and Susceptance. f u n d a m e n t a l r e s o n a n t f r e q u e n c y f = 8 4 . 5 MHz. A c c o r d i n g t o R e i l l y e t a l [ 6 5 ] , t h e s e c o n d r e s o n a n c e a t about 1 6 5 MHz i s due t o t h e g e n e r a t i o n o f b u l k s h e a r waves. The shunt c i r c u i t model was used i n c a l c u l a t i o n s , b ecause S m i t h 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 . The s i m p l e e q u i v a l e n c e ( 3 . 3 4 ) between s h u n t and s e r i e s models does n o t a p p l y t o t h i s t r a n s d u c e r , s i n c e 8 . 2 4 N 2 A 2 * .5 , so t h a t ( 3 . 3 3 ) i s n o t s a t i s f i e d . F i g u r e 3 . 1 0 shows the t r a n s d u c e r a d m i t - t a n c e i n t h e v i c i n i t y o f f Q . The c i r c l e d p o i n t s a r e e x p e r i m e n t a l and the c u r v e s 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 ( 3 . 2 8 ) - ( 3 . 3 1 ) and 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 LiNbO^ f r o m T a b l e 3 . 2 . M i c r o s c o p i c e x a m i n a t i o n o f t h e IDT r e v e a l e d t h a t t h e m e t a l l i z a t i o n f a c t o r a was 0 . 6 . The e x p e r i m e n t a l v a l u e s = 1 0 . 9 p f and G Q = 4 . 2 0 m i l l i m h o s compare 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 4 . 1 7 3 . By comparing the magnitude o f R q and R^ i n t h e s e r i e s model, i t can be shown t h a t 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 i n t h i s t r a n s d u c e r ( t h e s e r i e s model, a l t h o u g h i n a c c u r a t e , i s adequate f o r a rough c o m p a r i s o n ) . 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 cm a p a r t was c o n n e c t e d as a d e l a y l i n e . B o t h I D T 1 s were c o n n e c t e d w i t h 3 . 3 uH s l u g - t u n e d i n d u c t o r s i n p a r a l l e l . These c a n c e l l e d t h e c a p a c i t i v e t r a n s d u c e r s u s c e p t a n c e a t the r e s o n a n t f r e q u e n c y f Q , t h e r e b y r e d u c i n g t h e i n s e r t i o n l o s s f o r SAW g e n e r a t i o n . A grounded aluminum s h i e l d was c o n n e c t e d a c r o s s t h e c e n t e r o f the d e l a y l i n e t o m i n i m i z e s t r a y r f c o u p l i n g between i n p u t and o u t p u t . The i n p u t t r a n s d u c e r was 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 the o u t p u t was d e t e c t e d w i t h a low c a p a c i t a n c e o s c i l l o s c o p e p r o b e . Use o f 80. f ( M H z ) Fig. 3.10 Transducer Admittance near Resonance. .81. pulse excitation and a storage oscilloscope gave a propagation delay of 5 ys, in close agreement with the predicted value of 5.17 ys. With sine-wave excitation at the center frequency, the measured power in- sertion loss at the center frequency was 26 dB. Both transducers were matched with parallel inductors. The equivalent drive and IDT circuit is shown in fig 3.11. The electrical power into the device is given by 2 P = P ( 1 - p ), where P is the forward power on a matched traris-e o r o mission line, and the reflection coefficient is given by p =(Y -Y)/(Y +Y) r c c Y c is the characteristic line impedance, and Y is the IDT equivalent . circuit admittance. 50 OHM COAX IAL CABLE L P<3 C. T G. Fig. 3.11 Radiating IDT Equivalent Circuit The acoustic power in the forward direction is half of P . The ratio e of acoustic forward power to matched electrical power is thus given by 2Y G c a (Y + G ) 2 + (B + QC + i/(£2L ) ) 2 c a a T p (3.53) The power insertion loss is then given by I.L. = -10 log(P /P ) a o (3.54) At the center frequency , with Y = G , the minimum insertion C ci 82. 0 I I 1— 1 1 1 60 8 0 100 f ( M H z ) F i g . 3.12 Power I n s e r t i o n L o s s . l o s s w o u l d be 3 dB. The c a l c u l a t e d i n s e r t i o n l o s s o f t h e i n p u t t r a n s - 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 . 3.12 as a f u n c t i o n o f f r e q u e n c y . A t t h e c e n t e r f r e q u e n c y , t h e i n s e r t i o n l o s s i s 5,.4 dB. The l a r g e o v e r a l l l o s s o b s e r v e d e x p e r i m e n t a l l y was due t o impedance mismatch o f t h e o u t p u t t r a n s d u c e r and p r o b a b l y a l s o due t o a l a c k o f a c c u r a t e a l i g n m e n t between t h e i n p u t and o u t p u t t r a n s d u c e r s . SAW wavelength and amplitude measurements were made by diffrac- ting light from propagating waves. In the limit of small deflection angles, the Raman-Nath theory [66] of light diffraction gives [67] I d - I0J1(2AK) °- . 2 * I (AK) o (3.55) for the deflected light intensity. The approximation is 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] for small angles. The e = — = < L A cosiji x experimental values d = 3.3 ± .1 cm, (3.56) x - 154.8 ± .5 cm, d, = 45° and X = .6328 ym, give A = 42 ± 1.5 ym . for the sound wavelength. This agrees well with spacing of 20.5 ym. 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. Figure 3.14 shows the I/I 0 ratio plotted against the square of the driving voltage. As expected from (3.55), a straight line is obtained. 30 - X l O " 20 10 100 V0Z (volts*) 200 Fig. 3.14 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 is better than the 10% claimed by Smith et al, possibly because Auld and Kino's expressions for G and R , and Engan's C take into account variation of the trans- 3. cl X ducer metallization factor. In the next chapter, the series equivalent circuit modelddeve- loped in section 3.4 will be used for- the analysis-of-interdigi- tal arrays in acoustooptic light deflectors. V 86.. 4. BRAGG BEAM-STEERED SURFACE WAVE ACOUSTO-OPTIC LIGHT DEFLECTORS 4.1 Introduction The phenomenon of light diffraction by ultrasonic bulk waves was first 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 difficult 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 acousto- optic 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 is 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 in recent years [72-76]. A treatment applicable to non-uniform waveguides in anisotropic piezoelectrics will be given here which combines features of the above references. Consider a light wave propagating in the mth guided mode inci- dent on an acoustic surface wave of width L and wavelength A. Let 8 6 mo be the angle of incidence between the light wavevector k and the mo planes of constant phase of the sound wave, which produces a phase dif- fraction grating in the solid by means of a periodic perturbation of the refractive index. When a suitable phase matching condition is met, the light'swill in general be deflected into a diffracted beam of order £ propagating in the nth guided mode at an angle ^n^- The interaction may be regarded as a collision where conservation of energy and momentum obey the relations t o . = <o ± &fi i I o and " k = £ ± UK . (4.1) nil mo -> -> Here to and k refer to the incident light wave, and fi and 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) will be used; however, when tensor properties of the acousto-optic medium are needed, this is to be considered equivalent to (x^jX^,^ -). 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 prob- lem, so this approach will be taken here. In a non-magnetic, non-conducting medium with dielectric permittivity tensor e(x, t) ,e,0 , Maxwell-'s- Equations^are \; V X 8 =,|| " • (4.2) 89. VH = 0 and V-D = 0 -> * ->• -> Use of the relation D = e0e(x,t).E and elimination of H from the first two 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) - V2E and V.D = 0 = -> -> e Ve.E + e eV.E gives o o - ( i Ve-E) - V2E = - ^ p - (e E) (4.4) It can be shown [78] that the first term on the left is 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 first term in (4.4) may be ignored. The wave equation is then V2E = - f ^ r (e(x,t).E) (4.5) In the interaction region, the permittivity is e = e + Ae (x,t) , where £ s is the unperturbed value. In an anisotropic, piezoelectric solid, the SAW consists of a mechanical strain wave with up to six com- ponents and an associated electric field 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 field (linear electro- optic 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 9 0 . terms f o r LiNbO^, so i t w i l l be n e g l e c t e d h e r e , so t h a t e = e + A£& + A g P (4.6) s 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 e f f e c t s , r e s p e c t i v e l y . The change i n p e r m i t t i v i t y may be e v a l u a t e d by use o f t h e i n v e r s e d i e l e c t r i c p e r m i t t i v i t y t e n s o r B, w h i c h i s d e f i n e d by B- e = 1 T a k i n g 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 = - e AB e I n s u b s c r i p t n o t a t i o n , t h i s r e l a t i o n i s A e i * " " £ i j A B j k ek£ • ( 4 ' 7 ) C o n s i d e r a t i o n o f e q u a t i o n s ( 3 . 1 ) , (3.9) and (3.10) e n a b l e s us t o w r i t e t h e r e a l p a r t o f the e l a s t i c s t r a i n and e l e c t r i c SAW f i e l d s i n the f o r m ^ ( x , t ) = S ^ ( y ) cos ( f i t - Kz) (4.8) and ET ( x , t ) = E* (y) c o s ( f i t - Kz) , (4.9) W H 6 R E E ^ y ) , R e { E . ( y ) } The change i n the 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 f i e l d i s [75,79] where P.,., 0 a r e the p h o t o e l a s t i c c o n s t a n t s a t c o n s t a n t E . S i m i l a r l y , t h e change 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 91. where r^^yi a r e t n e l i n e a r e l e c t r o o p t i c c o e f f i c i e n t s a t c o n s t a n t s t r a i n . The change i n p e r m i t t i v i t y 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 - Kz) (4.10) i n ' i n where Se. = e. . (p 0Sr. + r . l n E^) e. (4.11) i n i j rjk-J,m £m -jk£ r k n S i n c e the 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 t h a n t h e o p t i c a l f r e - quency co, t h e t i m e der£vat*iv.ea_of -Ag(x,-t)^ma35ebe..neglect'ed, and t h e wave e q u a t i o n becomes C d t C dt F o r t h e p e r m i t t i v i t y f u n c t i o n g i v e n i n ( 4 . 1 0 ) , t h i s i s a form o f M a t h i e u ' s e q u a t i o n . The r i g h t - h a n d t e r m may be r e g a r d e d as a s o u r c e o f t h e g u i d e d modes d e s c r i b e d by the e x p r e s s i o n on t h e l e f t . S o l u t i o n s a r e g i v e n by Chu and Tamir [80] f o r t h e s i m i l a r p r o b l e m o f d i f f r a c t i o n o f a p l a n e e l e c t r o m a g n e t i c wave by a c o u s t i c microwaves. The TE mode m o f a g r a d e d - i n d e x OWG p r o p a g a t i n g i n t h e xz p l a n e may be w r i t t e n as E ( x , t ) = U (y) exp j (tot - k x = k z) (4.13) n r m w r J mx mz v ' 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 s i g n i f i c a n c e . F o l l o w i n g Chu and Tamir, we may w r i t e s o l u t i o n s o f the wave e q u a t i o n ;(4.12) i n terms of an i n f i n i t e s e t o f c o u p l e d d i f f r a c t i o n modes, M 0 0 E ( x , z , t ) = I I • ( x ) U ( r i e r p j O ^ t - k ^ x - k z) (4.14) m=0 £=-<» where to^ = co^ + SLQ , 9 2 . A is the diffraction order, and ^(x) is a coupling constant dependent on Se and the width of the interaction region. Since the waveguide permittivity has a modulation of periodi- city A, the transverse wavenumber in isotropic materials satisfies kAz = k0z + £ 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 ,2 - Ax 0 - k = k 0 Az (k Q z + HQ' (4.17) Figure (4.2) shows the acousto-optic dispersion curves implied by this relation, for the case Se = 0. Only the incident (A = 0) mode propagates; Fig. 4.2. 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 intro- duces stop bands into the dispersion curves at their intersection points, which now become the Bragg regimes where coupling between diffraction modes is possible. For example, when k = K/2, coupling is strong U Z between the incident and first diffracted modes. In general, where R Q z = + £K/2, energy is coupled from the incident to the SL th order dif- fraction mode. These modes propagate at the angles sinG^ = k £ z/k Q » sin9 0 + £K/kQ (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. o f i n c i d e n c e 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 a d j a c e n t 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 . S t r o n g c o u p l i n g i s p o s s i b l e o n l y a t the B r a g g r e g i m e s , where 6Q = 6̂  = 6̂ , t h e B r a g g a n g l e . Then s i n e , = lK/(2kn) = W ( 2 n A ) , (4.19) b 0 where A i s t h e vacuum w a v e l e n g t h o f l i g h t and n i s t h e r e f r a c t i v e i n d e x . I n 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 c o m p l i c a t e d 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 d i f f e r e n t i n d i c e s o f r e f r a c t i o n . F o r B r a g g d i f f r a c t i o n i n t o t h e f i r s t o r d e r , D i x o n [82] g i v e s S l n 8 0 = 2n71 a + 4 ( n 0 - n l » ^ 2 0 > " X and s l n 9 i = _ ^ ( 1 _ i ! ( n 2 _ n 2 i ) ) ( 4 > 2 1 ) X A f o r t h e a n g l e s o f i n c i d e n c e and d i f f r a c t i o n . I n LiNbO^ m e t a l - d i f f u s e d w a v e g u i d e s , n^ and n ^ t y p i c a l l y d i f f e r by 1% o r l e s s , so t h e d i f f r a c t i o n may 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 l o s s o f a c c u r a c y . F o r 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 t h e f i r s t d i f f r a c t i o n o r d e r , 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 g r a t i n g be t h i c k enough to s u p p r e s s d i f f r a c t i o n i n t o h i g h e r o r d e r s . The c o n d i t i o n f o r t h i s g i v e n by A l p h o n s e [27] i s L > nA 2/X (4.22) 2 ( f r e q u e n t l y L > 5nA /(TTA) i s used i n s t e a d ) . I n t h e f o l l o w i n g d i s c u s s i o n , i t i s assumed t h a t (4.22) i s s a t i s f i e d , 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 i n the f i r s t - o r d e r B r a g g r e g i m e . Only two terms r e m a i n i n t h e s o l u t i o n (4.14) o f t h e wave e q u a t i o n : the i n c i d e n t wave, E Q(x,t) = <j>0(x)U0(y)expj(u)0t - k^x - k^z) (4.23) and the diffracted wave, E 1(x,t) = (),1(x)U1(y)expj(a)1t - k^x - k^z) (4.24) We require the coefficients ̂ (x) and <j>̂ (x), which describe continuous coupling between the waves in the interaction region. The acoustic surface wave is assumed to be propagating along the z coordi- nate 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 field variation of TE optical guided modes with depth. They are normalized by the integral k. °° 2 1 / U^(y)dy = 1 . (4.25) —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 n0 "0 k„ + YT. + U - " = 0 (4.26) Ox Oz 2 c a n d ..2 . 2 2 2 n r . • w i c In general, the two waves will propagate in different directions, and may be in different guided modes, so the unperturbed permittivities 2 2 nQ' and n^r will differ. (cSubstMutiowdtc'c^^a') :°and- • (4.24) into the wave, .equation -(4.1,2)' "such that .each ,wave:lis ̂ regaEded?tas: the source of the. other gives--, ;r.*-d as 'chc . • •• c..hi-. ( 4 . 2 8 ) c and - — AfeU^O e x p j ( o ) 0 t - k Q x x - k 0 zz.) . ( 4 . 2 9 ) The p r i m e s denote 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 x. The s e c o n d d e r i v a - t i v e s may be n e g l e c t e d , i f we assume t h a t <Kx) v a r i e s s l o w l y . When the c o s ( f i t - Kz) f a c t o r i n Ae i s w r i t t e n as t h e sum of e x p o n e n t i a l s , i t becomes c l e a r t h a t t h e s e e q u a t i o n s d e s c r i b e d i f f r a c t i o n i n t o b o t h t h e 1 = 1 and I = - 1 o r d e r s . I f d e f l e c t o r o p e r a t i o n i s w e l l i n t h e B r a g g r e g i m e , i t i s p o s s i b l e t o d e f l e c t l i g h t i n t o o n l y one o f t h e s e a t a t i m e , so t h e & = - 1 o r d e r may be dropped. M u l t i p l y i n g ( 4 . 2 8 ) by UQ and ( 4 . 2 9 ) by U^, i n t e g r a t i n g o v e r a l ] y} and u s i n g ( 4 . 1 1 ) and {A.25) gives two coupled-mode equations in <j>0 and c j ^ , 0 = - J o i g ^ e x p ( j B ) ( 4 . 3 0 ) and K = - J a-,<r n e x p ( - j B ) , ( 4 . 3 1 ) where B = Ak x + (Ak - K ) z - (Aw - fi)t, w i t h 0 1 0 2 In f u l l subscript notation the overlap integral r ^ is 7 rv,i = / u*n 5 e U 1 dy , (4.33) mOnl J mOp pq nlq 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 cos6 . For reasons discussed J6X J6 As earlier, coupling occurs only in the vicinity of Akz = K 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>Q + aQa-f (4.34) where p = Ak /2. x Similarly, i t can be shown that d)£ + 2jp<j)1 + aQa^.- =• 0. (4.35) If we assume solutions of the form exp(j£Lx), substitution gives quadra- tic equations in B Q and g^, B2 T 2p3Q - = 0 (4.36) and 3 2 + 2pB-L - = 0 , (4.37) from (4.34) and (4.35), respectively. Ti Setting q = /p + CSQCX^ , we obtain the solutions B0 = P ± q (4.38) and 3n = -(p ± q) . (4.39) 98.. and Solutions for <$>Q and <j)̂  are d>0(x) = A± exp (jx(p+q)) + A2 exp (jx(p-q)) (4V40) ^(x) = A 3 exp(-jx(p+q)) + A 4 exp(-jx(p-q)) (4.41) The boundary conditions to be satisfied are <f>0(0) = 1 , *!«» = 0 , <!>Q(0) = 0 , d)̂ (0) = - j a 1 , (4.42) where the last two are obtained by applying the first 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>0(x) = exp(jpx) (cos(qx)- j ̂ ; sin(qx)) (4.43) and a l <j>1(x) =-j exp(-jpx) sin(qx) . (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 2 2 = C I Q C I ^ 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 , (4.46) x ' 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 is 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. Using this, along with (4.46) and CL the definition of q, the diffraction efficiency becomes n = g 2sinc 2[g 2 + (KA6L/2) 2] 1 7 2 , (4.47) .100. w i t h 4 2 2 r P a L g = 2 . • (4.48) 4 X COS9Q c o s 0 ^ F o r l i g h t i n c i d e n t a t t h e B r a g g a n g l e , A9 = 0, so (4.47) becomes 2 n Q = s i n g . (4.49) Whe g = TT/2, 100% d i f f r a c t i o n e f f i c i e n c y i s o b t a i n e d . The a c o u s t i c power r e q u i r e d f o r t h i s i s a p p r o x i m a t e l y p i o o = V r - ' (4-50> n" r L s i n c e u s u a l l y COS6Q ^ c o s 0 ^ ^ 1. The d e f l e c t i o n a n g l e i s a l t e r e d by v a r y i n g t h e a c o u s t i c f r e - quency f . The d i f f r a c t i o n e f f i c i e n c y f a l l s o f f , s i n c e l i g h t i s no l o n g e r i n c i d e n t a t the Br a g g a n g l e . The u s a b l e l i m i t s w i l l be s e t by e i t h e r t he a n g u l a r s p r e a d o f t h e sound beam, o r by t h e l i m i t e d b a n d w i d t h o f the i n t e r d i g i t a l t r a n s d u c e r . I n e i t h e r c a s e , the h a l f - p o w e r 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 argument o f the s i n e changes by 1.3916 away fr o m the c e n t r a l maximum. U s u a l l y t h e Br a g g a n g l e i s o f the o r d e r o f one degr e e , so the change i n d e f l e c t i o n a n g l e as a f u n c t i o n o f f r e q u e n c y change Af i s o b t a i n e d f r o m (4.19) A 6 b = 2nv <4'51> Thus, KA6L, 1 _ irrXLf Af * 2 2nv2 = 1.3916 a t t h e h a l f - p o w e r p o i n t s . D e f l e c t o r s a r e u s u a l l y t r a n s d u c e r b a n d w i d t h l i m i t e d , so t h i s e q u a t i o n g i v e s h a l f thermaximum u s a b l e i n t e r a c t i o n l e n g t h as a f u n c t i o n o f d e f l e c t o r b a n d w i d t h , i(u: Thus, L • !; 8 A; v 2 . (4.52) max Af Af o 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 is much greater., than the corresponding spread Acb of the light beam. Since the angle of deflection is twice the Bragg angle, the number of resolvable spots to which light can be focussed is Ng = 2A0,/A(f) (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 = Af A/v = Af T , (4.54) s m m where T is the transit time of the acoustic wave across the light beam and Af is the half-power bandwidth. v m r 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 will 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 al [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 fre- quency is 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 is 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 inte- ger ( P ) multiple of AQ/2, where waves with A = will propagate straight ahead after excitation by the array. If the transducers are driven out of phase, P must be odd; i f they are in phase, P is even. Fig. 4.5 Beam Steering Transducer. 103. In the far radiation field, the sound wave amplitude is given by the Fresnel-Kirchoff integral [85], MG expj (Kx sincj) - y(x)) dx (4.55) , 0 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 field, because the Fresnel distance is excessively long,. However, the analysis is done in the far radiation field for convenience. Consider the effect of one of the steps (Fig. 4.6). The phase change across each aperture is 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 is K(P | - H)/cos4> -v £| (A - AQ) = PTT(1 - K/K ) 104. for small angles <j>. Thus, the diffraction integral becomes M-l nD+G S(<f>) = I / expj(K<j>x - nPTT(l - K/K_) ) dx n=0 nD U M-l . e*p(JK»G) - 1 £ e x p j n f K ^ D - P.Cl - K/K.)] J 9 n=0 . sin{y M [KD(j> - PTT(1 - K/K ).]} = S sinc(y KG*) — ^ ' M s i n v y [KD* - PTT(1 - K/K Q) ] } where i s a constant. ( 4 . 5 6 ) The sound wave intensity is I(4>) = S 2 (cj,) = I 0A 2 ( < j.)B 2(<(>) , ( 4 . 5 6 ) where I Q is a constant, A((f>) = sincOj KG<j>) is the aperture function, and B(<j>) is the array function. The principal maximum of the array function is at the angle <)>Q, determined by | M [ K D * 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 terms o f ty^, s i n [ - 5 — (<f> - <J) )] B C * ) = ^ ~ v (4.59) M s i n [ ^ «, - <frQ)] The maxima a r e a t the a n g l e s *Bmax= * 0 + q D » ( 4 ' 6 0 ) 105. where q is an integer. The minima of A(<}>) are at Âmin q G (4.61) Figure 4.7 shows the array and aperture functions where A = and G 'v* D. The latter condition is desirable, since most of the transducer width is then utilized for SAW generation. This gives a longer acousto-optic interaction length and higher deflector efficiency (4.50). Furthermore, the secondary maxima of B(<}>) then occur near the zeros of A(<j>) , so l i t t l e acoustic power is carried outside the central peak. When A 4 AQ, 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 F i g . 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 ^ . I n a n i s o t r o p i c m a t e r i a l s , t h e d i r e c t i o n o f power f l o w i s n o t i n g e n e r a l c o l l i n e a r w i t h t h e p r o p a g a t i o n v e c t o r . I f t h e a n g l e s i n (4.56) a r e a l l changed by 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 c o n s t a n t d e f i n e d i n 3.15, we o b t a i n t h e d i s r i b u t i o n o f a c o u s t i c power f o r a n i s o t r o p i c p r o p a g a t i o n , . 2rMKDa , .1 _ _ . 2 ,KGa* v 8 i n - l — < » - *0>J . ^ I a n ( * ) = V" 1 0 <—> M 2 . 2f"KDa, , H ( 4 ' 6 2 ) M sxn [ - -y -Cf r - <f>0)J The maximum power i n t h e SAW i s i n t h e v i c i n i t y 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 B, where <J> = cj)̂ . Thus, t h e peak i n t e n - s i t y 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 t o t h e e q u a t i o n 107. I = I . s i n c 2 ( i KGa<J)). (4.63) an 0 2 To apply these r e s u l t s to Bragg d e f l e c t o r s , we require that the Bragg angle be matched at two frequencies, f ^ and f ^ ( F i g . 4.9). The s t e e r i n g angle (4.58) i s a hyperbolic function of frequency, and the Bragg angle with respect to an a r b i t r a r y d i r e c t i o n i s [85] A = c - ^ f , (4.64) where c i s a constant. (The angle of SAW propagation relevant to the Bragg i n t e r a c t i o n i s <J>Q (4.58), even f o r anisotropic materials, since i t i s the sound wavevector rather than the power flow d i r e c t i o n that matters.) Since 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 function of f, F i g . 4.9. Bragg-Angle Tracking. 108. f - f = T~-TJ ( 4 - 6 5 ) h £ h U s i n g - ( 4 . 5 8 ) , we o b t a i n Z U rh rO TSL V h The angle of intersection of light and sound waves deviates from the Bragg angle by the steering error, A0 = 4>b(f) - <|>0(f) (4.67) Let us define an, .array frequency f^ such that Ae ( f p is a maximum. The condition dt 4r (A6) = 0 (4.68) gives ''1 ~ "h f 2 = f 0 fu • (4.69) T h i s d i f f e r s f r o m Pinnow's [86] 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 by use 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 er p e r f o r m a n c e , r a t h e r t h a n the more n a t u r a l c o n d i t i o n ( 4 . 6 8 ) . U s i n g ( 4 . 6 9 ) , t h e s t e e r i n g e r r o r becomes A0(f) = ~ ( y - + j - - j - ^ ) . (4.70) h £ E q u a t i n g t h e s l o p e s i n e q u a t i o n s (4.64) and (4.66) f i x e s t h e P/D r a t i o , P Xfl nv 109. 4.4 D i f f r a c t i o n E f f i c i e n c y of Beam-Steered Transducers I t i s now p o s s i b l e to w r i t e expressions f o r the d i f f r a c t i o n e f f i c i e n c y of a beam-steered d e f l e c t o r as a f u n c t i o n of SAW frequency. With reference to equation (4.47), define , KA6L TrPLf ,1 . 1 1 f . .. ... h = — 2T- ( T ^ T ~ - f - - 2 ) ' (4-72> h Jl f ̂ The amplitude of the c e n t r a l maximum of the array f u n c t i o n i s s i n c O j KGOI4>Q) from (4.63), so the SAW a c o u s t i c power must i n c l u d e the 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 transducer to which the s e r i e s model a p p l i e s i s used f o r SAW generation, the a c o u s t i c power i s given by (3.48), P a = R a v 2 / ( 2 ^ Z ' | 2 ) 2 f o r a d r i v i n g v o l tage V. Thus, g (4.48) becomes 2 ^ r 2 p a L 2 g b = jr s i n e (irfGa(j>0/v) , .4 X COS8Q cos8^ where <f>Q i s given by (4.58). The anisotropy parameter a enters i n t o t h i s e x p r ession, although i t i s not i n (4.72), since d i f f r a c t i o n i s from the planes of constant phase of the SAW. The d i f f r a c t i o n e f f i c i e n c y of a beam-steered d e f l e c t o r i s the r e f o r e 2 . 2 , 2 , L 2 v l / 2 ,, .,„. n b = g b s i n e ( g b + h > ' , (4.73) where h = - 2 D ~ FI " f f? (4.74) n o . and w i t h 4 r 2 p 2 _ a . 2 rTrPGa _ ._ __. g b r - : — s x n c [-p^- (1 - f / f g ) ] , (4.75) 32c COSOQ c o s 0 ^ V 2 R P a ( 4 . 7 6 ) and 3 2[:(.R + R ) 2 + ($1L - l / ( f i C T ) + X Q ) 2 ] a. 6 S X cL 2 R = R n s i n e x , a 0 X a = R Q [ s i n ( 2 x ) - 2 x ] / ( 2 X 2 ) , where x = N i r ( f - fn)/fn. The IDT a r r a y c i r c u i t e l e m e n t s R , R , L and u (j a e s Cj, a r e o b t a i n e d f r o m t h e a p p r o p r i a t e s e r i e s / p a r a l l e l c o m b i n a t i o n o f t h e i n d i v i d u a l t r a n s d u c e r s e c t i o n s . 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 v o l t a g e cL Y g j ^ t i g a p p i i e ( i t o t h e s e r i e s e q u i v a l e n t c i r c u i t ( F i g . 3.8) r e p r e s e n t a - t i o n o f the b e a m - s t e e r i n g t r a n s d u c e r . I f t h e d e v i c e i s d r i v e n by a c o a x i a l t r a n s m i s s i o n l i n e o f c h a r a c t e r i s t i c impedance Z^, t h e e q u a t i o n s t a k e a d i f f e r e n t form. We w i l l assume t h a t t h e r f g e n e r a t o r i s i s o l a t e d f r o m the 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 on t h e l i n e . Then t h e e l e c t r i c a l power d i s s i p a t e d i n the IDT w i t h impedance Z = R + j X i s g i v e n by p e = p 0 a - ig2), 2 where P = V,/Z i s the 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 were o f c r matched and V f i s t h e f o r w a r d r . m . s . . v o l t a g e on.the l i n e . T h e r e f l e c t i o n c o e f f i c i e n t p-̂ s i s g i v e n by After some algebra, we obtain. P = e 4V2R 2" 2 * (R + Z ) + X c The current flowing in the series equivalent circuit is given .2 by P e = i R. The usable acoustic power is half the power dissipated in the IDT radiation resistance, that i s , P = | i 2 R = a 2 a 2 2V R f a [(Ra + Re + Z c ) 2 + < f l L s + X a - ^ ) 2 ] (4.77) 4.5 Acousto-Optic Overlap Integral Calculation The overlap integral (4.33) was evaluated for the TE guided modes shown in Fig. 2.35 interacting with a Z ± 21.8° propagating SAW on Y-cut LiNbO^. The permittivity, electro-optic and photoelastic tensors are given in Appendix I in matrix form for the principal axes system (X,Y,Z). To calculate these in the system ( X ^ J X ^ J X ^ ) , which is rotated by 21.8° about the Y = x^ axis, i t is preferable to revert to tensor notation in order to use the usual transformation laws [88], a. a. e lm jn mn i r i j k a. a. a, r lm jn kp mnp (4.78) p'. -i n = a. a. a, a. p rijk£ lm jn ko £p mnop The transformation matrix is 13 C03t 0 -sine 0 sinQ 1 0 0 C0s9 (4.79) 1 1 2 . 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 is 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 2 (y) SE' dy (4.80) —00 The rotated permittivity, electro-optic and photoelastic tensors are given in Appendix I, along with the permittivity change factors Se'ij E 0 . p . M n E-o and e 0. r.., E . „ . 3I rijkJl j3 3i ljk 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 is possible to find the acoustic strain and electric fields using , 3u. 3u. and s (4.82) 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 along x^, the actual magnitudes are x 2 = x̂ /S , N i U i l = ^ > $ l = |$ N |/v^ , '3M *The primes are omitted in the rest of this section. A l l quantities are in the rotated coordinate system. Fig. 4 . 11 114. |S..|= /P,_n |SN.| , 1 i j 1 3M 1 i j 1 and |E.| = ^ ~ f i |E^| , (4.81) where N indicates the normalized values. For SAW propagation along x^, i t is clear from (3.11) and (3.12) that there is no x^ variation, and that the operator identification S/̂ x̂  -»• jfi/v can be made. Differentia- tion with respect to x̂  must be done numerically; since only the magni- tudes of the fields are plotted, complex phase factors and C^i? defined by a l u . l a u . 2 1 9x2 8x2 (4.82) and C 7"4iL 2 3x2 3x2 are missing. 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, , 8u„ 8u 23 2 ^3x3 3x ' = i ( ^ f U2 + C 3 2 ^ T ) ' (4'83) At y = 0, Ii 2 (0) = 2.656 x 10_6(.0366 + .999j) and S23 ( 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 l » l n I N 1 | U2> A l U3l R e S N 2 3 = i ( .999 - ^ - + G 2 3 — f ) (4.84) A x 2 a t y = 0. T h i s g i v e s C 2 3 = 1.102. The e q u a t i o n s o b t a i n e d i n t h i s 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 and e l e c t r i c f i e l d s a r e S l l " ° » Ni A|uL Re Snn = sL = .0593 vffi"-22 " °22 . N ' A x 2 Soo = 0 , r t- A ' U 3 I ' | m2> S 2 3 = " ^ ( 1 ' 1 0 2 — ¥ - + - V - } ' X 2 I N l S j 3 = "-352 AT — , N, AI u sf„ = .0583 1 2 — N ' Ax 2 E r . > 2 5 6 ^ A I ^ L , A x 2 U N I a n d E* = -.293 & -L2-L ,. (4.85) 3 v I n F i g . 4.12, the 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 a r e shown as a f u n c t i o n o f de p t h f o r an a c o u s t i c f r e q u e n c y o f 165 MHz, and i n F i g . 4.13 the c o r r e s p o n d i n g e l e c t r i c f i e l d s a r e shown. B e t t e r a c c u r a c y w o u l d have been o b t a i n e d by s o l v i n g e q u a t i o n (14) o f r e f e r e n c e [40] f o r the 3™ c o e f f i c i e n t s , s i n c e t h e d i s p l a c e m e n t and p o t e n t i a l f i e l d s c o u l d n o t 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 2 = 0. The p e r m i t t i v i t y change 6~e 3 3 i s g i v e n b 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. 6 e 3 3 = =:3.96S* 2"^4 :25S* 2 - . 2 7 9 S 2 3 +.63S^ 3 +1.29x10 9E^ -7.15x10 8E^' (4.86) The p r i n c i p a l s t r a i n components a r e S 2 3 and S ^ 3 > and t h e p r i n c i p a l e l e c t r i c f i e l d component i s E.^. F i g u r e 4.14 shows t h e r e l a t i v e c o n t r i - b u t i o n s w i t h d e p t h t o t h e o v e r l a p i n t e g r a l by t h e p h o t o e l a s t i c and e l e c t r o - o p t i c p a r t s f o r a TE g u i d e d wave w i t h a c o n s t a n t e l e c t r i c f i e l d . C omparison w i t h F i g . 4-10 shows t h a t the 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 i n | | , where d | u 3 | / d x 2 = 0. T h i s i s a t about o n e - f i f t h t h e a o o u s t i c w a v e l e n g t h . S i m i l a r l y , t h e 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 about A/2, s i n c e |$| has a minimum t h e r e , and |E 3| i s p r o p o r t i o n a l t o |$|. These r e s u l t s a r e s i m i l a r t o t h o s e o b t a i n e d by T s a i e t a l [ 7 4 ] , e x c e p t t h a t t h e s e a u t h o r s appear t o have s c a l e d the e l e c t r i c f i e l d i n c o r r e c t l y ( i n f i g . 4) b y a f act-on Of: .2. .a's. .a f u n c t i o n o f d e p t h . T h i s c o u l d l e a d t o c o n s i d e r a b l e e r r o r i n T, p a r t i c u l a r l y f o r g u i d e d o p t i c a l modes p r o p a g a t i n g v e r y n e a r the s u r f a c e . 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 M = 1 w a t t and L = 1 meter f o r the t h r e e g u i d e d TE modes o f t h e OWG. A t low a c o u s t i c f r e q u e n c i e s , 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 o v e r l a p s the s h a l l o w 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. Above about 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 h i g h e r o r d e r modes, the f a c t o r g i s f l a t w i t h f r e q u e n c y . T h i s i s a d e s i r a b l e c h a r a c t e r i s t i c f o r d e v i c e a p p l i c a t i o n s , s i n c e i t g i v e s a f l a t t e r d e f l e c t o r f r e q u e n c y r e s p o n s e . I n c a l c u l a t i o n s , i t i s e x p e d i e n t t o w r i t e e q u a t i o n s i n terms o f the t o t a l a c o u s t i c power P , r a t h e r t h a n ^^H' ^ o r y ~ ^ ± 21.8° p r o p a g a t i o n on L i N b 0 3 > P^ = 1.05 P^ M [ 4 0 ] , so t h e f a c t o r g must be d i v i d e d by / l . 0 5 i n t h i s c a s e . Fig. 4.14 Relative Electro-optic and Photoelastic Contributions to Overlap Integral for f = 165 MHz. 120. Fig. 4.15. 121. 4.6 E x p e r i m e n t a l Work S e v e r a l b e a m - s t e e r e d l i g h t d e f l e c t o r s were made. The d e v i c e s were f a b r i c a t e d on the 6-mode N i / L i N b O ^ d i f f u s e d OWG c h a r a c t e r i z e d i n C h a p t e r 2. The f i r s t d e f l e c t o r t e s t e d was made b e f o r e t h e beam s t e e r i n g t h e o r y o f S e c t i o n s 4.3 and 4.4 had been c o m p l e t e d , so i t s d e s i g n was f a r f r o m o p t i m a l . I t c o n s i s t e d o f a 2 - s e c t i o n phased IDT a r r a y w i t h P = 1 ( F i g . 4.16). Each s e c t i o n was 1.55 mm w i d e and had 2 1/2 f i n g e r p a i r s . T r a n s d u c e r s w i t h a c e n t e r f r e q u e n c y o f 155 MHz were made by p h o t o l i t h o - g r a p h y o f vacuum d e p o s i t e d aluminum 0.3 ym t h i c k . (See C h a p t e r 5 f o r d e t a i l s ) . The d e v i c e was o r i e n t e d so t h a t s u r f a c e waves p r o p a g a t e d a t an a n g l e o f 21.8° f r o m the Z a x i s o f Y - c u t L i N b O ^ W i t h t h e two t r a n s - d u c e r s e c t i o n s c o u n e c t e d i n p a r a l l e l and d r i v e n o u t o f phase, impedance measurements a t the c e n t e r f r e q u e n c y gave C^ = 4 pF, and R + R = 40 fi. F i g . 4.16 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 Mask. 1 2 2 . Vinyl electrical tape was used to minimize acoustic reflections. These values are in reasonable agreement with the parameters of the IDT series equivalent circuit, which are C T = 3 . 4 6 pF, R 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. The IDT is on the right, and SAW propagation is from right to left. 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. Electrical 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 d e f l e c t o r o p e r a t i o n was o b t a i n e d , owing p a r t l y t o t h e d i f f i c u l t y o f f i n d i n g a s u i t - a b l e t u n i n g i n d u c t o r . ( W i t h o u t t h i s i n d u c t o r , t h e SAW g e n e r a t i o n e f f i c - i e n c y i s v e r y l o w ) . The c o r r e c t v a l u e was o b t a i n e d by t r i a l and e r r o r t o be 0.19 uH r a t h e r t h a n the a n t i c i p a t e d 0.31 uH r e q u i r e d t o r e s o n a t e w i t h C j . The d i f f e r e n c e 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 l e a d w i r e s and p o s s i b l y by a d d i t i o n a l s t r a y c a p a c i t a n c e . A t t e m p t s t o o b s e r v e the s u r f a c e waves by Raman-Nath d i f f r a c t i o n o f l i g h t f r o m t h e LiNbO^ s u r f a c e were s u c c e s s f u l ; however, t h e h i g h r . f . d r i v e power needed 1 w a t t ) b e f o r e the d i f f r a c t e d beams became e a s i l y d i s c e r n i b l e r e s u l t e d 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 . F i g s . 4.18 and 4.19 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 caused by a r c i n g between f i n g e r s , and t h e second by o v e r h e a t i n g due t o f i n g e r r e s i s t a n c e . F i g . 4.18 T r a n s d u c e r f a i l u r e 124. F i g . 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 as shown i n F i g . 4.17, 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 waveguide was h a r d t o f i n d , owing t o t h e d e l i c a t e a d j u s t m e n t s r e q u i r e d . The measured -3 dB d e f l e c t o r b a n d w i d t h was about 26 MHz; as a r e s u l t , t h e a n g u l a r range o v e r w h i c h the i n t e r a c t i o n was v i s i b l e was o n l y about 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 eq. 4.51. A t a n g l e s o f i n c i d e n c e n e a r l y n o r m a l t o t h e a c o u s t i c w a v e v e c t o r , the Raman-Nath d i f f r a c t i o n regime was c l e a r l y v i s i b l e ( F i g . 4.20). The d i f f r a c t i o n e f f i c i e n c y was v e r y l o w , as a n t i c i p a t e d . F i g s . 4.21 and 4.22 show B r a g g d i f f r a c t i o n o f t h e T E q and T M Q g u i d e d modes w i t h t h e r f d r i v e o f f ( u p p e r photo) and on ( l o w e r p h o t o ) . The T E q mode appears on t h e l e f t , s u r r o u n d e d 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 the TE mode was about 40%. 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 f o l l o w e d by a Boonton 230 A r f power a m p l i f i e r c a p a b l e o f d e l i v e r i n g 5 w a t t s i n t o a 125. F i g . 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 upper and 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, and the s m a l l s p o t on t h e r i g h t i s a TM mode) 50 fi l o a d . 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 f r e q u e n c y r e s p o n s e was c o m p l i c a t e d 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 and the 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 h i g h - i m p e d a n c e r f v o l t a g e measurements. These problems were overcome by c o n n e c t i n g a r e s i s t i v e v o l t a g e d i v i d e r a c r o s s t h e d e v i c e and m e a s u r i n g the v o l t a g e a t t h e m a t c h i n g i n d u c t o r w i t h a low 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 o s c i l l o s c o p e . F i g . 4.23 shows a c o m p a r i s o n o f t h e measured and c a l c u l a t e d r e s p o n s e . 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 e f f i c i e n c y a t 165 MHz w i t h t h e B r a g g a n g l e matched a t t h a t f r e q u e n c y . The r e s p o n s e was c a l c u l a t e d u s i n g e q u a t i o n s (4.73) - (4.76) w i t h t h e v a l u e s P = 1, 1/2 L = 3.1 mm, D = 1.55 mm, G = 1.9 mm, a = 1.374, f = v(nP/XD) = 163.4 MHz R = 38.6 fi, R = 4.9 fi, C = 3.16 pF and L = 0 . 3 1 uH. The e x p e r i m e n t a l Fig. 4.22 Same with rf Drive Switched On (n ** 0.4) 127. p o i n t s a r e t h e average o f s e v e r a l r u n s , and a r e f o r a l l t h r e e TE modes. The p r i s m c o u p l e r was a d j u s t e d f o r o p t i m a l c o u p l i n g o f t h e TE^ mode, so the o v e r l a p i n t e g r a l o f t h i s mode was used i n the c a l c u l a t i o n s . W i t h i n 2 about 2%, g^ can be a p p r o x i m a t e d by g 2 = 0.040 / f P L s i n e 2 b a I n t h e c a l c u l a t i o n , i t was found t h a t t h e magnitude o f t h e measured rms d r i v e v o l t a g e had t o be i n c r e a s e d by almo s t 20% i n o r d e r t o match t h e mag- n i t u d e o f t h e t h e o r e t i c a l c u r v e t o the e x p e r i m e n t a l r e s u l t s . E i t h e r SAW p r o p a g a t i o n l o s s e s o r i n a c c u r a c y i n t h e v o l t a g e measurements c o u l d have been t h e s o u r c e o f t h e d i s a g r e e m e n t . F i g . 4.24 shows t h e r e s p o n s e when the i n d u c t o r i s detuned t o a h i g h f r e q u e n c y and the B r a g g a n g l e matched a t a f r e q u e n c y b e l o w 100 MHz. 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 expense o f d i f f r a c t i o n e f f i c i e n c y . The b a n d w i d t h i n F i g . 4.23 i s l i m i t e d b y t h e e l e c t r i c a l b a n d w i d t h o f t h e t r a n s d u c e r e q u i v a l e n t c i r c u i t , about 24 MHz. I n F i g . 4.24 , the 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 l i m i t i n g f a c t o r . 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 (4.52) i s 146 MHz, so i t i s c l e a r l y t h e i n t e r d i g i t a l t r a n s d u c e r t h a t l i m i t s t h e o v e r a l l d e f l e c t o r p e r f o r m a n c e . 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 - p a t e d number o f r e s o l v a b l e s p o t s i s N g = A f x = 15, w h i c h i s r a t h e r a s m a l l number. I t c o u l d be i n c r e a s e d by u s i n g a w i d e r l i g h t beam a t t h e expense o f speed and g r e a t e r d i f f i c u l t y i n c o u p l i n g t o t h e OWG. A c c o r d i n g t o t h e c o n d i t i o n ( 4 . 2 2 ) , 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 t h e B r a g g regime by a f a c t o r o f 1.7. The second o r d e r d i f f r a c t e d beam was c l e a r l y v i s i b l e a t h i g h e r d r i v e power, and a c c o u n t e d f o r up t o 4% of t h e d i f f r a c t e d l i g h t . n P G n 2D (1 - f / f j ) (4.87) 128. 129. .06 Ob .04 .02 120 140 F i g . 4.24 160 f (MHz) 180 The d i f f r a c t i o n e f f i c i e n c y was measured as a f u n c t i o n o f d r i v e v o l t a g e a t 165 MHz. When the v o l t a g e i s c o r r e c t e d by t h e same amount as b e f o r e , good agreement i s f o u n d between e x p e r i m e n t and t h e o r y . ( F i g . 4.25) The s m a l l d i s c r e p a n c y t h a t e x i s t s may be due t o d i f f r a c t i o n i n t o t h e s e c o n d o r d e r beam. The a c o u s t i c d r i v e power was 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 c i r c u i t . The maximum d i f f r a c t i o n e f f i c i e n c y i n t o t h e f i r s t o r d e r beam was 70% a t a c o r r e c t e d e l e c t r i c a l d r i v e power o f 1.35 w a t t s , which corresponds to P = 600 mW. At higher voltage, the device burnt out. el 130. TOO 200 300 4 00 500 ACOUSTIC DRIVE POWER (mW) Fig. 4.25 600 700 Beam steering was not clearly observable with this device. It is most likely that the response obtained in Fig. 4.24 is 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 its final 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 have been o b t a i n e d ) , b u t 4 ym l i n e w i d t h s were c o n - s i d e r e d t o be the r e s o l u t i o n l i m i t o f the p h o t o l i t h o g r a p h y mask making p r o - c e s s used. S i n c e the f i n e s t l i n e s t h a t c o u l d be r u l e d on r u b y l i t h a r t w o r k m a t e r i a l were found to be 0.5 mm, the maximum p h o t o r e d u c t i o n r a t i o was 125:1. The maximum w i d t h o f a r t w o r k t h a t c o u l d be h a n d l e d was about 1.25 m e t e r s , owing t o t h e d i f f i c u l t y o f c u t t i n g and u n i f o r m l y i l l u m i n a t i n g g r e a t e r w i d t h s . C o n s e q u e n t l y , the maximum w i d t h o f the t r a n s d u c e r was about 1cm. On the b a s i s o f f r e q u e n c y r e s p o n s e c a l c u l a t i o n s , i t was con- c l u d e d t h a t the 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 the a r r a y f r e q u e n c y f ^ was g r e a t e r t h a n the IDT c e n t e r f r e q u e n c y f ^ , so t h e s e l e c t i o n f ^ = 205 MHz was made. T h i s f i x e d t h e P/D r a t i o o f t h e t r a n s d u c e r a r r a y ( 4 . 7 1 ) . F o r r e a s o n s d i s c u s s e d i n S e c t i o n 4.3., t h e r a t i o D/G was chosen t o be 0.9. A minimum o f f o u r t r a n s d u c e r 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 a p p r o x i m a t e d i f f r a c t i o n t h e o r y w o u l d be r e a s o n a b l y a c c u r a t e . The f i n a l r e q u i r e m e n t i n the d e s i g n was a t r a n s d u c e r impedance 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 v a l u e s o f P, the w i d t h and impedance of each t r a n s d u c e r s e c t i o n were c a l - c u l a t e d , and v a r i o u s s e r i e s / p a r a l l e l c o m b i n a t i o n s were t r i e d u n t i l l s u i t a b l e v a l u e s were f o u n d . The f i n a l d e s i g n p a r a m e t e r s were P = 2, D = 2.54 mm, L = 10.2 mm (4 s e c t i o n s ) and R + R = 50 ft. The t o t a l r e s i s t a n c e was o e f o r c e d t o 50 ft by s e l e c t i n g s u i t a b l e v a l u e s o f the aluminum e l e c t r o d e t h i c k n e s s and m e t a l l i z a t i o n f a c t o r a. C o n t r o l o f t h e forme r was d i f f i c u l t w i t h the p h o t o l i t h o g r a p h y p r o c e s s u s e d , b u t the l a t t e r c o u l d be v a r i e d a t w i l l . T hree f u n c t i o n a l d e v i c e s were f a b r i c a t e d u s i n g t e c h n i q u e s d e s c r i b e d i n the n e x t c h a p t e r . A l l h a d s i m i l a r c h a r a c t e r i s t i c s . The 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 ut on one d e v i c e . 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. The transducer had the dimensions D = 2.54 mm and G = 2.83 mm. Comparison with the mask artwork gave a reduction ratio of 1:118.2, which implied that A = 17.19 ym and f = 199.4 MHz for propagation along the Z-21.8° direction on Y-cut LiNbOQ. The calculated parameters of the equivalent series circuit model 133. are C T = 3.55 pF, RQ = 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 is reasonably high, the total equivalent series capacitance would be expected to be comparable in magnitude to the sum of the parallel capaci- tances. 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. Figure 4.27 shows the circuit used to drive the device. A 10 dB power attenuator was used to reduce reflections on the transmission line. SWR measurements were made 500 MHz LOW -PASS FILTER HP 230 AMP 10 dB ATT EN. BNC I ADAPTOR- G R 8 7 4 - L B A SLOTTED LINE GR1216A IF AMP Fig. 4.27 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 hand-wound 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 fo u n d . T h i s i n d u c t o r h ad 3i t u r n s 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 , 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 was 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 l e a d s . T h i s c o u l d n o t be measured d i r e c t l y , b u t a p i e c e o f w i r e o f about t h e same t o t a l l e n g t h had a measured i n d u c t a n c e o f .035 uH. The minimum SWR o b t a i n e d a t 200 MHz was about 2.2; i t was sub- s e q u e n t l y d i s c o v e r e d t h a t t h e GR t o BNC a d a p t o r and the BNC c o n n e c t o r had an SWR o f 2, so t h e t r a n s d u c e r was a p p a r e n t l y matched. However, i t was n e c e s s a r y t o c o n s i d e r t h e e f f e c t o f s t a n d i n g waves on t h e t r a n s m i s s i o n l i n e between t h e c o n n e c t o r and the IDT. An SWR o f 2 c o r r e s p o n d s t o a r e f l e c t i o n c o e f f i c i e n t o f magnitude | p j = (2-1)/(2+1) = 0.33. A t t h e l i g h t d e f l e c t o r h a l f - p o w e r p o i n t s , t h e magnitude o f t h e r e f l e c t i o n c o e f f i c i e n t p2 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 matched IDT e q u i v a l e n t c i r c u i t was l e s s 2 than 0.5. U s i n g P = P Q(1 - |p| ) t o f i n d t h e f o r w a r d power, i t can be shown ( F i g . 4.28) t h a t n e g l e c t o f th e m u l t i p l e r e f l e c t i o n s l e a d s t o a maximum e r r o r o f o n l y a few p e r c e n t . F i g u r e 4.29 shows a s c a n a c r o s s . t h e t h r e e g u i d e d TE modes o f the d e f l e c t e d l i g h t beam, u s i n g a Gamma S c i e n t i f i c M o del 2900 S c a n n i n g A u t o - P h o t o m e t e r . A t an a c o u s t i c f r e q u e n c y o f 200 MHz, co m p a r i s o n w i t h 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 i n d i c a t e d t h a t a l l t h r e e modes had com- p a r a b l e d i f f r a c t i o n e f f i c i e n c y , w i t h t h e TE.̂ .. mode b e i n g somewhat more e f f i c i e n t . T h i s i s i n agreement w i t h t h e c a l c u l a t e d r e l a t i v e magnitudes o f t he o v e r l a p i n t e g r a l s f o r r t h e t h r e e modes. When t h e b e a m - s t e e r i n g IDT was d e s i g n e d , t he a n i s o t r o p y p a r a - m e ter a was e r r o n e o u s l y i n c o r p o r a t e d i n t o (4.71). As a r e s u l t , t h e c a l - c u l a t e d a r r a y f r e q u e n c y was a c t u a l l y 180.4 MHz r a t h e r t h a n t h e 205 MHz 135. 1 3 6 . expected. Figure 4.30 shows the frequency response of the diffraction 150 T H E O R Y E X P E R I M E N T fh o r te 1 6 0 250 Q 270 2 0 0 f(MHz) Fig. 4.30 Diffraction Efficiency vs. Frequency. 250 efficiency for three values of f or f , the frequency at which the Bragg JO * i angle is matched. The theoretical curves were calculated with (4.73)- (4.75.) and (4.78), using the values C T = 3.55pF, L g = 1.8uH, R q = 33.2 fi, R = 32 ft, V = 3.8 Vvrms, f n =177 MHz, D = 2.54 mm, G = 2.83 mm, e f 1 L = 10.16 mm, P = 2, = 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 m a t c h i n g c i r c u i t . A d d i t i o n a l s e r i e s r e s i s t a n c e i s 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 , and a d d i t i o n a l i n d u c - t a n c e d e r i v e s from t h e w i r e s and f e e d t h r o u g h s ( e s t i m a t e d t o be . 1 uH) . The l o w e r v a l u e o f f ^ used (2% below 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 f i t t o t h e d a t a . 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 i n t h e t r a n s d u c e r geometry. 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 gave t h e b e s t f i t t o the o b s e r v e d f r e q u e n c y r e s p o n s e . T h i s a p p r o a c h p e r m i t s a b e t t e r c o m p a r i s o n 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 p a r a m e t e r s w o u l d have g i v e n s i m i l a r r e s u l t s , w i t h m i n o r d i f f e r e n c e s i n d i f f r a c t i o n e f f i c i e n c y , b a n d w i d t h and o v e r a l l r e s p o n s e 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 . 4.30. When t h e d i f f e r e n c e between f ^ and t h e B r a g g f r e q u e n c y i s l a r g e , beam s t e e r i n g becomes more pronounced; t h e d e f l e c t o r b a n d w i d t h i n c r e a s e s from 51 MHz when f„ = 160 MHz t o 68 MHz when f, = 270 MHz. T h i s i n c r e a s e I h i s a t t h e expense 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 drops f r o m 0.34 t o .04 a t t h e IDT c e n t e r f r e q u e n c y . F i g u r e 4.31 shows t h e d i f f r a c t i o n e f f i c i e n c y a t f = 200 MHz as 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 , and F i g . 4.32 shows d e f l e c t o r b a n d w i d t h v s . Brag g f r e q u e n c y . These a r e combined i n F i g . 4.33, w h i c h i l l u s t r a t e s 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 e f f i c i e n c y and b a n d w i d t h . These c h a r a c t e r i s t i c s a r e a l l p r e d i c t a b l e from F i g . 4.9. I n c r e a s i n g f ^ o r d e c r e a s i n g f ^ has t h e e f f e c t o f r a i s i n g t h e c u r v e , 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 range 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 e f f e c t i v e . F i g u r e 4.34 g i v e s a c o m p a r i s o n between t h e c a l c u l a t e d r e s p o n s e o f a c o n v e n t i o n a l and a beam-steered d e f l e c t o r . The l a t t e r has t h e B r a g g f r e q u e n c y matched a t 160 MHz and t h e f o r m e r a t 200 MHz. The d r i v e v o l t a g e 138. -3 -6 -12 -e—-e—er j L TOO 200 f (MHz) 300 Fig. 4.31 Diffraction Efficiency at f = 200 MHz vs. Bragg Frequency with V = 3.8 V rms. 70 60 ' N §50 40 0 o o -I —I 1 - J I ' I I J I 100 200 BRAGG FREQUENCY (MHz) 300 Fig. 4.32 Deflector Bandwidth vs. Bragg Frequency with V = 3.8 V rms.  140. i s 8 V rms, 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. The u n s t e e r e d 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 (4.74) and (4.77), u s i n g m o d i f i e d v e r s i o n s o f (4775) and (4.76). The a n g u l a r d e v i a t i o n f r o m t h e Bragg a n g l e i s A 9 b - 2nV A f > KA9 L so t h a t h = = - J L A _ F A F L . (4. 8 8) 2 2nv2 I n (4.76), the f a c t o r s i n e [-ZQ- (-1 ~ f / f 0 ) l must be r e p l a c e d by 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 can be shown t o be s i n c 2 ( K A 9 , L / 4 ) = s i n c 2 ( h /c) , (4.89) b c when f, = f . b o The b a n d w i d t h o f t h e bea m - s t e e r e d d e f l e c t o r i s a l m o s t t w i c e as g r e a t . The a c o u s t i c power ( 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 . The p r i n c i p a l r e a s o n f o r t h e d i f f e r e n t b a n d w i d t h s i s e v i d e n t i n F i g . 4.36, w h i c h shows the 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 . T h i s l i m i t s t h e b a n d w i d t h o f t h e u n s t e e r e d d e f l e c t o r t h r o u g h h i n (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 o f f somewhat more r a p i d l y on e i t h e r s i d e o f the IDT c e n t e r f r e q u e n c y . C a l c u l a t i o n s i n d i c a t e t h a t g r e a t e r b a n d w i d t h s c o u l d have been a c h i e v e d i f f ^ had been c l o s e t o f Q . F o r example, a s i x - s e c t i o n t r a n s - d u c e r w i t h D = 2.18 mm, P = 2 and f ^ = 195 MHz i s c a p a b l e o f g i v i n g 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 tQ = 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 in Fig. 4.37 for the device mode. With an rf 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 first 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 L 5 10 15 DEFLECTION ANGLE fmr) Fig. 4.39 Light Deflector Beam Profiles ( r ^ ̂  .9). 144. The deflector light beam profiles are shown in 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 is 2A9^ =5.05 x 10 radians. Using the Rayleigh criterion, the number of resolvable spots is Ng = 2A6̂ /A<J) = 43. The maximum number obtainable at either high drive voltage or reduced diffraction efficiency (due to beam steering) is over 50. Optical coup- ling was effected without a lens over the entire ̂ 2.5 mm width of the input coupling prism. The theoretical number of resolvable spots is N = Afx = 44 s for a bandwidth of 61 MHz, in good agreement with the observed number. The access time is limited by T, which is .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 in Sections 4.2-4.4 appears to agree within about 10% with the observed characteris- tics 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 is 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 in 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 is shown in 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 filter 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. Best results were obtained with an exposure time pf 6 minutes at f/4.5. 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 in 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 in 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 in acid etchants (Fig. 5.3) or alkaline etchants with strong F i g . 5.3 L i f t i n g o f P h o t o r e s i s t . 149. gas evolution. Good results were obtained in an alkaline ferricyanide etchant made with 7ig.K3 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. The negative contrast was increased by intensification. Chromium intensifier was tried first, 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. Figure 5.4 shows the typical result in an etched transducer. 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 Shorted Transducer. Fig. 5.5 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 in 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 its 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 Lifting of Aluminum Film. (2) A layer of aluminum a few tenths of a micron thick was deposited in a Veeco vacuum system using a tungsten coil as the evaporation source. The thickness was monitored with an Ificon 321 quartz crystal film thick- ness 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 photo- resist. 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 Correct Mask Alignment for Z-21.8° SAW Propagation (tan 21.8° = .4). (9) A postbake of 30 minutes at 100°C was given. (10) 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 in 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 dif- fraction efficiency was calculated using the known and measured proper- ties of LiNb03 and the SAW and OGW fields. Addition of a modified theory of acoustic beam steering suitable for the analysis of IDT arrays com- pleted the device model. Good agreement was found between theory and experiments. As pointed out in 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 ̂  in 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. 1 5 6 . The primary limitation in 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 fre- quencies, 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 the problem of stable, efficient coupling to the OWG must be solved. Use of grating couplers should improve the performance in 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 in the calculations. There is some variation in the published values of the refrac- tive indices. At A = 0.6328 ym, Kaminow and Carruthers [4] give n g = 2.214 and n Q = 2.294. These values give the best agreement between experiment and theory. The dielectric permittivity tensor used is therefore r 15 J;2;624 0 0 OD 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 8̂  6 0 0 30.8 0 28 0 28 0 0 -3.4 0 0 x 10 1 2 m/V in the principal axes system, used is The matrix form of the elastooptic tensor 2036 .072 .092 .055 0 0 .072 .036 .092 -.055 0 0 .178 .178 .088 0 0 0 .155 -.155 0 .019 0 0 0 0 0 0 .019 .11 0 0 0 0 .31 .048 158. Most of t h e s e a r e D i x o n and Cohen's v a l u e s [93]. However, t h e s e a u t h o r s do n o t g i v e P44 a n^ Pg6» s o K l u d z i n ' s numbers were used f o r t h e s e [9 3]. On t h e n e x t t h r e e p a ges, t h e d i e l e c t r i c p e r m i t t i v i t y , e l e c t r o o p t i c and e l a s t o o p t i c t e n s o r s 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 about t h e c r y s t a l , a x i s . The r o t a t i o n by 21.8° c o r r e s p o n d s t o t h e v a l u e s used i n t h e 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 . The p e r m i t t i v i t y / change f a c t o r s f o r aeoustiGcsferai-nv e,.P., , e, » and f o r t h e SAW e l e c t r i c f i e l d , e„.r., ,e, „, 33*3klm kn 3j j k l k3 a r e a l s o g i v e n . They a r e c a l l e d DEPSS(L,M) and DEPSE(L), r e s p e c t i v e l y . 159 to i c e 1 > c a c c . c- ci © i IT C1 C if • ^ c t ' \ '-I • j r- (\) c l c rvj • rv n.! f — — i - fx, I , f. fl F . n | a . i i i © o ^ : t o t n i u i u u l ! C C (TJ C : J O © rv © X © © tr> c, : ii ll i<, *- r— >~\ rv r*i j rv' A J I t r Q t r i v. VI a a n. I U i LU Ul c | o c cr c.! f*. «~ <-z ; i i:l ll <i ill - i I V ru ; rvj f • r ru; f\- rv rv) Ci f! o ©{ I I III II -J A.- A J fj I M • fi rd o © c c] «-J C — < iii II II it] *-' " -~- n — . A J f | i^i (V . f l r", r**. f It ci i ! t c-i c. c CS " °I ft lit it 11 M .-• ru f f r\ Aj rv *J ni a u Q "4 tr a a C! — 1 1 I f i a - c o -i II III i! I I _ C Z rv rvj fl C ll 11 rv A j o c o • If* c C 1 f>. © © I .'I; tl II 11 a l a a a j — ry -j ru rv H - 7 i i ! i I o c a a o d e r . c j c i © ©' © © c o c; iri c ir- ©J u'ir c o © r- o| r- o o j o © rvi 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 — RF| A , — fl| A J — f J rv c c © c c. ci c — o ̂  c c II II ll' II II II • — A i f. fl — • f A J — fi r\J a or a-, rr a a a a . a. a c i c c © c < C I/'. C. c e r- c-i r- c c ll it It: ii n li f\, — rvj — C £J £i ; C C c, c © c © o © ©. c \r | f . © e -a ll ll It ll <_< © o cl O — =3' =J < II II If. II II tl] r\ A . rvj f f f j I A , A j fi — — rvi r f» A i f A j • , n ( / , : c c © © © LT> LTi U" ' c i— r~~i T— - c r- AJ r- i t ti tl n. II 5 r r̂ j c o f, f] f. fi - a f " (- A j A J O © • < r- © ©- © • H ll 11' ll »- w-i m— *~ rvj A J rvj rvi A j f ! f - j r e rv, f< — | — f i A ; —• f - A ) — fi rv; — n ft iv; IT o nj ci f r fl' o | a a n. a o, oJ a a CL a. f f) r" — AJ f . f i A ) - © © IP d c. c o c. c c c c <; c o X c O C' M C C © I © c; c c c II 11 II M ll • r\j ru rv • f f a; cr.' ; n c/') f/: w, ( / I 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. 160 CV C, a O •J"' c. ir Ki o =3 o tr A J °_ cr - o It H II — rv m y. a- Cr rt cr. n.. CL a U-' L J U J o o ° Ci O c o c .c ~ rv °" ir- n ii I, 7- ru r\ A.< Ct c C C* o; c: I T a G ii a.' Lil > cr > C* o 1 1 O r. •- r- u-rv K X a rv o fV U.' • • -ft IT- •-- II H I I a. t- ... A f (_ -~ LL' _; X rr Ci" tv tr. IT> t— a. a . C UJ u. rri -r -a L/i —1 - K i at — r-j —• f\' fV f l A.! t*\ a j rv tv rvt —i c ! — r. oi o c <--! rv K> cr c;t o-oj ru" o II li tl it Zl rvi K I f\j -» J c ^ a ) a a' l i a.1 Or u. Or II II tl II as? . cu a a: a: c el tl i U W • - f l 1 r*i K 1 rv C rv t»; a a- • a l C —« — rv tr. :̂ — — -i — O i i 0 r\| A . K'I (VI 'J"' =Jt ^ r-n i\. f- rvt C, W N ^ I I 1 ; 11 It ll| I I c at e c y- r»- -j", r- - *7 AJ / ; a rv. y i IT. 3- rv ir> ! | — c •i II II II • rv r , ; i — rv , •, m K r r 11 tl rv rv • li I M II : rv rvif i"1. i a a . i , a a . a < I rvi rv fj-r K . n i i t i a a CJ c. C C -f c C" c - c- c *- u! cr. — i/ — rv -C — rv i I n ll n ll it i i^ 1 r", P̂ I r • C if' a ' * 1/1 r»: «. I (V A ! f , — rv j rv r̂t rv rvj rv — -| r\i — H r\. rv rvj tv I i . l . c c r • — r— a- - ' -c C rv - rvj —< ^ i I t! !I tl rv i/v J> II !l| 11 rv rv- r̂i f^. K. — -~ t* : rv rvi <v r\j • i a; cr : rx rxja ; • o o e r • r~ o a- r • a rv - i ; .C, rv ; II llj II : tv r v r-A i , — rv. • rv a a. n it II rv rvi '1 »- K1 f\" rv (V ru : rv rv rv - j rv .-o K I — ll ll ll it li I A rv iv K ** I ̂  «̂  rv — i " i --t rv rv < f C K l A l - M (J-I f l̂ a a am a. a\ I rv rv rvi rv rv rv| r\, r -c rvj r- rj (. K | O i — tr rv i C K C a if' c-| r- 7̂ —• n r •-! r\ N AJ ^ | li ti u it it I z z z' z N, rv r\ I a a rv n rv r4 . rvj rv - . (\ K ( • • i\. • ^ rv1 - rv M tl U| 11 II iv rv' r. rv a. a. a a a a a rv r\ rvi iv rv r\j | c c c. c c K\ t cj i rv v «— a I ^ -i — • l i tt it ti it rv u-' o' v r-J". O ' OL. - -o -L> Ln ^ i X rv- *- — rvj I ' I I j n ti ii ii ii n 1 rv rv rvi K ' N j rv rv — - . A,.* - r> t /: c a ^ r; -cr <x- ^ -x ci -C ̂ " r -tr C; c o Z" IT CI i j A . A ' A j A. — . r I.'' -f C - I 0 -f i rv: - ̂  >c =t m A . r, rv rv c o c c o <z c- d it ii it H It tr ll tl it n it • —•> rv • Ul u.: 11. <r| v.i i-j t/; (rj a. o a cJ tr. tO tT; 'JT. y a a a. a aj U, LiJ UJ ILI L c c o c; c q i O ' ANGLE OF R O T A T I O N IN XZ OLANE - R0.00 DEGREES D J E L E C T R T C P E R M I T T I V I T Y TEtiSnR F D S » ( 1 1 ) : a, _FP_SR('2)= 0. EPS* (13)= o", R n i s o no FPso(21) = n,ooooo - o i O O O O n - O ! E o S R ( ? 2 ) = _ 5 . ? J > 2 £ 0 » « _ OOOOD-OI ~ £ B S P ( 2 3 ) = 0." 110 Of: 0-61 F P S R t 3 1 ) = 0 . O O O O D - O l E P S 9 ( 3 2 ) =_ _ 0 ' . OOOOD-0 1_ E P s R ( 3 3 ) = " 5'. 2 h 2 ' l 0 00 E L F C T S n O o T I C TENSOR R R f 1 1 I ) = P R f ' ! ! ) = R » { 1 2 2 1 s i . o e o n - i i 0. o o 0 ti - n 1 1 . aoon-1 I R R ( 2 i 1 1 = »P ( 2 3 1 1 = R R ( 2 ? 2 i = R 9 I U J ) : O . O O O D-01 RPfl33j= l . a O O D - l i O.ooon -0 ! o. ooon-o i i j i o o n - 1 2 RR C 2 ! 3 1 = RR ( 233 l= 0 . 0 0 0 0 - 0 1 O.OOOD-01 RR(31 1 l : RRf 3^ 1 ): J L R J 3 2 2 J J R R f 3 1 3 ) : R » ( 3 3 3 ) : O.OOOD-01 B . a 0 01) - 1 2 3.UOOD-12 1 . u o o n - l 1 o. o o » o - n t RRM21 ) = PR(I12)= J?HC132„)^. RRf123)= RRf O.OOOD -01 P R f 2 2 1 ) = e . 60 O C - 1 2 RR. (321) = O.OOOD - 0 1 O.OOOD - o i R R ( 2 1 2 ) = j . f i O O D - 1 1 R R ( 3 1 2 ) = O .ncoD-01 n.oonp-ni Rgj_?32)= 3 . " Q O D - I 2 RR ( 332 )--i. a o o o -I 2 o.oobn -o i RR(223)= O . O O O D - 0 1 R « f 3 2 3 ) = O.OOOD-01 I t ASTOnPT'C TE ' JSP.R P R f i i i n - fl.aoon. 02 PRC 21 t i ) 0 . 0 0 0 0 -01 PR(3111) z 0 . fl 0 0 D -01 PR(12111 = 0 . 0 0 0 0 - 0 1 PR(22I1) = 9 , 2 0 0 [.) -0 2 PR (321 1 ) 0 . 0 0 0 D -01 P 3 f 1 5 1 1 ) - 0 . 0 0 0 n-01 P R ( P 31 1 1 0 . 0 0 0 0 -01 R R f 3 3 1 1 ) z Q . 2 0 0 0 - 02 PR C 1 1 ,! 1 ) 0 , 0 0 0 0 - 01 PR(2121 ) " , 7 5 0 ! i -0 3 P R ( 3 1 2 I ) 0 , 0 0 0 0 -P I . P P f 1 2 2 1 ) Z a . 7 5 0 0 - 03 RR ( 2 2 2 1 ) -2 . 75(,0 -02 R R C 3 2 2 1 ) z 0 . 0 0 0 0 -0 1 PR(1321) = 0 . 0 0 0 0 - 0 1 PR(2321 ) 0 . 0 0 0 D ~ 0 1 P R ( 3 3 2 1 ) 2 • 7 5 0 0-02 PR f 1 1 31 ) = 0 . 0 0 0 0 - 01 P P 0 1 3 1 ) - 0 . 0 0 0 0 -01 P R f 3 1 3 1 ) z 11 . 7 5 0 0 - 03 P B d ? : ? ] ) 0 .0000- 0 1 PR(2231 ) = 0 , 0 0 0 0 - 0 1 P P ( 3 2 3 1 ) = 1 . 3 7 5 0 - 0 2 PR<1331) - « ,75on- "3 PR >' 2 33 1 ) - l ' . 3 7 5 0 - 0 2 PR f 3 3 3 1 ) z 0 . 0.0 AO • 01 P P ( 1 1 1 2 ) _0 .0000- 0 1 PR ( 2 !|2) .31.7.5 n.o_- 0 3' PR r31 12) 0 . 0 0 0 0 -01 PR I i 2 I 2) , 7 5 0 0 - 0 3 PRf?2 l21 -2 . 7 5 0" -0? RR f32!2) Z 0 ,(11)00-0 1 P R f 1 3 ) 2 ) " 0 .0000-0 1 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 PR f1 ( 2 ? ) - I ,79no- 01 P R ( 2 1 2 2 ) -7 .750" - 0 2 P R ( 3 1 2 2 ) z 0 . 0 0 0 0 - 0! PRC 1222) = -7 .750n- 02 PI- ( 2 2 2 2 ) — 3 , 6 0 0 0 - »?. P R ( 3 2 2 2 ) 0 , 0 0 0 0 -01 P P ( 1 3 2 2 ) 0 , 0 0 0 0 - 01 P R i ? 3 2 2 ) z 0 . 0 0 0 0 -01 P R ( 3 3 2 2 ) z 7 . 2 0 0 0 - 0 2 P R ( 1 1 3 2 ) 0 . 0 0 0 0 - 1! I R»(2132) 0 , 0 0 0 D -01 P R ( 3 1 3 2 ) 7 , 5000- 02 »R(1232) - o . 0 0 0 0 - 01 PP f 2 2 3 2 ) z 0 . oooo - n i PR f 3 2 3 2 ) z 0 . 0 0 0 0 - 01 PR(1332) = 7 . 5 0 0 0 - 02 ' PR(23?2) = 0,0000- 0] R R ( 3 3 3 2 ) 0 , 0 000-01 P p f 1 1 3 3 ) - 0 . 0 0 0 0 -01 PR(21131 s 0 . 0 0 0 0 -01 P R ( 3 l 1 3 ) z 11 , 7 5 0 D - 0 3 PR(12!3) 0 . 0 0 0 0 - 0 1 P R ( 2 2 ! 3 ) = 0 , 0 0 0 0 -01 P R ( 3 2 1 3 ) - 1 . 3 7 5 0 - 0 2 PR f1513) - a . 7 5 0 0 - _ PR (?3l_3l z .1 .3750 -!!2_ „ P R f33! 3 ) Z _fl , 0 0 o 0 -0_L_ PRO 12?) _0 . 0 0 0 0 - 0 1 e ? ( ? ! 2 3 ) 0 . fn. 0 0 -0 1 .PRC 3 12 3.) =. _ 7 . SJlO.Or .02 P B f J 2 2 3 ) z 0 . 0 0 0 0 - in PR ( 2 2 2 31 z 0 . 0 0 0 0 -0 1 P R C 3 2 2 3 ) Z 0 , 00n(>-0 1 P R ( 1 3 2 3 ) = 7 , 5 0 0 0 - 02 P R ( 2 3 2 3 ) = 0,0006-01 PR(3323) z 0 . OOOO- 01 PO ( i t33 ) z 1 , 7 B 0 0 - 01 P R ( 2 i 3 3 ) = 7 .750O -02 PR(3133) z 0 . 0 0 0 0 - 01 PR(1233) 7 . 7 5 0 0 - 02 PR(2233) = 7.2000-02 PR ( 3233 ) z O.OOOO- 01 P R f i 3 3 3 ) s 0 , 0 0 0 0 - 0 1 PR (P.333) .!) . 0 0 n 0 -01 P R C 3 3 3 3 ) z 3 ,<>0flD-02 R R ( P E R M I T T I V I T Y CHANGE F A C T O R S FOR TE MODE WITH E L E C T R I C F I E L D ALONG R O T A T E D Z A V I S 0 E P 3 EI 1 ) = D E ° S F f 2 ) = _0EPSEI3)s__ D E R o S ( 1 i 1 = D E P S?(12)= D F R S S ( 1 3 ) = 0FPSSC21)= O F P S S(22 1 = P F P S ? ( 2 3 ) = 0 F P S S < 3 1 ) = 0 F . P S S ( 3 2 ) ^ O E P S S C33) = .0 . 2 3 6 1epD-OP 0.o«157ftO-10 - o . o o o o o o O on , ?5'i 77fio . 7hl5h"0 01 0 0 -o.noonnco 00 - n , 7hi 5>,«n .0,10 0 3 010 00 01 . 0.0 0 0 0 0 0 0 00 - o . o n o d o O O 0 0 - 0 , 0 0 0 0 0 0 0 0 0 - 0 Q 9 o R 5 n D 0 0 O N 162. APPENDIX II WAVE PROPAGATION IN ANISOTROPIC MEDIA An anisotropic, non-absorbing medium is characterized by the dielectric tensor e = E l l 0 0 0 E22 0 0 0 e33 in the principal axes system. The wave equation has the form V X (V X E)= - 32S C at (II.l) For plane harmonic waves of the type, e^^'r the following opera- tor identification can be made: rf — — -*• -10) 3t 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 in the same plane. Expansion of the triple cross product and use of £•$ = 0 gives 2 2 E. [)= ED cosG = >f-y-2 D = < ^- D , k c c (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 X H , 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) consti- tute a mutually orthogonal triad of vectors. This is illustrated in Fig. II.1. The ray velocity (defined as the velocity of power flow) is then given by u = v /cos9 . (11.4) The principal indices of refraction are defined by n. (II.5) Fig. II.1 When equation (II.2) is written in Cartesian components, three homogeneous linear equations in E x > E^ and E^ result. A nontrivial solution exists only i f the determinant of coefficients[85] 164. [ ( n ^ / c ^ - k y - k 2 ] k k y x k k z x k k x y [ ( n 2 o ) / c ) 2 - k 2 - k 2 ] k k z y k k x z k k y z [ ( n , w / c ) 2 - k 2 - k 2 ] 3 x y = 0, (11,-6) where the s u b s t i t u t i o n ( I I . 5 ) has been made. T h i s d e t e r m i n a n t r e p r e s e n t s a 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 . I n a l l d i r e c t i o n s b u t t h o s e a l o n g t h e o p t i c axes o f the medium, the s u r f a c e i s d o u b l e - v a l u e d . I t can t h e n be shown t h a t two k v e c t o r s can e x i s t , c o r r e s p o n d i n g t o two m u t u a l l y o r t h o g o n a l d i r e c t i o n s o f wave p o l a r i z a t i o n . I f t h e s u b s t i t u t i o n i u k. = v. —7T i i 2 v i s made i n ( I I . 6 ) , the d e t e r m i n a n t assumes t h e f o r m r 2 4. 2 2 2 n Ln,v /c -v -v ] 1 y z v v y x V V Z X V V x y r 2 4. 2 2 2, [ n 2 v Ic - v x - v z ] V V z y v x v X z V V y z r 2 4. 2 2 2 n [ n 0 v /c -V -v ] 3 x y = 0, ( I I . 7 ) w h i c h r e p r e s e n t s t h e d o u b l e - s h e e t e d phase v e l o c i t y s u r f a c e . Two v a l u e s o f phase 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 t r a v e l - l i n g i n the same d i r e c t i o n . When I) i s e x p r e s s e d i n terms o f i t s p r o j e c t i o n s a l o n g E~ and u, -4. n -v .T\ ( I I . 8 ) 3 = E . f cose + u ^ W r i t i n g t h i s i n terms o f C a r t e s i a n components and s e t t i n g t h e d e t e r m i n a n t o f c o e f f i c i e n t s e q u a l t o z e r o g i v e s 165. r 2. 2 2 -2, Ic /n -u -u / 1 y z u u y z u u Z X u u x y , 2. 2 2 2, {c /n2-ux-uz} u u z y u u x z u u y z / 2/ 2 2 2\ {c /n„-u -u } 3 x y = 0, ( H - 8 ) the equation for the ray velocity surface. These equations may be utilized as follows. If propagation is, for example, in the xy plane, setting k , v z and u z equal to zero gives three sets of equations: k =0: k 2 + k 2 = (n.u/c)2 z x y 3 • k 2/(n l W/c) 2 + ky/(n 2u/c) 2 = 1 2 2 2 v = 0: v + v = c /n0 z x y 3 2. 2 . 2. 2 4.2 Vx / n2 + V y / n l = V / C n 2 , 2 2.2 u = 0 : u + u = c /n, z x y 3 2 2 , 2 2 2 n.u + n.u = c , 2 x 1 y which describe the propagation characteristics of waves in the xy plane. Waves with E = E ẑ  propagate with refractive index n , whilst waves with E = Exx + E^y propagate with a refractive index with magnitude between n^ and n2- 16.6. REFERENCES 1] D. Mar c u s e , Theory o f D i e l e c t r i c O p t i c a l Waveguides, Academic P r e s s , N.Y., C h a p t e r one (1974). 2] P.K. T i e n , A p p l . Opt. 10,2395(1971). 3] I.P. Kaminow and R.V. Schmidt, A p p l . Phys. L e t t . 25,458(1974). 4] I.P. Kaminow and J.R. C a r r u t h e r s , A p p l . Phys.. L e t t . 22,326 '(1973). P.K. T i e n , S. R i v a - S a n s e v e r i n o , R . J . M a r t i n , A.A. B a l l m a n and H. Brown, Appl:.- Phys>s:Lett,. 24, 503'(1974) . E. C o n w e l l , Appli^-Physv-Lett... 23,328"(1973) . . .S' . E. C o n w e l l , Appl:. P h y s v : L e t t . 25,40(1974) .' L. S c h i f f , Quantum M e c h a n i c s , M c G r a w - H i l l , 178(1949). D. Mar c u s e , IEEE J. Quantum E l e c t r o n . QE-9,1000(1973). G.B. Hocker and W.K. B u r n s , IEEE J. Quantum E l e c t r o n - . QE-11,270 (1975) J.M. White and P.F. H e i d r i c h , Appl:.:.Opt. .15, 151 (1976) . J.M. Hammer and W. P h i l l i p s , Appl:. Phys. .Lett..-24-,^;545'( 1974) . W. P h i l l i p s , F e r r o e l e c t r i c s 10, 221 (1976). V. Ramaswamy and R. S t a n d l e y , A p p l . Phys... L e t t . 26, 1 0 ( 1 9 7 5 ) . J . Noda, A p p l . Phys.. L e t t . , 2 7 , 1 3 1 ( 1 9 7 5 ) . J . M i d w i n t e r , IEEE J . Quantum E l e c t r o n . QE-6,583 (1970) . J . H a r r i s and R. S c h u b e r t , IEEE ^Trans'. ,MTT-19:, 269(1971). M. Dakss, L. Kuhn, P.F. H e i d r i c h and B. S c o t t , A p p l . Phys. L e t t . 17," 265(1970)": R. U l r i c h and R. Tor g e , A p p l .,_0pt.' 12, 2901 (1973). W. Tsang and S. Wang, A p p l i - P h y s . , L e t t . _ 2 4 , . 1 9 6 ( 1 9 7 4 ) . .-. J.R. C a r r u t h e r s , I . P . Kaminow and L.W. S t u l z , A p p l . Opt. 1 3 ( 1 9 7 4 ) . 2333 L. Brown, M e t a l l u r g y Department, UBC, p r i v a t e c o m munication. 167. A. Warner, M. Onoe and G. Coquin,J. Acoust. Soc. Am. 42,1223(1966). G.B. Brandt, E.P. Supertze and T. Henningsen, Appl. Opt. 12,2898 (1973). P.A. W h i t e and S.E. Smith, I n e r t Atmospheres, Butterworth and Co., 125 (1952) . Per Kofstadt, Oxidation of Metals, J . Wiley and Sons, N. Y., 170(1966). W. Ballman and M. Gernand, Phys. Stat. Sol. (a) 9, 301 (1972). R. Swalin, Thermodynamics of Solids, J. Wiley and Sons, 303(1962). Gmelin's Handbuch G57bL2, 414. ibid., 496. ibid., 1004. A.S. Grove, Physics and Technology of Semiconductor Devices, John Wiley and Sons, 45(1967). D.L. Staebler and W. Phillips, Appl. Phys. Lett.-24, 268(1974). . J. Powell and B . Craseman, Quantum Mechanics, Addison-Wesley Co., Reading, Mass., 140-147 (1965). M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, 298 (1972) ibid.,.446. R. White and Voltmer, Appl. Phys. Lett. 7,314(1965). J. Nye, Physical Properties of Crystals, Oxford, 99(1969). G.A. Coquin and H.F. Tiersten, J . Acoust. Soc. Am. 41,921(1966). A.J. Slobodnik, E.D. Conway and R.Tv.Delmonico, Editors, Microwave Acoustics Handbook, AFCRL-73-0597 (1973). J . Campbell and W. J o n e s , .IEEE T r a n s . , . SU-15,209(1968). R. Spaight and G. Koerber, IEEE :Trans., SU-18,237(1971) M.G. Cohen, J . Appl. P h y s . 38,3821(1967). R. Weglein, M. Pedinoff and H. Winston, Electron. L e t t . 6,654(1970). J.C. Crabb, J.D. Maines and N.R. Ogg, Electron. L e t t . 7,253(1971). 168. [46] T.L. Szabo and A.J. Slobodnik, IEEE Trans., SU-20, 240(1973). [47] M. Kharusi and G. Farnell, J. Acoust. Soc. Am. 48, 665(1970). [48] R.M. De LaRue, C. Stewart, C.D.W. Wilkinson and I.R. Williamson, Electron. Lett. 9, 326(1973). [49] R.F. Milsom and M. Redwood, Proc. IEE 118, 831 (1971). [50] C. Tseng, IEEE Trans., ED-15, 586(1968). [51] S. Joshi and R. White, J. Acoust. Soc. Am. 46, 17(1968). [52] H. Engan, IEEE Trans., ED-16, 1014(1969). [53] B. Auld and G. Kino, IEEE Trans., ED-18, 898(1971). [54] J. Collins, H. Gerard and H. Shaw, Appl. Phys. Lett. 13, 312(1968). [55] K. Ingebrigsten, J. Appl. Phys. 40, 2681(1969). [56] P. Emtage, J. Acoust. Soc. Am. 51, 1142(1972). [57] "W.R. Smith, H. Gerard, J. Collins, T. Reeder and H. Shaw, IEEE Trans. MTT-17, 856 and 865 (1969). [58] A. Bahr and R. Lee, Electron. Lett. 9, 281(1973). [59] R. Weglein and G. Nudd, IEEE Trans., ED-16, 1014(1969). [60] F. Marshall, C. Newton, and E. Paige,-.IEEE Trans., MTT-21,206(1973). [61] R.V. Schmidt, J. Appl. Phys., 43, 2498(1972). [62] M. Itaniel and P. Emtage, J. Appl. Phys. ,43, 4872(1972). [63] K. Lakin, IEEE Trans., MTT-22, 418(1974). [64] A. Warner, M. Onoe and G. Coquin, J. Acoust. Soc. Am. 42,1223(1966). [65] N. Reilly, R. Milsom and M. Redwood, Electron. Lett..9, 419(1973). [66] C. Raman and B. Nath, Proc. Ind. Acad. Sci., 2A, 406, 1933. [67] W.G. Mayer, G. Lamers and D. Auth, J- Acoust. Soc. Am. 42, 1255(1967) [68] L. Brillouin, Ann. Phys. (Paris), 9th ser., V17, 88 (1922). [69] P. Debye and F. Sears, Proc. Nat. Acad. Sci (USA) 18, 409 (1932). [70] L. Kuhn, M. Kakss, P. Heidrich and B. Scott, Appl. Phys. Lett. 17, 265(1970). 1 6 9 . [71] [72] [73] J . White, P. H e i d r i c h and E.G. Lean, E l e c t r o n . L e t t . 10,510(1974). [74] C S . T s a i , M.A. A l h a i d e r , L.T. Nguyen and B. Kim, P r o c . IEEE 64, 318(1976). [75] K.W. Loh, W.S.C. Chang, W.R. Smith and T. Grudkowski, A p p l Opt 15, 156(1976). * [76] R.V. Schmidt, IEEE Trans., SU-23, 2'2('l976). [77] G.A. Alphonse, RCA Review 33,543(1972). [78] .M. Born and E. Wolf, P r i n c i p l e s of O p t i c s , Pergammon P r e s s 569(1965). [79] J . Nye, P h y s i c a l P r o p e r t i e s of C r y s t a l s . Oxford (1969). [80] R.C. Chu and T. Tamir, IEEE Trans., MTT-17, 1002(1969). [81] C o l l i n , F i e l d Theory of Guided Waves, McGraw-Hill,368(1960). •[•82] R. Dixon, IEEE 3. Quantum E l e c t r o n . 3, 85(1967). [83] E. Gordon, P r o c . IEEE 54, 1391(1966). [84] A. K o r p e l , R. A d l e r , P. Desmarais and W. Watson, A p p l . Opt. 5, 1667 (.1966). [85] G.R. Fowles, I n t r o d u c t i o n t o Modern O p t i c s , H o l t , R i n e h a r t and Winston, Chapter 4 (1968). [86] D.A. Pinnow, IEEE Trans., SU-18(1971). [87] C S . T s a i , L. Nguyen, S. Yao and M. A l h a i d e r , A p p l . Phys. L e t t . 26 140(1975). [88] B. Kim and C. T s a i , P r o c. IEEE 64, 329 (1976). [89] G.I. Stegeman, IEEE Trans., SU-23, 43(1976). [90] B l a c k and White P r o c e s s i n g Data Book, American P h o t o g r a p h i c Book P u b l i s h i n g Co. I n c . , N.Y. (1975). [91] E . J . Walker, IEEE Trans., ED-22, 464(1975). [92] H. K o g e l n i k , IEEE Trans., MTT-23, 2(1975).

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