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 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 H^ , g i v e t h e e i g e n v a l u e e q u a t i o n 2 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 = 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 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 (2-14) 8. K 2(y) = n 2 ( y ) k 2 - f3 2 , (2.15) 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 (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 ±n^ 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 + -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 = 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 3^., 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 3^ . 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 = -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/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. ~ 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 = NIT2 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° - 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 . = -> 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 " 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) = 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 ^ (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 0 and c j ^ , 0 = - J o i g ^ e x p ( j B ) ( 4 . 3 0 ) and K = - J a-,^ and Q + aQa-f (4.34) where p = Ak /2. x Similarly, i t can be shown that d)£ + 2jpQ and 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 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 0(x) = exp(jpx) (cos(qx)- j ^ ; sin(qx)) (4.43) and a l 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 -Q0 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 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 . Thus, the diffraction integral becomes M-l nD+G S() = I / expj(Kx - 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) is the aperture function, and B() 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 — ( - ) are at A^min 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() , 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 - 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 = 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 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 KGaQ (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 f2 = 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 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 = x^ /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/^ x^ -»• jfi/v can be made. Differentia-tion with respect to x^ 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. 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A,.* - r> t /: c a ^ r; -cr c =t m A . r, rv rv c o c c o rv • Ul u.: 11. 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. 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