NONLINEAR ELECTROACOUTICS OF DIELECTRIC CRYSTALS : FUNDAMENTALS AND APPLICATIONS

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NONLINEAR ELECTROACOUTICS OF

DIELECTRIC CRYSTALS : FUNDAMENTALS AND APPLICATIONS

D. Nelson

To cite this version:

D. Nelson. NONLINEAR ELECTROACOUTICS OF DIELECTRIC CRYSTALS : FUNDAMEN- TALS AND APPLICATIONS. Journal de Physique Colloques, 1979, 40 (C8), pp.C8-149-C8-163.

�10.1051/jphyscol:1979826�. �jpa-00219531�

JOURNAL DE PHYSIQUE Colloque C8, supplement au n°ll, tome 40, novembre 1979, page C8-149

NONLINEAR ELECTROACOUSTICS OF DIELECTRIC CRYSTALS : FUNDAMENTALS AND APPLICATIONS D.F. Nelson

Bell Laboratories Murray Hill, New Jersey 07974.

Résumé.- Les équations dynamiques et les relations constitutives des phénomènes élastiques, élec- triques et électrostatiques dans les cristaux sont entièrement déduites d'une théorie électrody- nasrique Lagraangienne. Les équations s'appliquent à des cristaux de symétrie, de complexité ou de' propriétés non linéaires quelconques, incluant les matériaux pyroélectriques, diélectriques et piézoélectriques. Nous avons porté une attention particulière aux non linéarités d'ordre le plus bas c'est-à-dire aux termes croisés ou quadratiques des variables élastiques et électriques. Une relation entre le tenseur d'électrostriction et la limite basse fréquence du tenseur élastoopti- que, différente de celles existant dans la littérature est obtenue et discutée. A titre d'appli- cation nous déduisons les équations couplées décrivant l'interaction paramétrique de trois ondes acoustiques. Ces équations s'appliquent â la propagation des différents modes pour des directions quelconques dans des matériaux diélectriques, piézoélectriques ou pyroélectriques. Une expression condensée décrivant l'interaction additive (génération du 2ème haromonique comprise) ou soustrac- tive de deux ondes acoustiques de fréquences différentes est alors calculée. Le coefficient d'in- teraction du matériau qui pilote l'interaction est donné sous une forme générale différente des résultats connus dans la littérature. Nous déduisons ensuite les équations gouvernant les modes couplés décrivant l'expérience de Thompson-Quate, c'est-à-dire l'interaction paramétrique de deux ondes acoustiques de sens opposé et d'un champ électrique uniforme oscillant à la fréquence de second harmonique de la fréquence acoustique. Nous en déduisons une expression explicite générale du coefficient d'interaction des matériaux qui détermine l'intensité du phénomène. Nous montrons également que nos équations sont différentes de celles de Thompson-Quate, nous utilisons en effet la composante de la vitesse de groupe normale à la surface du cristal au lieu de la vitesse de phase.

Abstract.- Based on a fully electrodynamic Lagrangian theory of elastic dielectrics, a completely deductive derivation of the dynamical equations and constitutive relations for elastic, electric, and electroelastic phenomena is presented. The equations apply to crystals of arbitrary symmetry, structural complexity, and nonlinearity. These include pyroelectrics as well as dielectrics and piezoelectrics. Emphasis is placed on the lowest order nonlinearities, that is, ones depending either bilinearly or quadratically on the elastic and electric variables. A relation between the electrostriction tensor and the low frequency limit of the elastooptic tensor, different from any previous relation in the literature, is derived and discussed. As an application, we derive the coupled mode equations governing the parametric interaction of three acoustic waves. This deriva- tion applies to the propagation of any mode types in any directions (consistent with being close to or at phase matching) in any dielectric, piezoelectric or pyroelectric crystal. A. compact ex- pression describing the mixing of two input acoustic waves to produce a sum frequency (including second harmonic) or difference frequency output is then calculated. The material interaction coef- ficient that governs the mixing process is obtained in a general form ; it differs from previously published forms. Next we derive the coupled mode equations governing the Thompson-Quate experiment, that is, the parametric interaction of two counter-propagating bulk acoustic waves and a spatially uniform electric field oscillating at the second harmonic of the acoustic frequency. A new result of the derivation is a general but explicit expression for the material interaction coefficient that governs the strength of the process. We also find that the equations differ from the generic equations assumed by Thompson and Quate by the replacement of a phase velocity with the component of the group velocity normal to the crystal surface.

1. INTRODUCTION.- The work of Thurston and cowor- kers ' " ' and Torguet and Bridoux ' ' laid a sound basis for interpreting measurements of elastic non- linearities in dielectric orystals. Since then seve- ral papers ^ ~ ' have proposed generalizations to cover piezoelectric crystals. Such crystals have electric and elastoelectric nonlinearities in addi- tion to the purely elastic nonlinearity and must be described by electric as well as elastic equations.

There is little agreement between tne cited papers

^ " ' and, with the exception of one related pa-

per * ' they present no intercomparison. It thus seems worthwhile to present a derivation sufficien- tly careful and explicit in its method and general in its formulation that the various ambiguities in this field can be resolved.

The theory we present here is based on a long wavelength Lagrangian (wavelengths » unit cell dimensions) which allows a completely deduc- tive treatment once the Lagrangian is Known. It has been previously constructed ^ ' ' from a rai-

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Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979826

~ 8 - 1 5 0 JOURNAL DE PHYSIQUE croscopic viewpoint f o l l o w e d by a long wavelength

l i m i t . The l a t t e r produces a macroscopic, continu- um f o r m u l a t i o n . The theory i s completely e l e c t r o - dynamic, t h a t i s , i t i s n o t r e s t r i c t e d i n i t s f o r - m u l a t i o n t o t h e q u a s i - e l e c t r o s t a t i c (low frequency) regime as a l l the previous treatments (5-13) have been. We w i l l , however, a l s o l i s t the quasi-elec- t r o s t a t i c approximation o f our f i n a l r e s u l t s .

Our f o r m u l a t i o n a p p l i e s t o c r y s t a l s o f a r b i - t r a r y symmetry, anisotropy, s t r u c t u r a l complexity, and order o f n o n l i n e a r i t y . A f t e r t h e i n i t i a l formu- l a t i o n , however, we w i l l concern ourselves here o n l y w i t h the f i r s t l e v e l o f n o n l i n e a r i t y which gives r i s e t o b i 1 in e a r o r q u a d r a t i c n o n l i n e a r terms i n the e l a s t i c o r e l e c t r i c equations. We w i l l n o t consider l o s s o f any k i n d ; thus we w i l l exclude e l e c t r i c a l conduction. Our treatment w i l l apply t o o r d i n a r y d i e l e c t r i c s , p i e z o e l e c t r i c s , and t o pyro- e l e c t r i c s . The l a t t e r category was n o t covered by any o f t h e previous s t u d i e s . (5-13) I n t e r e s t i n g l y , we f i n d t h a t t h e n o n l i n e a r e l e c t r o a c o u s t i c equa- t i o n s are i d e n t i c a l f o r p y r o e l e c t r i c s and piezo- e l e c t r i c ~ (except, o f course, f o r the presence o f a spontaneous p o l a r i z a t i o n i n t h e former) provided t h a t t h e spontaneous e l e c t r i c f i e l d o f t h e pyro- e l e c t r i c has been c a n c e l l e d by e x t r i n s i c surface charge., We w i 11 n o t concern ourselves w i t h t h e ad- d i t i o n a l n o n l i n e a r i n t e r a c t i o n s made p o s s i b l e by t h e presence o f t h e spontaneous e l e c t r i c f i e l d s i n - ce experiments are seldom done on p r i s t i n e pyro- e l e c t r i c c r y s t a l s which possess t h e i r spontaneous e l e c t r i c f i e l d s .

An i m p o r t a n t technique .of t h e present work i s t h e use o f m a t e r i a l frame electromagnetic f i e l d s . (17-'0) ~ o t o n l y are r o t a t i o n a l l y i n v a r i a n t measu- r e s o f t h e m a t e r i a l p o l a r i z a t i o n ( o r t h e e l e c t r i c f i e l d ) e s s e n t i a l f o r expressing t h e s t o r e d energy o f t h e c r y s t a l , b u t i t i s u s u a l l y t h e m a t e r i a l mea- sures o f t h e f i e l d s t h a t are a c t u a l l y measured.

This occurs because e x c i t a t i o n and d e t e c t i o n d e v i - ces i n e l e c t r o a c o u s t i c s are u s u a l l y attached t o the c r y s t a l under study and so make the boundary c o n d i t i o n s known i n t h e m a t e r i a l frame, n o t t h e s p a t i a l frame. This i s t y p i c a l l y t r u e i n t h e quasi- e l e c t r o s t a t i c regime and remains t r u e f o r the e l a s - t i c f i e l d above t h a t regime.

I n s p i t e o f t h e use o f a m a t e r i a l measure

o f t h e e l e c t r i c f i e l d i n t h e book o f Born and Huang (21) ( c a l l e d by them t h e " c o n t r a v a r i a n t e l e c t r i c f i e l d " ) and a m a t e r i a l measure o f t h e po- l a r i z a t i o n i n Toupin's w i d e l y recognized paper,

most o f t h e workers i n t h i s f i e l d o f n o n l i - near e l e c t r o a c o u s t i c s have f a i 1 ed t o use m a t e r i a1 frame f i e l d s . T i e r s t e n and Baumhauer ( I 3 ) are an important exception t o t h i s . t*!e w i l l postpone a comparison o f our work t o these o t h e r s u n t i l t h e concluding s e c t i o n o f t h i s paper.

Our Lagrangian theory has been formulated i n a completely electrodynamic manner since our i n i ti a1 i n t e r e s t s were o p t i c a l phenomena. (23,24) The f o r m u l a t i o n thus includes magnetic f i e l d s , mo- v i n g media electromagnetic e f f e c t s , and an accoun- t i n g of a22 t h e modes o f mechanicalmotion of the c r y s t a l t h a t i n c l u d e o p t i c modes as we1 1 as acous- t i c modes. T h i s i s done v i a t h e use o f a complete s e t o f i n t e r n a l coordinates o f t h e c r y s t a l . I t i s e s s e n t i a l i n a theory capable o f accounting f o r o p t i c a l as w e l l as a c o u s t i c a l phenomena t h a t t h e s t o r e d energy o f the c r y s t a l be a f u n c t i o n o f r o - t a t i o n a l l y i n v a r i a n t measures o f a l l t h e i n t e r n a l coordinates ( n o t j u s t t h e e l e c t r i c f i e l d as i s p o s s i b l e when o n l y a c o u s t i c modes o f t h e c r y s t a l are considered) as w e l l as a r o t a t i o n a l l y inva- r i a n t measure o f t h e s t r a i n . The resonances o f t h e i n t e r n a l motions produce t h e frequency d i s u e r s i o n o f a l l t h e i n t e r a c t i o n tensors i n t h e i n f r a r e d and u l t r a v i o l e t regions. The frequency r e g i o n of i n t e - r e s t f o r coherent generation o f a c o u s t i c ( u l t r a s o - n i c , hypersonic, etc.) waves i s w e l l below those regions. Thus f o r e l e c t r o a c o u s t i c s t u d i e s t h e i n - t e r n a l motions o f the c r y s t a l f o l l o w t h e e l a s t i c motion o f the c r y s t a l i n an i n e r t i a l e s s manner. !Je thus use a procedure c a l l e d a d i a b a t i c e l i m i n a t i o n o f t h e i n t e r n a l coordinates from t h e Lagrangian i n t h i s study.

The previous use o f t h i s f o r m u l a t i o n t o s t u - dy t h e e l a s t o o p t i c e f f e c t (23) ( a l s o c a l l e d t h e p h o t o e l a s t i c o r p i e z o o p t i c e f f e c t ) l e d t o an ex- p r e s s i o n f o r t h e e l a s t o o p t i c tensor i n terms o f t h e s t o r e d energy constants. The present study ob- t a i n s a s i m i l a r s o r t o f expression f o r the elec- t r o s t r i c t i o n tensor and so produces a r e 1 a t i o n between these two tensors. S u r p r i s i n g l y , i t d i f - f e r s from a l l such previous r e l a t i o n s . (25-30) A discussion o f these previous r e l a t i o n s w i l l be

D.F. Nelson

presented i n c o n j u n c t i o n w i t h the discussion o f o u r r e s u l t i n a l a t e r s e c t i o n .

A f t e r o b t a i n i n g t h e general r e s u l t s we w i l l consider a few a p p l i c a t i o n s . F i r s t we w i l l o b t a i n t h e coupled mode equations o f t h r e e i n t e r a c t i n g a c o u s t i c waves. These equations a r e then solved f o r sum o r d i f f e r e n c e frequency generation. The two i n p u t and one o u t p u t a c o u s t i c waves can be o f any mode type, pure o r h y b r i d . They can propagate i n any d i r e c t i o n ( c o l l i near o r n o n c o l l in e a r ) r e 1 a- t i v e t o a c r y s t a l o f a r b i t r a r y symmetry as l o n g as t h e t h r e e propagation v e c t o r s s a t i s f y o r approxi- mately s a t i s f y phase matching. The r e s u l t a p p l i e s t o d i e l e c t r i c s , p i e z o e l e c t r i c s and p y r o e l e c t r i c s (provided i n t h e l a t t e r the spontaneous e l e c t r i c f i e l d has been cancel l e d by e x t r i n s i c surface charge). The e f f e c t i v e t h i r d order s t i f f n e s s go- verning a c o u s t i c wave m i x i n g t h a t we f i n d i s n o t o n l y more general i n i t s range o f a p p l i c a t i o n than any previous r e s u l t b u t i t contains new terms n o t p r e v i o u s l y obtained. T h e i r absence i n previous t h e o r i e s ( 6 y 7 ) can be t r a c e d i n each case t o t h e l a c k o f r o t a t i o n a l i n v a r i a n c e o f t h e s t o r e d energy w i t h r e s p e c t t o the e l e c t r i c v a r i a b l e .

The o t h e r a p p l i c a t i o n we t r e a t i s the d e r i - v a t i o n o f t h e coupled mode equations governing t h e Thompson-Quate experiment, t h a t i s , the i n t e r a c - t i o n o f two counterpropagating a c o u s t i c waves and a homogeneous time-varying e l e c t r i c f i e l d i n a p i e z o e l e c t r i c c r y s t a l . An i m p o r t a n t r e s u l t of t h i s d e r i v a t i o n i s t h e f i r s t c o r r e c t expression f o r t h e e f f e c t i v e n o n l i n e a r p i e z o e l e c t r i c c o e f f i c i e n t go- verning t h i s i n t e r a c t i o n .

For more d e t a i l e d d e r i v a t i o n s and comments on the r e s u l t s o f t h i s paper see r e f s . 31,32,33.

2. MATERIAL FRAME LAGRANGIAN DENSITY.- Our theory o f n o n l i n e a r e l e c t r o a c o u s t i c s o f d i e l e c t r i c c r y s - t a l s begins from, and i s completely determined by, t h e m a t e r i a l frame Lagrangian d e n s i t y LM whose c o n s t r u c t i o n has been discussed i n d e t a i l pre- v i o u s l y . (15*16) 1 t i s given by

where t h e f i r s t two terms are the electromagnetic f i e 1 d Lagrangian expressed i n m a t e r i a l frame e l e c - t r i c and magnetic f i e l d s , ( I Q ) t h e t h i r d term i s t h e electromagnetic f i e l d - m a t t e r i n t e r a c t i o n La- grangian i n t h e e l e c t r i c d i p o l e approximation, ( 3 4 ) and the l a s t t h r e e terms a r e the m a t t e r Laoranaian.

The n o t a t i o n used i n Eq. ( 1 ) i s d e f i n e d as f o l l o w s : The mechanical motion o f the c r y s t a l i s completely described i n t h e continuum 1 im i t (wave- l e n g t h s l o n g compared t o p r i m i t i v e u n i t c e l l d i - mensions) by a s e t o f

N

v e c t o r s p a t i a l coordinates

(where N i s t h e number o f p a r t i c l e s p e r p r i m i t i v e u n i t c e l l o f t h e c r y s t a l ) . The s p a t i a l system co- o r d i n a t e vectors c o n s i s t o f the center-of-mass co- o r d i n a t e v e c t o r and N-1 i n t e r n a l c o o r d i n a t e vec- t o r s ;Tv(v= 1, 2,

. . . ^,

^N-1)

.

The l a t t e r i n combination, describe a l l t h e o p t i c modes o f t h e c r y s t a l . These pl s p a t i a l coordinate vectors a r e f u n c t i o n s o f a s i n g l e m a t e r i a l system c o o r d i - nate v e c t o r

3

which i s equal t o t h e center-of-mass coordinate v e c t o r i n t h e absence o f deformation.

I n t h e presence o f a deformation

where

6

i s the displacement v e c t o r . Note t h a t l o - wer case L a t i n l e t t e r s l a b e l t h e ( r e c t a n g u l a r Car- t e s i a n ) components o f

2

w h i l e upper case L a t i n l e t t e r s 1 abel t h e ( r e c t a n g u l a r Cartesian) compo- nents o f

2

even though the coordinate axes o f t h e two systems a r e c o i n c i d e n t as i n d i c a t e d by tiiJ i n Eq. ( 2 ) . The i n t e r n a l coordinates $Tv have values o f

Tv

i n t h e absence o f deformation ( e s s e n t i a l t o e x p l a i n a spontaneous p o l a r i z a t i o n ) , t h a t i s ,

JOURNAL DE PHYSIOUE

where ;v i s t h e change i n t h e i n t e r n a l coordinate induced by a deformation, an e l e c t r i c f i e l d , o r a magnetic f i e l d . The m a t e r i a l frame mass d e n s i t y associated w i t h t h e c e n t e r o f mass i s p0 and t h a t associated w i t h t h e v- i n t e r n a l coordinate i s mv.

(15*16) The deformation g r a d i e n t axi/aXJ and i t s i n v e r s e aXJ/axi are denoted by and XJ,i r e s - p e c t i v e l y . Gradients o f t h e i n t e r n a l coordinates do n o t appear i n t h e l o n g wave-length l i m i t . The Jacobian o f t h e t r a n s f o r m a t i o n from m a t e r i a l t o s p a t i a l coordinates i s

J : d e t {xi ,J3

.

^{( 4 )}

The Green deformation t e n s o r and the Green f i - n i t e s t r a i n tensor r a r e d e f i n e d by

The r o t a t i o n a l l y i n v a r i a n t measures o f t h e i n t e r - n a l coordinates, which we use f o r expansion o f t h e s t o r e d energy p " ~ , a r e d e f i n e d by

where t h e constant Y: i s s u b t r a c t e d so as t o make 1; zero i n t h e n a t u r a l s t a t e (no deformation, no a p p l i e d f i e l d s ) .

The s t o r e d energy has been c o n s t r u c t e d t o be the most general f u n c t i o n o f t h e c o n f i g u r a t i o n o f t h e c r y s t a l t h a t i s c o n s i s t e n t w i t h conservation o f energy, momentum, angular momentum and p a r i t y . The q u a n t i t i e s EAB and IIF a r e a minimal s e t o f r o - t a t i o n a l 1 y i n v a r i a n t v a r i a b l e s needed t o speci f y t h e c o n f i g u r a t i o n . Since they each vanish i n t h e n a t u r a l s t a t e , a s e r i e s expansion o f the s t o r e d energy i n terms o f them i s p o s s i b l e . Up through terms o f t h i r d o r d e r we have

where the c o e f f i c i e n t s , c a l l e d m a t e r i a l descrip- t o r s , are frequency independent constants. We have dropped t h e term l i n e a r i n o n l y EAB by assuming, f o r s i m p l i c i t y , t h a t t h e c r y s t a l has no sponta- neous s t r e s s and t h e term l i n e a r i n o n l y

nz

by assuming, a l s o f o r simp1 i c i t y , t h a t t h e c r y s t a l , i f p y r o e l e c t r i c , possesses no spontaneous e l e c t r i c f i e l d i n t h e n a t u r a l s t a t e . This corresponds t o t h e commonly met s i t u a t i o n when e x t r i n s i c s u r f a c e charge has c o l l e c t e d on t h e surfaces o f a c r y s t a l i n such a way as t o cancel t h e spontaneous e l e c - t r i c f i e l d .

I n Eq. ( 1 ) t h e m a t e r i a l forms o f t h e e l e c - t r i c f i e l d , the maqnetic i n d u c t i o n , and t h e pola- r i z a t i o n ,

t, if,

and

p,

are g i v e n i n terms o f t h e

-L *

conventional s p a t i a l forms E , B , and by

where i n t h e l a s t e q u a l i t y qv i s t h e m a t e r i a l f r a - me charge d e n s i t y associated w i t h t h e v - i n t e r n a l coordinate. (15'16)

f

and

if

a r e f u n c t i o n s o f t h e s c a l a r p o t e n t i a l

4

and t h e v e c t o r p o t e n t i a l

A

^by

D.F. Nelson

where

?x

denotes t h e g r a d i e n t w h i t h r e s p e c t t o

3

and the m a t e r i a l time d e r i v a t i v e d/dt, i n which

3

_po

i

i s h e l d constant,is i n d i c a t e d by a dot. The s p a t i a l time d e r i v a t i v e ,

a / a t ,

on t h e o t h e r hand, holds constant d u r i n g d i f f e r e n t i a t i o n .

K The Lagrangian d e n s i t y o f Eq. ( 1 ) i s a func-

t i o n o f t h e general i z e d coordinates

2 , yTv

(v= 1, 2,.

. . ,

N- 1)

, 3

and 4. The independent v a r i a b l e s o f t h e problem i n t h e m a t e r i a l d e s c r i p t i o n are

2

and t. Thus, t h e q u a n t i t y

scat

appearing i n the La- grangian d e n s i t y should be i n t e r p r e t e d as a shor- thand n o t a t i o n f o r t h e r i g h t s i d e o f t h e i d e n t i t y

i n which X i s regarded as a f u n c t i o n o f x

1,j k,L

3. MATTER EQUATIONS.- The Lagrange equation o f mo- t i o n f o r t h e v- i n t e r n a l motion y i e l d s

The resonant frequencies o f t h e i n t e r n a l motions o f t h e c r y s t a l ( o p t i c modes) described by t h i s equation are a t i n f r a r e d o r h i g h e r frequencies.

Thus, throughout t h e frequency r e g i o n i n which coherent generation o f a c o u s t i c waves can presen- t l y be accomplished, t h e i n e r t i a l term i n Eq. (15) i s n e g l i g i b l e and may be dropped. T h i s leads t o a1 1 t h e m a t e r i a l i n t e r a c t i o n tensors t h a t e n t e r t h e d i f f e r e n t i a l equations being frequency inde- pendent i n t h e frequency r e g i o n considered. The s o l u t i o n of Eq. (15) f o r y y ( x ~ , ~ , E ~ ) o r i t s r o t a t i o n a l l y i n v a r i a n t measure I I ~ can then be s u b s t i t u t e d back i n t o t h e Lagrangian, d e n s i t y befo- r e the center-of-mass equation i s found, a proce- dure c a l l e d a d i a b a t i c e l i m i n a t i o n o f t h e i n t e r n a l coordinates.

The Lagrange equation o f motion f o r t h e cen- ter-of-mass p o s i t i o n x y i e l d s d i r e c t l y t h e equa- t i o n o f conservation o f momentum r a t h e r than New- t o n ' s f o r c e equation f o r x. The l a t t e r can be ob- t a i n e d from t h e former by combining i t w i t h t h e electromagnetic momentum c o n t i n u i t y equation found from t h e m a t e r i a l frame electromagnetic equations.

The r e s u l t i s

where Q

D

i s t h e m a t e r i a l frame d i e l e c t r i c (bound) charae,

fD

i s t h e m a t e r i a l frame d i e l e c t r i c c u r r e n t ,

Ti

A i s t h e P i o l a - K i r c h o f f s t r e s s tensor,

and' t h e m a t e r i a1 frame p o l a r i z a t i o n

'lf

i s regarded as depending on Xi,A and EK from t h e a d i a b a t i c e l i - m i n a t i o n procedure. The body f o r c e terms i n Eq.

(16) a r e the m a t e r i a l frame form o f t h e Lorentz electromapnetic f o r c e s a c t i n o upon t h e d i e l e c t r i c charge and t h e d i e l e c t r i c c u r r e n t .

4. MATERIAL FRAME POLARIZATION TO BILINEAR ORDER.

-

I n o r d e r t o o b t a i n t h e m a t e r i a l frame p o l a r i z a t i o n from Eq. (11) t h e i n t e r n a l motion equations (15) must be solved f o r

pTV.

The a d i a b a t i c e l i m i n a t i o n procedure described i n t h e l a s t s e c t i o n i s used.

The i n t e r n a l m o t i o n equations can be solved t o b i - l i n e a r and q u a d r a t i c order by i t e r a t i o n , t h a t i s , t h e l i n e a r s o l u t i o n i s s u b s t i t u t e d i n t o the b i l i - near and q u a d r a t i c terms. The f i n a l r e s u l t f o r t h e m a t e r i a l frame p o l a r i z a t i o n i s

Where PK i s t h e spontaneous p o l a r i z a t i o n , xKLis S t h e l i n e a r e l e c t r i c s u s c e p t i b i l i t y , eKAB i s t h e p i e z o e l e c t r i c s t r e s s tensor, bLMN i s t h e e l e c t r i c f i e l d m i x i n g tensor ( t h e low frequency l i m i t of t h e o p t i c a l m i x i n g tensor, t h a t i n o p t i c s i s deno-

C8-154

JOURNAL DE PHYSIQUE

t e d by dKLM),

lF>AB

i s t h e r e l a t i v e e l e c t r o s t r i c - Use o f t h e s t o r e d energy i n Eq. (24) and t h e t i o n tensor, which vanishes f o r a vacuum, and p o l a r i z a t i o n i n Eq. (20) i n t h e center-of-mass eLABCD i s t h e n o n l i n e a r p i e z o e l e c t r i c s t r e s s t e n - equation (16) y i e l d s

sor. T h e i r d e f i n i n g r e l a t i o n s i n terms o f t h e ma- t e r i a l d e s c r i p t o r s have been recorded elsewhere.

(31) The interchange symmetries o f these tensors

-

have r e s u l t e d from t h e dvnamics o f the i n t e r a c - t i o n s . For the n o n l i n e a r i n t e r a c t i o n tensors they

+ G~ P.~BJCD ⁺ G ~ + G ~~ ~ CC ~ ~ ~ ~ ~

are

l r e l = l r e l KLAB (KL) (AB)'

where parentheses i n d i c a t e interchange symmetry, t h a t i s , Z(AB)

=

(ZAB+ZBA)/2. The symmetries shown i n Eqs. (22) and (23) i n d i c a t e t h a t these two ten- sors can couple o n l y t o s t r a i n and n o t t o r o t a t i o n .

5 . ELASTICITV EQUATION TO BILINEAR ORDER.- Adiaba-

t i c e l i m i n a t i o n o f t h e i n t e r n a l coordinates from t h e s t o r e d energy makes i t e f f e c t i v e l y a f u n c t i o n o f t h e Green f i n i t e s t r a i n t e n s o r and t h e m a t e r i a l frame e l e c t r i c f i e l d ,

Here cABCDEF i s t h e t h i r d o r d e r s t i f f n e s s t e n s o r ( f o r i t s d e f i n i t i o n i n terms o f m a t e r i a l d e s c r i p - t o r s see r e f . 31). I t s d e f i n i t i o n gives i t i n t e r - change symmetry o f t h e form

f o r t h e b i l i n e a r i z e d dynamic e l a s t i c i t y equation.

As i s w e l l known t h e o r d i n a r y (second o r d e r ) s t i f - fness t e n s o r cABCD c o n t r i b u t e s t o t h e e f f e c t i v e t h i r d order s t i f f n e s s c o e f f i c i e n t s . These addi- t i o n a l terms, though possessing I ,J ++ A,B

*

C,D p a i r interchange symmetry, do n o t possess i n t e r - change symmetry w i t h i n t h e i n d i v i d u a l p a i r s . These terms, t h e r e f o r e , can couple t o r o t a t i o n s as w e l l as t o s t r a i n s . S i m i l a r remarks can be made concer- n i n g t h e c o n t r i b u t i o n t h a t t h e l i n e a r piezoelec- t r i c s t r e s s t e n s o r makes t o t h e n o n l i n e a r piezoe- l e c t r i c e f f e c t w i t h i n t h e divergence o f s t r e s s term, t h a t i s , interchange symmetry between p a i r s , 1,J ++ A,B, e x i s t s b u t n o t w i t h i n A,B. P.gain t h i s means t h a t t h i s term couples t o r o t a t i o n as w e l l

as t o s t r a i n . The existence o f a body f o r c e i n Eq.

(26) t h a t i n v o l v e s t h e o r d i n a r y p i e z o e l e c t r i c s t r e s s t e n s o r f u r t h e r complicates t h e s i t u a t i o n ; however, we w i l l show momentarily t h a t t h e l a t t e r term drops o u t o f an a l t e r n a t e form o f t h e equa- t i o n .

An a l t e r n a t i v e t o Eq. (26) i s the b i l i n e a r i - zed momentum conservation l a w which c o n s i s t s o f a sum o f Eq. ( 2 6 ) and t h e electromagnetic momentum c o n t i n u i t y equation and i s g i v e n by

The symmetry w i t h i n each p a i r i n d i c a t e s t h a t i t can couple o n l y t o s t r a i n s and n o t t o r o t a t i o n s .

D.F. Nelson

+

- ( C 1 +6 c +6 c 2 IJABCD I A BJCD I C DJA3

The advantage o f t h i s equation i s t h a t a l l f o r c e s a r e divergences o f stresses, none a r e body forces;

i t s disadvantage i s t h e e x t r a time d e r i v a t i v e term on t h e l e f t side. We w i l l f i n d t h e quasi-

e l e c t r o s t a t i c l i m i t o f t h i s equation t o be more convenient than t h a t o f Eq. (26). We have i n t r o - duced t h e e l e c t r o s t r i c t i o n tensor lLMIJ i n Eq. (27).

It i s defined by

t h a t i s , i t i s equal t o t h e r e l a t i v e e l e c t r o s t r i c - t i o n tensor (produced b y t h e d i e l e c t r i c m a t e r i a l ) p l u s vacuum e l e c t r i c s t r e s s terms.

I f P d e n o t e s t h e s t r e s s whose divergence appears on t h e r i g h t s i d e o f Eq. (27),

fi

t h e u n i t m a t e r i a l frame normal o f a body surface, and double brackets the jump i n t h e enclosed q u a n t i t y , then t h e s t r e s s jump c o n d i t i o n i s

6. MATERIAL FORM OF MAXWELL EQUATIONS TO BILINEAR ORDER.- The m a t e r i a l form o f Maxwell's equa- t i o n ~ ( ~ ~ - * ~ ) a r e

t h e f i r s t and second eguationsbeing t h e Lagrange equations f o r

A

and $I r e s p e c t i v e l y and t h e t h i r d and f o u r t h equations f o l l o w i n g d i r e c t l y from t h e p o t e n t i a l d e f i n i t i o n s (12) and (13). B i l in e a r - i z i n g these equations r e q u i r e s f i n d i n g

Zf

and

fi

t o b i l i n e a r o r d e r i n

P, 8, xc

^and

6 .

From t h e general expression(19) f o r

d

we o b t a i n

and from t h e general e x p r e ~ s i o n ( ~ ~ ) f o r

Ff

f o r a non- magnetic d i e l e c t r i c we o b t a i n

H

_K = - poJ K L L 1 C

B

+ E o J & ~ ~ ~

ax^

( C - l ) M N b +

a?

^x

3 ]

N

The s i x equations o f t h i s s e c t i o n t o g e t h e r w i t h Eq. (26) o f t h e l a s t s e c t i o n form a complete s e t of d i f f e r e n t i a l equations t o b i l i n e a r o r d e r f o r e l e c t r o e l a s t i c phenomena i n d i e l e c t r i c s i n &the duUy &e&odynamic heghime.

The jump c o n d i t i o n s a t moving, deforming, d i e l e c t r i c (charge-free and c u r r e n t - f r e e ) body sur- faces f o r t h e m a t e r i a l frame electromagnetic f i e l d s (19) a r e

7. QUJSI-ELECTROSTATIC APPROXIMATION.- The m a j o r i t y of e l e c t r o a c o u s t i c i n t e r a c t i o n s a r e s t u d i e d e x p e r i - m e n t a l l y under a c o n d i t i o n where t h e quasi-

e l e c t r o s t a t i c approximation may be used t o t r u n c a t e and hence s i m p l i f y t h e electromagnetic equations.

The c o n d i t i o n i s wL/c << 1 where w i s t h e angular frequency o f t h e e l e c t r i c wave, L t h e r e l e v a n t dimension o f t h e i n t e r a c t i o n r e g i o n

,

and c t h e v e l o c i t y o f l i g h t i n vacuum. For example, a ( c i r c u l a r ) frequency o f w / 2 ~ Q, 1 GHz and a l e n g t h o f L % 1 cm j u s t b a r e l y s a t i s f y t h i s c o n d i t i o n . When s a t i s f i e d , a1 1 t i m e d e r i v a t i v e terms except the i n e r t i a l term i n t h e e l a s t i c i t y equation may be dropped.

D.F. Nelson

I n t h e equations o f fineah p i e z o e l e c t r i c i t y t h e q u a s i - e l e c t r o s t a t i c approximation causes t h e disappearance o f a l l magnetic terms from the e l e c - t r i c a l and e l a s t i c equations and so obviates t h e need o f c o n s i d e r i n g t h e magnetic f i e l d . I n t h e b i l i n e a r i z e d equations i n t h e q u a s i - e l e c t r o s t a t i c approximation m a g n e t o s t r i c t i v e terms remain. A consequent d i f f e r e n c e between

fi

and

8

( a p a r t from t h e constant P O ) a l s o remains. T h i s i s n o t s u r - p r i s i n g since i t merely s t a t e s t h a t the b i l i n e a r m a g n e t o s t r i c t i v e e f f e c t can be important even i n a nonmagnetic d i e l e c t r i c i f t h e a p p l i e d magnetic f i e l d i s s u f f i c i e n t l y strong.

The important t h i n g t o be r e a l i z e d i s t h a t t h e magnetic f i e l d must be a p p l i e d s i n c e t h e r e i s no c o u p l i n g between e l e c t r i c and magnetic f i e l d s i n t h e q u a s i - e l e c t r o s t a t i c approximation i n a non- magnetic d i e l e c t r i c . Since magnetic f i e l d s a r e n o t t y p i c a l l y a p p l i e d i n s t u d i e s o f e l e c t r o e l a s t i c phenomena, we w i l l assume t h e i r absence i n t h e f o l l o w i n g equations.

We conclude t h a t i n t h e q u a s i - e l e c t r o s t a t i c regime a complete s e t o f d i f f e r e n t i a l equations i n t h e m a t e r i a l frame i s

p0Gl = TIJ,j ³ (36)

D ~

'

,O ~ (37)

accompanied by t h e b i l i n e a r i z e d c o n s t i t u t i v e r e l a - t i o n s ,

where t h e e l e c t r i c p o t e n t i a l has been introduced v i a =

-

-+

^V4.

Equations (36) and (37) c o n s i s t o f f o u r d i f f e r e n t i a l equations i n t h e f o u r unknown

functions: and

4 .

The jump c o n d i t i o n on s t r e s s (29) s t i l l a p p l i e s b u t w i t h 7 given by Eq. (38).

The e l e c t r i c a l boundary c o n d i t i o n u s u a l l y takes t h e form o f s p e c i f y i n g t h e value o f 4 on an equipoten- t i a l surface S.

8. RELATION BETWEEN ELECTROSTRICTION AND ELASTO- OPTIC TENSORS.- Because t h e b a s i c f o r m u l a t i o n o f t h e present work i s f u l l y electrodynamic, i t has been p r e v i o u s l y a p p l i e d t o t h e e l a s t o o p t i c e f f e c t e3) and an expression has been found f o r t h e e l a s t o - o p t i c s u s c e p t i b i l i t y . Thus, t h e low frequency l i m i t o f t h e e l a s t o o p t i c s u s c e p t i b i l i t y can be compared t o t h e d e f i n i n g expression f o r t h e r e l a - t i v e e l e c t r o s t r i c t i o n t e n s o r w i t h t h e r e s u l t

where t h e r e l a t i o n between t h e e l a s t o o p t i c suscep- t i b i l i t y x ~ ~t h a t couples t o s t r a i n and t h e ( ~ ~ ) Pockels' t e n s o r pIJKL has been used. Equation (40) i s a new r e l a t i o n which s t a t e s t h a t t h e nor- m a l l y measured e l e c t r o s t r i c t i v e e f f e c t d i f f e r s from t h e low frequency l i m i t o f t h e e l a s t o o p t i c e f f e c t by c o n t r i b u t i o n s from e l e c t r i c stresses caused by t h e presence o f a d i e l e c t r i c medium.

Several comments should be made about Eq. (40). F i r s t , t h e presence o r absence of minus signs and f a c t o r s o f two a r e dependent on t h e de- f i n i n g equation; t h a t d e f i n i t i o n c o u l d be changed p r o v i d i n g corresponding changes a r e made i n

Eqs.

(26), (27), and (28). Second, i t should be r e - membered t h a t t h e Pockels' t e n s o r i s n o r m a l l y measured w i t h o p t i c a l frequency e l e c t r i c f i e l d s and so i t s v a l u e can d i f f e r from i t s low frequency l i m i t because o f frequency d i s p e r s i o n . T h i r d , t h e d i f f e r e n c e between t h e r e l a t i v e e l e c t r o s t r i c t i v e t e n s o r and t h e e l a s t o o p t i c t e n s o r a r i s e s because t h e former i s measured w i t h m a t e r i a l frame e l e c t r i c f i e l d s and t h e l a t t e r w i t h s p a t i a l frame e l e c t r i c f i e l d s . Fourth, n o t e t h a t t h e e l a s t o o p t i c e f f e c t t h a t couples t o r o t a t i o n does n o t e n t e r t h e r e l a - t i o n (40). F i f t h , t h e l a s t t h r e e terms i n Eq.

(40) t h a t i n v o l v e t h e l i n e a r e l e c t r i c s u s c e p t i -

D.F. Yelson

b i l i t y cause a measurably l a r g e d i f f e r e n c e

(% 50%) between the r e l a t i v e e l e c t r o s t r i c t i v e t e n s o r

l;;bL

and t h e low frequency 1 im i t o f e l a s t o - o p t i c t e n s o r XIJ(KL). Last, Eq. (40) can be con- v e r t e d t o a r e l a t i o n f o r t h e e l e c t r o s t r i c t i o n ten- s o r lIJKL w i t h t h e use o f Eq. (28). We m i g h t say here t h a t we regard t h e r e l a t i v e e l e c t r o s t r i c t i

r e 1

t e n s o r lIJKL, which vanishes f o r a vacuum, and t h e e l e c t r o s t r i c t i o n tensor lIJKL, which does n o t , b o t h as u s e f u l q u a n t i t i e s j u s t as t h e l i n e a r elec- t r i c s u s c e p t i b i l i t y , which vanishes f o r a vacuum, and t h e d i e l e c t r i c tensor, which does n o t , a r e b o t h regarded as u s e f u l .

It has l o n g been r e a l i z e d t h a t t h e r e l a t i v e e l e c t r o s t r i c t i v e t e n s o r and t h e low frequency e l a s t o o p t i c s u s c e p t i b i l i t y are r e l a t e d . Osterberg and ~ o o k s o n , ( ~ ~ )

as on,(^^)

and Iansaken e t a1 .(29) show t h a t these tensors a r e equal based simply on t h e e q u a l i t y o f second mixed p a r t i a l d e r i v a t i v e s o f a thermodynamic p o t e n t i a l . Such a r e l a t i o n , though mathematically c o r r e c t , does n o t address t h e f a c t t h a t t h e e l e c t r i c f i e l d i n t h e two e f f e c t s i s normally measured r e l a t i v e t o d i f f e r e n t frames ( s p a t i a l and m a t e r i a l ) . Also, these treatments a r e n o t capable o f showing t h a t t h e e l a s t o o p t i c e f f e c t c o n s i s t s o f two p a r t s , one c o u p l i n g t o s t r a i n and one t o r o t a t i o n .

Maradudin and ~ u r s t e i n ( ~ ' ) and ~ e l d ! ? ~ a n ( ~ ~ ) have found r e l a t i o n s between t h e e l e c t r o s t r i c t i v e and e l a s t o o p t i c tensors d i f f e r e n t from ours. The t r o u b l e s w i t h t h e i r d e r i v a t i o n s have been d i s - cussed e l sewhere. (31)

9. COUPLED MODE EQUATIONS FOR THREE-WAVE INTER- ACTIONS.- Coupled mode equations a r e f i r s t order d i f f e r e n t i a l equations governing t h e para- m e t r i c i n t e r a c t i o n o f t h r e e waves. I f two waves a r e i n p u t waves, t h e equations can describe sum and d i f f e r e n c e frequency generation i n c l u d i n g second harmonic generation; i f one wave i s an i n p u t pump wave, t h e y can describe generation o f two o u t p u t waves, t h e s i g n a l and i d l e r waves.

I n a parametric i n t e r a c t i o n an exchange o f energy between t h e waves occurs w i t h o u t a p p r e c i a b l y a f f e c t i n g t h e i r propagation v e c t o r s o r t h e i r eigenvectors.

The t h r e e waves s a t i s f y a frequency con-

d i t i o n

Thus we may expand t h e displacement i n t o terms a t these t h r e e frequencies (and t h e i r complex conju- gates)

w i t h amplitudes s l o w l y v a r y i n g i n time

61 .

dz(i)(d,t)

,, ,;b1 . ^5")

( x , ~ )

d t (43)

where $'(I = a , ~ , y ) i s t h e orthonormal d i s p l a c e - ment eigenvector associated w i t h t h e wave a t f r e - quency wi(i = 1 , 2 , 3 ) . I f t h e terms a t each f r e - quency a r e s e p a r a t e l y equated t o zero, we g e t i n t h e q u a s i - e l e c t r o s t a t i c regime

f o r t h e t e r n s a t frequency wl. Here t h e second time d e r i v a t i v e o f has been dropped by Eq.

(42), t h e l i n e a r terms have been placed on t h e l e f t sides, and t h e n o n l i n e a r terms grouped on t h e r i g h t sides. Note a l s o t h a t Eq. (44) i s a s c a l a r product o f t h e e l a s t i c i t y equation w i t h t h e eigenvector o f t h e mode a t frequency wl;

o n l y t h i s component o f t h e equation can be s a t i s f i e d i n a p a r a n e t r i c i n t e r a c t i o n between t h r e e f r e e l y propagating eigenmodes.

?4ext we assume t h e displacement i s a plane wave i n space,

w i t h an amplitude Ui(X,t) whose f r a c t i o n a l change i n space can v a r y s l o w l y compared t o i t s wave- number,

D.F.

I n these equations

tf

i s t h e wavevector o f t h e 1 -

eigenmode and K: i s i t s wavenumber and the space dependence o f t h e amplitude Ui i s n o t on

d

b u t r a t h e r on a s c a l a r coordinate X

- ^{3 .}

^where

⁹

^{i s}

t h e u n i t m a t e r i a l frame normal t o t h e plane i n p u t s u r f a c e o f the n o n l i n e a r medium.

The meaning o f t h i s dependence on X i s t h a t surfaces of constant amplitude i n a parametric i n t e r a c t i o n a r e p a r a l l e l t o t h e i n p u t s u r f a c e a t which t h e n o n l i n e a r i n t e r a c t i o n began even though t h e s u r f a c e o f c o n s t a n t phase f o r each wave i s perpendicular t o i t s wavevector. A l t e r n a t i v e l y stated, t h e growth ( o r decay) d i r e c t i o n o f t h e waves i s normal t o t h e i n p u t s u r f a c e regardless of t h e i r propagation d i r e c t i o n s .

I t i s important t o r e a l i z e t h a t t h e s l o w l y v a r y i n g ampl i t u d e Ui i n Eq. (46) i s t h a t o f a p i e z o e l e c t r i c a l l y s t i f f e n e d mode whose eigen- v e c t o r i s $ ( I ) . I n o t h e r words, we should n o t i n t r o d u c e separate v a r i a t i o n s o f t h e d i s p l a c e - ment and e l e c t r i c p o t e n t i a l amplitudes i n t h e two coupled equations (44) and (45) b u t r a t h e r o n l y a v a r i a t i o n o f t h e displacement amplitude i n t h e e l a s t i c i t y equation a f t e r e l i m i n a t i o n o f t h e e l e c t r i c p o t e n t i a l from t h a t equation. The e l e c t r i c p o t e n t i a l , thus, should be found from Eq. (45) as a f o r c e d plane wave d r i v e n by the a c o u s t i c wave displacement and t h e n o n l i n e a r i t y

v . 0

^Y

where i s a u n i t v e c t o r i n t h e d i r e c t i o n o f

We now s u b s t i t u t e Eqs. (48) and (46) i n t o Eq. (44) and drop second space d e r i v a t i v e s o f U1 and so o b t a i n

where

-

^CIJAS i s t h e p i e z o e l e c t r i c a l l y s t i f f e n e d s t i f f n e s s tensor

Since the wavevector and eigenvector o f a wave i n a parametric i n t e r a c t i o n a r e e s s e n t i a l l y unaffected, t h e bracketed terms i n Eq. (49) a r e t h e unpertur- bed wave equation and so nay be s e t equal t o zero.

The c o e f f i c i e n t o f the space d e r i v a t i v e term may be r e l a t e d t o t h e a c o u s t i c group v e l o c i t y of the a-mode, l e a d i n g t o 9

a 8 y-

*

iAKNX K K K c U U e -

- -

1 2 3 3 2

I n t h e second f o r n o f Eq. (51) t h e nornal com- ponent of t h e phase mismatch

A K N

given by

AZ

^{= AK@,} where

has been introduced s i n c e t h e t a n g e n t i a l p a r t s a t i s f i e s ( ~ 2 ) ~ = 0 by i n p u t s u r f a c e boundary c o n d i t i o n s . The m a t e r i a l i n t e r a c t i o n c o e f f i c i e n t ( t h e e f f e c t i v e s t i f f n e s s constant)

c,

which w i l l be described i n d e t a i l p r e s e n t l y , has a l s o been introduced. By analogous d e r i v a t i o n s t h e coupled node equations f o r the displacenent ampl i t u d e s a t t h e o t h e r two frequencies a r e found t o be

These equations apply t o a c o u s t i c waves each having any mode t y p e and t r a v e l i n g i n gny d i r e c t i o n (con- s i s t e n t w i t h b e i n g a t o r c l o s e t o phase matching) i n a c r y s t a l of any symnetry. The c r y s t a l may be a simple d i e l e c t r i c , a p i e z o e l e c t r i c , o r a pyroelec- t r i c ( i f i n t h e l a t t e r case t h e spontaneous elec- t r i c f i e l d has been c a n c e l l e d by e x t r i n s i c surface charge). The i n t e r a c t i o n w i l l be s t r o n g e s t a t

D.F.

Nelson

C8-159

exact phase matching

( A K N =

0 and therefore n t

= 0 ) .

10. tlATERIAL II!TERACTIOY COEFFICIENT.- I t i s apparent from the development t h a t c i s the mate- r i a l interaction c o e f f i c i e n t of three acoustic waves. I t may be called an e f f e c t i v e t h i r d order s t i f f n e s s constant. The development gives i t the following form:

This expression applies t o c r y s t a l s of any symmetry and t o acoustic waves each of any mode type and propagating in any direction (consistent with being close t o or a t phase matching). I f the crystal i s nonpiezoelectric a l l odd rank ten- sors should be s e t equal t o zero.

The expression f o r the material interaction c o e f f i c i e n t Z contains a wealth of information about the acoustic mixing process. As expected, the expression i s symmetric upon interchange of any two acoustic waves, e.g. :(I), its

⁺

x ( ~ ) ,

p. W e see t h a t s i x types of processes contribute t o the overal

I

interaction. They are experimen- t a l l y distinguishable fron one another by t h e i r dependence upon the three unit propagation vectors and upon t h e i r coupling t o rotation in some cases.

The f i r s t t e r n i s the t h i r d order s t i f f n e s s tensor t h a t coup1 es three s t r a i n f i e 1 ds and represerits the most d i r e c t coupling of the t h r e e acoustic waves.

The second through fourth terms a r e

e f f e c t i v e t h i r d order s t i f f n e s s contributions t h a t involve only the second order s t i f f n e s s tensor and the Kronecker d e l t a . They represent a nonlinear e l a s t i c e f f e c t required of a l i n e a r e l a s t i c medium and are sonetimes called a "geometric nonlinear- i t y . "(35) These e f f e c t i v e t h i r d order s t i f f n e s s terms can couple t o both s t r a i n and rotation in contrast t o the t r u e t h i r d order s t i f f n e s s tensor term.

The f i f t h through seventh terms in c ^re-

present two-step e f f e c t s of two types. In one type the longitudinal e l e c t r i c f i e l d generated by one of t h e acoustic waves i n t e r a c t s with the s t r a i n produced by each of the other acoustic waves throtgh tlhc nonlinear piezoelectric s t r e s s tensor.

In the other type the sane e l e c t r i c f i e l d inter- a c t s with both the s t r a i n and rotation produced by the other two acoustic waves through an e f f e c t i v e nonlinear piezoelectric e f f e c t a r i s i n g fron the l i n e a r piezoelectric s t r e s s tensor. This i s a nonlinearity

heyuhed

of a l i n e a r piezo- e l e c t r i c medium and i s a second example of a

"geometric nonlinearity." I t i s missing in the work of McMahon(6) and Mathur and ~ u ~ t a ( ~ ) The new terms can

be

seen t o have the form of piezo- e l e c t r i c s t i f f e n i n g of the second order s t i f f n e s s tensors in the second through fourth terms of

Eq.

(55). Since piezoelectric s t i f f e n i n g can change t h e e f f e c t i v e s t i f f n e s s tensor components by a s much as 40% i n LiNb03, these new terms will be numerically s i g n i f i c a n t .

The eighth through tenth terms represent

three-step contributions. These represent the

interaction of two longitudinal e l e c t r i c f i e l d s ,

generated piezoelectrically by two of the acoustic

waves, with the s t r a i n of t h e t h i r d acoustic wave

through the e l e c t r o s t r i c t i o n tensor.

D.F. Nelson

The l a s t type o f i n t e r a c t i o n i s represented by t h e eleventh and l a s t t e r n i n

c .

It represents a f o u r - s t e p i n d i r e c t e f f e c t whereby t h r e e l o n g i - t u d i n a l e l e c t r i c f i e l d s generated p i e z o e l e c t r i c a l l y by t h e t h r e e a c o u s t i c waves i n t e r a c t through t h e e l e c t r i c f i e l d m i x i n g tensor.

11. ACOUSTIC WAVE MIXING.- As an example o f t h r e e - wave a c o u s t i c i n t e r a c t i o n s consider steady s t a t e sum frequency generation i n t h e regime where de- p l e t i o n o f t h e two i n p u t waves i s ignored. The formula f o r t h e generated a c o u s t i c f l u x w i l l thus be a p p l i c a b l e f o r i n t e r a c t i o n l e n g t h s i n which o n l y a few percent o f t h e i n p u t f l u x i s converted i n t o o u t p u t w a v e f l u x .

By t h e above assumptions we need consider o n l y Eq. (54) f o r U3 w i t h t h e time d e r i v a t i v e s e t t o zero and U1 and U2 taken as constants. The equation i s r e a d i l y i n t e g r a t e d and t h e i n t e g r a t i o n constant evaluated by using t h e approximate

boundary c o n d i t i o n U3(3) = 0. The displacement o f t h e o u t p u t f i e l d i s then

where t h e phase matching f u n c t i o n ~ ( 8 ) d e f i n e d by s i n 8 2

a ( @ ) 5

7

e

has been i n t r o d u c e d and where we have approximated ( q

+

^{+ 2!)/2 =}

q

i n t h e phase f a c t o r s i n c e t h e s o l u t i o n a p p l i e s near phase matching,

AR

^{= 0.}

The t i m e average ( i n d i c a t e d by o ) o f t h e a c o u s t i c energy f l u x ( i n t e n s i t y ) i s now found t o be

where

??

i s a u n i t v e c t o r i n t h e d i r e c t i o n of t h e group v e l o c i t y (energy propagation),

cos6' = f * T t ( 2 = a,@,y), and CS'> (I = a,@) i s

t h e i n t e n s i t y o f t h e I-input \%lave. A f a c t o r o f (CJ/2l2 where

= 1 i f o l = w 2 and a = B

has been introduced so t h a t t h e expression w i l l a1 so apply t o second harmonic generation.

T h i s expression, though very compact, i s v e r y general i n t h a t i t a p p l i e s t o c r y s t a l s o f any symmetry and t o a c o u s t i c i n p u t and o u t p u t waves each having any node t y p e and propagating i n any d i r e c t i o n ( c o n s i s t e n t w i t h being c l o s e t o o r a t phase matching). The formula a p p l i e s d i r e c t l y t o t h e m i x i n g o f two a c o u s t i c waves o f d i f f e r e n t f r e - quencies t h a t produce a surn frequency o u t p u t obey- i n g Eq. (41). I t a l s o a p p l i e s t o a c o u s t i c second harnonic generation i f t h e one i n p u t a c o u s t i c i n - t e n s i t y i s entered f o r each <s"> and <S > and i f 6 one p u t s D = 1. We n i g h t add t h a t Eq. (58) can be a p p l i e d d i r e c t l y t o a c o u s t i c d i f f e r e n c e frequency m i x i n g f o r wl

#

w2(D = 2 ) ,

provided t h e wavevectors a r e c l o s e t o obeying t h e ( d i f f e r e n t ) phase matching c o n d i t i o n

Note t h e l a r g e e f f e c t s caused by t h e n o n c o l l i n e a r i t y o f wavevector and t h e group velo- c i t y i n Eq. (58). These e f f e c t s on a c o u s t i c wave mixing, deri,ved here a p p a r e n t l y f o r t h e f i r s t time, r e s u l t from a n i s o t r o p y and a r e embodied i n t h e f a c t o r s c o s 6 " c o s 6 6 c o s 6 y / ( ~ ~ ~ ) 2 . Any o f these f o u r f a c t o r s can e a s i l y have a value o f o r d e r 1 / 3 and thus be n u m e r i c a l l y very s i g n i f i c a n t .

12. PARAMETRIC IZTERACTION OF AN ELECTRIC FIELD WITH TWO ACOUSTIC !4AVES.- I n t h e Thompson- Quate experiment (36) two equal -frequency counter- propagating a c o u s t i c waves produced, o r were pro- duced by, a s p a t i a l l y constant e l e c t r i c f i e l d a t the second harmonic frequency i n a parametric i n t e r a c t i o n . The i n t e r a c t i o n r e q u i r e s a piezo- e l e c t r i c medium; i t may a l s o be p y r o e l e c t r i c . Our

D.F.

Welson

treatment includes the l a t t e r type provided i t s spontaneous e l e c t r i c f i e l d i s zero. The coupled mode equations f o r t h i s interaction need consist of only the two amplitude equations of the acoustic waves. Their derivation i s analogous t o t h a t f o r the three acoustic wave parametric i n t e r - action j u s t studied except t h a t one of the input f i e l d s i s a s p a t i a l l y constant c l e c t r i c f i e l d hose frequency i s w3 and whose potential has a Fourier amplitude of t h e form

where s ( ~ ) i s a unit vector. Because t h i s f i e l d i s s p a t i a l l y constant, we can assume t h a t any acoustic displacement generated a t frequency w3 i s nogligible,

!de take the frequencies of the two acoustic waves t o be w and w f o r generslity and assume t h a t the ₁ 2 frequencies s a t i s f y Eq. ( 4 1 ) .

I t i s apparent from the s i m i l a r i t i e s of t h i s and the previous coupled mode problem t h a t the amplitude

U1

of the acoustic wave a t frequency wl s a t i s f i e s Eg. (51) except t h a t

?P 8")

^{must be}

evaluated f o r an acoustic wave input a t w2 and an e l e c t r i c f i e l d input a t wg of the form (61).

where t h e s c a l a r material interaction c o e f f i c i e n t

-

h i s given by

In expressing

h

we have s e t f ( l ) =

- 5")

z and (K;/K;)' = 1 from t h e phase matching condition t h a t we will obtain momentarily; these should be ade- quate approximations whenever the coupled mode equations a r e applicable.

t

2 s ~ b ~ ~ ~ s ~ s ~ e ~ ~ ~ s ~ e ~ ~ ~ + *

^-+

E o ( s * K ' S12

-

The material interaction c o e f f i c i e n t h contains several contributions experimentally distinguishable by t h e i r dependence on unit pro- pagation vectors and upon t h e i r coupling j u s t t o s t r a i n o r t o s t r a i n and t o t d o n . The f i r s t term may be called the d i r e c t interaction of the uni- form e l e c t r i c f i e l d and the s t r a i n f i e l d s of the two acoustic waves through the f i f t h rank non- l i n e a r piezoelectric s t r e s s tensor. In the second term the uniform e l e c t r i c f i e l d i n t e r a c t s with both the s t r a i n and rotation of t h e two acoustic waves through a c o e f f i c i e n t involving the l i n e a r piezo- e l e c t r i c s t r e s s tensor (and the Kronecker d e l t a ) . The next two terms represent two-step i n d i r e c t contributions a r i s i n g via the e l e c t r o s t r i c t i o n tensor from t h e interaction of the uniform e l e c t r i c f i e l d , t h e s t r a i n f i e l d of one acoustic wave, and the p i e z o e l e c t r i c a l l y generated longitudinal elec- t r i c f i e l d of the other acoustic wave. The l a s t term represents a three-step i n d i r e c t contribution via the e l e c t r i c f i e l d mixing tensor of the two piezoelectrically generated longitudinal e l e c t r i c f i e l d s of the acoustic waves with the uniform e l e c t r i c f i e l d .

bas b ~ s

L I J A B ' (64)

Substitution of Eq. (63) into Eq. (51) r e s u l t s i n the appearance of a phase mismatch

Since the plane input surface boundary conditions force the tangential components

(at)-,

to vanish, we may express

AR

^{a s} A K ~ ~ .

Phase matching t h e interaction (A? = 0) requires arranging AKN = 0. When done, Eq. (65) implies =

- Z!.

This indicates counter- propagating beams, $ ( I ) =

-

d 2 ) , with equal wavenumbers,

K:

=

K;.

If the two acoustic waves have equal frequencies,' wl = w2, then the l a t t e r r e l a t i o n requires t h e i r v e l o c i t i e s t o be equal.

This can be accomplished i f the mode types a r e the same,

a

= 6, o r i f they a r e degenerate in the

f

( l ) direction.

The final equation f o r the amplitude

U1

i s given by

D.F. Nelson

A corresponding development f o r t h e amplitude U2 gives

These a r e t h e two coupled mode equations f o r t h e two a c o u s t i c waves i n a Thompson-Quate t y p e e x p e r i - ment. These equations a l l o w f o r any d i r e c t i o n o f propagation i n a p i e z o e l e c t r i c c r y s t a l o f any symmetry and f o r a c o u s t i c waves o f any mode t y p e and frequency, s u b j e c t , o f course, t o t h e com- b i n a t i o n being near t o o r a t phase matching.

Further, t h e equations a l l o w t h e waves t o propa- gate and c a r r y energy i n d i r e c t i o n s o t h e r than normal t o t h e i n p u t surface. Loss terms o f alUl and-a2U2 may be added phenomenologically t o t h e l e f t sides o f Eqs. (66) and (67).

When we compare our coupled mode equa- t i o n s (66) and (67) w i t h t h e generic forms assumed by Thompson and ~ u a t e , ' ~ ~ ) two r e s u l t s of t h i s d e r i v a t i o n a r e seen t o be new and d i f f e r e n t . F i r s t , we have d e r i v e d f o r t h e f i r s t time t h e general expression f o r t h e m a t e r i a l i n t e r -

-

a c t i o n c o e f f i c i e n t h, Eq. (64), t h a t governs t h e s t r e n g t h o f the i n t e r a c t i o n . Second, when a general d i r e c t i o n o f propagation r e l a t i v e t o t h e c r y s t a l axes and t o t h e normal o f t h e i n p u t sur- face i s considered, t h e component o f the group v e l o c i t y normal t o t h a t surface,

fi $

r e p l a c e s a phaje v e l o c i t y i n t h e Thompson-Quate equations. g' When t h e phase v e l o c i t y d i r e c t i o n ( t h e wave- v e c t o r d i r e c t i o n ) i s normal t o t h e i n p u t sur- face, t h e phase v e l o c i t y and

f i e

$ become i d e n t i -

9

c a l . Since t h a t was t h e case i n t h e Thompson-Quate experiment, t h e i r a n a l y s i s i s c o r r e c t f o r t h e i r geometry. When t h e phase v e l o c i t y d i r e c t i o n i s n o t normal t o the surface, these two f a c t o r s w i l l d i f f e r and t h e d i f f e r e n c e can be l a r g e .

momentum conservation. Our work g e n e r a l i z e s t h a t o f T i e r s t e n and Baumhauer by being f u l l y e l e c t r o - dynamic and by a p p l y i n g t o p y r o e l e c t r i c s .

Because our theory i s f u l l y electrodynamic, i t has been a p p l i e d t o t h e e l a s t o o p t i c e f f e c t p r e v i o u s l y . (23) Comparison o f t h e expression f o r t h e low frequency l i m i t o f t h e e l a s t o o p t i c sus- c e p t i b i l i t y and t h a t f o r t h e r e l a t i v e e l e c t r o - s t r i c t i o n t e n s o r d e r i v e d here r e v e a l s t h a t they d i f f e r by e l e c t r i c s t r e s s c o n t r i b u t i o n s from a p o l a r i z a b l e medium. This i s a new r e s u l t d i f f e r i n g from a1 1 previous proposals

.

^(25-30) The o r i g i n of t h e d i f f e r e n c e i n these two tensors was shown t o be the d i f f e r i n g frames o f reference, s p a t i a l and m a t e r i a l , o f t h e measured e l e c t r i c f i e l d s .

Based on t h e equations i n t h e quasi- e l e c t r o s t a t i c regime we have d e r i v e d coupled mode equations f o r t h e n o n l i n e a r i n t e r a c t i o n o f t h r e e a c o u s t i c waves and from them d e r i v e d a very general formula f o r t h e generated f l u x from a sum o r d i f f e r e n c e frequency i n t e r a c t i o n . The expres- s i o n f o r t h e e f f e c t i v e t h i r d order s t i f f n e s s t h a t governs the i n t e r a c t i o n d i f f e r s from a1 1 previous expressions s i n c e t h e r o t a t i o n a l l y i n v a r i a n t formulation o f o u r theory causes new terms t o a r i s e i n t h e expression when a p p l i e d t o piezo- e l e c t r i c c r y s t a l s .

Coupled mode equations f o r t h e Thompson- Quate experiment have been obtained from a b a s i c d e r i v a t i o n f o r t h e f i r s t t i m e and an expression f o r t h e e f f e c t i v e n o n l i n e a r p i e z o e l e c t r i c tensor governing t h e i n t e r a c t i o n has been obtained.

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1. R. N. Thurston and K. Brugger, Phys. Rev.

133A, 604-1610 (1964).

2 . K. Brugger, Phys. Rev. 133A, 1611-1612

( 1 964).

13. CONCLUSIONS.- When o u r r e s u l t s a r e s p e c i a l i z e d 3' R' N. and ^{J '} Shapiroy ^J. AcOust- t o t h e q u a s i - e l e c t r o s t a t i c regime and compared Soc. Am. 41, 1112-1125 (1967).

t o previous t t ~ e o r i e s , ( ~ - l ~ ) ~ we f i n d t h a t we agree 4. R. Torguet and E. Bridoux, Rev. Phys. Appl.

o n l y w i t h T i e r s t e n and Baumhauer. ( I 3 ) Examination 7, 291-302 (1972).

o f t h e o t h e r s (5-12) r e v e a l s t h a t by n o t expanding 5. P. H. Carr, Phys. Rev. 169, 718-729 (1968).

t h e i r energy f u n c t i o n i n t h e m a t e r i a l frame 6. D. H. McMahon, J. Acoust. Soc. Am. 44, 1007- e l e c t r i c f i e l d they have l o s t r o t a t i o n a l i n v a r i a n - 1013 (1968).

ce, and so have equations t h a t v i o l a t e angular 7. S . S. Mathur and

P.

N. Gupta, Acustica 23, 160-164 (1970).

D.F. Nelson

V. E. Ljamov, J. Acoust. Soc. Am. 52, 199- 202 (1972).

E. A. Kraut, T. C. Lim, and B. R. Tittman, IEEE Trans. Sonics Ultrason. SU-19, 247-255 (1972); Ferroel

.

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43, 2510-2518 (1972).

Y. Nakagawa, K.Yammanouchi, and K. Shibayama, 3. Appl

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Acoust. 20, 259-263 (1974)l.

H. F. T i e r s t e n and J. C. Baumhauer, J. Appl.

Phys. 45, 4272-4287 (1974).

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SOC. Am. 54, 1017-1034 (1973).

M. Lax and D. F. Phys. Rev. @,

3694-3731 (1971 )

.

M. Lax and D. F. Nelson, Phys. Rev. B13, 1759-1 769 (1976).

J. B. Walker, A. C. P i p k i n , and R. S. R i v l i n , Accad. Naz. Lin., Ser. 8, 38, 674-676 (1965).

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