EXPERIMENTAL EVIDENCE OF DISTORTED RIPPLING PROFILES OF FINITE AMPLITUDE ACOUSTIC SUFACE-WAVES

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EXPERIMENTAL EVIDENCE OF DISTORTED RIPPLING PROFILES OF FINITE AMPLITUDE

ACOUSTIC SUFACE-WAVES

A. Alippi, A. Palma, L. Palmieri, G. Socino, E. Verona

To cite this version:

A. Alippi, A. Palma, L. Palmieri, G. Socino, E. Verona. EXPERIMENTAL EVIDENCE OF DIS-

TORTED RIPPLING PROFILES OF FINITE AMPLITUDE ACOUSTIC SUFACE-WAVES. Journal

de Physique Colloques, 1979, 40 (C8), pp.C8-249-C8-255. �10.1051/jphyscol:1979843�. �jpa-00219549�

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EXPERIMENTAL EVIDENCE OF DISTORTED RIPPLING PROFILES OF FINITE AMPLITUDE ACOUSTIC SURFACE-WAVES A. Alippi, A. Palma, L. Palmieri, G. Socino, E. Verona

Lstituto di Aoustiaa "O.M.Covbino", C.N.R., via Cassia 1216, Rome, Italy and I-stituto di Fisioa, Universitd di Perugia, Perugia, Italy

Abstract. - Waveform distortion of acoustic surface waves is experimentally analyzed through testing the modification of rippling profiles. An optical probing method has been used, which allows one to measure the phase shift between fundamental and second harmonic through the asymmetries of the light diffracted in second orders by the distorted waves. Several configu- rations of the rippling profiles have been investigated of surface waves propagating on LiNbOo, LiTa03, Bii2Ge020 an d SiOg. Cases have been found where phase relation between fundamental and second harmonic waves is such as to produce straight forward or backward sloping profiles.

1.INTRODUCTION. - It is well known that the gro- wing of harmonic along the propagation path of finite amplitude acoustic bulk waves in condensed media produces characteristic sawtooth distortions of the velocity profiles. The waveforms develop along the propagation direction into forward or backward sloping waves. These most stable configu- rations correspond to the two allowed phase rela- tions between fundamental and second harmonic waves, which differ by a IT angle one with respect to the other. In case of surface waves, the inhomogeneous nature causes various waveform distortion to be expected. The final configuration of the wave, indeed, depends upon the relative amplitudes and phases with which nonlinear partial waves grow along the propagation direction.

In the present work, waveform distortion of acoustic surface waves is experimentally analyzed through testing the modification of the rippling profiles along the propagation path. An optical method has been used, which allows one to measure the phase shift between fundamental and second harmonic through the asymmetries of the light diffracted into second orders by the distorted waves.

2. THE NON-LINEAR PROBLEM. - The analytical approacn to tne problem of nonlinear propagation of acoustic surface waves has been variously attem- pted in recent times, through either some kind of phenomenological approach /1,2/ or simplifying

assumptions /3,4/. A number of papers by Stegeman et al. /5/ recently obtained valuable theoretical results which match very closely the experimental date available.

The analytical problem is to solve the nonlinear equation of motion :

subjected to the stress-free boundary condition on the propagation surface. Here, p is the density of the medium, u the 'Hsplacer'ent vector and a., =

' ' Tk

= 3U/3(3ui/axk), with U density of the internal energy.

To our purpose, we presently discuss the problem in a very simplified qualitative way, by re- ferring to the characteristics approach proper to a wide range of problems dealing with wave propagation. According to it, and limiting our atten- tion first to the case of bulk waves in isotropic media, a family of curves can be traced on the plane (x,t) which are called characteristic curves and have the property that the acoustic perturbation is constant along anyone of them /6/. The slope of the curves gives the velocity with which the perturbation propagates at any point in space and instant of time. The curves run as parallel straight lines in case of linear propagation and may have different inclinations otherwise. In the former case, one obtains the well known result JOURNAL DE PHYSIQUE Colloque CQ, supplément au n° 11, tome 40, novembre 1979, page C8-249

Résumé. - La distorsion des ondes acoustiques de Rayleigh a été expérimentalement étudiée en analysant l'asymétrie du spectre de la lumière diffractée produit par les changements du profil de déformation de la surface sur laquelle se propagent les ondes.

On présente ici une méthode optique qui a permis la détermination expériementale de la relation de phase existant entre l'onde à présuence fondamentale et l'onde à la fréquence de l'harmonique 2 pour plusieurs échantillons de cristaux piézo-êlectriques (LiNb03, LiTa03, Bi'i2Ge020 et Si02)•

Article published online by EDP Sciences and available at

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979843

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JOURNAL DE PHYSIQUE

t h a t t h e shape o f t h e wave does n o t d i s t o r t along t h e propagation d i r e c t i o n (see f i g . l a ) , w h i l e d i s - t o r t i o n takes p l a c e i n t h e l a t t e r case (see f i g . l b ) i n a way which i s completely d e f i n e d by t h e shape on t h e c h a r a c t e r i s t i c curves.

Fig. l a

F i g l b Fig. 1

-

Representation o f t h e c h a r a c t e r i s t i c

curves i n t h e x,t plane, f o r t h e case o f a) l i n e a r ( u n d i s t o r t e d wave) and b) non- l i n e a r propagation ( d i s t o r t e d wave).

By t h i n k i n g o f a s i n u s o i d a l p e r t u r b a t i o n , one sees t h a t i n t h e l a t t e r case, indeed, energy i s f l o w i n g toward those p o i n t s where t h e c h a r a c t e r i s t i c curves cross on t h e (x,t)plane, g i v i n g r i s e t o f r o a t s . o f maximum energy concentration.

A F o u r i e r ' s a n a l y s i s of t h e d i s t o r t e d wave- f r o n t suggests t h a t a l l t h e harmonic waves match t h e i r phases a t these p o i n t s of accumulation. It i s i n s t r u c t i v e t o note t h a t , by i n v e r t i n g t h e time flow, one simply o b t a i n s a reverse o f the wave pro4 pagation d i r e c t i o n and proper " u n d i ~ t o r t i ~ ~ ~ o f a l - ready d i s t o r t e d waves.

Symmetry considerations of homogeneous plane waves propagating i n any e l a s t i c a l l y a n i s o t r o p i c m a t e r i a l r e q u i r e t h a t wave propagation i n t o oppo- s i t e d i r e c t i o n s ought t o take p l a c e i n i d e n t i c a l way, t h a t i s , t h e wave propagating i n one d i r e c t i o n ought t o be t h e m i r r o r image o f t h e other. T h i s means t h a t accumulation p o i n t s should be p o i n t s o f zero p e r t u r b a t i o n , where t h e wave propagates w i t h t h e v e l o c i t y proper t o t h a t o f t h e l i n e a r medium.

Surface a c o u s t i c waves, however, are i n t r i n s i d c a l l y inhomogeneous waves and have a d i 1 a t a t i o n a l s t r a i n component t h a t l a g s some a/2 radians behind the shear component along t h e propagation d i r e c t i o n Consequently, t h e r e are no wavefronts c a r r y i n g no p e r t u r b a t i o n . Something could be said, howeverain t h e case t h a t propagation takes p l a c e i n a d i r e c - t i o n which i s normal t o a m i r r o r plane o f simmetry o f the c r y s t a l . Then, t h e shear component o f t h e wave cannot generate h i g h e r shear harmonics and, t h e r e f o r e , i t propagates w i t h o u t producing, by i t - s e l f , d i s t o r t i o n . D i s t o r t i o n i s i n some way d e a l t w i t h o n l y by t h e d i l a t a t i o n a l component and i t i s subjected t o t h e c o n d i t i o n above mentioned f o r b u l k waves.

I n t h e most general case o f surface propaga- t i o n , n o t h i n g i s r e a l l y p r e d i c t a b l e and d i s t o r t i o n may be d i f f e r e n t f o r opposite propagation d i r e c t i - ons.

3. THE OPTICAL METHOD.

-

D e t e c t i n g t h e amount o f d i s t o r t i o n o f a wave r e q u i r e s a method t h a t should be s e n s i t i v e t o t h e phase between t h e fundamental and i t s harmonics. T h i s c o u l d be performed o p t i c a l - l y by a method which has been described elsewhere /7/ and which avoids s u b t l e e l e c t r o n i c procedures, l i k e those used i n r e f . 8. The method i s here out- l i n e d , 1;mited t o t h e a n a l y s i s o f t h e f i r s t two op- t i c a l d i f f r a c t i o n orders produced by t h e fundamen-

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PLIPPI A.

e t a l . c8-251

t a l wave and i t s f i r s t harmonic alone.

L e t t h e transverse displacement o f an acous- t i c surface wave on t h e f r e e surface o f propagation be :

where lower indexes r e f e r t o space coordinates (1=

propagation d i r e c t i o n , 3=normal t o t h e f r e e surface) and upper indexes t o harmonics (l=fundamental 2 = f i r s t harmonic) ; u and a are displacements and displacement amplitudes

,

respectively, and k are angular frequency and wavenumber, $211 i s t h e r e l a - t i v e phase angle t o be determined.

When a plane l i g h t wavefront w i t h wavelength A impinges l o c a l l y on t h e propagation surface a t an angle

3

w i t h respect t o t h e normal, i t i s r e - f l e c t e d i n t o mu1 t i p l e d i r e c t i o n plane wavefronts, angulary spread by amounts equal t o

"X-=mXk

^(n, ^{i n -}

A 2n

t e g e r ) around d i r e c t i o n

- g .

The i n t e n s i t y r a t i o o f t h e m-th t o t h e 0 - t h o r d e r wavefront is:/9,7/

where J ' s a r e Bessel's f u n c t i o n s of t h e f i r s t kind. By s p e c i f y n g eq. (2) f o r t h e f i r s t and second order, r e s p e c t i v e l y , one o b t a i n s :

ween f i r t s negative and p o s i t i v e orders i s o f t h e order o f $), w h i l e i t i s of t h e o r d e r o f u n i t y f o r t h e second orders. It w i l l be, t h e r e f o r e , most convenient t o analyze t h e second o r d e r d i f f r a c t e d l i g h t asymmetries i n o r d e r t o gain i n f o r m a t i o n about second harmonic d i s t o r t i o n .

F i g u r e 2 schematically shows t h e asymmetrical be- haviour o f t h e l i g h t d i f f r a c t i o n p a t t e r n s produced i n cases of forward and backward s l o p i n g p r o f i l e s o f d i s t o r t e d surface waves.

l o

F i g . 2 Schematic r e p r e s e n t a t i o n o f a s y m ~ l e t r i c a l d i f f r a c t i o n spectrum of l i g h t r e f l e c t e d by a) forward and b) backward s l o p i n g surface r i p p l i n g p r o f i l e s .

i n s p e c t i o n of eqs. (4) and (5) shows t h a t ,

4.EXPERIMENTAL. -

Measurements have been p e r f o r - under t h e usual experimental c o n d i t i o n s t h a t : med o f t h e i n t e n s i t y o f t h e l i g h t d i f f r a c t e d i n t o a ( 2 ) a a ( l ) < c A , t h e l i g h t i n t e n s i t y asymmetry bet- second orders by a c o u s t i c surface wave pulses ge-

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C8-252 JOURNAL DE PHYSIQUE

nerated through i n t e r d i g i t a l transducer technique on several c r y s t a l samples, i n t h e 20-80 MHz range o f frequencies. The l i g h t beam scans t h e propagation surface along t h e a c o u s t i c propagation d i r e c t i o n and t h e s c a t t e r e d l i g h t pulses a r e detected by a p h o t o m u l t i p l i e r and processed through usual s i g n a l processing techniques. I n t h e c o n d i t i o n s o f acous- t i c frequencies and powers i n v o l v e d i n our e x p e r i - ments, the approximation a1 ready mentioned t h a t a(2)<<a(1) i s c o r r e c t l y followed. Ik i s then possi- b l e t o assume t h a t d i s t o r t i o n i s c o n t r i b u t e d by second harmonic alone. The amplitude o f t h e second harmonic wave grows 1 in e a r y along the propagation d i r e c t i o n x i , p r o p o r t i o n a l t o t h e fundamental ' s a c o u s t i c power, according t o :

a(2) (XI) = lr2111 a ( 1 ) 2 ~ 1 = ~ 2 1 1 K~ a(1j2x1 (6)

wheve

lr2111

i s t h e magnitude o f the a c o u s t i c wave n o n l i n e a r c o u p l i n g c o e f f i c i e n t and a ( l ) i s suppo- sed t o remain p r a c t i c a l l y constant. G211is t h e e f f e c t i v e n o n l i n e a r c o u p l i n g c o e f f i c i e n t characte- r i z i n g the mater used

/ l o / .

By i n t r o d u c i n g eq. (6) i n t o eq. ( 5 ) and by approximating t h e Bessel f u n c t i o n s w i t h t h e f i r t s term o f t h e i r s e r i e s expansion, one o b t a i n s :

where

acos g a ( 1 ) 4 I 2 ( 0 ) = 4 I 0 (

h

1

and

a c o s 3 1

The upper signs a r e t o be taken f o r p o s i t i v e o r - ders, and t h e lower ones f o r t h e negative. The l i g h t i n t e n s i t i e s It2 v s . x l represent a p a i r o f parabolas symmetrically d i s p l a c e d around t h e transducer p o s i t i o n ( x l = 0) w i t h t h e v e r t i c a l axes a t a d i s t a n c e

xlsin = i xo s i n +211

*

( 10)

where the It2 f u n c t i o n s a t t a i n t h e i r minimum value, equal t o :

Figures 3 and 4 g i v e the experimental r e s u l t s r e l a - t i v e t o yxLiNb03 and z - c u t ( l l 0 ) propagation

B i 12Ge020 specimens. It should be noted t h a t , f o r experimental convenience, a doppler n o t a t i o n has been used f o r t h e d i f f r a c t i o n orders. According t o i t , t h e p o s i t i v e and negative orders m u t u a l l y ex- change one i n t o t h e o t h e r across t h e transducer po- s i t i o n x l = 0.

Both experimetal cases represent s t r a i g h t forward s l o p i n g p r o f i l e s o f t h e surface r i p p l i n g ($211 = 270°), s i n c e negative orders s t e a d l y grow along t h e propagation d i r e c t i o n , w h i l e t h e p o s i t i v e ones pass through a n u l l minimum value. The knowledge o f $ 211, t o g e t h e r w i t h t h e xlminvalue, permits t o d e r i v e t h e xo value through eq. (10) and subsequen- t l y t h e $ 211 value through t h e use o f eq.(9).

I n t a b l e I t h e $ 211 and Gzll values are summarized f o r t h e present experimental cases and f o r t h e f o l l o w i n g ones.

Figures 5 and 6 give t h e experimental r e s u l t s r e l a - t i v e t o I t 2 l i g h t i n t e n s i t y measurements f o r yzLiNb03 and yz-LiTa03 c r y s t a l s . Both cases s t i l l represent forward s l o p i n g r i p p l i n g p r o f i l e s , s i n c e I+2 curves a t t a i n a minimum value. Since, however, these minima d i f f e r from zero,

li-2111

angles wi 11 a c c o r d i n g l y be d i f f e r e n t from 270' by q u a n t i t i e s A$ 's, o b t a i n a b l e from eq. (11) and equal t o :

A = t s i n - ' [ ~ + ~ ( x ~ m i n ) / ~ ~ ( o ) ] l / ~ (12) I t should be noted t h a t i n these l a s t two cases examined, t h e c r y s t a l l o g r a p h i c planes xy

,

normal t o t h e propagation d i r e c t i o n , are n o t m i r r o r planes o f symmetry and, t h e r e f o r e , a $211 angle d i f f e r e n t from 270' was t o be expected.

Conversely, i n t h e previous two cases examined, t h e propagation o f t h e waves took p l a c e along d i - r e c t i o n s normal t o m i r r o r planes o f symmetry and, t h e r e f o r e , angles e x a c t l y equal t o 270' ( o r 90') were t o be expected. It can be seen from eq. (12) t h a t t h e method j u s t described leaves an ambigui- t y on t h e s i g n o f A$ angle. This ambiguity c o u l d be removed o n l y through a d d i t i o n a l experiments performed i n l i g h t transmission o r on d i s p e r s i v e substrates.

F i g . 7 i s r e l a t i v e t o t h e case o f xy-Si02, where t h e surface wave d i s t o r t s toward a s t r a i g h t

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ALIPPI A. e t a l .

TABLE 1

t

¹²² ^(mv)

1 6 Y - c u t LiNbO, f = 4 6 M H z Material

LiNb03

B j 12Ge020 LiNb03

Fig.3 I l i g h t i n t e n s i t y versus propagation d i r e c t i o n r e l a t i v e t o y-x LiNb03 c r y s t a l sample

k2

frequency f l (MHz)

46

24

45

YZl1

270'

270' 270' +49 270' -60 c r y s t a l

c u t

Y

z

Y

prop.

d l r e c t

.

f x

(110) +z

-

z

GZl1

.08

.08 .32 .32

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-254 JOURNAL DE PHYSIQUE

.

*!g. 4.1k2 l i g h t i n t e n s i t y versus propagation I i r e c t i o n r e l a t i v e t o y-z LiNb03 c r y s t a l sample

A

j ~ i c j .

5

Iq

l i g h t i n t e n s i t y versus propagation d i - r e c t i o n r e l a t i v e t o y-z LiNB03 c r y s t a l sample.

Fig. 6 It li g h t i n t e n s i t y versus propagation

!

r e c t i o n r e f a t i v e t o y - z LiTa03 c r y s t a l sample

( m V )

Y - c u t LiTaO, f=36 M H z G2,i=.1 6

~ 2 1 7 2 7 0 ~ 5 7 ~

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ALIPPI A. e t a l . c8-255

/5/ V e l l a P.J. Stegzman G . I . and R i s t i c V.M., J. Appl. Phys.

-

44 (1977) 85

25 f = 7 9 ~ ~ ~ /6/ Whithr~m G.b;., "Linear and non l i n e a r waves", Wiley

,

I n t e r s c i e n c e N.Y. (1974)

/7/ A l i p p i A., Palma A., P a l m i e r i L. and Socino G.

J. Appl. Phys.

-

48 (1977) 2182

/8/ Bains J.A.Jr and Breazeale M.A., J. Acoust.

Soc. Am. 5 7 (1975) 745

/9/ Neighbors T.H. and Meyer W.G., J. Appl. Phys.

42 (1971) 3676

/ l o / -

Lean E.G. and Powell C.G., Appl. Phys. L e t t . 19 (1971) 356

-

1 0 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 0

Fig. 7 I k 2 1 i g h t i n t e n s i t y versus propagation d i - r e c t i o n r e l a t i v e t o x-y Si02 c r y s t a l sample

backward s l o p i n g p r o f i l e o f the surface corruga- t i o n . Indeed, t h e Lk2 curve passes through a n u l l value, thus assuring t h a t a 90' angle i s a t t a i n e d by $211, coherently w i t h t h e m i r r o r symmetry o f t h e propagation d i r e c t i o n .

I n conclusion, several cases o f surface a c o u s t i c wave n o n l i n e a r propagation have been pre- sented, which r e f e r t o d i f f e r e n t c r y s t a l s a n d propagation geometries. I n s p i t e o f t h e inhomogeneous c h a r a c t e r i s t i c s o f s u r f a c e waves, a common f e a t u - r e was e x p e r i m e n t a l l y found w i t h a c o u s t i c b u l k waves, when t h e propagation d i r e c t i o n o f surface waves i s normal t o a m i r r o r plane o f symmetry o f t h e c r y s t a l . I n t h i s case, t h e d i s t o r t i o n o f t h e r i p p l i n g p r o f i l e o f t h e wave i s e i t h e r i n the s t r a i g h t forward o r backward s l o p i n g shape.

I n t h e l a c k o f these symmetry c o n d i t i o n s , angles '4211 may d i f f e r , even w i d e l y , from t h e two e x t r e - me cases o f 270" and 90"

AKNOWLEDGMENTS.

-

The authors wish t o thank P. Verardi f o r h i s h e l p i n performing SAW t r a n s - ducer f a b r i c a t i o n .

References

/1/ Lean E.G.H. and Tseng C.C., J. Appl. Phys.

41

(1970) 3912

/2/ Gibson J.W. and M e i j e r P.H.E., 3 . Appl. Phys.

45 (1974) 3288

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/3[ A d l e r E.C., Bridoux E., Coussot G. and D i e u l e s a i n t E., I EEE Trans. Sonics U l t r a s o n i c s SU-20 (1973) 14

/4/ Ljamov V.E., Hsu T.H. and White R.M., J. Appl.

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