STRENGTH OF NONLINEAR INTERACTION BETWEEN COLLINEAR SURFACE ACOUSTIC WAVES

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STRENGTH OF NONLINEAR INTERACTION BETWEEN COLLINEAR SURFACE ACOUSTIC

WAVES

A. Adamou, A. Alippi, A. Palma, L. Palmieri, G. Socino

To cite this version:

A. Adamou, A. Alippi, A. Palma, L. Palmieri, G. Socino. STRENGTH OF NONLINEAR INTERAC-

TION BETWEEN COLLINEAR SURFACE ACOUSTIC WAVES. Journal de Physique Colloques,

1979, 40 (C8), pp.C8-262-C8-266. �10.1051/jphyscol:1979845�. �jpa-00219551�

STRENGTH OF NONLINEAR INTERACTION BETWEEN COLLINEAR SURFACE ACOUSTIC WAVES A.Adamou, A. Alippi, A. Palma, L. Palmieri and G. Socino

jstituto disAouetica "O.M. Covbino", .C.B.R.., Via Cassia, 1216 -Rome, Italy and Istituto di Fisiea, Universita di Perugia, Italy.

Abstract. - An optical method has been developed for the determination of the coupling coeffi- cient in the case of nonlinear mixing of two collinear surface acoustic waves.

The method is simple enough and permits the evaluation of the amplitude and phase of the cou- pling coefficient without requiring any knowledge of the acoustic power flows. The value of the coupling coefficient has been measured for YZ LiNbOg in the cases of sum- and difference- frequency generation. It has been found that in the former case the magnitude of the coeffi- cient is greater than that of second harmonic generation by a factor of two and is not very sensitive to the variations of the frequency ratio fg/f^ of the two interacting waves. However, in the case of difference-frequency generation the magnitude of the coefficient is smaller than the one measured for the sum-frequency generation and also strongly dependent on the ratio fg/f].- 1. INTRODUCTION. - The present paper deals with

the mixing of two collinear surface acoustic waves propagating on a nonlinear medium. In the past, Lean and co-workers /1,2/ reported mixing of colli- near surface waves on a -quartz and YZ LiNbOg, whereas others investigated the mixing of non collinear waves /3,4/. The efficiency of this type of nonlinear interaction can be measured in terms of a phenomenological coupling constant, the know- ledge of which is important for the design of prac- tical devices / 5 / .

The novelty of the present work is the intro- duction of a simple and accurate method for measu- ring the amplitude and phase of the complex cou- pling coefficient between two collinear surface waves. First indicative measurement of the value of the coefficient for YZ LiNb03 crystals have been also performed. The theory presented here is a combination of the coupled system of equations introduced by Shiren for the explanation of the parametric amplification of bulk waves /6/ and the optical theory of Alippi et al. HI which has been successful in the determination of the coupling

coefficient in the case of surface wave second harmonic generation. An exact analytical treatment of the surface wave mixing problem is difficult and cumbersome because of the inhomogeneous nature of the waves involved in the interaction / 8 / and compares unfavourably with the present phenomenolo- gical approach which is capable of determining the strength of the nonlinear interaction with an esti- mated error of the order of 5 per cent.

2. ANALYSIS. - Consider two surface waves propaga- ting along the direction and decaying in the -z- direction of an elastically nonlinear medium.

Because of the elastic nonlinearity, the two waves will generate, apart from their own second harmo- nics, surface waves at their sum- (up conversion) and difference-frequencies(down conversion). Follo- wing Shiren /6/ or Lean and Tseng / 2 / , one can write

(la) JOURNAL DE PHYSIQUE Colloque Cg, supplément au n° 11, tome 40, novembre 1979, page C8-262

Résume. - Une méthode optique a été développée pour la détermination du coefficient de couplage non-linéaire dans la propagation colinéaire de deux ondes acoustiques de surface. Cette méthode simple fournit sans exiger la connaissance de la puissance acoustique, la valeur de l'amplitude et de la phase de ce paramètre non-linéaire.

On présente ici les résultats expérimentaux concernant la détermination du coefficient de couplage non-linéaire sur des échantillons de yZ -LiNbÛ3 relativement aux cas de génération paramétrique d'ondes acoustiques aux fréquences addition (f^+fg- et différence (fi~f2) des fré- quences fondamentales fi et f2

Dans le premier cas on a trouvé que l'amplitude du coefficient non-linéaire ne dépendait pas du rapport des fréquences f2/fi et était deux fois supérieure à la valeur du coefficient non-linéaire mesuré dans le cas de génération du second harmonique.

Au contraire, dans le cas de génération d'ondes acoustiques de fréquence différence (fi-f2) on a trouvé que l'amplitude du coefficient non-linéaire dépendait fortement du rapport f2/fi et avait une valeur toujours inférieure à celle mesurée dans le cas de génération paramétrique à la fréquence addition fj + f2

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:1979845

JOURNAL DE PHYSIQUE C8-263

where Al(x) and A2(x) a r e t h e surface values o f t h e transverse e l a s t i c displacement o f t h e two i n t e r - a c t i n g waves whose frequencies are fl and f 2 (fl>

f2), r e s p e c t i v e l y ; A3(x) and A4(x) a r e t h e sur- f a c e values o f t h e t r a n s v e r s e e l a s t i c displacement o f t h e p a r a m e t r i c a l l y generated waves a t frequen- c i e s f3 = fl

+

^f2 and f4 = f l

-

f p , r e s p e c t i v e l y .

r j 1 2 (j = 3,4) i s t h e complex c o u p l i n g c o e f f i c i e n t and j' and K j stand f o r t h e a t t e n u a t i o n c o e f f i c i - e n t and wavenumber, r e s p e c t i v e l y , o f t h e Aj(x) wave It can be shown t h a t t h e general s o l u t i o n o f eqns.

( l a ) and ( l b ) , under t h e assumption t h a t A ~ ( X ) and A2(x) are a f f e c t e d l i t t l e by t h e presence o f t h e parametric waves, i s give; by :

r312 A1 (0)A2(0) A3(x) =

'3- ( 9 + 9 ) + i (K3-(K1+K2))

exp

-

[ ( a l + 9 ) + i (K3-(K1+K2))1 x

.-

(pa)

and

exp

-

[ ( ~ ~ + ~ ~ ) + i ( K ~ - ( K ~ - K ~ ) )

I

^x

-

^(2b)

where i t has been assumed t h a t t h e waves are vary- i n g according t o exp i (yt-Kjx). For o p t i c a l l y p o l i s h e d s i n g l e c r y s t a l m a t e r i a l s i n t h e MHz r e g i o n o f frequencies t h e wave d i s p e r s i o n and a t t e n u a t i o n a r e small, and t h e above equation can be s i m p l i f i e d t o

A 3 ( x ) = r312A1(0)A2(0)x (3a)

and

A41x) =

-

r412~lf~)~f2(0)x (3b)

even f o r r e l a t i v e l y l a r g e propagation distances x.

Equations (3a) and (3b) show t h a t t h e amplitudes o f t h e p a r a m e t r i c a l l y generated waves a r e propor- t i o n a l t o t h e ' p r o d u c t o f t h e amplitudes o f t h e i n - t e r a c t i n g waves and grow l i n e a r y w i t h t h e t r a v e l l e d distance.

One o f t h e most powerful techniques f o r moni- t o r i n g t h e amplitude o f surface waves along t h e i r propagation p a t h i s by probing t h e c r y s t a l s u r f a c e

w i t h a narrow monochromatic l i g h t beam /7,9/. I n the subsequent a n a l y s i s t h e d i f f r a c t i o n spectrum o f a l i g h t beam r e f l e c t e d by t h e surface of a c r y s - t a l on which two s u r f a c e waves and t h e sum-frequen- cy wave a r e propagating w i l l be examined.

The b a s i c r e f l e c t i o n geometry i s shown i n Fig.1 A l i g h t beam impinges a t a small incidence a n g l e s on t h e c r y s t a l s u r f a c e which i s corrugated by t h e surface waves. The r e f l e c t e d l i g h t wavefront, E, i s phase modulated, so t h a t i t r e p l i c a t e s t h e sur- face c o r r u g a t i o n . Thus, one can w r i t e / 7 /

E = E o e x p [ i 2 k c o ~ [ ~ ~ l ( x ) \ cos (wlt-K1x@ 1)

where Eo and k a r e t h e amplitude and wave-number o f t h e r e f l e c t e d l i g h t wavefront, and t h e f a c t o r 2 takes account o f t h e doubling o f t h e phase corruga- t i o n r e s u l t i n g from t h e r e f l e c t i o n geometry.

Aj ( x )

,

wj

,

^{Kj and $}

.

a r e t h e amp1 i tude, angular J

frequency, propagation constant and phase o f t h e A j ( j = 1,2,3) a c o u s t i c wave, w i t h w3 =wl+w2 and K3 = K1+K2.

Equation (3a)shows t h a t t h e phase angle $3 can be w r i t t e n as @3=$1+$2+$312 where $312 i s t h e phase o f t h e complex c o u p l i n g c o e f f i c i e n t

312.

I f t h e diameter o f t h e scanning l i g h t beam i s small enough, so t h a t w i t h i n t h e i l l u m i n a t e d p a r t o f t h e c r y s t a l s u r f a c e t h e e l a s t i c displacements- can be taken t o be constant, t h e F o u r i e r spectrum o f t h e r e f l e c t e d wavefront i s composed o f a d i s c r e t e number o f terms s p a t i a l l y d i s p l a c e d by amounts p r o p o r t i o n a l t o t h e r a t i o K./k between t h e sound

J

and l i g h t wave-numbers. Using t h e i d e n t i t y

t d

exp(ircosc$)

=z

i n d n ( r ) e x p ( i n $ )

- *

where Jn i s t h e n - t h o r d e r Bessel f u n c i o n o f t h e f i r s t kind, can be shown e a s i l y t h a t t h e i n t e n s i t y o f t h e d i f f r a c t i o n o r d e r s which c o n t a i n i n f o r m a t i o n about t h e sum-frequency wave i s given by

I i 3 = I0 ~ - ~ ~ ( 2 k c o s i 7 ~ ~ ~ ~ ) J1(2KcosS1A2

1

⁾

~ ~ ( 2 k c o s g

I A ~ I

⁺

( 5 ) iJ0(2kcosBI~11 ) ~ ~ ( 2 k c o s 3 1 ~ ~ 1 )

c8-264

A. ADAMOU et al.

From the physical point of view, the first term on the right-hand side of eqn. (5) represents the part of which results from the double scattering of the impinging light caused by the two fundamen- tal surface waves, whereas the second term repre- sents the contribution to 123 of the light scatte- red directly by the parametric wave A3. In the series expansion of Bessel functions all terms higher than the first can be ignored because, at ultrasonic frequencies, 2k \Aj l a 1, even for a lar- ge acoustic power flows. Hence, substitution of eqn. (3a) into eqn. (5) results in

where IO

=

E$ is the amplitude of the intensity of' the reflected light. Following a similar analysis for the difference frequency wave, it can be shown that the intensity of the corresponding diffrac- tion orders, I+4, is given by

The magnitude and phase of the coupling coeffi- cient can be calculated directly from the last two equations, each of which represents a pair of para- bolas, using the relations

cos 2

*jI2 =

I-j(~min)/Ij(0) (7a) and

1

^rj12

I= kcos9sin*j12/xmin (7b) where xmin is the distance at which

I-j

attains its minimum value and Ij(0) is the value of Ij at the origin (I+3(O)

=

L3(0)

=

1+4(0)

=

I-4(0)).

It must be noted that the method described here has an intrinsic uncertainty of

IT

in the calcula- tion of the phase angle,

i

.e. the latter can be

*jl2 or 8-*j12. This uncertainty can be removed by using dispersive crystals

or,

in the case of transparent media, by performing light transmi- sion measurements. /7/

Fig. 1 Schematic representation of the diffraction orders caused by two collinear surface acous- tic waves and their sum frequency wave.

3.RESULTS AND DISCUSSION. - The interdigital trans- ducers of central frequencies fl

=

89.9 mHz and f2

=

18 MHz have been constructed on a dispersion- less Y-cut Z-propagation LiNb03 sample. Each trans- ducer was composed of five finger pairs whose width was 150 and 30 wavelengths long, respectively.

Two collinear surface wave trains were launched at the above frequencies and the high frequency pulses were properly delayed so that they were overlapping with those of low frequency launched by the rear transducer. The propagation surface has been scan- ned by a narrow collimated He-Ne laser beam

( A =

6328 a) which was impinging at a small incidence angle3

:

5O.A photomultiplier, whose output was measured using a box-car integrator, has been emplo- yed for the detection of the various diffraction orders. No absolute measurements of the light inten- sity have been made since this is not required by the method presented here.

The two propagating waves interactirgvia the elastic nonlinearity of the medium produce a sur- face acoustic wave at the sum-frequency f3=fl+f2=

107.9 MHz and the second at the difference-frequen-

cy f4= fl-f2= 71.9 MHz. The measured values of the

light intensity diffracted by the two parametric

waves at various positions along the propagation

direction are shown in Fig. 2. The theoretical

JOURNAL DE PHYSIQUE

curves obtained by b e s t - f i t t i n g eqns. (6a) and (66) on t h e measured values are a l s o p l o t t e d on the same f i g u r e , f o r comparison. The c a l c u l a t e d values o f t h e amplitude and phase o f t h e c o u p l i n g c o e f f i c i e n t f o r the sum-frequency wave arelr3121 = 2.28x10~mm'~ and

1

312 = 300°, r e s p e c t i v e l y

.

Simi

-

l a r l y , f o r t h e difference-frequency wave i t has been found t h a t

I ^rq121

⁼ 1 . 7 6 ~ 1 0 ~ m m - ~ and $412 = 312' The estimated e r r o r i n t h e value o f

Ir

^j121 and $ j12 i s o f t h e order o f 5 per cent and 4O, r e s p e c t i v e l y , f o r every case r e p o r t e d here. I n t h e case o f d i s p e r s i o n l e s s media t h e amplitude o f

rj12

i s p r o p o r t i o - n a l t o t h e product K1K2 o f t h e wave vectors o f t h e i n t e r a c t i n g waves /6,7/. A more meaningful measure o f t h e s t e n g t h o f t h e n o n l i n e a r i n e r a c t i o n i s t h e q u a n t i t y G j l 2 =

rj12

/K1K2, which depends o n l y on t h e m a t e r i a l constants and p o s s i b l y on t h e pro- f i l e s o f t h e waves which p a r t i c i p a t e i n t h e i n e r - a c t i o n . Thus, i t has been found t h a t G312=0.44 and 6412=0.34 f o r t h e up and down conversion, r e s p e c t i - v e l y . I n t h e present case, f o r which t h e frequency r a t i o f2/fl=0.2 i s small

,.

t h e value o f 6312 i s 23 per cent g r e a t e r than t h a t o f 6412. T h i s r a t h e r small d i f f e r e n c e i n t h e s t r e n g t h o f t h e i n t e r a c t i o n between t h e sum- and difference-frequency g e n e r a t i - on cases can be seen d i r e c t l y from F i g . 2 which show t h a t t h e r a t e o f growth w i t h d i s t a n c e i s s i - m i l a r f o r both parametric waves.

I n o r d e r t o check t h e e f f e c t o f t h e frequency r a t i o f2/fl on t h e s t r e n g t h o f t h e n o n l i n e a r i n t e r - a c t i o n , t h e measurements have been repeated f o r f l=90.9 MHz (170 wavelengths wide, f i v e elements transducer) and f 2 = 54.6 MHz (100 wavelengths wide f i v e elements transducer), which g i v e a frequency r a t i o f 2 / f l = 0.6 considerably g r e a t e r than t h e previous one. The experimental r e s u l t s , p l o t t e d i n f i g . 3, a r e again i n good ageement w i t h those pre- d i c t e d t h e o r e t i c a l l y . Comparison between Figs. 2 and 3 shows t h a t i n t h e l a t t e r case t h e sum-frequen- cy wave grows w i t h d i s t a n c e much f a s t e r than t h e difference-frequency wave, whereas i n t h e former case t h e d i f f e r e n c e between the r a t e o f growth o f t h e two waves i s r a t h e r small. I n f a c t , f o r t h e case shown i n F i g . 3 t h e caTculated values o f t h e amplitude and phase o f t h e c o u p l i n g c o e f f i c i e n t a r e

6312'1

r3121

/KlK2=0.43 13i2=333O and ~ ~r4121 ~ ~ = 1 /K1K2=0.19 $412 = 328' f o r t h e up- and down-conver-

sion, r e s p e c t i v e l y , i . e . t h e d i f f e r e n c e between 6312 and G 4 i 2 i s 56 per cent, t w i c e as much as t h e c a l c u l a t e d f o r f2/f1 = 0.2.

0 5 10 15

P R O P A G A T I O N D I R E C T I O N ( m m l Fig.2 T h e o r e t i c a l curves and measured values 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 by t h e sum and d i f f e r e n c e frequency waves as a f u n c t i o n o f propaga- t i o n distance, f o r a frequency r a t i o f 2 / f l = 0 . 2

Second harmonic generation can be considered as mixing o f two i d e n t i c a l waves. I t i s t h e r e f o r e i n - t e r e s t i n g t o compare t h e value o f t h e c o u p l i n g coef- f i c i e n t G211=)I' 211

1

/K1 f o r t h e second harmonic 2 generation w i t h those d e r i v e d here f o r t h e up-con- version case. F o l l o w i n g t h e method described i n r e f . /7/, i t has been found t h a t G211=0.23 and

,

.

$211=3170 f o r b o t h low (fl=18MHz) and h i g h ( f l = 89.9 MHz) frequencies. I n r e f . /7/ has been r e p o r - t e d 6211=0.31 and $211=31g0, f o r t h e same m a t e r i a l and o r i e n t a t i o n . Although t h e measured phase angle i s s i m i l a r t o t h e one r e p o r t e d here a considerable d i f f e r e n c e e x i s t s between t h e two values o f t h e magnitude o f t h e c o u p l i n g c o e f f i c i e n t . T h i s d i f f e - rence i s t o o l a r g e t o be j u s t i f i e d by experimental errorsi o r phenomena l i k e a t t e n u a t i o n , d i s p e r s i o n o r t h e i n f l u e n c e o f F r e s n e l ' s f i e l d , which have n o t been i n c l u d e d i n t h e theory. It i s more l i k e l y t h a t t h i s d i f f e r e n c e i s due t o i m p e r f e c t i o n s i n t h e crys- t a l l i n e s t r u c t u r e o f LiNb03 and, also, t o t h e d i f - f e r e n t p a s t h i s t o r y o f t h e samples used i n t h e r e - p o r t e d experiments

/lo/.

Comparison between t h e present values o f t h e amplitude o f t h e c o u p l i n g c o e f f i c i e n t f o r t h e sum-frequency and second harmo-

c8-266 A. ADAMOU e t a l .

P R O P A G A T I O N D I R E C T I O N ( m m l Fig. 3 T h e o r e t i c a l curves and measured values 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 by t h e sum and d i f f e r e n c e frequency waves as a f u n c t i o n o f propagation distance, f o r a frequency r a t i o f 2 / f l = 0.6

n i c generation shows t h a t t h e former i s g r e a t e r than t h e l a t t e r by a f a c t o r o f two, o r so. The p h y s i c a l meaning o f t h i s r e l a t i o n s h i p i s t h a t second harmonic generation i s a m i x i n g process i n which t h e one h a l f o f t h e a c o u s t i c energy o f t h e funda- mental wave i n t e r a c t s p a r a m e t r i c a l l y w i t h t h e remaining h a l f .

4. CONCLUSION.

-

An o p t i c a l method f o r t h e determi.

n a t i o n o f t h e c o u p l i n g c o e f f i c i e n t f o r t h e up- and down- conversions o f two c o l l i n e a r surface a c o u s t i c waves propagating on a e l a s t i c a l l y n o n l i n e a r medi- um has been developed. The method i s simple and accurate and does n o t r e q u i r e any knowledge o f the a c o u s t i c power f l o w s ; i t uses o n l y t h e r e l a t i v e values 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 by t h e surface waves. The v a l i d i t y o f t h e propo- sed phenomenological t h e o r y has been v e r i f i e d e x p e r i m e n t a l l y u s i n g a YZ LiNb03 c r y s t a l , f o r two d i f f e r e n t values o f t h e r a t i o f2/fl between t h e frequencies o f t h e two i n t e r a c t i n g waves. The r e - s u l t s i n d i c a t e t h a t i n t h e case o f sum-frequency

generation t h e amplitude o f t h e c o u p l i n g c o e f f i c i - e n t i s p r a c t i c a l l y constant and g r e a t e r than t h a t o f second harmonic generation by a f a c t o r o f two, whereas i n t h e c a m o f d i f f e r e n c e frequency genera- t i o n t h e amplitude o f t h e c o e f f i c i e n t i s smaller and appears t o decrease as t h e r a t i o f2/fl (fl f2) increases. F u r t h e r measurements a r e needed f o r t h e determination o f t h e f u n c t i o n a l behaviour w i t h f 2 / f l o f t h e amplitude and phase o f t h e c o u p l i n g c o e f f i c i e n t f o r both, up- and down-conversions.

An accurate knowledge o f these f u n c t i o n s i s neces- sary f o r t h e c o n s t r u c t i o n o f several n o n l i n e a r devices, such as parametric a m p l i f i e r s /11/ and a c o u s t o e l e c t r i c c o r r e l a t o r s /5/.

References

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15

(1969) 10-12

2. E.G. Lean and C.C. Tseng, "Nonlinear E f f e c t s i n Surface Acoustic Waves", J . Appl. Phys.,G(l970) 3912-3917

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.

^Phys.

,

42(1971)5330-5332.

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~ ~ p l

.Phys.

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21

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5. M. Luukkala and J.S. Surakka, "Acoustic Convolu- t i o n and C o r r e l a t i o n and t h e Associated N o n l i - n e a r i t y Parameters i n LiNbOg", J .Appl .Phys., 43 (1972)2510-2518.

-

6. N.S. Shiren, ' U l t r a s o n i c Travelling-Wave Parame- t r i c Amp1 i f i e r "

,

Proc. IEEE, 53(1965) 1540-1546.

-

7. A.Alippi, A.Palma, L.Palmieri and G.Socino,

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2182-2190.

-

8. P.J. V e l l a , T_G. Padmore, G . I . Stegeman and V.M. R i s t i c , Nonlinear Surface Wave I n t e r a c - t i o n s : Parametric M i x i n g and Harmonic Genera- t i o n " , J.App1. Phys.,

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