RAYLEIGH SURFACE WAVES (RSW) GENERATED BY PHOTOTHERMAL PROCESS : THEORY

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RAYLEIGH SURFACE WAVES (RSW) GENERATED BY PHOTOTHERMAL PROCESS : THEORY

S. Huard, D. Chardon

To cite this version:

S. Huard, D. Chardon. RAYLEIGH SURFACE WAVES (RSW) GENERATED BY PHOTOTHER- MAL PROCESS : THEORY. Journal de Physique Colloques, 1983, 44 (C6), pp.C6-91-C6-96.

�10.1051/jphyscol:1983614�. �jpa-00223173�

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JOURNAL DE PHYSIQUE

Colloque C6, suppl6ment au nalO, Tome 44, octobre 1983 page C6- 91

R A Y L E I G H SURFACE WAVES ( R S W ) GENERATED B Y PHOTOTHERMAL PROCESS :

THEORY

S . J . Huard and D. Chardon

I n s t i t u t dfOptique ThBorique e t AppZique'e, Laboratoire d'Expe'riences PondamentaZes en Optique, Betiment 503, Centre Universitaire d f @ s a y , B.? N046, 91406 Orsay Cedex, Frrmce

Resume : Une etude theorique de l ' e x c i t a t i o n d'ondes de Rayleigh i l a s u r f a c e de s o l i d e s i s o t r o p e s par voie thertroelastique e s t pr6sent6e. Nous donnons e n s u i t e une estimation du c o e f f i c i e n t de conversion thermique acoustique.

A b s t r a c t : Thermoelastic e x c i t a t i o n of RSW i s t h e o r e t i c a l l y i n v e s t i g a t e d on i s o t r o p i c s o l i d s . Thcrmal-.acoustic conversion e f f i c i e n c y i s evaluated.

I t i s well known t h a t Rayleiph s u r f a c e waves (RSW) can propapate on t h e s u r f a c e of s o l i d s . Generally, suck waves a r e e x c i t e d by p i e z o e l e c t r i c couplinc on non-isotro- pic s u b s t r a t e . On i s o t r o p i c m a t e r i a l s a thermoelastic coupling appears a s a s u i t a - b l e way t o generate RSW. Recently, experimental i n v e s t i c a t i o n s have been c a r r i e d out with heat sources c r e a t e d by o p t i c a l absorption / I , Z / , but no complete theory of t h i s e f f e c t e x i s t s .

In t h i s paper, we p r e s e n t a ~ o n o c h r o m a t i c approach of t h i s problem which l e a d s t o t h e exact determination of t h e d i s p l a c e c e n t vector f i e l d , s t r e s s e s and a l l o t h e r s thermal ahd e l a s t i c v a r i a b l e s . l!hile we have rnainly d e a l t with RSW, t h e method y i e l d s a l s o bulk waves [compressional and shear waves). From thermoelastic

equations / 3 / , which a r e coupled, s o l u t i o n s a r c expressed i n terms of thermal mode and of c o m p r ~ s s i o n a l and s h e a r modes a s usual.

A o n e dimensional s p a t i a l Fourier t r a n s f o r n connected w i t h s u i t a b l e outgoin? ( o r decaying) wave condition i s used t o solve t h e problem. In o r d e r t o prove t b e small influence of t h e gas above t h e s u b s t r a t e on RSW, we have compare? t h e s o l u t i o n s in two c o n f i g u r a t i o n s : s o l i d ln vacuum and sol id in a ?as ( a ~ r o r be1 ium). C'e have a l s o t h e o r e t i c a l l y s t u e i e d t b e amplitude of RSW on d i f f e r e n t i s o t r o p i c s o l i d s f o r t b e same beat source amplitude.

Morecver, n u r e r i c a l ccmputaticns show t h a t t h e thermal RSV conversion e f f i c i e n c y can be e a s i l y cval uated.

When a l i g h t beam 'ir focused'on t h e s u r f a c e o t an absorbin? s o l i d , a p a r t of t h e e l e c t r o m a g n ~ t i c power i s converted i n t o heat. Foreover, i f t h e l i g h t beam i s s i n e mcdulated, e l a s t i c waves a r e rcnerated by thermoelastic processes 141. In order t o s i m p l i f y t h e n ~ a t h e a a t i c a l t r e a t m e n t , only a l i n e source along t h e z-axis i s considered [Fig. I j .

When a s o l i d i s highly absorbing such a s a metal o r when t h e heat source appears on account of s u r f a c e plasmcn wave phenomena, t h e heat source can be reduced t o a surface l i n e source 6 ( 3 j 6 ( y ) . $very thernloelastic p r o p e r t i e s csn be derived from

&he+ t e m p e ~ a p r e f i e l d T ( r , t j = T ( r ) . e x p ( i w t ) and form t h e displacement v e c t c r u ( r , t j = ~ ( r ) . e x p ( i w t ) . (As usual in harmonic time-dependent processes, complex

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983614

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JOURNAL DE PHYSIQUE

.J t o c u s e d b e a m

F i c u r e 1 : N o t a t i o n s : @ power p e r u n i t l e n g t h a l o n g izhe z . a x i s .

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amp1 i t u d e s o f f i e l d s a r e used). I n o r d e r t o e x h i b i t c o m p r e s s i o n a l + w a ~ e s and+ shear waves, t h e d i s p l a c e m e n t vec$or+is s p L i t i n iir 10nciturJinal p a r t u L ( r ) = n r a d ^@ a n d i n i t s t r a n s v e r s e p a r t u ~ ( r ) = c u r l v where @ and v a r e t h e well-known e l a s t i c p o t e n t i a l s .

Because o f t h e symmetry o f t h e l i n e source on t h e s o l ~ d , e v e r y f i e l d i s z-indepen- d e n t ; i t comes t h a t o n l y t h e z-component o f t h e t r a n v e r s e p o t e n t i a l $ i s s i c n i f i - c a n t , so we s h a l l n o t e 142 = v.

D e a l i n g w i t h harmonic t h e r m o e l a s t i c processes i n i s o t r o p i c s o l i d s , t e m p e r a t u r e and e l a s t i c p o t e n t i a l s a r e c o u p l e d by t h e f o l l o w i n g e q u a t i o n s /3/ :

( I . a ) A T - ^T- i w n A @ = -A 6 ( x ) 6 ( y ) s

where AS i s t h e t h e r m a l d i f f u s i v i t y o f t h e s o l i d , kL and k ~ a r e r e s p e c t i v e l y t b e wave number o f t h e l o n p i t u d i n a l ( o r c o ~ p r e s s i o n a l ) and o f t h e t r a n v e r s e ( o r s h e a r ) wave, n and m a r e t h e c o u p l i n r c o e f f i c i e n t s b o t h of them b e i n g p r o p o r t i o n a l - t o t h e t h e r m a l t x p a n s i n n c o e f f i c i e n t / 3 / . A i s t h e m a ~ n i t u d e o f t h e l i n e h e a t source. It must be c u o t e d t b a t T and @ a r e d i r e c t l y c o u p l e d by e q . ( l . a j and e q . ( l . b ) when v

i s r e l a t e d t o @ o n l y t h r o u c h boundary c o n d i t i o n s on t h e y = 0 p l a n e . F o r s o l v i n g t h e r m o e l a s t i c e q u a t i o n s ( I ) , s u i t a b l e c o n d i t i c n s have t o be chosen on t h e p l a n e y = b. C e n e r a l l y , wben s u r f a c e wave c e n e r a t i c n i s s t u d i e d , t h e s u r r o u n d i n s a i r i s

i g n o r e d : f o r example, when s u r f a c e e l a s t i c waves a r e induced on a s o l i d by weans o f a p i e ~ o e l e c t r ~ c t r a n s d u c e r , t h e power l o s s i n t h e a i r i s n e g l i r i b l e . B u t i n t h e p r e s e n t case, a p a r t o f t h e h e a t d i f f u s e s i n t o t h e f l u i d . I t becomes necessary t o t a k e i n t o account t h e thermal p r o p e r t i e s o f t h e upper medium t h r o u g h t k e thermodynamic e q u a t i o n s o f t h e f l u i d / 5 / :

which a r e snalogous t o eq. ( 1 ) . ( I t n u s t be quoted t h a t n o v i s c o s i t y e f f e c t s a r e considered, so shear waves can n o t e x i s t i n t h e f l u i d ) . I n ep. (21, AF i s t h e f l u i d t h e r m a l d i f f u s i v i t y , y i s t h e r a t i o o f s p e c i f i c h e a t s Cp/CV, a = (ap/aT), and k ~ i s t h e wave number o f t h e sound waves.

The u s u a l boundary c o n d i t ~ o n s a r e t h e c o n t i n u i t y o f t h e t e m p e r a ~ u r e , o f t h e h e a t flmw, o f 0 l e n o m l d i s p l a c m t uy a n d t h e c o n t i n u i t y o f t h e s t r e s s e s uxy, uyy. A t t h i s p o i n t , i t r u s t be n o t e d t h a t t h e i n f l u e n c e o f t h e f l u i d ( a i r ) i s weak because o f t h e r a t i o o f thermal c o n d u c t i v i t i e s K ~ / K F , much g r e a t e r t h a n I ( = 104 f o r aluminum- a i r system). TG e v a l u a t e t h e i r x f l u e n c e o f t h e gas on t h e e l a s t i c s u r f a c e waves g e n e r a t i o n , tws r e s c l u t i c n s have been c a r r i e d o u t . F i r s t , we c o n s i d e r t h a t vacuum

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i s i n the y Z O h a l f space and then we s o l v e t h e r e a l problem ( s o l i d - g a s ) . 1-1 SOLIC IN VACUUM

I n t h i s case, the stresses ax? and uyf vanish on y = t! as a l s o t h e heat f l o w towards t h e $as. To solve eq. 1 ) a x- o u r l e r t r a n s f o r m i s used accordinp the d e f i n i t i o n :

2inux dx etc.. .

For a ~ i v e n u, t h e p r o b l e ~ i s reduced t o one dimension. The system'(1) becomes :

a z i 2 2 2

(3.a) , - ( 4 n u + i W, T t i u n kL $= - ^A6(y)

aY A

i n which we have used eq. (1.b) t o s u b s t i t u t e A $ i n eq.(l.a). I n t h i s transforma- t i o n we s e t

I n t r o d u c i n y aL 2 = 4n2 u2 - k: and a: = 4n2 u2 + i f , i and 4 can be e a s i l y expressed f o r y < 0 :

- "9

( 4 ) T-(u,y) = a ( u ) e ⁺as B ( U ) e OLY

where oL and a a r e chosen so t h a t only y decaying waves o r outgoing waves e x i s t i n t h e s o l l d . as tand tS - a r e r e l a t e d t o c o u p l i n g c o e f f i c i e n t s by

and

2 .

because f >> kL 7n t h e XPz frequency range. I n a s i m i l a r manner : (3.c) $(u,y) = y ( u J e OTY

4ni u2 - k 2 : o w i l l be chosen t o s a t i s f y the o u t c o i n ? wave c o n d i t i o n . :k:r&;~itudes a i u ) , & ( u ) 2nd v [ u ) o f t h e thermal mode (eYtY) , 1 o n p i t u d i n a l mode

( e U i y ) and transverse rcode (euTY) a r e derived from boundary c o n d i t i o n s . We o b t a i n :

. ,

2 2 2 4 i a u t S ( 8 n u -kT) (ut

( u ) = --- ^--

A

- "L' A

w i t h A = ot a. - a s t S ~ L ~ l

i n which do = (8,' uZ - ^kT2)2- 16n2 u2 uLuT

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C6-94 JOURNAL DE PHYSIQUE

2 2 2 2

and bl = ( 8 n 2 uZ kT ) - 16n u utuT

Because >> k L ~ and l a s t S / << ^1, i t appears t h a t t h e f i r s t t e r r in A i s preponde- r a n t unless no vanihes. I t i s important t o note t h a t A, = 0 y i e l d s t b e R a y l e i ~ h + phase v e l o c i t y alono t h e x.axis V R /6/. Usin? t h e expression u ⁼grgd :d + c u r l y, t h e x. Fourier t r a n s f o r n fix and fiy of ux and uy a r e e a s i l y obtained a s a l s o Sxy and Syy.

1-2 SOLIC IN -- A CAS

The main modification r c f e r r e d t o t h e precedinc case l l e s in t h e boundary condi- t i o n s . Now, t h e c o n t i n u i t y of temperature, of heat flow and of normal displacement f o r y = 0 must be w r i t t e n . I'oreover, t h e n o r m a l - s t r e s s uyy equals t h e cas pressure p a t y = G .

From t h e l i k e n e s s of eq. 1 and eq. 3 a thermal mode and an a c o u s t i c a l rode a r e z l s o introduced in t h e pas ; outgoinc ( o r decayin?) wave c o n d i t i o n s a r e n a t u r a l l y taken i n t o account. The boundary i o n d i t i o n s lead t o f i v e l i n e a r equation which w i l l y i e l d t h e amplitude of each mode e i t h e r i n t h e s o l i d o r i n t h e cas. The s c l u t i o n i s numerically derived because no simple a l e e b r a i c expression can be obtained. The x-Fourier transform of t h e displacement i s c a l c u l a t e d a s in 5 1-1.

I 1 - NUPIERICAL RESULTS 11-1 RSK AYPLITUPE

Fio.2: Amplitude of normalized Iu^,,I-for vacuum and two gases

In order t o s i m p l i f y t h e physical i n t e r p r e t a t i o n , we have c a l c u l a t e d t h e d i s p l a c e - ment ux and uy on t h e y = 0 plane. Both \fix/ and Ifiy 1 e x h i b i t a sharp peak f o r

u = U R such-that ao(Up) = 0 correspondinc t o t h e Rayleirh phase v e l o c i t y .

J

k h i l e only 112 1 i s p l o t t e d on Ficure 2 n u ~ e r i c a l computations give t h e same shape f o r f i x f and in p a r t i c u l a r t h e same width a t half~naximum. I t appears t b a t t h e width A U i s rinimum when no gas i s p r e s e n t 1 t h i s confipuration corresponds a l s o t o t h e h i c h e s t peak. When a gas covers t h e s o l i d , t h e amplitude of t h e RSll decreases a s i t propagatesalonc t h e x-axis because of therrral l o s s e s . Accordino t o Fourier trans for^ p r o p e r t i e s , t h e damping length i s around I / A U .

4.

0.5 .

In every cases t h e narrowness of t h e peak near up j u s t i f i e s t h e vacuum model even f c r thermoelastic e x c i t a t i o n of RSW. fro^ now on we s h a l l use t h i s ~ o d e l . Because of t h e outgoino s u r f a c e wave c o n d i t i o n , only p o s i t i v e s p a t i a l frequencies u must be considered f o r x > ⁰when t h e inverse Fourier transform i s c a l c u l a t e d . This o p e r a t i c n c i v e s t h e amplitude of ux and uv ( f o r exanple a t t = 0) versus x(Fi?. 3 ) .

nIurj

ax

lii

-- --

- -

u h m ^{- 4}

328.5 328.6 128.3.

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rnm

F i q . 3 : Amplitude o-F ux and u versus t h e d i s t a n c e x f r o r r t h e source Y

As usual a a/i o u t o f phase e x i s t s between ux and u , and t h e wavelenpth observed i s t h e R a y l e i a h wavelenoth. We have compared two s u g s t r a t e s (aluminum and c o p p e r ) . A small d i f f e r e n c e appears on t h e w l d t h o f t h e 10yl and lOxl peaks ; on copper RSK decay s l o w e r t h a n on aluminum.

11-2 THERMAL-RSN CCF!VEP.SICK EFFICIENCY

F o r e v a l u a t i n o t h e therrral-RSI! c o n v e r s i o n e f f i c i e n c y i t i s necessary t o c a l c u l e t e t h e f l u x 6 f t h e P l a s t i c P o y n t i n a v e c t o r t h r o u g h a x = c s t e p l a n e f o r a narrow s p a t i a l f r e q u e n c y band AU around UR.

The x corrponent of t h e t i m e averaged P o y n t i n n v e c t o r i S

1 a u

i s PX = [oXY (-3- a t ⁺^c.c.] because ox,

-

⁰

i e 1

P = [-iw'u*

X ^Oxy + C ' C '1

Because I O / and 10 / a r e narrow peaks, t h e y can be approximated w i t h 6 f u n c t i o n s so t b a t becomes

where Bu(y) and B G ( y ) a r e t h e a c p l i t u d e s o f O ( u ) and 6 ( u ) f o r u = uR.

Y XY

The x c o ~ p o n e n t of t h e P o y n t i n c v e c t o r i s d e r i v e d f r o m z ⁼^gr$d ^;d ⁺^{c z r l}^$and f r o m e x p r e s s i o n ( 5 ) and ( 6 ) . An e v a l u a t i o n o f t h e Px f l u x p e r u n i t l e n c t h a l o n g t h e z. a x i s

1 eads t o 2

e ;d w i t h e z lo-' w-I, f o r aluminum. T h i s r e s u l t s shows t b a t a Great p a r t ;!'li Fhe l ? r h t absorbed i s used t o h e a t t h e s u b s t r a t e .

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C6-96 JOURNAL DE PHYSIQUE

CONCLUSION

Ue have s t u d i e d t h e KSW e x c i t a t i o n by thermoelastic processes on i s o t r o p i c substra- tes. When the s o l i d i s covered w i t h a gas a r e s o l u t i o n o f rhermoelastic equations and thermodynamic equations i n t h e gas shows a weak i n f l u e n c e o f t h i s l a s t f l u i d . This a n a l y s i s j u s t i f i e s t h a t the gas i s u s u a l l y i c n o r e d when RSK are s t u d i e d even i n t h e case o f thermoelastic e x c i t a t i o n .

E h i l e e x p e r i s e n t a l i n v e s t i p a t i c n o f t e n use p u l s e sources, we have d e a l t w i t h harmonic sources. The knowinc of t h e phenomena versus t h e modulation frequency then a l l o w s a s p e c t r a l a n a l y s i s o f t h e pulsed case.

REFERENCES

/ I / A.M. AINDOIJ, R.J. DEWFURST and S.B. PALMER Opt. Com. - 42, ( 1 9 8 2 ) , 116 /2/ A.C. TP,M and H. COUFAL Appl. Phys. L e t t . - 42, ( 1 9 8 3 ) , 33.

/5/ Irl. NOWACKI : " T h e r m o e l a s t i c i t y " Pergamon Press, Oxford New-York, London, P a r i s 1962.

/4/ M.A. BICT J. Appl. Phys. - 27, (1956) 24C

j 5 f P.M. FCORSE and K.U. IPIGARD : " T h e o r e t i c a l Acoustics", Kc Graw F i l l , New-York 1968.

/ 6 / F.S. LOCKETT Jour. Pech. Phys. Sol. - 7, ( 1 9 5 8 ) , 71.