ACOUSTO-OPTIC INTERACTION OF SECOND HARMONICS IN LAMB WAVES

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ACOUSTO-OPTIC INTERACTION OF SECOND HARMONICS IN LAMB WAVES

N. Brower, W. Mayer

To cite this version:

N. Brower, W. Mayer. ACOUSTO-OPTIC INTERACTION OF SECOND HARMON- ICS IN LAMB WAVES. Journal de Physique Colloques, 1979, 40 (C8), pp.C8-175-C8-178.

�10.1051/jphyscol:1979829�. �jpa-00219534�

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A C O U S T O - O P T I C INTERACTION OF S E C O N D H A R M O N I C S IN LAMB W A V E S N.G. BROWER and W.G. MAYER

Physics Department, Georgetown University, Washington, D.C. 20007

Résumé. - La possibilité de génération d'harmoniques d'ondes de Lambse propageant dans une plaque isotrope est examiné. Les conditions de la génération d'harmoniques sont discutées. Un système de détection utilisant l'interaction acousto-optique permet de mettre en évidence l'existence de deu- xièmes harmoniques en mesurant les disymétries de la figure de diffraction. Les résultats expérimen- taux sont présentés.

Abstract. - The possibility of harmonic generation in Lamb waves propagating on isotropic plates is investigated. The conditions for harmonic generation are discussed. A detection scheme using the acous-' to-optic interaction is designed by which asymmetries in the diffraction pattern are measured to de- termine the existence of second harmonics. The results of the experiments are presented.

1. Introduction. - Finite amplitude mechanical vi- brations in an elastic solid are nonlinear in natu- re which may give rise to a number of interesting interactions. One such interaction is the generation of harmonics as a finite amplitude acoustic wave is propagating in an elastic solid.

Acoustic bulk second harmonic generation was observed by Gedroits and Krasil'nikov' .'When an ini- tially monochromatic longitudinal ultrasonic bulk wave was launched in an elastic medium, a secondary longitudinal wave was observed. The second harmonic amplitude increased linearly with the interaction length, i.e., with the distance of travel.

Harmonic generation of acoustic surface waves in crystals was observed by Ltfpen / 2 / . His experimental results agreed with the theoretical analysis.

As in the case of bulk waves, the amplitude of the second harmonic surface wave varied linearly with the interaction length and quadratically with the fundamental surface wave amplitude.

The possibility of harmonic generation in elastic isotropic plates is investigated in the present experiment. As an introduction, general features of second harmonic generation are presented as a theoretical background. Next, conditions for generation are discussed. The detection scheme utili- zing a laser optical probe is outlined. A brief des- cription of the experimental design is given. Final- ly results of the experiment are presented.

2. Theoretical Background. - Using a perturbative approximation / 3 / , the nonlinear equations of motion for an isotropic elastic solid bulk medium re-

duce to

2-E - cf V 0 . U) + d V x v x U = Q, (1) 3r

where U is the particle displacement amplitude and C, and C|- are the longitudinal and transverse bulk wave velocities. The term on the righthand side of eq.(l), Q, is the source or forcing term. In the case of harmonic generation, Q consists of terms quadratic in the fundamental plane wave amplitude.

An iterative solution/4/ to the nonlinear equation of motion yields

U2 = C Uj x. (2)

The amplitude of the second harmonic, U«, is pro- portional to the square of the fundamental wave amplitude, U,, and also to the interaction length, x, with C a proportionality constant. This result is also obtained by Ltfpen / 2 / for surface waves in which case U, is the fundamental surface wave amplitude. Similarly, for plate mode interactions, eq.

(2) is valid, where U, is the Lamb mode fundamental amplitude. The characteristics implied by eq. (2) will be used to identify a generated harmonic.

In order that a harmonic be generated, the velocities of the fundamental and the harmonic must be equal, that is

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:1979829

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JOURNAL DE PHYSIQUE

3. O p t i c a l D e t e c t i o n of Second Harmonics.

-

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A. E3asjs-Thggry.

The o p t i c a l d e t e c t i o n system i s based on t h e T h i s i s a resonance c o n d i t i o n . Two means o f s a t i s -

f a c t t h a t a l i g h t beam ( l a s e r ) w i l l be d i f f r a c t e d f y i n g eq. (3) are p o s s i b l e f o r Lamb mode i n t e r a c -

upon r e f l e c t i o n from a p e r i o d i c surface Corugation.

t i o n . The curves i n Fig. 1 show t h e Lamb mode velo-

T h i s c o r r u g a t i o n may be an acoustic surface wave

t i o n may a l s o be s a t i s f i e d f o r d i s p e r s i v e regions

provided t h e r e i s mode coupling. F o r example, con- i n t e r a c t i o n . The l i g h t i s i n c i d e n t i n t o t h e s u r f a c e sider a fundamental in mode at 5 MHz (point 3 ) a t an angle

0

and i s d i f f r a c t e d i n t o d i s c r e t e d i f - t h e second harmonic i s i n t h e A2 mode ( p o i n t 4). f r a c t i o n orders w i t h angular spacing 8, centered Whenever Lamb modes couple, eq. (3) i s s a t i s f i e d . around the r e f l e c t i o n angle 0. The l i g h t i n t e n s i t y T h i s coupling has been observed by Brower and i n t h e mth d i f f r a c t i o n o r d e r due t o a s i n g l e - Mayer /5/ f o r t h e " i s o t r o p i c three-phonon i n t e r a c - frequency Lamb mode o f amp1 i tude U1 i s g i v e n by t i o n " .

4-

2.

The two cases o f p o s s i b l e resonance are e x p e r i - 2 I +1 = Jm (y1)9 m e n t a l l y i n v e s t i g a t e d here. Table I l i s t s t h e funda-

.=

>

&

²

&

( a Rayleigh wave) o r a p l a t e v i b r a t i o n a l mode (a Lamb wave). A t h e o r e t i c a l treatment o f t h i s acousto- o p t i c i n t e r a c t i o n f o r a Rayleigh wave has been g i - ven by Mayer, Lamers, and Auth /6/. This treatment can be extended t o i n c l u d e Lamb waves /7/. Fig. 2

A, S, A1 S, i s a schematic diagram of the l a s e r - s u r f a c e wave

5.3

0 6

2 4 6 8 10 12

mentals used and t h e p o s s i b l e harmonics generated.

where

O fd

F i g . 1.

-

V e l o c i t y d i s e e r s i o n curves f o r

a

f r e e brass p l a t e as a f u n c t i o n of product frequency times p l a t e thickness ( f d i n u n i t s o f MHz.mm). Selected p o i n t s f o r harmonic generation are i n d i c a t e d by numbers 1 t o 4

-

see t e x t .

c i t y d i s p e r s i o n f o r an unloaded brass p l a t e . One

means o f s a t i s f y i n g eq. ( 3 ) i s by e x c i t i n g a 1 0 ~ a l l ~ F i g . 2.

-

Schematic diagram f o r acousto-optic i n - non-dispersive Lamb mode. For example, i f t h e funds- teraction caused by Lamb waves' refers to the

wave v e c t o r o f t h e l i g h t i n c i d e n t a t an angle

0 ,

mental were e x c i t e d a t 7 MHz i n t h e A1 mode ( p o i n t 1 ) 8 i s the angular spacing o f t h e f i r s t l i g h t d i f - then a second harmonic at 14 M H ~ (point 2) will sa- f r a c t i o n order, and x denotes t h e d i r e c t i o n of

t r a v e l o f t h e Lamb wave on t h e p l a t e . t i s f y t h e resonance c o n d i t i o n . The resonance condi-

y1 = 2KU1 cos

0 ,

TABLE I. Fundamental Lamb Modes and Harmonics

w i t h K t h e wavenumber o f t h e l i g h t and Jm t h e mth Fundamental Mode Second H a n o n i c o r d e r Bessel f u n c t i o n . It i s e v i d e n t from eq. ( 4 ) t h a t the d i f f r a c t i o n p a t t e r n i s symmetric w i t h r e s - Al a t 7 MHz A1 a t 14 MHz p e c t t o t h e c e n t r a l o r d e r r e f l e c t e d a t angle

0 .

A1 a t 5 MHz A2 a t 10 MHz Now consider a s u r f a c e c o r r u g a t i o n caused by a Lamb wave which, i n a d d i t i o n t o i t s fundammental

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frequency, contains a second harmonic component o f amplitude U2. I n t h i s case t h e p a r t i c l e displacement w i l l no l o n g e r be pure s i n u s o i d a l and a d i s t o r t e d

Lamb wave w i l l i n t e r a c t w i t h t h e i n c i d e n t l i g h t beam.

The r e s u l t i n g d i f f r a c t i o n p a t t e r n w i l l be asymmetric w i t h respect t o t h e c e n t r a l order. The amount o f Bsymmetry w i l l depend on t h e magnitude o f U2 r e l a t i - ve t o U1.

This process i s s i m i l a r t o t h a t described by Neighbors and Mayer /8/ f o r nonsinusoidal Rayleigh waves. Using t h e same approach f o r nonsinusoidal Lamb waves and assuming t h a t o n l y t h e fundamental and a second harmonic component are present i n t h e wave, t h e l i g h t i n t e n s i t i e s i n t h e p o s i t i v e and ne- g a t i v e f i r s t orders a r e g i v e n by

t h e Lamb wave generation, one can measure t h e chan- ge i n harmonic content as a f u n c t i o n o f distance.

Due t o t h e method o f generating t h e funda- manta1 Lamb mode, i.e., by means o f t h e " l i q u i d wedge" method /9/, t h e p o s s i b i l i t y e x i s t s t h a t second harmonic components are generated i n t h e li- q u i d p r i o r t o incidence on t h e p l a t e surface. Ho- wever, i f such components should be present and should continue t o propagate along t h e p l a t e they would a t best decrease i n amplitude as a f u n c t i o n o f d i s t a n c e t r a v e l e d on t h e p l a t e . T h e i r amplitude c o u l d n o t increase as a f u n c t i o n o f d i s t a n c e on t h e p l a t e unless harmonic generation on t h e p l a t e i s possible. Thus eq. (2) w i l l be s a t i s f i e d o n l y i f harmonic generation on t h e p l a t e i t s e l f occurs.

i f harmonics propagate on t h e p l a t e , asymmetry i n t h e d i f f r a c t i o n p a t t e r n w i l l be observed; t o ob- t a i n numerical values, a new parameter, Ip, i s i n - troduced, d e f i n e d as

-

J 3 ( ~ 1 ) J 2 ( ~ 2 )

...I

2 ⁹ ⁽⁶⁾ Using eq. (2), i t may be shown t h a t I i s a l i n e a r f u n c t i o n o f d i s t a n c e from t h e source o f t h e P Lamb wave /7/ i f the p l a t e i s considered t o be a where y2 = 2KU2 cos

0 .

d i s s i p a t i o n l e s s medium. Therefore, I can be measu- r e d as a f u n c t i o n o f distance. P

The s e r i e s o f products o f Bessel f u n c t i o n s i n above equations can s a f e l y be t r u n c a t e d t o t h e terms shown s i n c e b o t h yl and y2 are u s u a l l y small so t h a t i n s i g n i f i c a n t e r r o r s i n the .values o f I+1 and

I-1 are i n t r o d u c e d by n e g l e c t i n g terms i n JrJs w i t h r,s > 3.

Equations (5-6) a r e t h e bases i n t h e o p t i c a l d e t e c t i o n o f t h e presence o f second harmonics. I f U2 # 0, one f i n d s t h a t I-1 > I+1, g i v i n g r i s e t o an asymmetric d i f f r a c t i o n p a t t e r n . It should be no- t e d t h a t eqs. (5-6) reduce t o eq. ( 4 ) i f U2 = 0. I n t h i s case a l l terms vanish except J1(y1)JO(y2) which kquals J ( y ) .

1 1

T h i s a n a l y s i s provides a method o f determining t h e presence o r absence o f second harmonics i n a v i - b r a t i n g s o l i d p l a t e . Probing t h e surface w i t h a l i g h t beam and measuring t h e i n t e n s i t y d i s t r i b u t i o n i n t h e

g i f f r a c t i o n p a t t e r n y i e l d s i n f o r m a t i o n about t h e ma- Fig' 3'

-

setup for O p t i c probing of harmonic generation i n Lamb waves.

g n i t u d e o f U2 compared t o U1. Moreover, probing t h e Surface a t d i f f e r e n t distances from t h e souree o f

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JOURNAL DE PHYSIQUE

B. Experimental Design and Results.

The design f o r the optical probing system i s shown i n Fig.3. Light from a He-Ne l a s e r i s expan- ded and coll imated (detai 1 s of the necessary optical components not shown i n Fig. 3.) The collimated l i g h t beam i s reflected from a highly polished brass p l a t e which supports the Lamb wave. The d i f f r a c t e d - reflected l i g h t i s focussed approximately 8 m from the plate. This distance i s s u f f i c i e n t t o properly separate the d i f f r a c t i o n orders so t h a t i n t e n s i t y measurements of I,1 and Iel can be made with a pho- todi ode.

Measurements of I were made a t various distances

x

on the p l a t e f o r the d i f f e r e n t fundamental P frequencies and modes l i s t e d in Table I. Excitation of the modes was accomplished by adjusting the quartz transducer t o the required angular position f o r the generation of the desired Lamb mode.

The polished brass p l a t e , excited i n the A1 mode a t 7 MHz, was probed with the He-Ne l a s e r . A d i f f r a c t i o n pattern was observed and t h e asymmetry between the f i r s t d i f f r a c t i o n orders was measured. A t a r e l a t i v e interaction length of 0.8 cm, the percent asymmetry was approximately 15%. The l a s e r probe was then moved along the plate in the positive x-direc- tion and the asymmetry increased. A t a distance of 1.2 cm, a 29% asymmetry was observed. Likewise, f o r the p l a t e in the fundamental Al mode, excited a t 5 MHz, asymmetry was observed. In t h i s case the asymmetry also increased with interaction distance.

The r e s u l t s a r e l i s t e d in Table 11.

TABLE 11. I as a function of

x

P

cond harmonic content i n the detected mode. The measured d i f f r a c t i o n pattern showed t h i s t o be true:

no detectable asymmetry i n the f i r s t orders could be found.

In conclusion, second harmonic generation in Lamb modes was detected. Harmonics were observed f o r both a l o c a l l y dispersionless mode and a l s o f o r t h e mode coupling case, provided the appropriate conditions f o r harmonic generation in plates were s a t i s f i e d .

Acknowledgment.

-

This work was supported by the Office of Naval Research, U.S. Navy.

REFERENCES

/1/ Gedroits, A.A., and Krasil 'nikov, V . A . , Sov.

Phys. JEPT

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(1963) 1122.

/2/ Ldpen, P . O . , J . Appl. Phys.

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(1968) 5400.

/3/ Jones, G . L . , and Kobett, D.R., J . Acoust. Soc.

Am.

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(1963) 5.

/4/ Hikata, A., Chick, B . B . , Elbaum, C . , J . Appl.

Phys.

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36 (1965) 229.

/5/ Brower, N . G . , and Mayer, W.G., J . Appl'. Phys.

49 (1978) 2666.

/6/ Mayer, W.G., Lamers, G.B., A u t h , D.C., J . Acoust Acoust. Soc. Am.

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(1967) 1255.

/7/ Brower, N.G. Ph.0. Thesis (1976) Georgetown University, Washington, D.C.

/8/ Neighbors, T.H., and Mayer, W.G., J . Appl

. ,

Phys. 42 (1971) 3670.

-

/9/ Hallermeier, R.J., and Diachok, O.I., J . Appl.

Phys.

-

41 (1970) 4763.

0.8 cm 15%

1.2 cm 29% Al mode a t 7 MHz

0.9 cm 14% *Asymmetry will be influenced or be absent under

A1 mode a t 5 MHz c e r t a i n conditions of t h e phase r e l a t i o n ship

1.3 cm 39% between t h e fundamental and the second harmonic,

a s pointed out by ALIPPI A . , e t a l . J . Phys.

48

(1977) 2182.

As a control, the So mode a t 3 MHz was excited (point 5 i n Fig. 1 . ) . As can be seen in Fig. I . , there i s no second harmonic which s a t i s f i e s Eq. (3) because the required location (point 6) does not f a l l on a curve, Therefore, there should be no se-