Scattering of guided optical beams by surface acousticwaves in thin films

R. Normandin, V. C-Y. So, N. Rowell, and G. I. StegemanDepartment of Physics, University of Toronto, Toronto, Canada M5S 1A7

(Received 24 August 1978)

A new approach to the problem of light scattering by surface acoustic waves in thin-film opticalwaveguides is presented. In this analysis electromagnetic fields are found to rigorously satisfy boththe polarization driven wave equation and the electromagnetic boundary conditions at the frequen-cies of interest. A component of the solution field is found to evolve with propagation distance intoa linearly growing guided wave mode that dominates the solutions under conditions that can bedefined in terms of waveguide mode parameters. Analytical formulas for the growth coefficient areobtained for both the elasto-optic and corrugation scattering mechanisms for the geometriesTEm, TE,,., TEr TMm,, ., T TM,, and TMm 4TEm .. The present results are found to be consist-ent with previous normal mode analyses of this phenomenon for the cases in which comparison ispossible. Numerical calculations for thin films of As2S3 and Corning 7059 glass on fused silica sub-strates indicate that the elasto-optic effect does not always dominate the scattering cross section andthat the corrugation mechanism must often be taken into account.

INTRODUCTION

The scattering of guided optical beams by surface acousticwaves (SAW) has been a subject of continuing interest 1 -20primarily because of its potential application to signal pro-cessing. Generated surface acoustic waves are used and theratio K/k < 0.1 is usually satisfied where K and k are theacoustic and optical wave vectors, respectively. Schmidt' inan excellent review article has presented the current state ofthe art in this field, especially with respect to theoretical de-velopments. He discussed the two basic scattering mecha-nisms, namely, the elasto-optic (and electro-optic where ap-plicable) and corrugation effects. The former' is usuallytreated via a normal mode analysis' 2' 21' 22 which leads to anexpression for the scattered field amplitude in terms of anoverlap integral. The corrugation effect is treated in a similarway.1 " 2 1 5 However, it is generally believed that the corru-gation effect can be neglected and most analyses (with thenotable exception of Refs. 12 and 15) have included only theoverlap integral (i.e., the elasto-optic and electro-opticmechanisms). In this paper we present a different andcomplimentary approach to this problem in which both theelasto-optic and corrugation effects are treated for all possiblescattering geometries involving TE and TM modes. In par-ticular it will be shown for isotropic films and substrates thatthe overlap integral calculation alone is not adequate in manycases, even for K/k = 0.1 and that the corrugation effect shouldbe included.

Brillouin scattering from thermal surface2 3 and bulk24' 25

phonons (acoustic noise) in thin film23' 24 and in-diffusedwaveguides25 has recently been reported. These experimentsare characterized by large scattering angles, i.e., Kik > 1, aregion that has not been investigated before in integratedoptics. In this paper we examine the acousto-optic interactionin this limit also.

Wave interaction phenomena can be analyzed in a numberof ways, i.e., via a normal mode' 2' 21' 22 approach, by usingGreen's functions,14'1 5 or by computing the total fields createdby the interaction. We explore the latter approach in a seriesof three papers. The present work deals with a wave inter-action in which the generated guided wave is phase matchedto the polarization source. In the companion paper2 6 har-

monic generation is analyzed as a non-phase-matched inter-action. The third paper will contain a detailed comparisonof the total field approach with a normal mode analysis basedon a Green function formalism.

The goal in the present formulation of the scatteringproblem is to evaluate the electromagnetic fields that satisfy(a) the polarization driven wave equation in both the film andsubstrate and (b) the electromagnetic boundary conditionsacross both acoustically corrugated interfaces. First the fieldsgenerated by the acousto-optically created polarizations areevaluated and it is shown that the driven fields producediscontinuities in the usual- electromagnetic boundary con-ditions at both interfaces. Next, the tangential electric andmagnetic fields are evaluated on both sides of each oscillatinginterface and the resulting discontinuities in these quantitiesare calculated. Finally a linear combination of the appro-priate fields that satisfy the wave equation is found to ensurethat the total fields do satisfy the appropriate boundaryconditions.

The acousto-optic diffraction efficiency was treated nu-merically for two different films, As 2S3 and Corning 7059deposited on fused silica. The first system corresponds to afilm with a refractive index vastly different from that of thesubstrate, and the second, to materials with comparable in-dices of refraction. Particular attention is paid to the con-tribution of the corrugation effect to the total scattering crosssections.

1. INCIDENT FIELDS

A. Surface acoustic waves in thin filmsThe propagation characteristics of surface acoustic waves

in this films deposited on semi-infinite media are wellknown.27 In general the acoustic disturbance must satisfy (a)the wave equation in both the film and the substrate and (b)the acoustic boundary conditions at both interfaces. Themechanical displacements are of the form27

Uj = - f Aj(P)e-a(P)z+i(nt-xx-KYY) + c.c. (1)

1153 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979 0030-3941/79/081153-13$00.50 9 1979 Optical Society of America 1153

evaluated from the electromagnetic boundary conditions atboth interfaces, i.e., from the continuity of the electric andmagnetic fields. Finally, the fields are normalized so that I,,,

GIUE/ where Im is the optical power associated with the mthmode in watts/meter and Gm is a constant for a given geom-etry.

ii. TM incidenceThe incident optical field associated with a TM mode is

written

air:

FIG. 1. Acousto-optic interaction geometry.

and

Lu = tf B(q) e3(h(z )+i($ht-xxx-syy) + C.C.

2q=1

H = (1/2)JHaei(-0ot-]xx)+Soz + c.c., (6)

film:

H = (1/2)XHfei(wt(-kxx)(eikzz + e-ikzhz+b) + c.c. (7)

substrate:(2) H = (1/2);Hsei(Ot-hxx)-S2(-z1) + c.c. (8)

in the film and substrate, respectively, for an air-film surfaceat z = 0 and a film-substrate boundary at z = h (see Fig. 1).The acoustic frequency and wave vector are Q and K

= (KxKYO), respectively, and the parameters at (P) and A)P) and/3 (q) and B(q) are evaluated partially from the wave equationin the film and substrate, respectively, and partially from theacoustic boundary conditions. In isotropic media (whichcorresponds to the case under consideration here), po = qo= 2 and the acoustic displacements are confined to the planedefined by X and z. (The assumption of isotropic media isused to simplify the expressions for the acoustic displacementsand does not represent an inherent limitation on this anal-ysis.)

The SAW parameters were evaluated for various thick-nesses of28 7059 and arsenic trisulfide glass3 on fused quartz.(Values of the longitudinal and shear velocities of 5590 and3092 m/sec were used for 7059 glass.) As discussed by Adlerand Farnell,27 this involved numerical iteration techniques.The variation of the SAW velocity (which will be required inorder to define the Bragg angles) with the parameter Kh isshown in Fig. 2. As expected,2 7 the velocity varies smoothlyfrom the silica value at xh = 0 to the film values as Kh a.

B. Optical fields

i. TE incidenceThe incident optical fields are assumed to propagate along

the x axis. For a TE mode of frequency 29 W0

air:

E = (1/2)jEaei(w1ok-xx)+Soz + c.c.,

The parameters Ha), Hf, Hf evl, H,, and wo/k, are calculatedby requiring continuity of the tangential electric and magneticfields at the two boundaries. Again the fields are normalizedvia the relation I = GmHf.

II. ELASTO-OPTIC EFFECT

This scattering mechanism is the dominant process in mostacousto-optic interactions. 1' 30' 31 An incident optical field Ejcouples via the elasto-optic tensor pUNl to an acoustic strainSkl = (1/2)(bu k/ax, + aul/oxh) to produce new polarization

(20s)

2200

1601

3400

(3)

film:

E = (1/2)JEfei(watk-hxx)(eikzz + e-ikzz+i) + c.c., (4)

substrate:

E = (1/2)jEsei(wOt- xx)-S2(Z-h) + c.c.,

32001

(is)s

30001(5)

where S2 = k2 - n2Wgic2 , S2 = k2 - n2Xw/c2 , k2 = n2W2/c2 -

kx, and na, nf and n8 are the refractive indices of the air, film,and substrate, respectively. Henceforth the superscripts orsubscripts a, f, and s refer to the air, film, and substrate, re-spectively. The parameters Ea, Ef, Efei¢, E, and coo/ks are

1154 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979

2800

(a)

As,S,/ SiW,

hh

7059 / si,

I Z 4r S ro e1i

FIG. 2. Surface acoustic wave velocity vs normalized film thickness.

Normandin et al. 1154

1U

K,, / - X~

FIG. 3. Wave-vector conservation parallel to the surface and rotation ofcoordinate axes.

fields Pi via

Pi = -Eon 4Pijk1Sk1Ei. (9)

[These fields are generated in both the film and substrate andthe parameters in Eq. (9) refer to the appropriate medium.]The polarization fields act as sources for the Hertz vector irvia the driven wave equation in each medium, 3 2 i.e.,

V27r - (n2 /c2 ) = - P/Eon2 (10)

Standard relations3 2 are used to recover the usual electro-magnetic fields via

E = V X (V X 7r)-P/Eon 2 , D = En 2 E + P, (I1)

H = (1/Mo)(n 2/c2)V X *, B = gioH. (12)

These solutions contain transverse electromagnetic fields aswell as the longitudinal fields associated with a scalar poten-tial. Note that these fields are driven fields, i.e., they existonly in the acousto-optic interaction region and do not cor-respond necessarily to radiation or guided modes. Further-more, when evaluated at the air-film surface and on both sidesof the film-substrate interface, there is no reason for the usualtangential boundary conditions to be continuous. (If theappropriate fields are all continuous, there would be no"scattering" via the elasto-optic effect.) Additional fields,usually containing a growing guided mode, are then requiredto ensure that the total fields do satisfy the electromagneticboundary conditions. These additional fields correspond tosolutions of Eq. (10) with no driving polarization fields, i.e.,they are solutions to the homogeneous wave equation.

It proves convenient to define a new coordinate system inwhich to calculate the scattered fields. From Eqs. (1)-(5) and(6)-(10) it is evident that the polarization and therefore thedriven electromagnetic fields are proportional to expji (co± Q)t - i(kh + Kx)x -iKyy] and exp[(i(co - Q)t - i(k, - K.)x+ iKyy]. In order to simplify the subsequent analysis, thediscussion of the scattered fields will henceforth be limitedprimarily to the downshift case. (The intensity of the totalscattered light will be the same for the upshift case as for thedownshift when the acousto-optic interaction is evaluated atthe equivalent Bragg angle.) As indicated in Fig. 3, the x' andy' axes are defined so that the wave vector associated with thepolarization fields in the plane of the surface is coincident withthe x' axis. Therefore the fields are now proportional toexp[i(coo - Q)t - ihxx'] and the scattering angle t (Fig. 3) isgiven by cos = (kx - Kx)/kx where 0, = (kx - KX)

2 + K 2

Since scattered TE modes are associated with Ey, and Hymnthese coordinate transformations facilitate the separation ofthe scattered fields into TE and TM modes.

Only the case of phase-matched scattering (Bragg scatter-ing) into guided wave TE or TM modes will be consideredhere. Therefore, key is associated with either a TE or a TMmode of frequency co( -Q = c 0o and the corresponding"Bragg angle" is given by

coso = (kW -1, + K2 )/2KkX. (13)

It is also convenient at this point to identify the mode pa-rameters for the scattered fields, i.e., SJ2 = k 2,-n22/c2,

h-2 = 4w/c2 and 1, = nf ;?wc2 -k

A. TE incidenceThe polarization fields are now calculated for TE incidence.

In the film these fields are

1 52po 2X -'P,Pi, = - E eonyKP(P)eil(w0-Q) t-kx'x >a z

[Thee feldsaregeneate inboththefilmandsubsrat an

X (eikzz + e-ikzz+i0) + C.C.,

where

PxP) = costP(P) - sintP§P),

p(g) = cos4P(P) + sin4P(P ),

pIP) = pfP) = 2inop

X (A3(P) + ia*(o)A(P)),

andP(P) = -in2Ef sinocos~pY{Aj 1(P),

P(P) =-iEf (Cos2op (QA;(P)2 2

+o ia(P)pA'(P) + sin (kpIA P))

(14)

(15)

(16)

(17)

(18)

(19)

The elasto-optic constants are written in the Voigt notationand the acoustic displacement is referred to the acousticsagittal plane, i.e., for A*(P), 1 corresponds to the SAW prop-agation direction and 3 to the coordinate down into the me-dium. In the substrate

I qoP,= con EOnbKP(92 q=l

X ei[(oO-)t-kx'x]-(f*(q)h+S2)(Z-h) + C.C.,

with

pxq)= cosP((q) - sintPgq),

pq) = cosP,( + sin$P(t),

pXq) -in,2E, sing cosopis)B ,

p(q) - insE8 (cos20ptqBjliq)2

+ jj3*()psIB3'() + sin2qp (ps)),

and

ptq) = p(9) - inE(q) +2 SInO4 4p(s (B) ± j3(e)B;I)

(20)

(21)

(22)

(23)

(24)

(25)

For an acoustically and/or optically anisotropic material theserelations are more complex but the method of analysis is stillvalid.

The driven fields are now calculated from Eqs. (9)-(25).

1155 J. Opt. Soq. Am., Vol. 69, No. 8, August 1979 Normandin et al. 1155

For example, the magnetic field in the substrate is found tobe

2 q0 P(,;)(S2 +0f3(qi)KH2 ' = -(1/2)icon,E( Z_ A

X eid('o0-1)t-kx'x'l-lS2+l3(Q),l(z-h) + c.c. (26)

with Al' = 3*(q)(2S + 3* (q)K) + (S2 - S2)/&. Similarly Hr,,E2 ,, and E,' are calculated in both the substrate and the film.These fields are subsequently evaluated at the two interfacesand the discontinuities in the tangential fields calculatedrelative to the film side. For example, AE', = [Ei (z = h)]film- [Ex (z = h)]Isubstrate where the superscripts b and t refer tothe bottom and top interfaces, respectively. In total AEb,,

AEt,, AE A, t,, AH' AHt AH, and AH', are evalu-ated.

Scattered TE modes are obtained if one of AE,, E',, AH'

or AH' are nonzero. The discontinuity in the magnetic fieldat the air-film surface is written

AH', = (1/2)AHIe i(wt-hx'x') + c.c.,

with

AH', = -iwnfljo 2Po - a*(P)K

p~l + +ik, + a -(P)Keo

A+ = a'(P)(2ikh - a*(P)K) + (ke-h9)K

and

A- = a*(P)(2ik, + a*(P)(K) - (kw-kz-)

Similarly,

2f p,(i a*(P)K

p= 1 A+

+ ik, + a*(P)K e .ik1^+io

-(P)xh + ~~2( 'o Pyq)(S2 + 13(q)g)

X e A/+ IcnfOE Z A

AE~. p=1 i+ e1/\yS= f2 f Py() eZ /)

and

AE', = n'k2fP(p) (-eizh - izh+i) ePKh

pq( LI q)+n 2k

2

q1 A

(27)

AH@. = iwnk~f [ |PX (- aZ(P e)ik

+ A e-ih+i0k) e-1P)Ah

+ ik PV' (ek - A ) e-(P)Ahj

+ 3o, f [ k - P( V(S 2 + f*(q)K)]q=1 A' ,(35)

A 2f = f ik, -a (P)K +iki + Ca(P)K

+ k (P/?+ -and

b 2po [ ik?2 al(P)K i~

AEb = [-ik'P(S) e

ik, + a*(P)K k h i)

eik h e-ih] h+ip )+ kI',PX8) z z e Kh

+ h , q (ik S +3*(q)K) -S2p)+ 2 (ikxP( )(S2 + AK' 9'P)

(36)

(37)

(28) In summary to this point, the elasto-optic interaction has ledto driven electromagnetic fields, which, when evaluated at thetwo interfaces, lead to discontinuities in the usual tangential

(29) electromagnetic boundary conditions.

B. TM incidenceThis case is more complex than the previous one since the

incident electric fields have two components, i.e., E2 and E,.The polarization fields are

2K2poi PZPi, = (1/2)Eonf Kc e

p=1

X (P!P')eik, + P(1_e-ikzz+i0) + c.c. (38)

and

Pi = (1/2)Eon2K

(31) X 2 ei(wt-k x')-(,*(q)K+S2)(Z-h) + C.C.q=1

(32) in the film and substrate, respectively. Furthermore,

PxP, ) = cosPP) - sintPyP),

PX(g) = costP(P) - sintPYP),

PfP,) = costPyP+) + sintPP),

(33)

TM modes are generated if one of E,, AE', AH> , or His nonzero. For this case it can be shown that

2Ht2po = ik - a*(P)K ik, + a* (P)K+, = iwfCO l, A+ A )_'

+ik2',Pzi' kP JJ (34)

P(P) = cosP(P) + sinWPVp2

I= I [icos2o p Cos )

+ p %2)(i sin20Ai(P) - a*(P)A (P))I,

P y(+) =Hz i sinX coslpVQA ,(P),COW

PXP = 2Ewcos~p~4(iA~(P) - ot*(P)Al*P)- Hk

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

1156 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979 Normandin et al. 1156

p) . HfkPyP) = H/sinop~4(ikA3(P) - a*(P)AI(P)),

Z+= Z'= B cos-p(iA,-(P)- a(P)AI-2foco

pP) = P(q) = C Sp()_ i*(P)p()AP))2 e0W

pxq) = costPq-)

= cosp) + inP(),

p(q) = - - [cos2 *pS2plBl(q) + S2p,2 e0C*

X (sin 2 OBI(q) + ij*(q)B3*(q))

- cos(-xp('4(iB(q)- *(q)B (q)

p(q) _ HPYq =2E p(41 [2 sino CS O2B,*q

+ sin kx(,3*(q)Bl j(q) -(q))]

p(q) H-[ [s s `2p(S)(B,(~) + ij*(q)B*(q))

- , q) + k3*(q)p *qBtl)].

exp[i(wo - Q)t - kX x'] and the discontinuities in the tan-(47) gential electric and magnetic fields are evaluated on the film

side. For TE scattered fields(48)

(49)

(.rnM

AH =-ifE ((P + YP)(Zik-a*(P)K)

+ (P,! - P?+))(ikZ + a*(P)K) (55)A+ po(P~ PY)(k eiOa*(PK)

(51) x P=-( 1 A+ eikzh

+ (PP! - PYP'+)(ik + a *(P) K) e-ikzh+iO) e-(P)Kh

(52)

(53)

(54)

The driven fields are calculated as outlined in Sec. II A.Again the fields at the two surfaces are proportional to

1

-AH' = con2E 2fo (P(,PxP') +p=l

+ 2 l P q) (S 2 + /3*(q)K)+ iwns eOE0 Y_

AE', = nk2k ((P + P ) Y- -- +) eio),

and

AE'b = n2k2fo ((P ±+ Ps¢!) eikzh

- (Y- e-*(P)Kh + ns2 Fm a qf A'

Similarly for coupling to TM scattered fields

PxP& ) (ikz - a*(P)K) + (PzP?) + P(P))ik,A+

(P(P) - PP+())(ik, + &*(P)K) - (P(P) - Pz))ikx

AHb, = iCnfl2EO 2 ((P1+4 + Pxg) )(ik - a!*(P)K) + ikx(P2 + p2)e

±((4xP) - PX/9)(ikZ + at*(p)K) -(P1'?!- P1'?2)ikx e.zhi .eA(P)

+ iYn E0 + P19 (-S2 e- 3()K) + jhXP~q)]

q1 A

AEt. =° +(P(P+! -P,)(-ikP)(ik, -a*(P)K) + (PX) PX2)kh

+ - P(P'))(-ikX)(ikz + 2*(P)K)-k2 (Px')- P')+ '+ Z

= 2po ((PzP?) +Pz(ft)')(-ikx)(ik_ -a*(P)K) +kz(Pk?+ RI'?!) h

+P(zP) -P-P+))(-ikx)(ikz + a*(P)K) -k 2(Px'? +- p ) e ikh+i

q0 b~)ih (S + *(q).K) - S~px~q)X e -c,*(P)kh + ' PZ (S2 + 2q=l 1

I11. CORRUGATION EFFECT

Sound waves present at an interface between two mediaof different refractive indices can scatter light via the acous-tically induced surface ripple. For the geometry under con-

1157 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979

Isideration the SAW produces the corrugations 3

t cos(Qt - Kxx

- Kyy) with

= 2po A (P)=lp=1

(63)

Normandin et at. 1157

and

(56)

(57)

(58)

(59)

and

(60)

(61)

(62)

S, cos(5t-KxX-KyY)

,Sbcnslt-KxX-KYY -l

Z=h

FIG. 4. Acoustically corrugated film surfaces.

and 6b cos(Qt - xx - Kyy - '7) with

bbe-i3e = 2PE e-o p).I, Beq) (64)p=l q1=1

at the air-film and film-substrate interfaces, respectively.These ripples act essentially like traveling sinusoidal dif-fraction gratings, and Doppler shifted light is scattered intothe usual ±1 diffraction orders.

The acoustically generated corrugations affect the elec-tromagnetic boundary conditions in two ways. First, theappropriate tangential and perpendicular fields must becontinuous across the moving surfaces indicated in Fig. 4. Forexample, the fields at the upper interface are evaluated at z= &, cos(Qt - Kix- Kyy) rather than at z = 0. Furthermore,the tangential fields must be defined as parallel to the in-stantaneous surface, i.e., they must be evaluated in a x "y "coordinate system fixed to the surface (Fig. 4). This lattereffect will mix the usual orthogonal field components associ-ated with the stationary surface. (It does not appear that thisparticular effect is included in the previous formulations1 2' 1 5

of the corrugation effect.) It can easily be shown that a vectorR - (Rx,RYR2 ) originally defined in the xyz coordinatesystem is described relative to the x",y",z" axes by(Rx ,Ry ,RT) where

Rx = R, + R, cosd'sin#,

Rya = RY + R2 sinrksini, (65)

R,,= -R, costksin4' - RJksintksin# + 14.

The surface tilt angle 4 in the SAW sagittal plane is assumedto be small and is given by

sinkt tanl/t = 6tKsin(Qt - KxX - KyY) (66)

and

sinVb, tanmk = 6bKsin(wt - KxX - Kyy - i7) (67)

at the air-film and film-substrate interfaces, respectively.Therefore total electromagnetic fields must be found for whichthe tangential electric and magnetic boundary conditions aresatisfied across a moving surface in the x", y", and z" coor-dinate system.

The approach used here utilizes the fact that the surfacecorrugation amplitude is small in comparison to an optical oracoustical wavelength. Thus the incident fields in thewaveguide are given essentially by Eqs. (3)-(5) for a TE modeand Eqs. (6)-(9) for a TM mode. The corrugation producesa small perturbation in these fields which is manifest in termsof discontinuities in the usual boundary conditions at thefrequencies wo *£ Q. Additional (scattered) fields are required

in order to assure that the appropriate total fields at wo ± £2are continuous across both interfaces.

A. TE incidenceConsider now the case of TE modes with the incident fields

expressed by Eqs. (3)-(5). The tangential magnetic andelectric fields are evaluated at the corrugated surfaces in thex", y", and z" coordinate systems. For example, from Eq. (3)and using V X E = -13 it can easily be shown that the mag-netic field in the air is given by

H = -(ISoi + kx/)ei(ot-k/xx)+Soz + c c. (68)H=2 wol-o ('o(8

When evaluated at the moving surface [i.e., at z = 6t cos(£t- KX - Kyy)] in the coordinate system fixed to the surface

H,, = - a ei(aOt-hxx)+SoltcoIs(Slt-sxx,-ay)2 wyo0o

X [-iSo + cosikx 6tKsin(Q2t - Kxx - Kyy)] + c.c. (69)

The standard expansion in terms of Bessel functions Jm isused for the exponential term to obtain

e Soet cos( s2t-kx-&vy) = Z (-j)mJm (iSo0 t)m

X e-iM<&lt-,Kxx-,YY) (70)

with the result that Hx- contains terms at the frequencies wo+ m Q. Assuming that /i&t << 1 and Kbt << 1, then the termsat V ± £ are

Hx- =- a at 2 oo i 2osoxhx)2 wo~po 2 2

X e il(WQ)t-(okx+&x)x TKYyI + C.C. (71)

[Note that the magnetic field at the frequency wo is essentiallythe same as that given by Eq. (68).] The fields Hy,,, Ex,, andEy are evaluated in a similar way at both sides of each sur-face.

The continuity of the tangential electric and magnetic fieldswas examined at each interface. Since the terms at.w0 in thisanalysis are given essentially by Eqs. (3)-(5), the boundaryconditions are automatically satisfied at this frequency.However, at the frequencies co I Q it can easily be shown thatthe corrugation results in

AE,- = AEy, = AHy = 0,

1ie5~kE, 2 2

2 2cg0 n

x eiI(wo±U)t-(kx±x)xw&,,yy + 6.c.,relative to the film side at z 0 and

AEx- = AEy, = AHy = 0,

1 ibkE,2 2AHx' 22 (n/ n,2

2 2cu+

X e il(wo±Q)t-ikX+a1X)XTFyYT-q] + c.c.,

(72)

(73)

(74)

(75)

at z - h when evaluated on the film side. (Note that theterms proportional to 6

K have canceled out for this case.)Therefore the presence of corrugations introduces new Fouriercomponents into the boundary fields and is manifest in termsof discontinuities in the usual tangential electromagnetic

1158 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979 Normandin et al. 1158

boundary conditions. Just as in the previous discussion onthe elasto-optic effect, additional solutions to the homoge-neous wave equation are required in order to ensure that thetotal tangential electric and magnetic fields are continuousacross both interfaces.

It again proves useful to express these discontinuities in thex', y', z' coordinate system in which the scattered wave vectoris parallel to the x' axis. For coupling to scattered TE modesat the frequency coO I Q,

AE'> = 0,

AH', = cosg(ibtkE 0 /2yoc)(n -n2), (76)

at z 0 and

AE b,= 0,

AH = cosW(ibseTifkEs/2goc)(n4 - ni), (77)

at z - h, all evaluated on the film side of the respectiveboundaries. TM scattered fields are produced via theterms

AE, = 0,

AH$, = sint(ibthEa/2poc)(n2 - n2), (78)

at z 0 0 and

AEX' = 0,

AH', = sint(i6be -"iJE,/2,2oc)(nf -n'),

at z , h, all relative to the film side.

(79)

B. TM incidenceThe analysis here parallels closely that for the previous TE

case. The electromagnetic fields [Eqs. (6)-(8)] were evaluatedat the moving surface in a frame of reference fixed to the os-cillating surface. Again it is found that this scatteringmechanism can be expressed in terms of discontinuities in theusual electromagnetic boundary conditions. It can be shownthat for scattering into TE polarized modes

AE't = 0,

AH, =-sinE SobtHG (ni - n2)

AE b, = 0, (80)

and

H = sin= 2 ) (81)

Similarly for coupling to TM modes

AEx, = 2cX xn2 (n2)

AHt= cost SobtHa (I)-n

-ihxkx56be-iqH 8 (a2 - n\AEx = 2 n, (82)

and

,~H,= -cosE ~b~n~ n- 2

2 2 J (83)

It is noteworthy that the terms proportional to 3K are con-tained in the kx5 term in AE5 '.

IV. STRUCTURE OF THE SOLUTION FIELDS

The acousto-optic interaction has been characterized in theprevious sections by driven fields and discontinuities in theusual optical boundary conditions at wo ± Q. At this pointthe problem is reminiscent of the classical calculation of theFresnel coefficients associated with the reflection and trans-mission of light incident on the interface between two media.In that case the incident field is considered to create tangentialelectric and magnetic fields at the interface, and a linearcombination of solutions to the wave equation in the twomedia is used to ensure satisfaction of the electromagneticboundary conditions. By analogy, four solutions to the ho-mogeneous wave equation are required to satisfy the boundaryconditions in this acousto-optic interaction.

A. TE scattered fieldsThe problem therefore is to find a total of four independent

solutions to the wave equation characterized by exp[i(woi Q) t - k,'x'] in the three media. Four such solutions are

air:

(1/2)D(1)jEaei(w'04-)t-hx'x'+soz' + c. c.,

film:

(1/2)D(2)lEfei[(.o 1t-kx'x'+kzz'] + C.C.,

(1/2)D(3)iEfei[(wo±U2t-kx'x'-kzhz'+k'] + c c.,

substrate:

(1/2)D (4)jEsei[(wot0)t-kx'x']-S2(z'-h) + c.c.

(84)

(85)

(86)

(87)

Since the coefficients D(i) are to be determined, then withoutloss of generality we choose El, Ea. EfeiO', and Es to be thenormal mode field amplitudes associated with a TE guidedwave characterized by exp[i(wo i Q)t - kx'x']. Since thesesolutions are synchronous with a TE guided wave, then therealways exists one linear combination that corresponds exactlyto the guided wave. However, a constant amplitude guidedwave does not by definition respond to the boundary condi-tions, and therefore effectively only three of the four modesrepresented by Eqs. (84)-(87) independently contribute tothe tangential electromagnetic fields. In mathematical terms,the determinant of the coefficients for the boundary valuematrix is zero, the usual indication of a guided wave. Thusonly three of the modes are useful in satisfying boundaryconditions.

A fourth mode is required that satisfies the homogeneouswave equation and contributes to the boundary conditions.This new mode is constructed as follows. First we formulatefields which are solutions to the homogeneous wave equationcharacterized by exp[i(wo + Q)t-kx,,x'i with klz3 # kh,, i.e.,

air:

(l/2).D/Eaeil(ao+Q)t-kxx']+Soz' + c.c., (88)

film:

(1X2)±D'Efei((e0++t-"x'' + cX (eik-,-zt + e-ik,,zh+i0') + C.C. (89)

1159 J. Opt. Soc. Am., Vol. 69, No. 8, August .1979 Normandin et al. 1159

substrate:

(lI2).DEseitl(-o4Q)t-kx-x'l-s2(z'-h) + C.C. (90)

Note that since these fields do satisfy the wave equation

h2 + k2t = n'k 2 , kh2 - S;2 = n2k2 , k2 - S; 2 = nk 2. (91)

Furthermore, the field amplitudes are associated with theguided wave at wo Q. The next step is to take the fields

described by Eqs. (84)-(87) with all of the Di = D' and

subtract from them the fields described by Eqs. (88)-(90).For example, in air

E = (1/2);D'Eaeilo-;Q)t-kx'x'l + S'z'

X (1 - ei(kx-kx)x'+(S S )z') + c.c. (92)

Noting from Eq. (91) that S - So = (khx - kx )(khx + khx)/(S;

+ S') and expanding the exponential in parenthesis in thelimit kx, - kxr, then Eq. (92) becomes

(1/2);DEa[ix' - (kx /So)z]

X eit(wo4Q)t-hkxx'l+Soz' + c.c. (93)

The parameter D = D'(kx,, - kx) remains finite (and is usu-ally nonzero) as krx - kx . [If Eqs. (88)-(90) were used in

conjunction with three of Eqs. (84)-(87) to satisfy the

boundary conditions, then it can be shown that D' a (kxn -kx,)-l.] Similarly the fields in the film and substrate are given

by

(1/2);DEfei[(wo+Q)tkx'x'l [i (x + kx, Z) eikz'z'

+ i (x' - xz') e-ihz'+ij + c.c. (94)

and

x (ix' + kx -h)) + c.c., (95)

respectively. Since these fields correspond to the differencebetween two sets of solutions to the wave equation, then they

also satisfy the wave equation. (This can also be verified by

direct substitution.) It can easily be shown that this modecontributes the tangential field discontinuities

Et,= 0,

at z = h across the interfaces as evaluated on the air and

substrate sides, respectively.

It is instructive to examine the spatial structure of thegrowing modes. In the limits x'>> k /So? x' >> kx'h/k,' andx' >> kd'/S", this mode becomes

air:

(1/2)JiDx'Eaei[(woJ:2)t-kx x'1+Soz' + c.c., (98)

film:

(I12)J'iDx1Efe i [(,o-4 ) t- kx x']

X (e ikz' + e-ikz2z'+i ') + C.C., (99)

substrate:

(lI2)JiDx'Esei[(.o-4)t-kx x'l-S2(z'-h) + C.C. (100)

Therefore, this field evolves with propagation distance intoa linearly growing normal mode guided wave. Since the di-vergence in the coefficients D i and Dkhx has been removed

and they are comparable in magnitude, this growing modedominates the solution fields in this limit. However, the zdependence in Eqs. (93)-(95) is a consequence of the re-

quirement that this mode satisfy the wave equation and it isresponsible for the contributions at the boundaries given byEqs. (96) and (97).

B. TM scattered fieldsThe analysis here is exactly the same as for the TE case.

Three steady-state solutions are required and are chosenfrom

air:

(1/2)ID(1)Haei(wooI)t-kx'l]+Soz' + c.c.,

film:

(1/2)D (2)Hfei[(wo4-Q)t-kxtx'+kzz'l + c.c.,

(1/2)ID (3)Hfe i[(wo4Qt-kx x'k Z'+0'l + C.C.,

substrate:(1/2);D (4)Hsei[(wo lQt-kx'x/]-S2(z'-h) + C.C.

(101)

(102)

(103)

(104)

Note that we again limit the discussion to Bragg angle scat-tering and therefore S', S2, kz,, Ha, Hf, HfeiO', and Hs areassociated with a guided mode. The appropriate growingmode has the form

air:

(96)ikxk 2Ea(n]-n2)wyoSok 2

Ebkx S'2h Es-ikxEs (h +( - n2)kHXo I kZ,,2S2 (97)

(1/2))DHa (ix' - X z) ei[.o4°)t-kxx]+Soz' + c.c (105)

film:

(1/2)jDHfei[(wo4O)t-kxx'l X [i (xi + d- Zi) eikZ'z'

+ i (xI - kX z') e-ikz'z'+io1 + c.c., (106)

1160 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979

at z 0 and

(112)IDEse i [(coo+ Q)t - kxx'l -S2(z'-h)

Normandin et al. 1160

substrate:

(112)j!DHI (x, + kX ,

X ei[(wo±Q)t-kx ]x'J-HS(zŽ-h) + C.C. (107)

It can easily be shown that these fields contribute to theboundary conditions via the z dependence and produce

H>s= 0,

St,- ik f / H2( -_/ a (108)2wnls~hk'co

at z = 0 and

- kx'S'hHl n 2

H = kxHs h + (n2-tn2)k2.X 2wco n 2 Sk2k n 2S

across both interfaces. For example, at z = X,

D(2 )Efeikzh - D(4)E- DE ±E, + AE, (110)

ensures continuity of the tangential electric field. (For con-venience we choose the steady-state fields described by Eqs.(84), (85), and (87).) Usually only the amplitude of thegrowing mode is of interest and solving gives

D = D1/D2, (111)

where

D= (1 + (wgoAH/' - iS4E',)

- eik2h (I - i k- (wgoAH%' + iSoAE>) (112)

and(109)

at z = h when evaluated on the air and substrate sides, re-spectively.

V. SCATTERED FIELD AMPLITUDES

A. TE modesThe field amplitudes D (i) and D are evaluated by ensuring

that the tangential electric and magnetic fields are continuousI

+ i) (wpoH> - iSsE'>)

- eikz ( (113)

Note that the AH% AE > At',, and AE'L terms are the sumof the elasto-optic and corrugation terms discussed in previoussections. This particular form for D is general and is valid forthe off-synchronous case as well. Substituting Eqs. (96) and(97) into (113) simplifies D2 into the form

(114)D 2 =- k 2 2Efe'(0/ 2) 2( 1 + i S Cos (1zh- 2' (I + h) (n 2 - n/) + eik'h Cos (k) ( Ni - j ) I (n2 -

It is tedious but relatively straightforward to compare theseanalytical results with those obtained from a normal modeanalysis. This will be done in a later paper and the results forthe elasto-optic and corrugation effects can be shown to beidentical with formulas (18) and (26) of Ref. 1 in which thisacousto-optic phenomenon has been reviewed. There doappear to be some differences when mode interactions areanalyzed in the off-synchronous case and only the dominantnormal mode is used. This will be discussed in the companionpaper2 6 which deals with harmonic generation under non-phase-matched conditions.

B. TM modesThe amplitude of the growing field is calculated by en-

forcing the continuity of the appropriate boundary conditions.It is easy to show that in this case (for D = DI/D2),

+ is( iSAH->-wo4±DikIh( + A) (/S2AH$E WoE%)(15

(f S O X

(iS' 1 ) t.So AHt

and

D2 = + t) ( H - WeoE)

n n 2h nf 2HX ce~ ) 16+ eik' (flth -~(if H' -w~oE~j (116)

Again it is assumed that the field discontinuities AH',, AEX,,ASfy and AEt, include both the elasto-optic and corrugation

I

terms. Simplifying D2 gives

-ik 2kH (,b'/2) { ± Ink ( In/ - n)co ( - ) 2 (nf ;- 2k;ik l hij

I ' I nf2- n2)k 2, lCOS lkz, h- '~2 _ l2- nt2 k h +

2 2Z n 2 cSknf n.k'Ie an Cos 2MJJ(117)

When just the elasto-optic effect is considered, these formulasagain agree with the results of normal mode analysis [Eq. (18)of Ref. 1]. However, for the corrugation effect that wastreated in Ref. 12 there seems to be a difference attributableto the terms discussed previously, i.e., the terms that arisefrom the oscillating directions associated with the tangentialfield components. This aspect will be discussed in more detaillater.

VI. NUMERICAL CALCULATIONS

The scattering cross sections for a variety of geometriesare evaluated numerically using the analytical formulas de-rived in the previous sections. In particular, the importanceof the corrugation mechanism is examined, since it has beenneglected in most calculations to date. Two film-substratecombinations are considered, a film with a refractive index (a)a little larger than that of the substrate (Corning 7059 glasson silica) and (b) considerably larger than that of the substrate(AS2S 3 on silica).

1161 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979

D2 = �1

Normandin et at. 1161

Most scattering experiments involve the detection of thescattered light outside of the acousto-optic interaction regionand it is therefore necessary to consider what happens to thescattered field at the boundaries of the acoustic beam. Thetwo regions differ in the sense that there are no driving fieldsor corrugations in the "free" waveguide region. The linearly(with x') growing fields satisfy the "free" waveguide boundaryconditions and therefore they are transmitted completelythrough the boundary defined by the edge of the sound waves.(The appropriate maximum value of x' corresponds to thewidth of the acoustic beam as measured along the directionof the scattered light.) As will be discussed in a later paper,the remaining terms in the total fields are partially reflectedand partially transmitted as normal modes at this boundary,as well as at the boundary x' = 0. However, here it will beassumed that the linearly growing mode dominates, andtherefore the remaining terms are neglected in the computercalculations.

The numerical results are expressed in a form convenientto experimental verification. First the SAW power is nor-malized according to

PSAW = K5pPR, (118)

where PSAW is the acoustic power per unit length of wavefrontand PR is a constant. Secondly, the scattered light was ex-pressed in terms of the incident light via

Is/ i = (Fk2X')2 (PsAw/K). (119)

In this equation *, and Ii are the scattered and incident opticalpowers per unit length along the respective wavefronts andthey can be either TE or TM polarized. The form of the crosssection parameter F was chosen so that its magnitude is ap-

F

100.01 0.1 1.0 10.0

FILM THICKNESS h (pm)

FIG. 5. Scattering cross sections for TEr -TE,,, for As2S3 films on silica.For the solid line (K/k = 0.1) and the dash-dot line (K/k = 1.0) both the el-asto-optic and corrugation effects were included and for the broken line(Klk = 0.1) and the dotted line (K/k = 1.0) only the elasto-optic mechanismcontributes. Only representative results are given for K/k = 1.0, i.e., TEo-- TEO and TE3 -E TE3. For TE2 -> TE2 and TE3 - TE3 the curves withand without the corrugation effect are indistinguishable on this scale.

1162 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979

F

10 _ 010.01 0.1 1.0

FILM THICKNESS h (>m)

10.0

FIG. 6. Scattering cross sections for TMm - TMm for AS2 S3 films onsilica. For the solid line (K/k = 0.1) and the dash-dot line (K/k = 1.0) boththe elasto-optic and corrugation effects were included and for the brokenline (K/k = 0.1) and the dotted line (K/k = 1.0) only the elasto-opticmechanism contributes. Only representative results are given for K/k= 1.0, i.e., TMo - TMo and TM3 - TM3. For TM2 TM2 and TM3 TM3the curves with and without the corrugation effect are indistinguishable onthis scale.

proximately independent of K/k for small values of thisratio.

A. As 2 S3 ISiO 2

In this section the acousto-optic interaction betwen surfaceacoustic waves and light guided by a thin film of arsenic tri-sulfide glass deposited on silica is evaluated for various filmthicknesses. The pertinent elasto-optic constants were takenfrom Ref. 33. It is assumed throughout that a He-Ne laseroperating at 0.63 Am is used as the light source.

There are two features common to all of the curves shownin Figs. 5-8. The scattering efficiencies as expressed by thecoefficeint F of Eq. (119) always tend to zero when theacousto-optic interaction region becomes a decreasinglysmaller fraction of the total illuminated volume. This occurs,for example, near the mode cutoff thickness, i.e., when theevanescent tail of the light in the substrate carries most of theoptical energy. Also, when the film becomes progressivelythicker, the cross section F decreases since the acoustic pen-etration depth into the film remains approximately con-stant.

The numerical results are shown in Figs. 5 and 6 for scat-tering that involves the same incident and scattered modes(intramode). For K/k = 0.1 the maximum cross sections occurat a film thickness that corresponds approximately to theSAW wavelength. This result is not surprising since (i) theSAW penetration depth is comparable to the acoustic wave-length and the acousto-optic interaction is maximized nearthat point, and (ii) the guided modes examined numericallyare all far above cutoff and their field distributions vary onlyslowly with thickness. However, for /k = 1 the maxima occurnear cutoff and the corresponding film thicknesses vary frommode to mode.

Normandin et al. 1162

The relative contribution of the corrugation mechanism isusually (but not always) small for intramode scattering. Itis typically 5%-10% for K/h = 0.1 and is important for filmthicknesses up to a few times the cutoff thickness: contri-butions of less than 5% are not shown in the diagrams. Forvery thick films the elasto-optic effect always dominates sincethe scattering volume is many wavelengths thick. Note,however, the TEO - TEO case in which complete interferencebetween the corrugation and elasto-optic mechanisms occursjust above cutoff and the corrugation effect remains importantover a large range of thicknesses. In the region of interest forBrillouin scattering from acoustic noise, i.e., K/k > 1, thecorrugation effect becomes progressively more important withincreasing mode number. For backscattering (K/k 2), botheffects must always be considered.

The results for scattering with mode conversion of TE in-cident modes are shown in Fig. 7. For small scattering angles(i.e., K/k = 0.1) the Bragg condition [Eq. (13)] cannot be sat-isfied until relatively large values of h are reached, andtherefore scattering does not occur near the cutoff thicknessassociated with the scattered modes. For example, for TEO- TE 1 , phase-matched scattering takes place for h > 0.7 gmif K/k = 0.1 although for K/k = 1 it can take place just abovethe TE1 mode cutoff. Furthermore, we note that for /k =0.1 the cross sections TEO - TErn decrease with increasingm and that the corrugation effect is an important scatteringmechanism: however, the small values of F make this typeof intermode scattering impractical in device applications. Atlarge values of K/k( = 1), the cross sections become comparablewith those of intermode scattering and the corrugationmechanism must be included for obtaining useful scatteringestimates.

Intramode scattering shown in Fig. 8 for TEO - TM,, ex-hibits many features similar to those discussed previously forTEBO TEr. The cross sections are smaller, especially for

i0'7r

lo- 8

tot-g

100.01

TEo- TE1

,1

!t

1.

11

i!

i!Iii' l li

t. . ..X

: : I-

:TE.-.-TE.. . I

0.1 1.0

FILM THICKNESS h (pm)

FIG. 7. Scattering cross sections for TEo - TErn for As2S3 films on silica.For the solid line (K/k = 0.1) and the dash-dot line (K/k = 1.0) both the el-asto-optic and corrugation effects were included and for the broken line(K/k = 0.1) and the dotted line (K/k = 1.0) only the elasto-optic mechanismcontributes. For TEo - TE2 only the case K/k = 0.1 is given.

1163 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979

10

10

F

10

10

-13

F

-14

-15

0.01 0.1 1.0 10.0

FILM THICKNESS h (pm)

FIG. 8. Scattering cross sections for TEo -TM,, for As2S3 films on silica.For the solid line (K/k = 0.1) and the dash-dot line (K/k = 1.0) both the el-asto-optic and corrugation effects were included and for the broken line(K/k = 0.1) and the doffed line (K/k = 1.0) only the elasto-optic mechanismcontributes. For TEo - TM, only the case K/k = 0.1 is given. The leftscale refers to K/k = 1.0 and the right scale to K/k = 0.1.

K/A = 0.1 in which case this interaction geometry is not ofpractical interest. At K/ = 1 the cross sections are larger thanfor K/k = 0.1, but they are still approximately two orders ofmagnitude less than the intermode cases shown previously inFigs. 5 and 6. We note that the corrugation effect plays a veryimportant role in these geometries.

B. 7059/SiO 2

The elasto-optic constants of 7059 glass were evaluated 2 8

by using Brillouin scattering from thermal phonons in bulksamples of 7059 glass. The details are discussed elsewhere 2 8

and for the present purposes we note thatpul = 0.17 and P12= 0.28.

The numerical results for this case are summarized in Figs.9 and 10 for x/K = 0.1 (only). Representative results are givensince (i) the trends are similar to those described for the pre-vious case, and (ii) the curves presented are typical of a largerange of geometries. For intermode scattering, i.e., TErn- TErn and TMrn TMrn, the corrugation mechanismcontributes typically 5%-10% to the cross sections; there areagain exceptions for which the inclusion of the corrugationeffect is mandatory. The cross sections for intramode scat-tering were again found to be too small to be of practical in-terest for K/ = 0.1, although we again found the corrugationmechanism to be important.

VII. DISCUSSION

The solution fields described in this paper rigorously satisfythe polarization driven wave equation and the electromagneticboundary conditions across the rippled film interfaces in thesource region. We note that these fields are determined onlyto within an arbitrary number of guided wave normal modes

Normandin et al. 1163

F

-910

-1010

F

-12 r j j 510 1' -s

0.1 1.0 10.0 100.0

FILM THICKNESS h (pm)

FIG. 9. Scattering cross sections for 7059 glass films on silica for K/k= 0.1. For the sohid line both the elasto-optic and corrugation effects wereincluded and for the broken line only the elasto-optic mechanism contrib-utes. The left scale refers to TEo - TMo and TE2 - TE2 and the right toTEO -> TE1.

of arbitrary amplitudes. (This is equivalent to the previousstatement that any three of the four nongrowing solutionfields, as described for example by Eqs. (84)-(87), were re-quired to specify the solution fields.) This ambiguity is re-moved and the normal mode amplitudes evaluated by en-forcing continuity of the electromagnetic boundary conditionsat the transverse extremities of the source region. For ex-ample, for small scattering angles, the acoustic wave is ap-proximately orthogonal to the incident optical beam and thesource region extends from x = 0 to x = L where L is theacoustic beamwidth; the transverse boundaries correspond

10

1a-10I

F

10

100.1 1.0 10.0

FILM THICKNESS h ("m)

10-1 to the y-z plane at x = 0 and L. However, it will be shown ina later paper that the component of the solution fields thatgrows linearly with propagation distance is the only field thatemerges through the transverse boundary at x = L with thenormal mode wave vector k.,,, providing that the scattering

-13 is phase-matched. The remaining normal modes correspond10 to the reflection and transmission of the other field compo-

nents at both transverse boundaries. Therefore the ampli-tude coefficient D calculated here does describe completely

F the amplitude of the dominant normal mode generated, evenoutside the acousto-optic interaction region.

The present calculation also provides an estimate of thetypical interaction distance required for the phase-matchednormal mode to be dominant. Assuming that the nongrowingfields are all of comparable magnitude, of the order of Dkx,the linearly growing normal mode dominates for x >> x',.Since x'm = k,'/SŽ' and kx'h/k',, this characteristic length canbe many wavelengths in extent near mode cutoff and for thickfilms.

The field analysis approach presented here is significantlydifferent in philosophy and computational methods fromnormal mode analysis. However, for the elasto-optic effectthe results are identical in the phase-matched case: differ-ences related to the corrugation mechanism will be discussedlater. Normal mode analysis usually deals only with thatcomponent of the polarization source characterized by V-P= 0 and ignores the remaining longitudinal component (andelectromagnetic fields). This is normally sufficient since thefields generated are detected outside the interaction regionand only normal radiative modes leave the source region. Thepresent calculation utilizes the Hertz vector which containsboth the vector and scalar potentials. Thus the driven fieldsare complete in the sense that they contain both the longitu-dinal and transverse parts of the electromagnetic field. Al-though only tangential field boundary conditions were usedat the two film interfaces, it can easily be shown for opticallyisotropic media that the usual perpendicular components arealso satisfied.

There do appear to be significant differences between thefield and normal mode analysis techniques when the corru-gation effect is evaluated (consider Fig. 4). The film materialperiodically crosses the flat boundary into the air and sub-strate, and vice versa. In normal mode analysis the region ofthe film, for example, on the substrate side of the boundaryis treated as a dipole source with P m (nf 2

- n 2 )WEE embeddedin a medium of dielectric constant eon2 and the radiation ofthese dipoles into the normal modes is calculated. This isequivalent to evaluating the filds at an oscillating interfacein the present formalism. However, we do not believe thatthe mechanism that corresponds to satisfying boundaryconditions in a fluctuating frame of reference has been in-cluded in previous'l2 5 normal mode treatments. As shownhere, this second mechanism is important only for TM inci-dence, since this effect does not contribute in the TE case.

The numerical results for the two film-substrate systemsexamined indicate that the corrugation effect can in somecases constitute an important scattering mechanism. Thisis usually true for scattering near the mode cut-off for thegenerated wave: note that for films that are very thick on thescale of an optical wavelength, the elasto-optic effect always

100.0

FIG. 10. Scattering cross sections for TMm - TMm for 7059 glass filmson silica for K/k =0.1. For the solid line both the elasto-optic and corru-gation effects contribute and the broken line only the elasto-optic mecha-nism is used.

1164 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979 Normandin et al. 1164

dominates. For most cases of intramode scattering, the crosssections for K/k = 0.1 contain contributions of at most 5%-10%from the corrugation mechanism. We did find, however, asignificant number of exceptions for which the effect wasimportant and had to be included in the calculations. For /k= 1, which is of recent interest in thermal Brillouin scattering,the corrugation effect appears to be important in most cases,even for intramode scattering.

The formalism presented here is applicable to a large rangeof wave interaction phenomena in the integrated optics: thecase of harmonic generation is treated in the following paper.26

Its range of validity is the same as for normal mode analysisto which it provides an alternative. It has recently been ap-plied to surface acoustic wave34 and surface plasmon 35 non-linear interactions.

ACKNOWLEDGMENTS

This research was supported by the National ResearchCouncil of Canada.

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10 C. S. Tsai, M. A. Alhaider, L. T. Nguyen, and B. Kim, "Wide BandGuided-Wave Acoustooptic Bragg Diffraction and Devices UsingMultiple Tilted Surface Acoustic Waves," Proc. IEEE, 64, 318-328(1976).

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