Thus Eq. (A10) does indeed satisfy Eq. (A7). It remains toevaluate the constants in Eq. (A9). At x = 0, H,, and(1/lYn)(dHm/dx) must be continuous. Since Tm(0) 0, Cm4= (.,..1. Furthermore,
-ifi.mCm1 = .m4Cm4 -BmdHo/dx Io+, (A 13)
where from (All), dTm/dx lo+ =-nBmdHo/dxlo+. Henceboth C,,,4 and Cm4 are determined in terms of Cmi. Next byinvoking continuity of H,, and (1/nf)(dHm/dx) at x = g, wedetermine the radiating field amplitudes Cmi and Cm3 .Specifically,
and these two equations, together with Eq. (A13) for Cm4 interms of C,,, suffice to yield Cmi and Cm3. Note that Tn, (g)and TM,(g) are given by Eqs. (A10) and (All), respectively,with x = g. Moreover, except in the case of a distributed
feedback resonance, the determinant of the coefficients isnonzero, thereby ensuring a unique nontrivial solution.
'A. A. Zleuiko, A. M. Proklhorov, A. A. Spikhal'skii, and V. A. Sychu-gov, "Emission of E Waves from a Corrugated Section of a Wave-gluide," Sov. J. Quantum Electron. 6, 565-567 (1976).
2E. M. Zolotov, V. A. Kiselev, V. M. Pelekhatyi, A. M. Prokhov, andE. A. Shcherbakov, "Diffraction Emission and Excitation of E andH Waves in Diffused Optical Waveguides," Sov. J. QuantumElectron. 7, 806-809 (1977).
IS. T. Peng and T. Tamir, "TM-Mode Perturbation Analysis of Di-electric Gratings," Appl. Phys. 7, 35-38 (1975).
4W. W. Rigrod and D. Marcuse, "Radiation Loss Coefficients ofAsymmetric Dielectric Waveguides with Shallow Sinusoidal Cor-rugations," IEEE J. of Quantum Electron. QE-12, 679-685(1976).
6Y. Yamamoto, T. Kamiya, and H. Yanai, "Improved Coupled ModeAnalysis of Corrugated Waveguides and Lasers-II: TM Mode,"IEEE J. Quantum Electron. QE-1 4, 620-624 (1978).
6W. H. Lee and W. Streifer, "Radiation Loss Calculations for Cor-rugated Dielectric Waveguides," J. Opt. Soc. Am. 68, 1701-1707(1978).
7W. Streifer, D. R. Scifres, and R. D. Burnham, "TM CouplingCoefficients in Guide-Wave Distributed Feedback Lasers," IEEEJ. Quantum Electron. QE-12, 74-78 (1976).
8H. Kogelnik and T. P. Sosnowski, "Holographic Thin Film Cou-plers," Bell Syst. Tech. J. 49, 1602-1608 (1970).
9W. Streifer, D. R. Scifres, and R. D. Burnham, "Analysis of GratingCoupled Radiation in GaAs:GaAIAs Lasers and Waveguides,"IEEE J. Quantum Electron. QE-12, 422-428 (1976).
Comparison of normal mode and total field analysistechniques in planar integrated optics
J. E. Sipe and G. I. StegemanDepartment of Physics, University of Toronto, Toronto, Canada M5S 1A7
(Received 26 March 1979)
Normal mode and total field analysis techniques for calculating the fields generated by a polariza-tion source are discussed. The amplitudes of the normal modes are evaluated using a Green's-functionapproach and the results are compared to a previous treatment by Yariv. The generated fields arealso calculated by solving the polarization driven wave equation subject to the usual electromagneticboundary conditions at all of the pertinent interfaces. It is shown that these two approaches yieldequivalent results for a simple case, and their relative merits for solving problems in different regionsof a waveguide are discussed.
INTRODUCTION
The generation of guided optical waves by polarizationfields (for example, via acousto-opticl nonlinear, 2 etc., in-teractions) occurs often3 in integrated optics. In order to testthe understanding of various phenomena or to optimize devicecharacteristics, it is frequently necessary to model the inter-action and calculate the amplitudes of the generated fields.There are essentially two analytical techniques for accom-plishing this: (1) via normal mode analysis, and (2) by findingthe total fields produced directly. In normal mode analysisthe generated fields are expanded in terms of normal modesand the mode orthogonality relations are used to evaluate theamplitude of the dominant normal modes. An alternatetechnique (henceforth called total field analysis) is to solveanalytically the polarization driven wave equation subject to
the usual electromagnetic boundary conditions. In this typeof calculation, one component of the total field evolves withpropagation distance into a growing normal mode guidedwave. These two approaches should, of course, yield identicalresults.
Normal mode analysis was developed in the early stages ofintegrated optics and has been discussed in detail by a numberof authors, the principal ones being Marcuse 4 and Yariv.5,6 (Atotal field analysis of the electromagnetic fields generated bya polarization source has only recently been reported.7' 8)Starting from Maxwell's equations, Marcuse 4 derived rigorousexpressions for the amplitudes of the normal modes generatedby an externally applied polarization field. In a differentapproach, Yariv5_6 obtained similar results by analyzing di-rectly the polarization driven wave equation for the TE elec-
tric fields. (As shown for example by Conwell,2 TM modesare treated by using the wave equation for the magnetic field.)It is usually assumed in this second approach that second-order derivatives of the normal mode amplitudes can be ne-glected. In this paper we calculate the normal modes gener-ated by a 5-function polarization source and then find theGreen's functions associated with an extended source We alsodemonstrate that neglect of the second-order derivatives inthe wave-equation approach is unnecessary.
The normal mode and total field techniques for analyzingwave interaction problems in integrated optics are comparedin this paper. In Sec. I we develop a Green's-function versionof normal mode analysis for a waveguide geometry which fa-cilitates comparison with the results of Yariv,5' 6 i.e., a thinisotropic film on an isotropic substrate. Since this particulargeometry has been discussed in detail recently 7' 8 using a totalfield treatment, we shall only briefly summarize that tech-nique in the next section with emphasis on those aspects notincluded in the previous work. In Sec. III we compare andcomment on the relative merits of the two methods of anal-ysis.
1. NORMAL MODE ANALYSIS
In many problems of interest, 1-4 it is necessary to calculateonly the amplitudes of a limited number of normal modesgenerated by an applied polarization field P(r,t). In suchproblems, a Green-function type of analysis may be employedto yield these amplitudes, thus reducing.the problem to oneof quadrature. Analyses of this nature have been presentedin the literature, 5' 6 usually in the context of coupled-modetheory; in such treatments, unnecessary approximations haveoften been made. By assuming the simple geometry in whicha specified polarization source exists in the region 0 < x < Lof a waveguide that consists of a film of thickness d (Fig. 1),it is easy to show where these earlier treatments have been inerror; we, therefore, do so below, in the course of presentingour analysis for this waveguide geometry. The extension tomore complicated geometries (and, of course, to fibers) isstraightforward and will not be given explicitly.
Writing all fields f(r,t) as
f(r,t) = 2 [f(r)eilt + c.c.]2
the total fields E(r) and' H(r) generated by the applied po-larization P(r) must satisfy the Maxwell equations,
V X E(r) + iwglH(r) = 0,
V X H(r) - iwE(z)E(r) = iwP(r), (2)
and the appropriate boundary conditions at infinity. A so-lution of Eqs. (2) valid over all space satisfies both the usualMaxwell saltus conditions at interfaces where E(z) is discon-tinuous,9 and the remaining two Maxwell equations,
V- e(z)E(r) = -V- P(r),
V- _,H(r) = 0, (3)
where the magnetic permeability u is taken as uniform overall space. Thus we need only consider Eqs. (2). Since we areinterested in fields that propagate along x and are indepen-dent of y, it is convenient to write the state of the electro-magnetic field in the form
Ex (xz) 1
F Hx (x,z)
Et (x,z) ,Ht (x,z)
(4)
where Ex = E-x, Et = E-(zi + y'), etc. Consider first a po-larization
P(x,z) = p(z)3(x - x'). (5)
Then, for x > x' and x < x', the field F is of the form F= F+ and F = F-, respectively, where
I Ex (x,z)
F Hx (x,z)Et- (x,z)
Ht (x,z)
(6)
and the components of F± satisfy the homogeneous form ofEqs. (2),
V X E+ + iwAH± = 0,
V X H+ - iWe(z)E+ = 0, (7)
and are outgoing propagating or evanescent fields as x- +coand x- -- , respectively. Eqs. (2) and (5) can in fact besatisfied at all points in space by putting
FIG. 1. Thin film waveguide with a polarization source confined to the filmbetween x = 0 and x = L. The aT(x) are growing normal mode amplitudesobtained from normal mode analysis and the aTo and a+L are the modeamplitudes generated at the boundaries in the total field approach.
1677 J. Opt. Soc. Am., Vol. 69, No. 12, December 1979
where 9(x) = 0, 1 as x < 0 or x > 0. Putting Eqs. (5) and (8)
in Eqs. (2) and using the fact that dO(x)/dx = W(x), we findthat
e(z) = -Px(z)/e(z), (9)
and that the components of F+ must satisfy
xX [E+(x',z) -E-(x',z)] =Yd [Px (Z) Ic(Z)]dz
x X [H+(x',z) - H-(x',z)] = icopt(z). (10)
From Eqs. (10) we can find the amplitude of any normal modedriven by p(z)b(x - x'); by superposition we may then de-
J. E. Sipe and G. I. Stegeman 1677
AIR :
FILM :
SUBSTRATE
x=o
-
termine the amplitudes generated by an arbitrary polarizationP(x,z).
We illustrate this by first considering bound TE modes,which for the waveguide geometry we consider are of theform, 6
= H'(z) e ipmx
iEHt(z)J
for forward and backward propagating modes, respectively,where Et(z) = 9E (z), Ht (z) = ifmEy(Z)/1w, H'(z)=-(i/lw)dE-(z)/dz, and
E' (z) = Cmeqmz air- Cmlcos(hmz) + (qm/hm)sin(hmz)} film (12)
Here hm = (W2 E 2 )1/2 , q. = (/2 - W2pe 1)'/2, and Pm= - C
2OE 3 )1/2
where
e(Z)= el
= E2
air
film
= E3 substrate
and [3m > 0 satisfies the implicit equation
tan hmd = (qm + pm)/hm (I1 pmm)
(13)
This gives the amplitudes of the modes F' that are generatedby the polarization p(z)b(x - x'). We now turn to an arbi-trary polarization P(x,z). Writing
P(x,z) = f p(x',z)b(x - x') dx', (20)
we use superposition to find, from Eq. (19), that the ampli-tudes at x of TE modes propagating forwards and backwards,a '+(x) and a' m(x), are given by
a+(x) = J eimx's-m(x')dxI,-F_
where
a'(x) = f e-iimx'sm(x/)dx/,
sm(X) = - I f- P(x,z) . Em(z)dz.4 f-
(21)
(22)
If, for example, the applied polarization P(x,z) extends be-tween x = 0 and x = L, then the amplitudes of the normalmodes which appear outside this region, a' = a" (x > L) anda'_n - a"'(x < 0), are given by
a- = fLe±bi..mx'sm(xI)dx/. (23)
Before turning to the equations analogous to Eqs. (21) and(22) for TM modes, we compare our results with those of amethod often used for deriving the amplitudes a.,(x).
5,6
Beginning with Eqs. (2), one finds(14) V X (V X E(r)) = W2 ,ue(z)E(r) + W2 MtP(r).
The modes are orthogonal and normalized in the sense that
(E7- X H-) * .dz = (1 W/m 2)6mn, (15)
if the constants Cm are chosen according to
Cm = 2h [-(id + - + (h2 + qm)] (1 W/m 2 )'/2.
(16)They are also orthogonal to the bound TM modes and theradiation modes,
I ( ) H dz2 - (E;`l X HO) -xdz
= - (E° X H7') * xdz = 0,2 f- (17)
where H"l', E2 refer to any of these other modes.4 Making theusual assumption of completeness, we write
F= F° + Z am F,m
(18)
where the sum is over TE modes, and F,: contains contribu-tions from all other types of modes. We now dot the first andsecond of Eqs. (10) into H't` (z) and El"(z), respectively; in-terchanging the dot and cross products, integrating over z andusing Eqs. (11) and (18) and the orthonormality relations (15and 17), we obtain two equations for a". Solving, we find
a =-(iw/4)e'inimx ' P(z) - E" (z)dz. (19)
1678 J. Opt. Soc. Am., Vol. 69, No. 12, December 1979
(24)
For fields that do not depend on y, f (r) = f (x,z), the y com-ponent of Eq. (24) is
± + E, E(xxz) =-W2/Ie(z)E (xz) - @2py (x z)
(25)
which is the starting point of this approach. 10 If we write
E,,.(x,z) = a"'}(x)e-i1m -E`m(z)
+ _ an' (x)ei1m3xEn '(z) + E (xz), (26)m'
where EY (x,z) is the contribution to E (x,z) from all but theTE modes, upon substituting Eq. (26) into Eq. (25), multi-plying the latter by E~'(z) and integrating over z, we find
e-itmx dCW (X) -eidmX dc (x) ,dx dx
where s (x) is given by Eq. (22) and
c 1 (x) = a' (x) + da ( )2 Om dx
(27)
(28)
Now these equations are exact; substituting our solutions (21)into Eqs. (27) and (28), we may verify that the latter equationsare correct. However, Eqs. (27) and (28) are usually solvedby assuming the amplitudes au±'(x) are "slowly varying," andapproximating Eq. (28) by
cl(x) -a(x),
which reduces Eq. (27) to
(28a)
J. E. Sipe and G. 1. Stegeman 1678
e-i 1,x da+(x) - ei13mX da- sm xi. (27a)dx dx
Then, if for example s m (x) has a spatial dependence that will
generate a large am, aT(x) is neglected in Eq. (27a) to write
e-Omx da+(x Sm(x), (27b)dx
and an integration yields
am+(x) ei0mx's m(x) dx'. (27c)
Equation (27c) is in fact exact [cf. Eq. (21)], but it has beenincorrectly derived from the correct Eqs. (27) and (28), and
is obtained in this approach only by the fortuitous cancellation
of errors. From Eqs. (21) we see that
da+(x) da-(x) sm(x). (29)e il~mx. =- i3,X (9
dx dx
Thus I da'+(x)/dx I = I daT(x)/dx |, and the neglect of, say, the
latter compared to the former, even if s " (x) has a spatial de-
pendence that will generate a large a' and only a small am, is
clearly incorrect. Furthermore, the approximation (28a) is,in general, incorrect. Consider, for example, a source term
that drives the forward propagating mode synchronously,
Sm(X) = 0 x<0 x>L,(30)
sm(x) = e-i0mx 0 < x < L.
Then, from Eqs. (28) and (29), we find that
a+(x)=x O<x<L, (31)
and so, from Eq. (28),
c+(x) = x + 0 < x < L, (32)
and we note that, indeed, c+ (x) a+(x) for essentially all of
the region where am (x) is increasing, if L >> 0'. And, in fact,
for this example of a source term driving a mode at resonance,dc (x)/dx = dam(x)/dx exactly. However, from Eqs. (29) and(30) we also find
a'(x) = 2 x (33)2io.
a rapidly oscillating function, since the backward propagatingmode is being driven far off resonance. Because of this,
Ida'(x)/dx - 2/3m Iam(x)j1 and Eq. (28a) is clearly invalidfor that mode. It is only because this error is canceled by the
incorrect neglect of da'(x)/dx, in going from Eq. (27a) to(27b), that the correct Equation (27c) is obtained. Otherexamples of sm (x) confirm the conclusion that, in general, Eq.(28a) is invalid for one or both of the modes. Because of this,
the correct derivation of Eqs. (21) from Eq. (27) is not
straightforward, and thus we feel a derivation of the type wegave preceding Eqs. (21), in which no terms were neglected,is to be preferred.
We conclude this section by giving the equations corre-
sponding to Eqs. (21) for the TM mode amplitudes. Thesemodes are of the form 6
TEn (Z)
Fm E=(z) eimxX (34)E` (z)1 (Z )
1679 J. Opt. Soc. Am., Vol. 69, No. 12, December 1979
where hm, qm and Pm are given by the expressions after Eq.
(12), and qm = e2 qm/El, Pm = e2Pm/E3; m > 0 satisfies theimplicit equation
tan hind = (Pmin + im)/hm (i -I hr) (36)
and the modes are normalized [cf. Eq. (15)] if the Cm are setaccording to
[ ( 2 h 2 + qmE
±1h, + Pm (1 W/M 2 )1/2 .
h~m'm PmEJIJ(37)
Proceeding as in the derivation of Eq. (21), and after an in-tegration by parts over the variable z, we find that the am-plitudes of the TM modes propagating forwards and back-wards, which we denote by b (x) and b m(x), respectively, aregiven by
b+(x) = ei mx's+(x')dx',+ -(38)
bm-(x) = e-ilmx'sm(x/) dx',
where
fs(x) =-J' [Px(xz)'t + P,(x,z)]- Em(z)dz. (39)
If the applied polarization P(x,z) extends between x = 0 andx = L, then the amplitudes of the normal modes that appearoutside this region, b- b+(x > L) and bm T b 2(x < 0), are
given by
b- = fqL e +i/3m5 x I (Xof) dx'. (40)
11. TOTAL FIELD ANALYSIS
It is sometimes useful to find the electromagnetic fieldsthemselves in the region of a waveguide in which the appliedpolarization P(r) #d 0. Of course, the fields can be obtainedby adding up the contributions at each x from all themodes-bound and radiation-that are being generated byP(r), and then by adding in the field that corresponds toe(z)W6(x - x') in Eq. (8) for a distributed P(x,z) as in Eq. (20).However, an alternative approach of directly solving Eqs. (2)to find the total electromagnetic field has been developed,largely out of a wish to avoid both the approximations that hadbeen made in previous normal mode analyses,5' 6 and the ne-cessity of having to sum over what in principle is an infinitenumber, and what in practice may be a large number of nor-mal modes. This direct approach to find the "total field" atall points in space is the solution we outline in this section; the"total field" and "normal mode" analyses will be compared
J. E. Sipe and G. I. Stegeman 1679
in Sec. III, where we also discuss the relative strengths of themethods.
We begin by writing the total electromagnetic field thatsatisfies Eqs. (2), for an applied polarization P(r) which isconfined to a region 0 < x < L, as the sum of three compo-nents:
(a) Fields in the region 0 < x < L which, in each me-dium, are a particular solution of Eqs. (2) and (3).In practice it is easiest to find this solution from aHertz vector analysis, 7' 8 where the Hertz vector 7rsatisfies the driven wave equation
V2 7r - (n2 /c2 )r -p/on 2 (41)
and n is the refractive index of the medium. Oncethis is solved (analytically or numerically), theelectromagnetic fields are recovered from thestandard relations
Ep = V X (V X 70)- 2 D =Eon 2E + P,n2'
Hp = 2 V X *, Bp =ALHp,
(42)where the subscript p denotes the particular solu-tion. This is done separately for each medium; theresulting fields obviously exist only in media whereP(r) is 0, and they vanish outside the region 0 <x < L. In general they do not satisfy the saltusconditions either at the film interfaces (z = 0 andz = d) or at the "transverse" interfaces (x = 0 andx =L).
(b) Fields in the region 0 < x < L which are solutionsof the homogeneous Maxwell equations [Eqs. (2)and (3) with P(r) = 0] in each medium, but whichdo not satisfy the saltus conditions at z = 0 and z= d. These solutions are added to the particularsolution so that the sum does satisfy the saltusconditions at z = 0 and z = d in the region 0 < x <L.
(c) Solutions of the homogeneous Maxwell equationsat all points except for x = 0 and x = L, where theyare discontinuous. These are chosen so that, whenadded to the above-mentioned sum, the resultingsum satisfies the saltus conditions at all interfaces.This final sum is the complete solution that weseek.
In practice we can only construct the component (c) by find-ing the amplitudes of the normal modes, discussed in Sec. I,that contribute to it. Thus we cannot actually write an ex-plicit expression for the total field. However, if the appliedpolarization is almost synchronous with a guided mode, oneof the solutions in (b) is a field whose amplitude grows as itpropagates through the active region (0 < x < L). Since thisgrowing field normally dominates the total field in that region,it is usually the quantity of interest.
Detailed solutions for the field components (a) and (b), forapplied polarizations of the form
P(r) - P(z)e-iJx, (43)
[recall Eq. (1)] which result from acousto-optic and nonlinear
1680 J. Opt. Soc. Am., Vol. 69, No. 12, December 1979
interactions, have been given earlier 7' 8; only the salient fea-tures required for comparison with the normal mode analysisJresented in Sec. I wvil he summarized here. The discon-tinuities in the tangential electric and magnetic fields at thefilm boundaries which are present in the particular solution(a) are of the form
(44)
where here the superscripts a and s refer to the air and sub-strate sides of the film, respectively. We find that TE fieldsof type (b) are necessary because of the discontinuities HaXand El',, and TM fields because of H. and E S. Thus theproblem of determining solutions (b), and in fact, we shallfind, that of determining (c), separates into a "TE problem"and a "TM problem." Considering first the TE fields, werecall the form of electric field in a TE normal mode of thewaveguide,
E(x,z) = yC,, exp(-iOmx + qnz), air
= 5CnE{n exp(-ifln,x)X [exp(ihnz) + exp(-ihnz + ikm)] film
= 9C,,IE', exp[-if0mx -p (z -d)],
(45)
substrate
where E{f = 1/2(1 - iqm/hm), 0bm = 2 tan'1(qm/hm) and E'm= cos(hmd) + (qm/hm) sin(hmd); the constants hm, qm andp,, are as defined after Eq. (12) and satisfy Eq. (14). Equa-tion (45) is equivalent to Eqs. (11) and (12) but, in this section,is a more convenient way of writing the normal mode field[note Eq. (1)]. For /3 3d Om, the TE fields required for (b) areof the form
E(x,z) = 9CmD(l)exp(-iox + qz), air
= yCmEfl exp(-ifx) (46)
X [D(2 )exp(ihz) + D(3)exp(-ihz + im), film
= yCmE D (4) exp[-i/x - p(z - d)], substrate
where q = (/2 - 2//1El)1/2, h = (W2 /gE2 - /2)1/2 and p =
(/2 - C02 /.z 3 )1/2. Note that there is no loss of generality inassuming amplitude terms normally associated with the mthnormal mode, since for any values of the D ( in Eq. (46) thefields satisfy the homogeneous Maxwell equations in the threeregions. Of course, the saltus conditions at z = 0 and z = dare not satisfied by the fields (46); this is, in fact, just what isrequired, for we set the D i so that the sum of the particularsolution (a) and the fields (46) does satisfy those saltus con-ditions. For / , Okm, we find that the D(i) so obtained arefinite, and our determination of the fields (a) and (b) is fin-ished. However, if /3 /m the D (M) obtained are proportionalto A/-1, where AO3 =/ - /m. Therefore, the field (46) di-verges as /3 - /,
The total fields in the active region, of course, do not di-verge; the source P(r) is finite and confined to a finite regionof space, and all the generated fields must, therefore, be finite.If we were to continue with (46) as the solutions (b), we wouldfind that the necessary solutions (c) also diverge as : - Omin such a way that the fields at all points in space are finite forall /3. However, since we are primarily interested here invalues of /3 near a /m, it is both simpler and physically moremeaningful to deal with solutions (b) that do not themselvesdiverge as :3 -S /m. We find that such a choice of the solutions(b) yields a field that grows in amplitude through the active
,J. E. Sipe and G. I. Stegeman 1680
a a a,, a,, -illx[EY,', Ex,', HY , HX le
region for L flm; the corresponding solutions (c) then yieldfields that describe the propagation of this generated energyto x = + in the normal mode of the waveguide characterizedby m, the "backscattering" contribution of these solutions(c) to the field in the active region being essentially negli-gible.
To determine the required nondivergent solutions (b), wefirst construct a new solution of the homogeneous equationsby subtracting a normal mode solution of amplitude D' froma particular combination of the solutions (46): we set all fourD(i) = D'. This gives the field
9CrnDexp(-iox + qz) 11 - exp[Af3(ix - qz)]/IA/3, air
9CmE D exp(-ix)[exp(ihz) 11 - exp[iAI3(x + hz)]I/IA1(47)
+ exp(-ihz + ihm) $1 - exp[iA/3(x - hz)]1/AO], film
where D = D'AO, 7 = (/ + Om.)/(q + qm),h = (13 + Am)/(h +hm), and = (O + /m)/(P + Pm).The field (47) grows with x as AO - 0; adding the field (47)and those fields associated with any three of the D i of Eq.(46) to the particular solution (a) we find that the saltusconditions at z = 0 and z = d can be satisfied with finite valuesof D and the appropriate D(i). In most cases of practical in-terest: >> AO, and the "growing field" (47) dominates thesolutions. Only the coefficient D is then of importance, andsolution gives D = D1/D 2Cm withDi = (1 + iq/h)(woH' - ipEE')
+ eihd(l - ip/h)(wAHx + iqEa),(48)
D2 = (1 + iq/h)(woAAh' - ipXE')
+ eihd(1 - ip/h)(cwtgAH' + iqAEa),
where AH', AE', AHa, and AEa are the discontinuities in thetangential components of the growing field (47) at the sub-strate and air interfaces, respectively. (Details may be foundin Refs. 7 and 8.) Of course, we have not yet satisfied thesaltus conditions on Hy and Ex at z = 0 and z = d. This mustbe done by adding on TM fields corresponding to Eqs. (46)and (47); we shall turn to this at the end of this section.
First, however, we consider the TE part of the component(c) of our total field; the fields calculated to this point, the sumof (a) and (b), exist only in the region 0 < x < L and are,therefore, discontinuous at x = 0 and x = L. Their tangentialcomponents are [AEy(O+,z), AH11(O+,z)] and [AEy(Lz),AH, (L-,z)] as x - 0+ and x - L, respectively; these termscontain contributions from both the components (a) and (b).The field (c) that we seek can be decomposed into normalmodes (bound and radiative) of the waveguide emanatingfrom x = 0, and normal modes emanating from x = L (Fig. 1);the amplitudes must be set so that the total field [(a) + (b) +(c)J satisfies the saltus conditions everywhere. Using theorthogonality relations discussed in Sec. I we find
a =-- [AEY(0+,z) + - AHz(0+,z)]EmI*(z)dz,2Zm 13m
am( = 1Z f' [Ey(0+,z) - AHz(O+,z)]En*(z)dz,2Zm O-/m
(49a)
1681 J. Opt. Soc. Am., Vol. 69, No. 12, December 1979
for the bound mode amplitudes; similar equations may bederived for the radiation mode amplitudes. We see that thetotal fields in the region 0 < x < L contain normal modes withamplitudes a'o and a'L, as well as the fields previously cal-culated. For x ' L the fields propagate in the form of normalmodes of amplitude
a+ = a+o + a+L, (51)
while for x ' 0 they propagate as normal modes of ampli-tude
a = a- 0 + a-L- (52)
The a" of Eqs. (51) and (52) correspond, of course, to the a f
of Eq. (23) in the normal mode analysis of Sec. I.
These results simplify considerably if we have synchronismwith one of the normal modes, 3 = /m The growing field (47)has the form
We note that this field contains a "linearly growing normalmode" of amplitude (-ixD), as well as a term with a compl}cated transverse (z) dependence. Assuming that [D D ()(as can be verified for our example in Sec. III), pnly the field(53), which grows with both x and z, can become large; fur-thermore, since the z variation is bounded by exponentiallydecaying terms or by the film thickness, the linearly growingnormal mode dominates the solution (53) for x >> XM,where
XM = Max(f3/q2 , 3d/h, f/p 2 ). (55)
Solving for the amplitudes of the m' modes outside the regionof' generation, we find
These will be compared in Sec. III with the correspondingexpressions obtained from the normal mode analysis of Sec.I.
For completeness we now give the TM components (b) and(c) that constitute part of the total field solution. We writethe TM normal modes in the form [cf. Eq. (45)]
.J. E. Sipe and G. I. Stegeman 1681
bH e= b'io + b'L, (62)
= SCmHf exp(-if8mx)1exp(ih,,x)
+ exp(-ihmz + jobm)), film (57)
= .CmH-n. exp[-iofx - Pm (z - d)], substrate
where O3m, q,,t, Pm, qm, and Tm are as defined after Eq. (35),
and Ha = -hm/qm, Hf = (1/2)(i - hm/1m), Om = 2tan-'(6m/h.), and H' = -(hm/lm) cos(hntd) - sin(hnd).The constant amplitude and growing fields corresponding toEqs. (46) and (47), respectively, are those expressions with theelectric field quantities replaced by the corresponding mag-netic field quantities. The amplitude coefficient D = D11D2Cm is here given by
D, = (1 + iqi/h)(-ipH' -weE')
-eilzz(l - ip/h)(iqH0 - wce E), (58)
D2 = (1 + ic/h)(-ioAeH- W 1AE')
-ei/lz(l - i j5/h)(iqjAH'a - WE2 AE'.),
where the field terms are defined in analogy with the TEanalysis and h = h/n2. The coefficients of the TM modesgenerated at x 0 and x = L are given, respectively, by
b -= -2- 4 [AHy(O+,z) - -AD,(0+,Z)]Hm*(z)dz,
b =2Yf J= [AHy(0+,z) + c ADz(0+,z)]Hm*(z)dz,
(59a)
and
e illLnSbrL = 2Y [J EAH(L.Jz) ADz(Lz)IHn2*(z)dz,
L 2Ym Oin
b"L = e-imL2et 2Y J
where
[cf. Eq. (40)]. If the applied polarization is synchronous witha bound TM mode, i = , we find
In this section we compare the two methods outlined inSecs. I and II; we begin by noting that for any problem, ifcarried out completely they should, of course, lead to the sameresult. However, this equivalence is not easy to demonstratefor a completely general P(r), since the normal mode solutionrequires the evaluation of integrals of the form (21) and (38),while the total field analysis requires instead the solution ofthe driven wave equation (41) and the evaluation of integralsof the form (49) and (59). Thus, in discussing the equivalenceof the methods, we shall restrict ourselves to a problem inwhich both techniques may be easily applied: we consideran applied polarization field in the film
P(r) = (xPx + 9P, + iPz)e-ix-Yz, (65)
[AH(L-z) + d AD,(L_,z)IHm*(z)dz, for 0 < x • L, and look at the fields generated in the regions01?1' x > L and x < 0. If we have phase synchronism with the m'
(59b) bound TE mode, i.e., : = a we can easily evaluate a" as inEqs. (23) and (51) by both methods; we find a" = -iDL,where
Ym = fHm(z)Hm*(z)dz. (60)
Thus, for x > L we have TM modes with constant ampli-tudes
b= b' + b'L, (61)
while for x < 0 the amplitudes are
) D = - c 2kthP,(Qy2 + h2 )(h2 + q2)(h2 + p2 )1/23C,
X Jfh2 + q 2 )(p )e-yd +(h2 + p2)(rq + i)j (66)
using either the normal mode or total field analysis. Simi-larly, if we have phase synchronism with the m' TM mode, wefind b' = - iDL, as in Eqs. (40) and (64), where
-whq(hI + V2)"2 [flp3(73 - y) + iP (h2 + -y))]e -- d - (h2 + p2)1/2[- f3p (q + _y) + iP (h 2 - )]
It is clear that the two methods will also give the same am-plitudes a"" and bl'- for these cases. A comparison of the
1682 J. Opt. Soc. Am., Vol. 69, No. 12, December 1979
generated fields away from synchronism (f3 $k (m) is moredifficult; for convenience we consider only the TE fields in the
J. H. Sipe and G. I. Stegeman 1682
(67)
�CmHa exp(-iflnx + q,,,z), air
region x > L. The normal mode calculation is simple andyields
a = -iDL exp(-iA( L/2) (AOL/2) (68)
where AO = 3- (m, D is given by Eq. (66), and all parametersare those associated with the mth mode. However, to calcu-late the amplitudes in the total field analysis we must make
a number of approximations. First, we expand the termexp[AO(ix - qz)] in the field (47) in the air as exp(iAl\x) [1- q (A3)z + ...], and note that the first term dominates when1 >> OA03/q2; expanding the corresponding terms in the fieldin the film and substrate we find that, if we also have 1 >>
O3A13/p2 and 1 >> OA(d/h. the field (47) may be written as
9CrnDexp(-i3x + qz) - exp(ix) airI AO 3
C9CmEf D exp(-i(x) [1- exp(iA(3x)' (69)
X lexp(ihz) + exp(-ihz + ijbm)}, film
.9CmE' D exp[-i(x - p(z - d)] - exp(iAI x)}
substrate
Uising Eq. (69) in Eqs. (49) and (51), and neglecting the con-tributions from the solution (a) of the inhomogeneous wave
equation and those from the fields (b) with coefficients D M,we obtain Eq. (68). All the above-mentioned approximationsmust, therefore, lead to errors that cancel each other.
We now discuss the relative merits of the two techniques.We see from the treatment in Sec. II that, even in the total
field analysis, a summation over bound and radiation normalmodes cannot be avoided. Since integrals such as (23) and(40) are generally easier to perform, either analytically or
numerically, than the solution of the inhomogeneous waveequation (41), it should generally be easier to determine the
fields outside the region of generation with the normal modeanalysis than with the total field analysis, as our exampleabove clearly illustrates.
The total field approach does appear to have some advan-tages in calculating the fields inside the generation region,
especially when they are coupled out via, say, a prism orgrating. For example, in the off-synchronous case ((3 F$ (m),
solutions (a) and (b) are proportional to exp(-i(x) and coupleout in the direction appropriate to the wave vector component(. The normal modes associated with the two transverseboundaries are of constant amplitude and couple out intodirections appropriate to (3m. On synchronism the normalmode associated with (3 = (m must also be evaluated as part
of the dominant field. This is in contrast to normal modeanalysis in which the mode amplitudes vary with distance andthe fields proportional to exp(-i(x) consist of a summationover all of the normal modes. However, if only the normalmode amplitudes (excluding ( = (m) are required, normalmode analysis again becomes the preferred technique.
An indication of the characteristic distance after which thegrowing mode (dominant normal mode) dominates the solu-
tion fields can be obtained from the total field analysis. Notethat in order to make a definitive calculation of this interac-tion length, all of the field amplitudes (a), (b), and (c) must
1683 J. Opt. Soc. Am., Vol. 69, No. 12, December 1979
be evaluated. However, if we assume all of these termscomparable in magnitude to OD, then the linearly (withpropagation distance) growing normal mode dominates thesolution fields for x >> xm. Since Xm / 3/p2 and 3d/h, thischaracteristic length can be many optical wavelengths nearmode cutoff (p 0) and for thick films (d -a , h - 0).
We, therefore, conclude that, for the generation of fields by
an applied polarization source, it is easier to determine thefields outside the region of generation with the normal modeanalysis than with the total field analysis; for a calculation of
the dominant field in the region of generation, total fieldanalysis presents a simpler approach, both off and on syn-chronism. Furthermore, we note that wave interaction andgeneration problems in integrated optics occasionally requiremore than just an obvious polarization source, as in Eqs. (2)and (13), for their description. An important example is theacousto-optic interaction that results when a sound wavecorrugates the surface. Two effects appear: the ripplecreates surface polarization sources, since the dielectric con-stants of the two media are different, and the corrugationrequires that the usual saltus conditions must be satisfiedacross a moving surface. Since both these effects may bedescribed by applied field discontinuities at the surface, 7 theymay easily be incorporated in a total field analysis similar tothat given in Sec. II. But, to our knowledge, only the first ofthese effects has been correctly described by a normal modeanalysis; the total field analysis 7 and the normal mode analysispresented in Refs. 1 and 11 agree for incident TE light, butdisagree for incident TM light. Thus, at least at the moment,total field analysis is the only technique available for the de-scription of phenomena in which surface corrugation plays animportant role. We hope to turn to a correct normal modeanalysis for surface corrugation problems in a future publi-cation.
1R. V. Schmidt, "Acoustooptic Interactions Between Guided OpticalWaves and Acoustic Surfaces Waves," IEEE Trans. Sonics Ultra-son. SU-23, 22-30 (1976).
2 E. M. Conwell, "Theory of Second-Harmonic Generation in OpticalWaveguides," IEEE J. Quantum Electron. QE-l0, 608-612(1974).
3P. K. Tien, "Integrated Optics and New Wave Phenomena in OpticalWaveguides," Rev. Mod. Phys. 49, 361-420 (1977).
4 D. Marcuse, Theory of Dielectric Waveguides (Academic, New York1974).
6A. Yariv, "Coupled Mode Theory for Guided Wave Optics," IEEEJ. Quantum Electron. 9, 919-933 (1973).
6A. Yariv, Quantum Electronics, 2nd edition (Wiley, New York, 1975),
Chap. 19.7 R. Normandin, V. C. Y. So, N. Rowell, and G. I. Stegeman, "Scat-
tering of guided optical beams by surface acoustic waves in thinfilms," J. Opt. Soc. Am. 69, 1153-1165 (1979).
8V. C. Y. So, R. Normandin, and G. I. Stegeman, "Field Analysis ofHarmonic Generation in Thin Film Integrated Optics," J. Opt. Soc.Am. 69, 1166-1171 (1979).
9M. Born and E. Wolfe, Principles of Optics (Macmillan, New York1964).
°0 Eq. (25) is derived in Yariv 5 6 from the equation
V'E(r,t) = AEW(z) E(r2t) + ,u (2 P(r,0,
which follows from the Maxwell equations only if V-E(r,t) = 0.Such an assumption is, as shown above, not necessary to derive Eq.(25).
"K. W. Loh, W. S. C. Chang, W. R. Smith, and T. Grudkowski, "Braggcoupling efficiency for guided acoustooptic interaction in GaAs,"Appl. Opt. 15, 156-166 (1976).
