Field analysis of harmonic generation in thin-film integratedoptics

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

(Received 24 August 1978)

Harmonic generation of light in thin-film optical waveguides is analyzed theoretically by evaluat-ing the total electromagnetic fields that satisfy the polarization driven wave equation and the electro-magnetic boundary conditions at both interfaces. Analytical expressions are given for the amplitudeof the growing wave component of the total fields. The results agree with those of normal modeanalysis only in the limit of phase-matched harmonic generation. This analysis is believed to beapplicable to a large range of wave interaction phenomena in integrated optics.

INTRODUCTION

It is well known1 -" that high optical power densities andhence nonlinear effects can easily be produced in thin-filmoptical waveguides. Harmonic generation in particular hasbeen observed experimentally2 '5 -1" by a number of investi-gators and has usually been analyzed theoreticallyl"3-5 by someform of normal mode analysis. In a comprehensive paperConwell 4 has treated harmonic generation using both a rayand a normal mode approach and found them to be essentiallyequivalent in the appropriate limit. In this paper we presenta different analysis of this problem which is valid even in thelimit that the doubled fundamental field is not phase-matchedto the harmonic. In particular, it will be shown that the re-sults of the present analysis can differ in some cases from thoseproduced by the usual normal mode treatment when thenonlinear driving fields are not phase-matched with a guidedwave at the harmonic frequency.

The theoretical formalism presented here is based onfinding the electromagnetic fields (hence the field nomen-clature) that satisfy both the nonlinearly driven wave equationand the usual electromagnetic boundary conditions at the filmsurfaces. (This technique has recently also been applied tothe harmonic generation of surface plasmons12 and surfaceacoustic waves.' 3 ) In the closely related preceding paper14the scattering of light in thin-film optical waveguides bysurface acoustic waves was treated in detail and it was foundthat the analytical results agree with normal mode analysiswhen the Bragg condition (which corresponds to phasematching) is satisfied. Since the approach used here is es-sentially an extension of the technique outlined in the previouspaper, the reader is directed to that work for details and onlythose aspects unique to the nonlinear problem will be treatedcomprehensively here.

The analysis presented in this paper differs from the pre-ceding work14 in two ways. In the acousto-optic interactionthe acoustically induced polarization fields were not syn-chronous with electromagnetic fields that satisfy the homo-geneous wave equation. However, under the Bragg anglecondition the wave-vector component parallel to the surfaceof the driving fields was equal to that associated with a normalmode guided wave of the same frequency. In the presentproblem the nonlinear polarization fields are assumed to besynchronous with solutions to the linear wave equation butare not necessarily phase matched to guided wave normalmodes at the harmonic frequency. (A similar situation would

occur in the acousto-optic interaction for scattering from bulkacoustic waves when the Bragg condition is not satisfied.)

This paper is structured as follows. The nonlinear polar-ization fields are defined in Sec. I and the driven electro-magnetic fields are derived. Next the TE modes generatedat the harmonic frequency are discussed (Sec. II). A similarcalculation is summarized for TM generated fields in Sec. III.A representative numerical calculation of nonsynchronousharmonic generation is given in Sec. IV and the principal re-sults of this work are discussed in Sec. V.

1. SOLUTIONS TO THE INHOMOGENEOUS WAVEEQUATION

The nonlinear polarization fields are written in the usualway' 5 as

Pi = dijkEjEk, (1)

where the dijk are elements of the nonlinear optic tensor in theappropriate medium. The terms E, and Eh represent electricfield components of the incident TE electromagnetic waveguided by a film of thickness h, i.e.,

air:

E = (112).E ei(1t--xx)+Soz + c.c., (2)

film:

E = (1I2 )jEjfei(t-kYx)(eikzz + e-ikzz+i) + c.c., (3)

substrate:

E = (1/ 2 ))Esei(,t-hxX)-S2(Z-h) + c.c., (4)

with k2 - SO = n'(w)k 2 , k2 + k2 = n2(w)k2 , and k2 - SI =

nS(w)k 2. Here n is the refractive index and the subscripts(and later superscripts) a, f, and s refer to the air, film, andsubstrate, respectively. Therefore, from Eqs. (1)-(4) thepolarization fields are given by

P= (1/ 4 )dfi22E2e

2i(wt-kxxx)(e2ikzz + 2eiO

+ e-2ikzz+2i0) + c.c. (5)

and

Pi = (1/4 )dq2 2 E2 e2i(.t-kxx)-2S2 (z-h) + c.c. (6)

in the film and substrate, respectively. (TM incidence canbe treated in exactly the same way, but, for the sake of brevityonly, the TE case will be discussed in this paper.) For i = 2,

1166 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979 0030-3941/79/081166-06$00.50 9 1979 Optical Society of America 1166

TE polarized harmonics are produced and for i = 1 and/or 3,TM waves are generated.

The polarization fields act as sources for the inhomogeneouswave equation and lead to driven electromagnetic fields. Theappropriate field quantity that is driven by a polarization fieldis the Hertz vector 7r which is calculated from

V2 7r -(n 2 /c7 = P/Eon 2

The tangential fields given by Eqs. (10)-(13) do not ingeneral satisfy the usual boundary conditions at the two in-terfaces. When evaluated on the film side at the boundariesz = 0 and z = h the discontinuities

Allt =- dt,2 2 E eiksinq,

AEt = - k df222E'eei4,(7)

in both the film and substrate. For example, evaluating Eq.(7) for y-polarized (TE) fields and subtracting appropriatesolutions to the homogeneous wave equation gives

1 d(222Ef e2i(t-hx) e2ikzz - e2

ikzz-z

Y 2 8con2(2co) \k'2 k 2

e-2ikzz -e-2ikz 2ei+ e 2io k_ _ _ _ _ - 1 ± c.c. (8)

k - k" 2 4j 2

and

1 d'22E 2e2i(t-k-x) e-2S2(Z-h) - e-2S2-(Z-h)

= S on S(2 w) 2 ±2 ~ . 9

Note that the solutions to the homogeneous wave equationat the frequency 2w are characterized by the wave vectors k X+ k%2, = n72(2w)k 2 and k 2 - S2," = n 2(2w)hk

2. Standard rela-tions are then used to calculate the usual electromagneticfields and the tangential components in the film and substrate,respectively, can be shown to be

1 wdf222Ef2e2i()t-kxx)

2 2

(k e2 ikzz - kz,,e 2ikz-z kze-2ikzz - hzke2ikz^zh2 - kz2, k2 -k2,

+ c.c., (10)

1 df22 2 E2e2i(wt kx)

= 2 2Eon7 (2w)

x ~k2n 2 (W) (e2ikzz + e-

2ikzz+2i4) - k2n2(2w)

(ikfz(z ± k2 -z2ik nf(2w)

X (e2ik-,"z + e-2ikz4>+2i0) _ 2e iPnf2(2(o)k 2

- (e2

ikzz + e-2ik2z+2it)) + C.C.

H. = - iwds222s e 2i(t-kxx)2 2

(S 2 e-2S2(Z-h) -S2,e-2S2-(z-h)

A l + c.c.,

1 ds222E 2Ey-- e 2i(.t-kxx)

2 2Eons(2w)

X (n()t2e-2S2(z-h) - n%2(2w)k 2e-

2S2,(zh) +

(11)

(12)

(13)

Note that these fields are valid for the general case which, forexample, could include dispersion in the material constantswith frequency. Of particular theoretical interest here is thecase of zero material dispersion, i.e., k2,, = k, and henceforththe fields will be discussed only in this limit. This assumptionallows us to study the case in which the driving fields aresynchronous with solutions to the homogeneous wave equa-tion.

1167 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979

(14)

(15)

AH' = - d2 d22E 2eik2k,

X [2hkz cos(2kzh - 4) + sin(2kh -0)] +- ± ds222E,482

(16)

and

AE' = -2 dW222E 2eib[hk sin(2k~h - )) + 11 (17)

are found to occur. Here the superscripts t and b refer to thetop (z = 0) and bottom (z = h) of the film, respectively, andthe coefficients described by Eqs. (14)-(17) are understoodto be the amplitudes of surface fields of the form

AH% = (1/2)XA2i(et-kxx) + C.C. (18)

The fields associated with TM generated harmonics areevaluated in the same way. Again zero material dispersionwith wavelength is assumed and it can be shown that thenonlinear interaction can be expressed in terms of driven fieldsand the tangential field discontinuities

AH t = f eio (2k dY', + idf2,2 sin2 ,- 2k2 k d2 I

AE% = EL - -k d ,,,2sin4) - 22eoni 2 k2 I

(19)

(20)

AH' = Ef ei" |2kxdg 2 (h sin(2kzh - d) ± )

-idjf2 2 [sin(2k~h - 4)

+ 2hcos(2kzh - 2)] - 4 E2d( (21)

AE' = Ž [ikxdyf2 2

X I-sin(2h~h - 4) + 2h cos(2hh - 4))

- d(f2)[2hkz sin(2kzh - 0) + 2 + itk 2E d 2)2.I48 2c0 4s

(22)

11. TE HARMONIC FIELDS

As discussed in detail in the preceding paper,14 this problemnow requires additional fields (which are solutions to thehomogeneous wave equation) in order to ensure the continuityof the appropriate boundary conditions at the harmonic fre-quency. These solution fields are characterized by exp2i(cot-kx) and four are required since there are four boundaryconditions to be satisfied. The appropriate steady-state fields

So et at. 1167

J)s2, - s2

in the absence of dispersion are

air:

E = (1/ 2 )jD(M)Eae2i(.t-hxx)+2soz + c.c.,

film:

E = (1/2)D(2)Efe2i(.t-kxx+kzz) + c.c.,

E = (1I2 )jD(3)Efe2i(t-kxx-kzz+ '/2) + c.c.,

substrate:

E = (1l/2 )]D(4)ESe2i(wt-kxx)-2S2(Z-h) + c.c.

+ k,')/2, So = (SO + So0 )/2, and S2 = (S2 + S2')/2. The

coefficient D = D'Ak. does not diverge when kx - kx' since

DI a Ak;-. These fields simplify in the case of phase(23) matching (k, = kx ) to give

air:

(24) E = (1/2 )j7DEae2i(wt-kxx)+2Soz (ix - k/Soz) + c.c., (33)

(25) film:

(26)

For reasons which will be evident later the amplitude coeffi-cients Ea, El, EfeiO', Es are those associated with a normalmode guided wave of frequency 2w and wavevector 2kx' 5:2kx. (There is no loss of generality in this choice since thecoefficients DM ... D(4)are still to be determined.) It can beshown that the coefficients D M are proportional to (2kx- 2kx,)-l when these fields are used to ensure continuity ofthe electric and magnetic boundary conditions at both sur-faces. Therefore the amplitudes diverge in the limit of phasematching which is an undesirable characteristic.

The solution fields can also be formulated in terms ofgrowing waves. We first note that guided waves of frequency2w (i.e., normal modes which do satisfy the boundary condi-tions) are

E = (1/ 2 )jD/Eae2i(,t-kxhx)+2So'z + c.c.,

E = (1/ 2 )jD'Efe2i(wt-k'x)

X (e2 ik,'z + e-2ikz'z+i0') + c.c., (28)

substrate:

E = (1/ 2 )IDIEse2i(wt-kx~x)-2S2'(z-h) + c.c., (29)

with k2, - S, = n2k2 k2, + k2 2 and k2,-S2, =n2k2.The growing fields are constructed mathematically by settingthe coefficients in Eqs. (23)-(26) equal to D' and subtractingthe normal mode solutions (27)-(29) in the respective media.This procedure givesair:

E = (112 )jDEce2i(wt -k x)+2S0z (eAkx(ix-kx/Soz)-1 + c.c.,Ak: /

(30)

film:

E = (1/2 )jDEfe2i(.t-kxx) [e2ikzz (eiAkx(x+i:z) - 1)e Akx~xz-

+ e-2ikzz+W eiAkx(x-z) . + c.c., (31)

substrate:

E = (1/2 )jDESe 2i(wt-kx)-2S2 (z-h)

leAkx[ix+(kx$/2(z-h)] -

x ) + c.c. (32)Akx

Here Ak,, = 2kx - 2kx', , kx/k-z, kx = (kx ± k.')I2, k, = (kz

E = (1I 2 )jiDEfe2i(.t-kxx)

X (ix(e2

ihzZ + e-2ikzz+di0d)

+ iniz(e2ikzz -e-

2ikzz+iO')) + c.c.,

substrate:

E = (1l2 )!DESe2i(wt-xx)-2S2(z-h)

(34)

X [ix + (kx/S2)(z -h)] + c.c. (35)

Note that for

x >> kX/s, x» (kx/k 5)h, and x > 2

the phase-matched growing fields evolve into linearly growingnormal mode waves that dominate the solutions. (The im-plications are discussed in more detail in the precedingpaper.14 For nonsynchronous harmonic generation the x andz dependences cannot be decoupled, but under the conditionsof Eq. (36) the x dependence does dominate the solutions formost values of x.

The fields given by Eqs. (30)-(32) and (33)-(35) contribute(27) to field discontinuities at the two interfaces. For the non-

phase-matched case the amplitudes of the discontinuitiesevaluated relative to the air and substrate sides, respectively,are

Et= 0,DEakx (kzkz + S0S0)

2co,4uoSOk~k

(37)

(38)

E b= DEf e 2ikz'h (e-i,?kxh -1

[ ( kx

+ e2ikz'h+i'fi (Akxh (39)

and

DEf - eikxhHb= Ikz'e~i'z h |k - kz e 2ikz h+iek

' Ako 2k /x(eiAkxh 1) kX 2ikz'h (iO 2 +- ieAkxh)

+ syenhroni- + ekZ 71 (40)

On synchronism these relations simplify considerably togive

Et= 0,Ht = iDEakxk2(n2 - n)/2 WtoSok ,

E' = DEskxS2 h/k 2

(41)

(42)

(43)

and

H = - (iDEskx/2wyuo)t2h + [(nI - n2)k2/k2S2J1. (44)

A linear combination of any three of the fields given by Eqs.

So et al. 11681168 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979

(36)

(23)-(26) and the growing field are used to ensure that thetangential boundary conditions are satisfied by the total fields.Solving for the amplitude coefficient D, which is usually thequantity of interest, gives

D = DI/D2, (45)

with

D = (1+ k ) (wgoA'x - iS2 AE')

- e2 ikzh (i - (wy0 AH' + iSoAE%) (46)

Et) = D [ke2ikz'h ( e x)Wcoflon

- kh,e-2ikz'h+i0k' eiAk,)

+ -ae 2ikzh lz| +-eiAkxnhI2 kS2 S2 k 2,

- e-2ikz'h+ih k' + - eakxihh)J

2 E,2 )Ikz J (54)

evaluated on the air and substrate sides, respectively. For thecase of phase matching, these expressions simplify to

D2= 1+ kl (wgoH' - iS 2E>)

- e 2ikzh 1-k-S2 (w#oHt + iSoE,).

H'= 0,

-i DHakxk2(nf2 - n2)Ext- 2w EoSok2 n2

Hb DHshkxS 2 (nT)2

Eb =iDH~kx | h ±(n-inf2)k2)(,),Eo W~ 2S2 Znr7 2

(47)

In the case of phase matching these formulas can be shown toagree with the results of a normal mode analysis of thisproblem as discussed by Conwell.4 Off synchronism someimportant differences occur which will be illustrated numer-ically in Sec. IV.

III. TM HARMONIC FIELDS

The analysis for TM generated waves is exactly the sameas in Sec. II. For the non-phase-matched case the growingfields are of the formair:

feAkx(ix-(kz/So)z) -H = (1½2)iDHae2i(wt-kxx)+2Soz (i kx + c.c.,

(48)film:

H = (1/ 2 )iDHfe2i(wt-kxx) fe2 ikzz l iAkxlx:z)

fe itx(X-z) - 1

+ e-2ikzz+i0' e Akx + c.c., (49)

substrate:

(55)

(56)

(57)

(58)

The amplitude coefficient D is evaluated by ensuring thatthe total TM polarized electromagnetic fields satisfy theboundary conditions. Evaluating D = D1 /D2 yields

DI= (1+ St4(- 2 AHb - WoOAE')

and

k kz nn n n

-e2ikzh (1-k iS2 1 2f iAH' - 6oAEx (59)

These results are identical to those predicted by normal modeanalysis4 in the case of phase-matched harmonic generation,but not off synchronism.

IV. NUMERICAL EXAMPLE

H 2)DHse 2i(t-kx)-2S2 (z-h) (a - The amplitude coefficient D was examined numerically forAkx a representative film and substrate. Refractive indices of 1.75

+ c.c. (50) and 2.27 were used for the substrate and film, respectively(which happens to correspond to a lithium niobate film grown

The parameters HI, Hf, Hfei&', Hs, 2hz, 2h,,, 2So', and 2S2' on a sapphire substrate1 6). An incident radiation field at 1.06are all associated with a TM polarized normal mode guided gm was chosen and dispersion in the material quantities withwave of frequency 2w. For the case of phase matching the wavelength was ignored: this allows us to examine phasefields reduce essentially to those given by Eqs. (33)-(35) with mismatches in a particularly interesting theoretical limit.the appropriate changes to magnetic field parameters. These The coefficients D were calculated using: (a) the exact non-fields (off synchronism) produce the discontinuities phase-matched formalism; (b) the approximate equations

Ht = 0 (51) (40)-(43) for E5, etc.; and (c) a normal mode analysis as out-Y lined by Conwell. The results were normalized to the value

-iDHa (x + that would be obtained for D if the driving fields, etc., were2wEona S kphka(52Et 2c~2 ts (52 kz hs-matched to the second harmonic at the frequency 2w

and wave vector 2k'.x[24S (i5Akxh - 1

H = DHfLIbz e J A relatively extreme example of approximate phase

- matching was selected for analysis. It can be shown in general+ e- 2 ihzh+io& (einAkxh - 1) (53) that a harmonically doubled TEo mode becomes progressively

1 LAk" I] better phase-matched [in the conventional sense that (kx

1169 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979 So et al. 1169

Cn0

Cd,C,,0

C-LU

3.0

1.0

0.3F-

0.1

* t: I l I

3.0 1.0 3.0 10.0

10

103.0

FILM THICKNESS h (pm)

FIG. 1. Relative phase mismatch vs film thickness.

- k.')/kh approaches zero] to a TE1 mode at 2w as the thick-ness of a film is increased. (For real materials the materialdispersion would produce phase matching at a finite filmthickness and the transverse mismatch effects of interest herewould not be present.) Two calculations were performed forthis case: in the first the nonlinear optical coefficients of thesubstrate were set to zero; and, in the second those of the filmwere neglected. The results are shown in Figs. 1-3.

Consider first the case of harmonic excitation via non-linearities in the substrate. As illustrated in Fig. 2, the overlapintegral for the dominant normal mode gives a good approx-imation to the theory presented here. The approximate

1.0- - -

0

C-,LUn

CnC,,0

C.3LU

I-

LU

0.3

0.1

0.03

0.1 0.3 1.0 3.0

FILM THICKNESS h (pm)

10.0

FIG. 2. Relative cross sections for harmonic generation via substrate

nonlinearities only versus film thickness. Solid line Is the exact total fieldanalysis; dotted line is the normal mode analysis, dominant mode only;broken line is the approximate total field analysis.

1170 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979

FILM THICKNESS h (Pm)

FIG. 3. Relative cross sections for harmonic generation via film non-linearities only as a function of film thickness. Solid line is the exact totalfield analysis; dotted line is the normal mode analysis, dominant mode only;broken line is the approximate total field analysis.

version of the total field approach, i.e., with the denominator(D2) of D simplified, also agrees well with the full theory.These results are reasonable in the conventional sense sincethe longitudinal phase-mismatch term (k. - k.')/k. decreasesrapidly with increasing film thickness. Furthermore, therelative cross section approaches unity asymptotically for largeh. Also shown in Fig. 1 is the ratio (S2 - S2 )/S2 which we notedecreases rapidly with increasing film thickness.

The calculation in which the film nonlinearities generatethe harmonics produces results that initially appear contra-dictory to those of the previous case. The relative amplitudecoefficients as predicted by the overlap integral and by thepresent field analysis differ radically and do not converge forlarge film thicknesses, despite the fact that the longitudinalphase-mismatch term tends rapidly towards zero. We note,however, that the approximate version (via a simplified D2)does converge to the full total field value as h increases. Thisindicates that it is not the mode normalization that createsthe difference between the two theories. The effect is ratherattributed to the D1 term (numerator of D) and to the integralS E*-Pdz which is a consequence of the normal mode treat-ment4 (where E is the generated normal mode wave). In-spection of Eqs. (14)-(17) and (46) shows that no normal modeparameters at 2w appear in the D1 term which is in contrastto the overlap integral case. The implications of this con-clusion are discussed further in Sec. V.

V. DISCUSSION

The formalism presented in this paper is believed to be validin general, even far from a phase-matching condition. It isrigorous in the sense that the total electromagnetic fieldsrigorously satisfy: (a) the nonlinear polarization driven waveequation in all media; and (b) the boundary conditions at bothinterfaces. Of interest in most cases is the growing wave thatcorresponds to a linearly growing guided wave normal modewhen the doubled fundamental and harmonic are phase-

matched. The principal result of this formalism is an ana-lytical expression for the amplitude of the growing wave.

So et al. 1170

1.0I..

.. "I

N 2(k, -k, )h

I-

InLU

C,10

lU.U1 E ,

. _S'. (S2 2 I / S2

1Ln no

The theory proposed here agrees with normal mode analysisfor the dominant mode when the nonlinear fields are syn-chronous with a second harmonic guided wave. This is rea-sonable since the growing mode for this case does correspondto a linearly increasing (with propagation distance) normalmode which dominates the solutions under certain conditionson waveguide parameters that emerge from the total fieldanalysis.

There are, however, differences between the two theoriesfor non-synchronously generated harmonics. In a normalmode approach it is assumed that the total fields at the fre-quency 2w can be expressed as a sum over the complete set ofguided and radiation normal modes. Usually only the am-plitude of the nearest normal mode is evaluated and hence thedifference between the conventional overlap integral calcu-lation and the total field approach which by definition in-cludes all of the modes. (A sample numerical calculation hasindicated that large differences can occur in the field ampli-tudes predicted by the two theories.) In principle it shouldbe possible to reproduce the total field results by evaluatingthe amplitudes of all the normal modes.

The question now arises of how to identify geometries forwhich the differences in the two theories would occur. Wenote that the term 2(k, - kz,)h shown in Fig. 1 characterizesthe phase mismatch across the transverse dimension of thefilm and it remains relatively large, even as h continues toincrease. We suggest that the phase mismatch in both filmdimensions must be small for the two calculations (Fig. 3) toconverge to the perfect phase-matched case. This was veri-fied numerically by artificially forcing this condition on thecalculations. We further note that for the case of nonlinearexcitation in the substrate, the transverse mismatch term isgiven by (S2 - S')/S2 and indeed, when this term is small,then the two calculations do converge.

The theoretical formalism outlined here for the non-phase-matched interaction between guided wave fields isbelieved to be quite general. It can, for example, be used toanalyze the acousto-optic effect discussed in the precedingpaper for the case of scattering away from the Bragg condition.At present this approach is applicable primarily to situationsin which the inhomogeneous wave equation can be solvedanalytically. However, there is no reason to believe that it-erative computer techniques cannot be used to solve numer-ically the polarization drive wave equation.

In summary, we have analyzed harmonic generation in thinfilm optical waveguides by evaluating the electromagneticfields generated by the nonlinear interaction. The solutionsrigorously satisfy both the polarization driven wave equationand the electromagnetic boundary conditions at both film

interfaces. The amplitude of the growing wave agrees withthat predicted by normal mode analysis only if the drivingfields are synchronous with guided waves at the harmonicfrequency. Finally, it was shown that phase matching is im-portant both in the transverse dimension as well as along thepropagation direction.

ACKNOWLEDGMENT

This research was supported by the National ResearchCouncil of Canada.

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1171 J. Opt. Soc. Am., Vol. 69, No. 8, August 1979 So et al. 1171