2340 J. Opt. Soc. Am. B/Vol. 11, No. 12/December 1994

Optically induced grating in a directional coupler

Ewa Weinert-Rqczka

Institute of Physics, Technical University of Szczecin, Al.Piast6w 17, 70-310 Szczecin, Poland

Received February 23, 1992; revised manuscript received June 16, 1994

An optically induced grating in an asymmetric nonlinear directional coupler, formed by the interference oftwo strong external fields in a nonlinear medium, is analyzed. The grating constant and the amplitude ofthe grating depend on the parameters of the external waves, and thus they can be varied during coupleroperation. As a result external-power-controlled coupling of a chosen pair of normally mismatched guidedmodes can be realized.

INTRODUCTIONDirectional couplers are important elements of inte-grated optical systems. Efficient power exchange be-tween waveguides in a directional coupler occurs whenthe propagation constants of the guided modes of thewaveguides are equal. Coupling between nonidenticalwaveguides can be obtained with the help of a grat-ing formed by periodic changes of the refractive indexalong the coupler.'15 The grating constant K = 2/A(where A is the length of the grating period) should beequal to the difference between the propagation con-stants (K = /,, - 8,8) of the waveguide modes. Theimproved coupled-mode theory has been applied to de-scribe the interaction of the forward-propagating modesin the grating-assisted coupler.fr 0 A coupler with pa-rameters changing along the direction of propagation wasconsidered within the framework of the local-normal-modes theory.1 ' 2 A nonlinear grating-assisted coupler,with switching and filtering behavior obtained with thehelp of both the grating and the nonlinearity, was alsodescribed.1 3 The gratings in the traditional grating-assisted couplers are usually formed during fabricationand have constant, immutable parameters, which lim-its the operation of the coupler to one strictly definedfrequency.

In this paper an optically induced grating for a non-linear directional coupler is proposed. The grating isformed by spatially periodic changes of the refractive in-dex, which result from the interference of two counter-propagating external fields in a nonlinear medium. Theparameters of the grating depend on the external wavesand can be varied during operation of the device. In par-ticular, we can tune the grating constant to the differ-ence in the propagation constants of the guided modesfor the desired frequency by changing the propagation di-rections of the external waves. The interference methodenables us to create the long-wavelength grating that isnecessary to couple codirectional modes as well as theshort-wavelength grating needed to couple contradirec-tional modes. Hence the same device can drive the in-put power to any forward or backward niode, dependingon the properties of the external waves.

The approach presented here is based on the improvedcoupled-mode theory,'4' 7 which is applied for forward-

and backward-propagating modes in a directional couplerwith an optically induced grating.

FORMULATION OF THE PROBLEM

The considered coupler consists of N dielectric guides par-allel to the z axis of the coordinate system. To permitcreation of an optically induced grating, at least one ofthe coupler's components should be endowed with a Kerr-like nonlinear material. The general distribution of thedielectric constant is described by

E(X, y, z) = eL(x, y) + a(x, y)IE1 2 , (1)

where E is the total electric field in the coupler and a (x, y)describes nonlinear properties of some of the coupler'scomponents. The linear part of the dielectric constant'sdistribution in the whole coupler can be represented bythe sum of the dielectric constant's distribution of one,say the Pth, waveguide and the difference between thedielectric constants of the coupler and the waveguide:

eL(X, y) eL(x, y) + AeL (x, y), (2)

where e (x, y) represents the distribution of the dielec-tric constant of a single, separate waveguide.

According to the improved coupled-mode theory,14-7

a weak field at frequency co propagating in a couplerconsisting of weakly coupled monomode waveguides canbe approximated by a linear combination of the separatewaveguide modes:

NE = E A,(z, t)E.(x, y)exp(iwt - i,, 1z) + c.c.

,a =1

N+ B,.(z, t)E-,(x, y)exp(icst + i,8z) + c.c.,

!L=1

NH = A,(z, t)H,,(x, y)exp(iwot - i,8,z) + c.c.

N+ _ Bjt(z, t)H-,,(x, y)exp(icwt + ip,,z) + c.c.,

,a=l

(3a)

(3b)

where E., HA and E_., HA represent the transverseconfigurations of the forward- and the backward-propa-

0740-3224/94/122340-07$06.00 ©1994 Optical Society of America

Ewa Weinert-RgLczka

Vol. 11, No. 12/December 1994/J. Opt. Soc. Am. B 2341

gating modes and 3,x3 is a propagation constant of themodal field of waveguide ,u in isolation. The forwardand the backward fields' envelopes A,(z, t) and B,(z, t)are assumed to be slowly varying functions. For weakguided modes the power exchange between the two wave-guides occurs in the same way as in the linear coupler,with no effective coupling between modes with differentpropagation constants.

The grating, which is necessary to obtain coupling, canbe created by two external waves at a frequency Wexpropagating in different directions and overlapping in thenonlinear medium. These fields are approximated hereby plane waves with slowly varying amplitudes of thesame transverse distribution. We neglect reflections ofthe external waves at the boundaries to simplify the cal-culations. Longitudinal propagation vectors of the waveshave equal values, ex, and opposite directions,

E(X) = A(+)(x, y, z, t)exp(iC ext - iKexX - /exZ) + c-c-,

(4a)

E( = A(e-)(x, y, z, t)exp(iWext - KexX + i3exZ) + c.c.

(4b)

The intensities of the external waves are much higherthan the intensities of the guided modes. Hence the totalelectric field in the nonlinear interaction region can beapproximated by a sum of only the external waves. Thediameters of the external waves are much bigger thanthe width of the coupler. Thus their amplitudes insidethe nonlinear components of the coupler can be describedby a function Aex(x, y, z, t), which fulfills the followingcondition:

Y(x, y)Ae. (X, Y, Z, t) = a(x, y)A(j+)(x, y, z, t)

=ac(x, y)A(-x)(x, y, z, t).

that the coupled-mode differential equations for the pre-sented structure are given in matrix form by

C a a = -i[BC + Q(+) + f (z)G(+)]a - if (z)G(-)b, (8a)

C a b = i[BC + Q(+) + f (z)G(+)]b + if (z)G(-)a, (8b)az

where a(z, t) and b(z, t) are vectors describing amplitudesof the individual waveguide modes,

a,(z, t) = A,(z, t)exp(-i,83/z),

b,(z, t) = B,(z, t)exp(i3flz),(9a)

(9b)

f(z) = exp(iKz) + exp(-iKz), and B is a diagonal matrixof propagation constants B = ,l,. The elements of ma-trices C, Q(+), G(+), and G(-) are defined by the followingintegrals:

CA = ff(E. x H* + E,* x H,)zdxdy, (lOa)

Q(+) = cseO ff Ae6)[EtEt,* + (u),-iEi,,Ezi*]dxdy,

(lOb)

G (+) = eoff AeG[Et,,Etv,*/ _4 f

+ Ej)e C EZ,.,EZV*]dxdy,

(lOc)

The propagation directions of the external waves allow usto obtain an interference pattern along the z axis. Thedielectric constant of the coupler can be represented by

e(x, y, z, t) = L(x, y) + 2a(x, y)lAex(x, y, Z, t)12

x [1 + cos(Kz)], (5)

where the grating constant, K = 2 ex, depends on the fre-quency and the incidence angles of the external waves andcan be varied during operation of the coupler. Accordingto Eqs. (1) and (2) the dielectric constant distribution ofthe structure can be represented by

e(x, y, z, t) = (x, y) + A (x y, z, t), (6)

G( ) = eoff| AeFG[EtEt,,*-xI, f

- e2'6 elEz Ez*]dxdy,

(lOd)

where AeNT) = e + 2alAexI2 and AIEG = aIAexI 2.The coupling equations in the form presented above

are consistent with previously published ones for the cou-pling between codirectional6 i10 and contradirectional 1,12

modes. The difference is that the grating parameters(grating constant K and grating amplitude A EG) are vari-able here, and the same device is able to couple codirec-tional or contradirectional modes, depending on the actualvalue of K.

with Ael')(x, y, z, t) consisting of three elements,

-=EL + 2lAexI 2 + 2alAexl 2cos(Kz). (7)

The intensities of all interacting waves are assumed to berelatively low, so that the nonlinearity allows for gener-ating small index changes in the nonlinear medium butdoes not change the properties of the modes.

The derivation of the coupled-mode equations in itscomplete form is presented in Appendix A. It is shown

TWO-WAVEGUIDE COUPLER WITH ANONLINEAR CENTRAL LAYER

As an example, a coupler consisting of two different slabwaveguides separated by a nonlinear layer is examined.A two-dimensional scheme with a translation invariancewith respect to the y coordinate as well as TE polar-ized fields was chosen to simplify the calculations. Fortwo weakly coupled, mismatched waveguides (CM, = L,,),Eqs. (8) simplify to the following set:

Ewa Weinert-Rqezka

2342 J. Opt. Soc. Am. B/Vol. 11, No. 12/December 1994

idal(z)/dz = [ + Qjj2]al(z) + f(z)G12 a2(z)

+ f(z)G~llbl(z) + f(z)G1) b2 (z), (11a)

ida2 (z)/dz = [,82 + Q22 ]a2(z) + f(z)G21 al(z)

+ f(z)G2-)b2(z) + f(z)GHI~b1(z), (lib)

-idbl(z)/dz = [Pi + Qll]bl(z) + f(z)G12 b2(z)

+ f(z)G~llal(z) + f(z)G1) a2(z), (lic)

(+) H~~~(+-idb 2 (z)/dz = [P2 + Q22 ]b2(z) + f(z)G 2 i bi(z)

+ f(z)G22a2(z) + f(z)G21 al(z). (lid)

For an asymmetric coupler the terms describing so-callednatural coupling between different modes are not phasematched and have been neglected here. The mismatchof the other terms can be reduced with the help of thef(z) function.

An analytical solution is possible in the case of astrongly mismatched coupler when there is practically notunneling coupling between two different modes propagat-ing in the same direction and when the grating-inducedinteraction between each pair of modes can be consid-ered separately. When only one waveguide is excited,ai(O) = ao, a2(0) = 0, bi(L) = 0, and b2(L) = 0, the set ofEqs. (11) reduces to three independent pairs of equations:

idAil/dz = G12 A2 exp[i(yi - y2 )z]exp(-iKz), (12a)

idA 2/dz = G .+)Al exp[-i(y, - y2)z]exp(iKz); (12b)

id-Al/dz = Gil B3 exp(2iylz)exp(-iKz), (13a)

-id3l/dz = Gil Al exp(-2iylz)exp(iKz); (13b)

id.A,/dz = G12 32 exp[i(y, + y2 )z]exp(-iKz), (14a)

-idB 2 /dz = G2 -Al exp[-i(yi + y2 )z]exp(iKz), (14b)

where y, = 8a + Q(+). Equations (12) describe couplingfor grating constant K close to K1 = 71 - 72, Eqs. (13)for K close to K2 = 2yl, and Eqs. (14) for K close toK3 = Yl + Y2. The functions A and 3 are defined bythe following relations:

PH = 1B2(O)12/I.Al(O)J2 , depending on the normalizedparameter T = K/yl, for the difference of the effectivepropagation constants Y2 - y7 = 0.02yi, is presented inFig. 1. The calculations have been performed for thecoupling coefficients Gil = G2 = G2 = 7r/2L (where Ldenotes the length of the coupler) to permit a completepower exchange between forward-going modes. The nar-row peaks in the figure illustrate the character of thecoupler's responses to the varying grating constant andconfirm the separation of Eqs. (11) into three independentpairs of equations. The intensities of the external wavesnecessary to yield coupling can be estimated from therelation G(+) = 7r/2L, where GH depends on the gratingamplitude and the coupler's parameters [see Eq. (10c)].

More-detailed calculations were performed for a couplerexploiting an excitonic nonlinearity of a GaAs-AlGaAsmultiple-quantum-well (MQW) structure. The systemis composed of two 2-gm-thick Al.GaliAs waveguideswith different dielectric constants, el = 10.3 (x 0.27)and e2 = 10.4 (x 0.24), coupled through a MQW struc-ture consisting of 40 periods of 6.5-nm-thick GaAs quan-tum wells with 21.2-nm-thick AlGal-xAs (x = 0.4)barriers. The nonlinear properties of similar struc-tures (samples consisting of 120 periods of the samewells and barriers as here) were examined in detail byMiller et al.9 They obtained a linear dependence ofthe refractive-index change on an average input in-tensity up to An = 0.016 by using short pulses of-600-fs duration with a pulse separation of 130 ns.The refractive-index change was dependent on the den-sity of optically generated electron-hole pairs, An=neh(co)N. The density of carriers generated by a shortpulse, N(x, z) = a/(hw) f I(x, z, t)dt, can be approxi-mated by N(x, z) = a/lCw) Ia(x, z)At, where Ia(x, z) =Io[(l/2) + (1/2)cos(Kz)]exp(-ax) is the average intensityduring the pulse and a denotes an absorption coeffi-cient of the MQW layer. The strong resonant absorp-tion, - 1.2 x 104 cm-' for the considered wavelength,20

does not spoil the operation of the coupler because of theshort path of the external waves through the nonlinearmedium (the thickness of the MQW layer is -1.1 /Lm).The guided-mode wavelength, Ag = 1.55 ,um, was chosenin the range of transparency of all involved semicon-ductors. The refractive-index grating, formed by thespatially modulated carrier density along the wells, iswashed out by the recombination and the diffusion pro-

fka(Z) = a(z)exp(iyaz),Ba(z) = b(z)exp(-iyaz).

(15a)

(15b)

A solution for each pair of Eqs. (12)-(14) for a constantvalue of K can be found in a standard textbook. 8

NUMERICAL RESULTS

The complete set of Eqs. (11) has been solved nu-merically for different values of the grating constantK. The relative output power in the different ports,

-) = B1B(0)1 2/j_. 1(0)12 , p2+) = I.A2(L)12/1_A,(0)12, and

a)

w0

.0a)

U)

1,0

0,8

0,6

0,4

P2(

0,00 0,02 0,04 0,06 0,08 1,96 1,98 2,00 2,02 2,04

Normalized grating constant Ky1

Fig. 1. Forward- and backward-propagating modes' relativeoutputpower,P2 =IA 2(L)12/Po - I3(0)12/PoandP )1B2(0)12 /Po as functions of the normalized grating constantK/y, for the coupler with a propagation-constant differenceof Y2 - y1 = 0.02y'.

Ewa Weinert-Rqczko,

Vol. 11, No. 12/December 1994/J. Opt. Soc. Am. B 2343

cesses. The recovery time has recombination, TR, anddiffusion, rD, contributions: 1T = 1/rR + 1/7-D, withrD = A2/4,T2Da, where A is the grating-period length.The recombination time and the diffusion coefficient formotion along the wells obtained by Miller et al.19 areTR = 86 ns and Da = 16.2 cm2 /s, respectively.

The TEo-mode propagation constants and the couplingcoefficients of the considered coupler are 813 = 1.298 107 m- 1, 02 = 1.304 X 107 m-1 , G~Ii/An = 11.2 X 104 m 1 Iand G)/lAn = G/An = 8.2 104 m-1. The depen-dence of the relative output power in different ports on thegrating amplitude for a 10-mm-long coupler is presentedin Fig. 2. The calculations were performed for the grat-ing constant tuned to the difference of the propagation

(+) (-constants, K = 2 - y for P , K = 2y, for P1 , andK = + 2 for P-). The dashed curve in the figureplots the periodic power exchange between two forwardmodes. It shows that for the constant coupling length theoutput power appears in one or the other guide, depend-ing on the grating amplitude. The result of increasingthe external waves' intensities is similar to that obtainedin the externally controlled couplers,21 22 where control ofthe power can change the coupling coefficient and drivethe symmetric coupler from the parallel to the crossedstate. The amplitude of the grating necessary to switchthe output power to the second guide is 0.0019. Therefractive-index change for a wavelength of 1.55 ,um isdue mainly to the polarizability of the three carriers.2 3 24

The efficiency, neh, is -1.1 x 10-20 cm 3 , and switching ispossible with pump pulses of IAt near 4 x 10-6 J/cm2 .The grating period length A calculated for the forwardcoupling is -0.13 mm. The diffusion decay time of sucha grating, - 2.5 x 10-7 S, is much longer than the car-rier recombination time, and the grating can be gener-ated by long pulses or even by a cw beams of an intensitynear 50 W/cm2. The angles of incidence of the externalwaves are given by the relation sin F = A/2A (where Aodenotes the free-space wavelength of the external waves),which is satisfied by F = 0.18°.

Generation of the backward modes is described by thesolid curves; 95% switching is obtained with the gratingamplitudes An 0.002 and An 0.003 for the first andthe second waveguides, respectively. A higher value ofthe grating amplitude is necessary for the second wave-guide's mode generation because of the difference in thecoupling coefficients. The grating periods are very short(-0.24 m) in that case, and the diffusion washes outthe gratings in 10-12 s. The generation of such grat-ings would require subpicosecond external wave pulseswith a pulse separation much longer than the carrier re-combination time. The longitudinal wave vectors neces-sary to yield backward coupling, /3ex = 13 x 106 m, arelonger than the external beams' free-space wave vectors,ko = 2 7r/Aex = 7.6 x 106 m. Hence proper insertion ofthe external beams into the coupler's central layer re-quires a prism coupler with the incidence angles B givenby the relation sin B = 3 ex/npko, where np denotes theprism's refractive index.

The dependence of the relative output power in differ-ent ports on the grating constant is presented in Fig. 3.The difference of the propagation constants is A/8 = 6 104 ml, and the generation of backward modes in dif-

ferent waveguides occurs for different grating constantvalues. The same dependence for backward modes ina weakly mismatched coupler ( 2 = 10.305) is shown inFig. 4. The propagation constants' difference is A =0.23 x 104 m-1 in that case, and the generation of onebackward mode is accompanied by generation of anotherone. The output power is divided between the backwardmodes of different waveguides.

The parallel grating-assisted couplers exhibit charac-teristic sidelobes in the curve of the dependence of theoutput power on grating detuning.2"10 Similar sidelobescan be observed in the figures presenting output powerversus the grating constant for the couplers with grat-ings induced by external waves with constant amplitudes

0.a.

00)

0)IW

1,0

0,8

0,6

0,4

0,2

0,0

Grating amplitude An

Fig. 2. Relative output power of different modes versus the am-plitude of the grating in the case of exactly tuned grating con-stants K = 2yj for P1 , K = + 2 forP 2 ),and K = 2 - 1for P2

0.:3

.00)

a)W

1,0

0,8

0,6

0,4

0,2

0,0

Grating constant ( 107m 1 )

(a)

1,0

a)3 0,80

-~0,6

.3 0,4a)

0,2

0)Cr: 0,0

596 2,598 2,600

Grating constant ( 107m1 )

(b)Fig. 3. (a) Forward mode's relative output power in the secondwaveguide versus the grating constant. (b) Backward mode'srelative output power in both waveguides versus the gratingconstant.

Ewa Weinert-Rqczka

2344 J. Opt. Soc. Am. B/Vol. 11, No. 12/December 1994

03U)

a)

2,5776 2,5780 2,5784 2,5788

Grating constant (107m-1)

Fig. 4. Backward mode's relative output power in both wave-guides versus the grating constant for the coupler with lowasymmetry.

a).

0.

a)

U)

1,0

0,8

0,6

0,4

0,2

2,5950 2,5955 2,5960 2,5965 2,5'

Grating constant

(a)

1,0

3 0,8

5 0,60.3 0,4a)

in 0,2

Mn n

2,5

(1 0 7m-

1)

950

Grating constant (107m 1)

(b)Fig. 5. (a) Relative output power of the backward mode inthe first waveguide, P1 , versus the grating constant for theexternal waves with the amplitudes constant inside the couplingregion. (b) Relative output power of the backward mode in thefirst waveguide, Pi , versus K for the external beams with theintensity having a Gaussian shape with the half-width w = 0.5L.

(Figs. 3 and 4). An optically induced grating allows usto reduce the sidelobes by varying the grating amplitudealong the coupler, which can be accomplished by usingexternal beams with radii smaller then the length of thecoupler. The dependence of the backward-mode outputpower on the grating constant for external waves withconstant amplitudes inside the coupling region of lengthL is presented in Fig. 5(a). The same dependence forexternal beams with the same average intensity but aGaussian shape with the half-width w = 0.5L is shown inFig. 5(b). It is easy to see that when the external beamradius decreases the peak becomes wider and the side

lobes diminish as a result of the z dependence of the cou-pling coefficient.

CONCLUSIONS

It has been shown that an optically induced grating pro-vides coupling between weak guided modes in the asym-metric directional coupler. Parameters of the grating,such as the grating constant and the grating amplitude,depend on the external waves' intensities and directionsof propagation. As a result the output port and the ef-ficiency of coupling can be varied during operation of thedevice. Grating-assisted couplers exhibit a strong wave-length selectivity because of the dependence of the differ-ence of the propagation constants on the frequency of theguided modes. The optically induced grating with a vari-able grating constant permits a choice of the wavelength.The grating obtained by interference of beams with smalldiameters gives rise to a z dependence of the coupling co-efficients that causes a reduction of the sidelobes. Thepractical realization of the proposed coupler requires alarge nonlinearity, which can be obtained with resonantnonlinear mechanisms in semiconductors or semiconduc-tors multiple quantum wells.

An additional advantage of the optically induced grat-ing arises from the possibility of all-optical switching oramplitude modulation by the time and space dependenceof the amplitudes of the external waves.

APPENDIX AThe field at frequency w propagating in the coupler isdescribed by the Maxwell equations, written as

V X E = -icoAOH,

V X H = ieoeL E + iP. (Al)

where e(^) represents the dielectric function of a singleseparate waveguide and P = e0Ae(')E, with Ae(V) de-scribing a difference between the grating-assisted cou-pler and the isolated vth waveguide dielectric constantdistributions. The coupled-mode equations can be foundwith a Lorentz reciprocity theorem,

V(E1 x H2 * + E2 * x HI) = -isPlE 2 * + icoP2 *El. (A2)

If P1 = e0 Aff('')El and P2 = 0, then E1 and H1 describe thetotal field in the coupler and E2 and H2 a guided mode ofthe isolated vth waveguide.

Integrating over the coupler cross section and using thedivergence theorem lead to

ff a (E1 x H2 * + E2* x Hl)2dxdy

-iso |ff AC(") EIE2*dxdy. (A3)

The transverse parts of fields E1 and H1 (Ref. 25) are

PI

B l | l |

| l | l S l |

Ewa Weinert-Rqczka

Vol. 11, No. 12/December 1994/J. Opt. Soc. Am. B 2345

NElt = E t(X, y)

X [A,,(z, t)exp(-i,z) + B,2(z, t)exp(i,8,z)],N

Hit = HAt(x, y)

x [A,(z, t)exp(-i,81z) - Bm(z, t)exp(i,8z)]. (A4)

The longitudinal part of E is

N(AE = 7 6L') eEAZ(X, y)

,=1

X [A,(z, t)exp(-i,8,z) - B,(z, texp(i, 1z)]. (A5)

If we now choose E2 to be a forward-propagating mode,

E = E,(x, y)exp(-iBvz), H, = H,(x, y)exp(-i3,z),

substitute all fields into Eq. (A3), take AE(v) = eN +AeGf(z), and neglect fast-oscillating terms, then the fol-lowing equation will be obtained:

Y CAP ut [A,. (z, t) + i(,3 - ,)A, (z, t)]

X exp(i/3,z - iz)N

= -i A,,(z, t)Q(+) exp(i,/3^z - i,6,z)

N- i >3 A,(z, t)G(+)f (z)exp(i/,3z - i,81z)

,=1

N- i >3 B,(z, t)G(-)f(z)exp(i,83z + i,8z),

,a=l

where f(z) = exp(iKz) + exp(-iKz),

CAP = ff(E X H,* + E* Hl,)2dxdy,

(A6)

(A7)

Q(^) = aeOJJ LXEN [EtLEt* + e6E6 Ez,*]dxdy

(A8)+xo

AV= coeO ff AeG[Et,Et,* e1)e-lEzHEzy*]dxdy.

(A9)

Similarly, if we substitute E2 as a backward-propagatingmode of the vth waveguide,

E- = E(x, y)exp(if3vz),

H-v = H-v(x, y)exp(i/3,z),

with E_ = E and H- = -Ha, we obtain

> CAP a B,(z, t) + i(-3, + ,8,)B,(z, t)]

X exp(-i8,z + i,z)N

= Y B,(z, t)Q(+) exp(-i3,z + iz),u=1

N+ i Y BA(z, t)G(+)f(z)exp(-i3,z + iz)

N+ i Y A,(z, t)G,(-)f(z)exp(-i,8,z - i,z). (A10)

a=l1

Dividing Eq. (A6) by exp(i3,z), Eq. (A10) by exp(-i3Pvz),and making the substitutions

a,(z, t) = AA(z, t)exp(-i,8Az),

b,(z, t) = B,(z, t)exp(i,8,z),

we obtain the following set of equations:

(All)

N NY c,~4 a,(z, t) = -Yi {[iC,. + Ql+j) + f(z)G(j)]

X aA (z, t) + f (z)G b(z, t)},N N

>3 Ca b,(z, t) = i 3 {[p8C + Q(+) + f(z)G(+)]

x bA(z, t) + f (z)G(-)a(z, t)}. (A12)

The results can also be presented in a matrix form:

azca a = -i[BC + Q(+) + f(z)G(+)]a - if(z)G(-)b,

c a b i[BC + Q(+) + f(z)G(+)]b + if(z)G(-)a, (A13)az

where a(z, t) and b(z, t) are vectors described byEqs. (All) and B is a diagonal matrix of propagation con-stants Bqq = /q. The elements of matrices C, Q(+), G(+),and GH-) are defined in Eqs. (A7)-(A9).

ACKNOWLEDGMENTS

This paper was supported by project 108/E330 of theTechnical University of Szczecin and Committee for Sci-entific Research grant 202779101.

REFERENCES

1. C. Elachi and C. Yeh, "Frequency selective coupler for inte-grated optics systems," Opt. Commun. 7, 201-204 (1973).

2. P. Yeh and H. F. Taylor, "Contradirectional frequency-selective couplers for guided wave optics," Appl. Opt. 19,2848-2855 (1980).

3. R. R. Syms, "Optical directional coupler with a grating over-lay," Appl. Opt. 24, 717-726 (1985).

4. D. Marcuse, "Directional couplers made of nonidenticalasymmetric slabs. Part II: grating-assisted couplers," J.Lightwave Technol. LT-5, 268-273 (1987).

5. G. Griffel and A. Yariv, "Frequency response and tunabilityof grating-assisted couplers," IEEE J. Quantum Electron. 27,1115-1118 (1991).

6. G. Griffel, M. Itzkovich, and A. A. Hardy, "Coupled modeformulation for directional couplers with longitudinal per-turbation," IEEE J. Quantum Electron. 27, 985-994 (1991).

Ewa Weinert-Rqczka

2346 J. Opt. Soc. Am. B/Vol. 11, No. 12/December 1994

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18. See, for example, A. Yariv, Quantum Electronics (Wiley, NewYork, 1975).

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20. D. A. B. Miller, D. S. Chemla, D. J. Eilenberger, P. W.Smith, A. C. Gossard, and W. Wiegmann, "Degenerate four-wave mixing in room-temperature GaAs/GaAlAs multiplequantum well structures," Appl. Phys. Lett. 42, 925-927(1983).

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23. S. Y. Auyang and P. A. Wolff, "Free-carrier-induced opticalnonlinearities in semiconductors," J. Opt. Soc. Am. B 6,595-605 (1989).

24. D. F. Prelewitz and T. G. Brown, "Optical limiting and free-carrier dynamics in a periodic semiconductor waveguide," J.Opt. Soc. Am. B 11,304-312 (1994).

25. H. Kogelnik, "Theory of dielectric waveguides," in IntegratedOptics, T. Tamir, ed. (Springer-Verlag, Berlin, 1979).

Ewa Weinert-Rqczka