68 OPTICS LETTERS / Vol. 13, No. 1 / January 1988

Dispersion relations and power-flow expressions for nonlinearmodes guided by a multilayer symmetric dielectric structure

Ajit Kumar*

Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi-110016, India

Received June 22, 1987; accepted October 18, 1987

The dispersion relations and the power-flow expressions are derived for nonlinear modes guided by a multilayersymmetric dielectric structure.

Theoretical as well as experimental studies of layereddielectric structures1 -I0 are important because of theirapplications in integrated optics,9"10 optical hystere-sis,10 signal processing, etc. Following the earlierwork in this connection, I present here the generaldispersion relations and power-flow expressions fornonlinear TM-polarized modes guided by a symmetricdielectric structure with an arbitrary number of layers.Basic equations and the forms of solutions are alsogiven.

Consider an N-layer symmetric dielectric structureconsisting of a series of linear and nonlinear filmsplaced one over the other in the (x, y) plane. Thestructure is bounded on the free sides by the nonlinearmedium. The films have finite thickness along the 2axis. Let the thickness be fl and 0 for the linear andnonlinear films, respectively. The coordinate systemis chosen such that the x axis coincides with the longi-tudinal axis of a nonlinear film, which we take as thecentral film. The linear films have the dielectric con-stant el, while the nonlinear films are characterized bythe diagonal dielectric tensor7

exx = eyy = eo, EZZ = eO + AIE21, EZ -2 E, (1)

where a characterizes the intensity of the transversenonlinearity. Note that, following Seaton et al.,7 wehave assumed that IEJI2 << IEZJ2, and hence only E,appears in the above expressions.

We describe the films on both sides of the centralnonlinear film as follows. The linear films above thecentral films are indicated by a [a = a1 , a 2 , .. ., a(n/2), nbeing the total number of linear films] while those thatare below it have the designations b [b = bl, b2, . . ., b(n/2)]Similarly, the nonlinear films above and below thecentral film have the designations c [c = cl, c2, ... ., C(m/2)m being the total number of nonlinear films] and d[d = di, d2, ... , d(m/2)I, respectively. Thus m = n + 1and N = 2n + 1 (including the ends).

For TM-polarized electromagnetic waves travelingin the x direction the independent field componentsare

E1,3 = 61,3 (z) exp[-i(ct -kx)],

H2 = 12(z) exp[-i(wt - kx)], (2)

where 1, 2, and 3 stand for the x, y, and z components,respectively. In the first approximation in a, usingEqs. (1), the basic set of Maxwell's equations can nowbe reduced to

d2 _ (ab) (k2- 2)(ab) = o

dz 2

d22 cH { 2 A 4 (c~dT 2 (c,d) = O.dz2 Eoj0 -oq [ko2 d)

(3)

where q2 = k2- eoko2. For a > 0 (self-focusing medi-

um) the solutions to Eqs. (3) for the modes that areconfined near the surface of the films and tend to zerofor Zi -k - can be written as

W 2a(Z) = a cosh[kl(z -z)]

w b(Z) = b cosh[kl(z -ZA,

Ab cos[k 2 (z -Zb,

W2°( = 2/af sech[q(z -z

14c(z) = 2/af sech[q(z -zc)]

y12 d(Z) = \/Taf sech[q(z -Zd)

where the subscript 0 stands for thecentral film and

n > n1

n < n,

n > n,n < n,

(4)

(5)

(6)

(7)

(8)

solution in the

n, = J , n = k/ko, ko = w/c,

k, = (k2-elk02)1/2, k2 = (ko 2e-k 2 )/ 2,

f = eOkOq/k2.

By writing down the matching conditions for El andD3 (which are determined from the equation" 3 kW2 =-kOD3) across all the interfaces and doing the algebraas in Ref. 3, we get 2n equations for the determinationof Za and Zb. For symmetric modes they have uniquesolutions:

Za = Zb, ZO = 0, Zc = Zd.

0146-9592/88/010068-03$2.00/0 © 1988, Optical Society of America

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January 1988 / Vol. 13, No. 1 / OPTICS LETTERS 69

In this case the dispersion relations for n > nj can bewritten as

tanh[q(IO + J13 + Zn)] = -bl tanh[k 1(IO + Je + Za)],

(10)

while for n < nj they have the form

tanh[q(IO + JO + Zc)] = -b2 tan[k2(10 + J3 + Za)]

(11)

where b1 = eokj1qcj, b2 = eok2/qcl, and I and J are the

The time-averaged power flow in the x direction perunit width in the 5 direction can be written asl1 3

8=7rK 2 [(z)dz

co (ko) \ 2(12)

The corresponding integrations yield the power-flowexpression for the symmetric mode as

Ps =PO-q 1 _ f2 (k 2) 1- (1 + rl - r12)jI(1 - r) {1 e 2 (k) [1- r2(I,])}

+ Po 4f2 cosh-2 [kl(10 + Jf + Za)] 11 - r(2)](IJ)~~~~~~~~~U

fkO + k (sinh[2k 1(I0 + JeO + Za)]

12 4k 1

- sinh{2kj[I0 + (J- 1)0 + Za}) - f2 /k 2 cosh 2[k1 (IO + JO3 + za)] [1 - r2,X sinhl3k1[10(ih2l JB+Z)-i2 + (J- + Z+]}

x [3kO3 + k (sinh[2k,(IO + JOf + za,)] - sinh{2kj[I0 + (Jl - 1)13 + zj)[-8 4k1

+ I sinh[4k,(IO + JB + Za)] - 8 sinh[4k1(IO + (J -8 a

coefficients before 0 and 13 at each interface in any onehalf of the z axis. In general,

UX J) = il Ji), (IUil Ji + 1), (Ij + 1, Jj + 1), . . . , (If Jf),(lla)

where the final values If and Jf are related to the initialvalues Ii (=1/2) and Ji (=0) through

If=Ii+ (n-2)/2, Jf = Ji + n/2. (llb)

Note that in Eqs. (10) and (11) a changes only if Ichanges and c changes only if J changes. For exam-ple, for N = 9 Eq. (10) will give (for n > nj) thefollowing dispersion relations:

tanh q2 = -bl tanh[ki + Za,)J

tanh[ (q + 1 + Z,)] = -bl tanh[ k,( + + Zal

tanh q( 2 + cj + zj, = -bl tanh[k,( 13 + + Za 2 )]

tanhq( 2 + 21 + Z 2)] = -bl tan k, 3( + 20 + Za2)1 (l0a)

Now if we replace the hyperbolic cosine and the cosinewith a hyperbolic sine and a sine, respectively, in Eqs.(4) and (5) and do the same algebra, we get the disper-sion relations for the antisymmetric mode given againby Eqs. (10) and (11), in which we have to replace thehyperbolic tangent and the tangent on the right-handside by a hyperbolic cotangent and a cotangent, re-spectively.

where

Po = c/8rko4 a,

r = -bl tanh[k1(IfO + Jfe + za)]l af = an/2=

(14)

r(Ij) = -bl tanh[k1(IO + JO + Za)]

and we have used

Aa2 Ab 2= 2f cosh 2[k1 (IO + JO + Za)] [1 -r2

a(15)

where Aa and Ab are the amplitudes of the magneticfield in the linear films. Note that in the above ex-pression we use a plus before Z_* if I changes and aminus if Jchanges. Further, E denotes that the sumis taken over all pairs (I, J) except (If, Jf). Thus for N= 9 the sum * contains three terms with (I, J) = (1/2,0), (1/2, 1), and (3/2, 1), while for N = 5 it contains onlyone term with (I, J) = (1/2, 0). Finally, ** denotesthat the sum is taken over all distinct pairs (I, J) (notwo pairs contain the same value of I and J) except (Ih,Ji). Thus for N = 9 the sum "'** has only two termswith (I, J) = (1/2, 1) and (3/2, 2), while for N = 5 it hasonly one term with (I, J) = (1/2, 1).

Similarly, one can obtain the power-flow expressionfor the antisymmetric mode by performing the samereplacements as for the dispersion relations.

Finally, we note that the power-flow expression forthe N = 5 case3 had some incorrect coefficients. Thecorrect expression is given by Eq. (13).

r(Ijs)

(13)1)0 + Za)]

70 OPTICS LETTERS / Vol. 13, No. 1 / January 1988

* Present address, Institute for Theoretical PhysicsI, University of Dusseldorf, 4000 Dutsseldorf 1, FederalRepublic of Germany.

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