Integral Equations Approach to TM-Electromagnetic Waves Guided by a (Linear/Nonlinear) Dielectric Film with a Spatially Varying Permittivity

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1 PIERS ONLINE, VOL. 5, NO. 8, Integral Equations Approach to TM-Electromagnetic Waves Guided by a (Linear/Nonlinear) Dielectric Film with a Spatially Varying Permittivity V. S. Serov 1, K. A. Yuskaeva 2, and H. W. Schürmann 2 1 Department of Mathematical Sciences, University of Oulu, P. O. Box 3, FIN-914, Finland 2 Department of Physics, University of Osnabrueck, D-4969 Osnabrueck, Germany Abstract A method is proposed for obtaining solutions of Maxwell s equations describing guided TM-waves in a three-layer structure consisting of a nonlinear dielectric film situated between two linear semi-infinite media. All media are assumed to be lossless and nonmagnetic.the linear part of the permittivity is modelled by a continuously differentiable real valued function of the transverse coordinate. The problem is reduced to a system of two integral equations. On the basis of the Banach fixed-point theorem, it is shown that the solution exists in form of a uniformly convergent sequence of iterations. The dispersion relation is presented. Some results are evaluated numerically. 1. INTRODUCTION The exact analytical solution of Maxwell s equations for a plane three-layer structure consisting of a central film between two half-spaces with a permittivity of the film depending on the (transverse) coordinate as well as on the field intensity is not known in the literature (to our knowledge). In particular, solutions representing guided waves are unknown. Compared with the TE-case the TM-case is more complicated leading to a system of two (nonlinear) differential equations that cannot be reduced to an exact differential equation if the permittivity of the film is a function of the coordinate. Thus, it seems reasonable to consider an integral equation approach [1] to this problem. 2. STATEMENT OF THE PROBLEM We consider electromagnetic waves guided by a homogeneous, anisotropic, nonmagnetic layer filled with a Kerr-type nonlinear dielectric medium situated between two linear semi-infinite half-spaces x < and x > h consisting of isotropic, nonmagnetic media without sources and having constant permittivities 1 and 3, respectively. Fig. 1 shows the geometry of the problem. The (real) electrical field satisfies Maxwell s equations E(x, y, z, t) = E + (x, y, z) cos ωt + E (x, y, z) sin ωt (1) roth = iωe (2) rote = iωµh, (3) E(x, y, z) = E + (x, y, z) + ie (x, y, z) (4) x z h 3, cladding 2, film 1, substrate y Figure 1: Geometry of the problem.

2 PIERS ONLINE, VOL. 5, NO. 8, is a complex amplitude. The permittivity inside the film is described by the Kerr law 2 = 21 + f(x) + a E x 2 + b E z 2, (5) 22 + f(x) + b E x 2 + a E z 2 21, 22, a, b are (real) constants and f(x) is a continuously differentiable real-valued function. It can be shown [2] that Maxwell s equations can be reduced to the system d2 Z 2 + γ dx = zz dz + γx = (6) x γ X. Here, the components of the electric field, E x, E z, are written as E x = ix(x)e iγz, E z = Z(x)e iγz and x, γ, have been normalized appropriately [2]. The problem is to find real solutions to (6) X, Z in whole space with real γ and 1, x < x = 21 + f(x) + ax 2 + bz 2, < x < h 3, x > h (7) 1, x < z = 22 + f(x) + bx 2 + az 2, < x < h 3, x > h. 3. INTEGRAL EQUATIONS AND ITERATIONS Separating the constant part 21, 22 in (7) system (6) can be rewritten as d 2 Z(x) 2 dx(x) + 22 Z(x) = γ dx(x) zz(x) = γ d2 Z(x) 2 and < x < h. Combination of (6) and (8) yields d 2 Z(x) 2 + ( γ 2) + d ( xx(x)) γ 2 21 { x = f(x) + ax 2 + bz 2, (8) z = f(x) + bx 2 + az 2, (9) = γ2 21 z Z(x) γ d ( xx(x)). (1) For further considerations we assume that γ 2 < 21. Equation (1) can be transformed to the integral equation applying the second Green s formula Z(x) = Z (x) ( γ 2) G(x, y) z (y)z(y)dy + γ G(x, y) x (y)x(y)dy. (11) Z (x) is given by Z (x) = E z () cosκx + E(h) z Z() cosκh sinκx. (12) sinκh Here E () z = Z() is prescribed and κ is given by κ = 22( 21 γ 2 ) 21. E (h) z = Z(h) must be determined.

3 PIERS ONLINE, VOL. 5, NO. 8, The Green s function G(x, y), corresponding to (1) is defined by [3] sinκx sinκ(y h), x y G(x, y) = κ sinkh, (13) sinκy sinκ(x h), y x κ sinkh κh πl, l = 1, 2,.... Integration by parts in (11) yields Z(x) = Z (x) 21 γ 2 21 G(x, y) z (y)z(y)dy + γ 21 G(x, y) x (y)x(y)dy. (14) y Using the second equation in (6) and (14) leads to the integral equation for the x-component of the electric field γ 21 dz (x) X(x) = + γ Z h G(x, y) γ 2 Z h 2 G(x, y) z(y)z(y)dy + x(y)x(y)dy. (15) x (γ 2 21) x x x (γ 2 21) y x The system (14) and (15) can be written in matrix form as follows v = v(x) = v (x) + ( ) Z (x), v(x) = and matrix M is given by 21 γ 2 G(x, y) z (y) M = 21 γ G(x, y) z (y) x (x) x M( v)(y) v(y)dy + L( v)(x), (16) ( ) Z(x), L( v)(x) = X(x) γ 21 x (γ 2 21 ) dz (x) (17) γ G(x, y) x (y) 21 y γ 2 2 G(x, y) x (x)(γ 2 21 ) y x. (18) x(y) We consider Equation (16) in the Banach vector space (C[, h]) of continuous complex-valued functions on the segment [, h] and solve it by iterations using the Banach fixed-point theorem. The iteration sequence v j (x) = v (x) + M ( v j 1 ) (y) v j 1 (y)dy + L ( v j 1 ) (x), j = 1, 2,... (19) yields the exact solution v (as the limit of v j ) of the Equation (16) if the conditions of the Banach fixed-point theorem are satisfied, namely I 2 4(1 f b ) 3 27a b, 1 f b >, (2) < R min R R max, (21) R min, R max are two positive roots of the polynomial a b R 3 (1 f b )R + I =, and < q < 1, (22) q = 3a b R 2 + κh2 γ a 2 b 2 R 4 (b sin κh ( 21 f ) a R 2 γ κh 2 ) sin κh ( 21 f ) 2 f + 2 γ 21a R 21 γ 2 ( 21 f ) 2 f = max f(x), a = max(a; b), { κh h 2 b = 2 sin κh max 21 γ 2 + h γ h γ γ 2 ; f γ 2 ( 21 f ) dz (x), (23) (24) (25) }, (26)

4 and The a priory error estimate [4] for v reads PIERS ONLINE, VOL. 5, NO. 8, γ 21 I = Z (x) + dz (x) 21 γ 2 ( 21 f ). (27) v v j qj 1 q v 1 v. (28) Figure 2: Plot of the dispersion relation for a =, b =, Z() = 1, m =.9, n =.6 (solid curves) compared with the linear case (dashed curves), Z() = 1. Figure 3: Plot of the dispersion relation for the pure Kerr-case (solid curves) compared with the exact (dashed) curves; a =.3, b =.2, m =, Z() = 1. Figure 4: Plot of the dispersion relation for a =.3, b =.2, m =.9, n =.6, Z() = 1 (solid curves) compared with the one for a =, b =, m =.9, n =.6, Z() = 1 (dashed curves).

5 PIERS ONLINE, VOL. 5, NO. 8, DISPERSION RELATION The dispersion relation can be obtained by combining (14), (15) with the boundary conditions. Z(x) and γx(x) dz(x) must be continuous at the boundaries x = and x = h leading to γ 1 x (X( +, γ, Z()), Z()) X ( +, γ, Z()) = Z(), (29) γ 2 1 γ 3 x (X(h, γ, Z()), Z (h, γ, Z())) X (h, γ, Z()) = Z(h, γ, Z()). (3) γ 2 3 Rewriting the left-hand side of (29) by using (15) one obtains, Z(h, Z, X) = Z lin (h)+ 21 γ 2 21 κ sinκ(y h) z (y)z(y)dy γ 21 cosκ(y h) x (y)x(y)dy, (31) Z lin (h) = Z()(cosκh + β sinκh) is the solution of the problem in the linear case (f(x) =, a = b = ) with β = 21 γ (γ 2 1). Equations (14), (15), (29), (3) are a representation of dispersion relation. It relates h, γ, Z() to the parameters a, b, 1, 21, 22, 3 and function f(x). Evaluation of the first iteration in (19) yields results shown in Figs The parameters are 1 = 4 (in substrate), 3 = 1 (in cladding), 21 = 16, 22 = 9 (in the film). We assume f(x) = m cos 2 nx. 5. CONCLUSION For the TM-case an iterative approach is presented to solve Maxwell s equations for a three layer structure with a permittivity given by (7). The solution Z(x), X(x) has been expressed as a uniformly convergent sequence of iterations of the corresponding vector integral equation (subject to the conditions of the Banach fixed-point theorem). For the problem in question, the Banach conditions are derived and used to estimate the quality of the approximation. The exact dispersion relation is presented and evaluated numerically in first approximation. The approach seems applicable to permittivities more general as (7). ACKNOWLEDGMENT One of us (K. A. Y.) gratefully acknowledges the support by the German Science Foundation (DFG) (Graduate College 695). REFERENCES 1. Serov, V. S., H. W. Schürmann, and E. Svetogorova, Integral equation approach to reflection and transmission of a plane TE-wave at a (linear/nonlinear) dielectric film with spatially varying permittivity, Journal of Physics A: Mathematical and General, Vol. 37, No. 1, , Yuskaeva, K. A., V. S. Serov, and H. W. Schürmann, TM-electromagnetic guided waves in a (Kerr-) nonlinear three-layer structure, Progress In Electromagnetics Research Symposium, Moscow, Russia, August 18 21, Stakgold, I., Green s Functions and Boundary Value Problems, 2nd Edition, , Springer- Verlag, New York, Zeidler, E., Applied Functional Analysis [Part1]: Applications to Mathematical Physics, 18 26, Springer-Verlag, New York, 1995.