PIERS ONLINE, VOL. 3, NO. 6, 2007 759 Diffraction by a Kerr-type Nonlinear Dielectric Layer Yu. V. Shestopalov 1 and V. V. Yatsyk 2 1 Karlstad University, Karlstad, Sweden 2 IRE National Academy of Science of Ukraine, Ukraine Abstract— The diffraction of a plane wave by a transversely inhomogeneous isotropic nonmag- netic linearly polarized dielectric layer filled with a Kerr-type nonlinear medium is considered. The diffraction problem is reduced to a cubic-nonlinear integral equation (IE) of the second kind and to a system of nonlinear operator equations of the second kind solved using iterations. Sufficient conditions of the IE unique solvability are obtained using the contraction principle. DOI: 10.2529/PIERS061009032848 1. STATEMENT OF THE PROBLEM OF DIFFRACTION BY A WEAKLY NONLINEAR LAYER Denote by  E ( r) ≡  E ( r, ω) and  H ( r) ≡  H ( r, ω) the complex amplitudes of the stationary electromagnetic field; the time dependence is exp (−iω t). Consider the problem of diffraction of a plane stationary electromagnetic wave by a nonmagnetic, isotropic, transversely inhomoge- neous, ε (L) (z )= ε (L) xx (z ), and linearly polarized,  E ( r)=(E x (y,z ) , 0, 0),  H ( r) = (0,H y ,H z ) (E-polarization), with vector of polarization  P (NL) =  P (NL) x , 0, 0  , and Kerr-like weakly non- linearity,   ε (NL)   <<   ε (L)   layered dielectric structure, see Fig. ?? and [1, 2]. Here P (NL) x = 3 4 χ (3) xxxx |E x | 2 E x , ε = ε (L) + ε (NL) at |z |≤ 2πδ is the permittivity of the nonlinear layer, ε (L) = 1+4πχ (1) xx (z ), ε (NL) =3πχ (3) xxxx (z ) |E x | 2 = α (z ) |E x | 2 , α (z )=3πχ (3) xxxx (z ), and χ (1) xx (z ) and χ (3) xxxx (z ) are components of the susceptibility tensor. Figure 1: Weakly nonlinear dielectric layered structure. One can show, similarly to [3, 4], that the total field E x (y,z )= E inc x (y,z )+ E scat x (y,z ) of diffraction of the plane wave E inc x (y,z )= a inc · exp {i [φy − Γ · (z − 2 πδ)] }, z> 2 πδ, by the weakly nonlinear dielectric layer (Fig. ??) satisfies the equation ∇ 2 ·  E + ω 2 c 2 · ε (L) (z ) ·  E + 4πω 2 c 2 ·  P (NL) ≡  ∇ 2 + κ 2 · ε  z, α (z ) , |E x | 2  · E x (y,z )=0, (1) and the following generalized boundary conditions: continuity of the tangential field components on the layer boundary, the spatial inhomogeneity condition E x (y,z )= U (z ) · exp (iφ y) , (2) and the radiation condition for the scattered field E scat x (y,z )=  a scat b scat  · exp (i · (φy ± Γ · (z ∓ 2 πδ))) , z > < ± 2 π δ. (3)