Application of the Banach Fixed-Point Theorem to the Scattering Problem at a Nonlinear Three-Layer Structure with Absorption
1 Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, Oulu 90014, Finland
2 Department of Physics, University of Osnabrück, Barbarastraße 7, D-49069 Osnabrück, Germany
Fixed Point Theory and Applications 2010, 2010:439682 doi:10.1155/2010/439682Published: 30 May 2010
A method based on the Banach fixed-point theorem is proposed for obtaining certain solutions (TE-polarized electromagnetic waves) of the Helmholtz equation describing the reflection and transmission of a plane monochromatic wave at a nonlinear lossy dielectric film situated between two lossless linear semiinfinite media. All three media are assumed to be nonmagnetic and isotropic. The permittivity of the film is modelled by a continuously differentiable function of the transverse coordinate with a saturating Kerr nonlinearity. It is shown that the solution of the Helmholtz equation exists in form of a uniformly convergent sequence of iterations of the equivalent Volterra integral equation. Numerical results are presented.