Multiple-Scale Analysis of Plane Wave Refraction at a Dielectric Slab with Kerr-Type Nonlinearity

TLDR: In this paper, multiple-scale analysis is employed for the analysis of plane wave refraction at a nonlinear slab, and it is demonstrated that the perturbation method will lead to a nonuniformly valid approximation to the solution of the nonlinear wave equation.

Abstract: Multiple-scale analysis is employed for the analysis of plane wave refraction at a nonlinear slab. It will be demonstrated that the perturbation method will lead to a nonuniformly valid approximation to the solution of the nonlinear wave equation. To construct a uniformly valid approximation, we will exploit multiple- scale analysis. Using this method, we will derive the zeroth- order approximation to the solution of the nonlinear wave equation analytically. This approximate solution clearly shows the effects of self-phase modulation (SPM) and cross-phase modulation (XPM) on plane wave refraction at the nonlinear slab. In fact, the obtained zeroth-order approximation is very accurate and there is not any need for derivation of higher-order approximations. As will be shown, the proposed method can be generalized to the rigorous study of nonlinear wave propagation in one-dimensional photonic band-gap structures.