Perturbation analysis of plane-wave transmission through a dielectric slab with Kerr-type nonlinearity

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. 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.