The Schrödinger approximation for the Helmholtz equation if the refractive index is a step function

Abstract

Time-harmonic electromagnetic waves in a medium are described by the Helmholtz equation Δu+ω2nu=0 for (x,y,z)∈R3, where n=n(x,y,z) is the refractive index. A Schrödinger equation can be derived through a multiple scaling ansatz for the evolution of such waves along the z-axis in case of a z-independent refractive index. It is the purpose of this paper to prove the validity of this formal approximation in case that the refractive index is a step function n=n(x). The proof uses mode filters and a spectral representation of the problem. The major difficulty comes from the fact that the Helmholtz equation is an ill-posed evolutionary system.