Outgoing wave conditions in photonic crystals and transmission properties at interfaces

Abstract

We analyze the propagation of waves in unbounded photonic crystals. Waves are described by a Helmholtz equation with x-dependent coefficients, the scattering problem must be completed with a radiation condition at infinity. We develop an outgoing wave condition with the help of a Bloch wave expansion. Our radiation condition admits a uniqueness result, formulated in terms of the Bloch measure of solutions. We use the new radiation condition to analyze the transmission problem where, at fixed frequency, a wave hits the interface between free space and a photonic crystal. We show that the vertical wave number of the incident wave is a conserved quantity. Together with the frequency condition for the transmitted wave, this condition leads (for appropriate photonic crystals) to the effect of negative refraction at the interface.