The junction of an open and a closed waveguide for periodic media

Abstract. In this paper we investigate the junction of a closed waveguide with an open waveguide where the refractive indices of both waveguides are periodic with respect to the axis of the waveguides. We allow also that the refractive index is locally perturbed. We formulate a proper radiation condition for this problem which follows from a limiting absorption principle and decribes the bahavior of the solution along the axis of the waveguides (which we take to be the $x_1$−axis) and also, for the open waveguide, normal to it. Away from the junction the solution consists of linear combinations of propagating modes travelling to the left or right, respectively, and a radiating parts which decays (exponentially fast in the closed waveguide and of order $\mathcal{O}(1/x_1^{3/2})$ in the open waveguide) along the $x_1$−axis. We show well-posedeness of the problem by introducing Dirichlet-to-Neumann operators and reducing the problem to a bounded region containing the junction of the waveguides. The Fredholm property of the problem is shown. By intruducing the fluxes of this problem along the axis we show (partial) uniqueness.