Radiation conditions, limiting absorption principle, and general relations in open waveguide scattering

Abstract

The purpose of this paper is to give solid ground to rigorous studies of open-waveguide-related scattering problems. In the core of the difficulties associated with them there lies the presence of infinite but penetrable boundaries. This calls for a revision of the condition of radiation, and to the necessity of justifying the limiting absorption principle. The analysis is based on the Fourier-transform approach and study of the analytic properties of transform functions with respect to a parameter. Except for simple 2-D geometries, regularized integral equations and operator theory theorems are used. The result differs from the free-space expression and involves the guided natural modes spectrum. It reveals also that the direction of power flow associated with each particular mode is to be taken into account. The obtained expressions validate the intuitively clear equations of power conservation and reciprocity. A numerical example is given and extension to more complicated open waveguides is discussed.