In this paper we consider the propagation of waves in an open waveguide in the half space R + 2 = { x ∈ R 2 : x 2 > 0 } under Dirichlet- or Neumann boundary condition for x 2 = 0 . The index of refraction n= n( x) is periodic along the axis of the waveguide (which we choose to be the x 1 − axis) and equal to one for x 2 > h 0 for some h 0 > 0 . We show first existence and uniqueness of a solution for the absorbing case, i.e. where the index of refraction is given by n( x)+ iεq( x) with ε>0 and some function q which is periodic with respect to x 1 , vanishes for x 2 > h 0 , and satisfies the angular spectral representation radiation condition. Then we prove convergence of the solution as ε tends to zero. We show that the limit solves the source problem for n( x) and satisfies a radiation condition which depends, first, on the choice of the absorption function q and, second, whether or not a cut-off value is critical with a non-evanescent mode. We show that the existence of non-evanescent modes is responsible for a slower decay of the solution along the axis of the waveguide. MSC: 35J05

We study the Helmholtz equation with periodic coefficients in a closed wave-guide. A functional analytic approach is used to formulate and to solve the radiation problem in a self-contained exposition. In this context, we simplify the non-degeneracy assumption on the frequency. Limiting absorption principles (LAPs) are studied and the radiation condition corresponding to the chosen LAP is derived; we include an example to show different LAPs lead, in general, to different solutions of the radiation problem. Finally, we characterize the set of all bounded solutions to the homogeneous problem.

We consider the scattering of a plane wave by a locally perturbed periodic (with respect to x_1) medium. If there is no perturbation it is usually assumed that the scattered wave is quasi-periodic with the same parameter as the incident plane wave. As it is well known, one can show existence under this condition but not necessarily uniqueness. Uniqueness fails for certain incident directions (if the wavenumber is kept fixed), and it is not clear which additional condition has to be assumed in this case. In this paper we will analyze three concepts. For the Limiting Absorption Principle (LAP) we replace the refractive index n=n(x) by n(x)+iε in a layer of finite width and consider the limiting case when ε tends to zero. This will give an unsatisfactory condition. In a second approach we require continuity of the field with respect to the incident direction. This will give the same satisfactory condition as the third approach where we approximate the incident plane wave by an incident point source and let the location of the source tend to infinity.

In this paper we consider the propagation of waves in an open waveguide in R^2 where the index of refraction is a local perturbation of a function which is periodic along the axis of the waveguide and equal to one outside a strip of finite width. Motivated by the limiting absorption principle (proven in an ealier paper by the author for the case of an open waveguide in the half space) we formulate a radiation condition which allows the existence of propagating modes and prove uniqueness, existence, and stability of a solution. In the last part we investigate the decay properties of the radiating part in the direction of periodicity and orthogonal to it.