Asymptotic Behavior of Solutions to Open Periodic Waveguide Problems

In this paper we consider the propagation of waves in an open waveguide in the half space [Formula: see text] under Dirichlet- or Neumann boundary condition for [Formula: see text]. The index of refraction [Formula: see text] is periodic along the axis of the waveguide (which we choose to be the [Formula: see text]axis) and equal to one for [Formula: see text] for some [Formula: see text]. Based on a limiting absorption principle, derived in a former paper, we formulate a radiation condition, prove existence and uniqueness of a solution and describe explicitely the asymptotic behavior along the axis of the waveguide. This behavior depends crucially on the existence of nonevanescent (normal to the axis of the waveguide) modes.