A scattering problem for a local pertubation of an open periodic waveguide

In this paper we consider the propagation of waves in an open waveguide in $\mathbb{R}^2$ where the index of refraction is a local perturbation of a function which 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 $h_0 > 0$. Motivated by the limiting absorption principle (proven in [17] for the case of an open waveguide in the half space $\mathbb{R} \times (0, \infty))$ we formulate a radiation condition which allows the existence of propagating modes and prove uniqueness, existence, and stability of a solution under the assumption that no bound states exist. In the second part we determine the order of decay of the radiating part of the solution in the direction of the layer and in the direction orthogonal to it. Finally, we show that it satisfies the classical Sommerfeld radiation condition and allows the definition of a far field pattern.