A limiting absorption principle for the scattering by a periodic layer in the case of a cut-off value

In this paper we consider the propagation of waves in an open waveguide in the half space $R^2_+ = \{x\in\mathbb{R}^2 : x_2 > 0\}$ 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 $n(x) + i\varepsilon q(x)$ with $\varepsilon > 0$ 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 $\varepsilon$ 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.