The trouble with the limiting absorption principle for periodic waveguides

In this paper we consider the open waveguide source problem $\Delta u+k^2\tilde{n}u=-f$ in $\mathbb{R}^2_+ := \{x \in \mathbb{R}^2 : x_2 > 0\}$ with homogeneous Neumann conditions for $x_2 = 0$. Here, $k > 0$ is the wavenumber and $\tilde{n}$ a local perturbation of a refractive index $n$ which is periodic with respect to $x_1$ and equals to one for $x_2 > h_0$. We derive radiation conditions by replacing $k^2$ or $\tilde{n}$ or the boundary condition by $k^2 + i\epsilon$ or $\tilde{n} + i\epsilon p$ (for some function $p$, periodic with respect to $x_1$ ) or a boundary condition of impedance type, respectively, and let $\epsilon$ tend to zero. We show that in general we derive different radiation conditions. They coincide only in the case that the spaces of propagating modes are one-dimensional.