On the scattering of a plane wave by a perturbed open periodic waveguide

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 $𝜀 → 0$. 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 approxi- mate the incident plane wave by an incident point source and let the location of the source tend to infinity.