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 when ε tends to zero. 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 approximate the incident plane wave by an
incident point source and let the location of the source tend to