  The purpose of this paper is to investigate the scattering by a nonlinear
crystal whose depth is about the wavelength of the impinging field. More
precisely, an infinite nonlinear slab is illuminated by an incident field which
is the sum of three plane waves of the same frequency, but with different
propagation vectors and amplitudes, in such a way that the resulting incident
field is periodic. Moreover, the height of the slab is of the same order of the
wavelength, and therefore the so-called slowly varying envelope approximation
cannot be used. In our approach we take into account some retroactions of the
scattered fields between them (for instance, we do not use the nondepletion of
the pump beam). As a result, a system of coupled nonlinear partial differential
equations has to be solved. To do this, the finite element method (FEM)
associated with perfectly matched layers is well suited. Nevertheless, when
using the FEM, the sources have to be located in the meshed area, which is of
course impossible when dealing with plane waves. To get round this difficulty,
the real incident field is simulated by a virtual field emitted by an
appropriate antenna located in the meshed domain and lying above the obstacle
(here the slab).
