Parabolic equation method and Fresnel approximation in Weinstein's problems

A problem of diffraction of a high frequency plane wave by a grating, consisting of absorbing screens is studied. Difficulties of a correct mathematical formulation of the problem are addressed. It is shown how this problem is connected with the classical Weinstein's problem of scattering by an open end of a planar waveguide. All results are derived by two different approaches: by the parabolic equation approach and by the method of Fresnel integrals. The equivalence of these approaches allows one to use Fresnel integrals for rigorous reasoning keeping the parabolic equation method for clear physical understanding of the results obtained.