Diffraction of a~high-frequency wave by a~grating with a~complicated period in the case of grazing incidence

A. V. Shanin

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 42, Tome 409 (2012), pp. 212-239

Résumé

A 2D problem of diffraction of a plane wave on a branched surface is studied. The configuration of branch points of the surface is periodic; these branch points play the role if a diffraction grating. The period of the grating is composed of two branch points. The incident wave travels at a grazing incidence angle. The consideration is held in the parabolic approximation; the axis of the parabolic coordinates is the edge of the grating. Edge Green's functions of the problem, i.e. the the fields generated by point sources placed near the branch points, are introduced. The embedding formula is proven. It expresses the coefficients of generation of diffraction orders in terms of the directivities of the edge Green's functions. A spetral equation is derived for the directivities of the edge Green's functions. This is an ordinary differential eqiation, the coefficient of which is unknown. Finally, for finding of this coefficient, an OE-equation is derived.