In the shadow theory, a new description and a physical mean at a low grazing limit of incidence on gratings in the two dimensional scattering problem have been discussed. In this paper, by applying the shadow theory to the three dimensional problem of multilayered dielectric periodic gratings, we formulate the oblique primary excitation and introduce the scattering factors through our analytical method, by use of the matrix eigenvalues. In terms of the scattering factors, the diffraction efficiencies are defined for propagating and evanescent waves with linearly and circularly polarized incident waves. Numerical examples show that when an incident angle becomes low grazing, only specular reflection occurs with the reflection coefficient -1, regardless of the incident polarization. It is newly found that in a circularly polarized incidence case, the same circularly polarized wave as the incident wave is specularly reflected at a low grazing limit.

The relative permittivity and permeability are discontinuous at the grating profile, and the electric and magnetic flux densities are continuous. As for the method of analysis for scattering waves by surface relief gratings placed in conical mounting, the spatial harmonic expansion approach of the flux densities are formulated in detail and the validity of the approach is shown numerically. The present method is effective for uniform regions such as air and substrate in addition to grating layer. The matrix formulations are introduced by using numerical calculations of the matrix eigenvalue problem in the grating region and analytical solutions separated for TE and TM waves in the uniform region are described. Some numerical examples for linearly and circularly polarized incidence show the usefulness of the flux densities expansion approach.

The scattering problem by metallic gratings has become one of fundamental problems in electromagnetics. In this paper, a thin metallic grating placed in conical mounting is treated as a lossy dielectric grating expressed by complex permittivity and thickness. The solution of the metallic grating by using the matrix eigenvalue calculations is compared with that of the plane grating by using the resistive boundary condition and the spectral Galerkin procedure, and the availability of the resistive boundary condition for thin metallic gratings in conical mounting is investigated. In order to improve the convergence of the solutions of thin metallic gratings, the spatial harmonics of flux densities which are continuous function instead of electromagnetic fields are used.