## 1. Introduction

[2] Electromagnetic wave scattering by metallic gratings is a fundamental problem in the engineering of radio and optical waves. Metallic gratings have often been analyzed in the design of various applications such as frequency selective screens (FSS), array antennas, and polarizers. In most analyses of thin metallic gratings, the grating is usually so electrically thin compared to a wavelength that thickness is disregarded, and calculations are carried out so that surface resistance and surface current distribution satisfy the boundary condition [*Sumitani et al.*, 2003; *Hwang*, 2000; *Hall et al.*, 1998] (and others). As disregarding the thickness parameter simplifies the calculation procedure, the resistive boundary condition is a convenient approximation. Therefore, an investigation into the limits of the thickness of gratings with rectangular profiles, which can be disregarded during analysis, is very important.

[3] Depending on the conditions, the surface profile might be not be uniform, as the grating groove varies along the boundary. At an optical wavelength, various analytical and numerical methods using the spatial harmonic of electromagnetic fields that can be applied to arbitrarily surface profiled and comparatively thick gratings have been reported [*Yamasaki et al.*, 1996; *Komatsu et al.*, 2005; *Ozaki et al.*, 2007]. We analyze extremely thin thickness-profiled gratings around the radio wavelength by using spatial harmonic expansions of the electromagnetic fields and the multilayered step method, on the two-dimensional scattering problem when the incident plane is perpendicular to the grating groove. Comparing the results of extremely thin gratings with those of plane gratings when disregarding the thickness parameter in calculations, yields results for both that are not in good agreement with an increased number of space harmonics for small surface resistance, or small periodicity in the TM case [*Wakabayashi et al.*, 1999]. Because, in the case of metallic conductors, the imaginary part of complex permittivity is very large at a radio wavelength, the electric field along the periodic direction exhibits discontinues. To resolve the problem in the TM case, we present a new method involving flux densities expansion. In addition, to improve convergence in the analysis of thin gratings with various thickness profiles, we investigate an effective scheme for partitioning gratings [*Wakabayashi et al.*, 2003].

[4] In these studies, however, the incident plane is assumed to be perpendicular to the gratings and we deal only with linear polarizations. Formulations for circular polarization are needed in array antennas for satellite communications, in FSSs for radio astronomy [*Irimajiri et al.*, 1991], and in polarizers [*Lerner*, 1965; *Munk*, 2003]. To the best of our knowledge, a situation in which the thickness profile is smaller than skin depth at a radio wavelength, for a case of conical mounting and circular polarization, has not been treated to date. Some of the reasons for this are that conical mounting analyses require complicated formulations and large calculations, and the convergence of the solutions for a thickness-profiled metallic grating is even slower at a radio wavelength in addition to when it is in a conical mounting. In this paper, our previous formulations using flux densities expansion for a thickness-profiled grating of two-dimensional scattering problem [*Wakabayashi et al.*, 2003] are extended to include a three-dimensional scattering problem and circular polarization, and we intend to demonstrate that thin metallic gratings with various thickness profiles placed in a conical mounting can be approximated by a plane grating with surface resistance as a function of the position parameter parallel to the boundary. In order to show the limits of the thickness of metallic gratings that can be disregarded during analyses, we consider circular polarized incidence represented as the combination of two linearly polarizations in addition to linear polarized incidence. As for the method of analysis for thickness-profiled gratings, a combination of the multilayered step method, in which the grating region is partitioned into multilayers having rectangular-profiled gratings, and the flux densities expansion approach based on the matrix eigenvalue calculations are applied to the gratings for oblique incidence. As for the method of analysis for plane gratings with a thickness of zero, a spectral Galerkin method based on the mixed form of current expansions in the spectral and spatial domains [*Wakabayashi et al.*, 2003; *Sumitani et al.*, 2003] is applied to the resistive boundary condition for oblique incidence. For clarity, both methods are formulated for the same analytical model on the three-dimensional scattering problem by linear and circular polarizations, and by using eigenvectors of unified derivations for all regions of uniform and grating regions. From a comparison of the numerical results of thickness-profiled and plane gratings, the limit of the thickness is investigated. In addition, from the results of the azimuthal characteristics of extremely thin and plane gratings, the asymmetricity caused by the structure is investigated.