## 1. Introduction

[2] Electromagnetic wave scattering by metallic gratings has been treated widely in the literature [e.g., *Cwik and Mittra*, 1987; *Jin and Volakis*, 1990; *Wakabayashi et al.*, 1994]. Metallic gratings with periodicities of the order of a wavelength have been used in various applications as polarizers and filters in antennas. Recently, interest in the analysis of resistive gratings has been growing [e.g., *Cwik and Mittra*, 1987; *Hall et al.*, 1988] in addition to perfectly conducting gratings. In the analysis of resistive gratings, the metallic grating is usually so thin electrically compared to a wavelength that thickness is neglected and calculations are carried out so that surface resistance and the surface current distribution satisfy the boundary condition. However, increases in the thickness of metallic gratings affect scattering properties, until thickness can no longer be neglected during analysis. The thickness of metallic gratings therefore have to be included in calculations. Some authors have applied periodic waveguide approaches to metallic gratings, with computations involving thickness as a parameter [*Lee et al.*, 1982; *Scott*, 1989]. Using this approach, however, only perfectly conducting gratings can be analyzed, and the situation where thickness is smaller than skin-depth cannot be treated. In addition, the surface profile is not uniform, as the thickness varies along the boundary and deformation of the surface may occur due to abrasion and use degradation of the grating. Thus, an investigation into the configurations of the thickness profiles of metallic gratings is very important.

[3] In previous works [*Wakabayashi et al.*, 1997, 1999], thin metallic gratings in which thickness is a function of position were analyzed by defining metallic gratings as lossy dielectric gratings with large complex permittivity and thickness. The two-dimensional scattering problem of metallic gratings has been considered using a combination of the Fourier series expansion method and the multilayered step method. Rigorous analysis of TE incidence was found to be possible using the Fourier series expansion method with an increased number of space harmonics. However, the convergence of solutions for TM incidence was found to be very slow because the electric field exhibits discontinuities when the permittivity profile of a metallic grating varies rapidly. Many authors have analyzed dielectric gratings around the wavelength of light using spatial harmonic expansion of the electromagnetic fields [e.g., *Yamasaki et al.*, 1991; *Matsumoto et al.*, 1996; *Popov and Neviere*, 2000; *Watanabe et al.*, 2002]. Other authors have made use of the “inverse rule” [*Li*, 1996] to improve the convergence of solutions [*Popov and Neviere*, 2000] and have analyzed deep dielectric or conducting gratings with a surface profile. However, the rule does not seem to explain the behavior of the electromagnetic waves. Another possible approach is the use of window functions in the Fourier series expansion of the permittivity profile [*Sheridan and Sheppard*, 1990]. In this approach, the geometry of the analytical model differs from that of the actual model being considered.

[4] In this paper, the derivation of a new method for resolving the problem in the TM case is presented. The derivation involves the use of spatial harmonic expansion of the flux densities *D*_{z} and *B*_{z} instead of the electromagnetic fields *E*_{z} and *H*_{z} normal to the surface profile of the approximated stratified grating, as the flux densities are continuous. To investigate the effectiveness of our method, the solutions are compared to those of the conventional method [*Wakabayashi et al.*, 1997, 1999].

[5] Next, the multilayered step method [*Peng et al.*, 1975; *Yamasaki et al.*, 1991] is applied to thin metallic gratings with sinusoidal and triangular profiles for TE and TM incidence. The grating region is partitioned into a set of stratified layers approximated by modulated index profile. Effective schemes for partitioning a grating into layers and for approximating each layer as a modulated index profile are investigated. Application of this approach to very thin metallic gratings is then compared with that of the spectral Galerkin procedure [*Itoh and Menzel*, 1981; *Cwik and Mittra*, 1987] for plane metallic gratings, and the validity of the present formulation is confirmed.