In this paper, we demonstrate that thickness-profiled metallic gratings placed in a conical mounting can be approximated by a plane grating with surface resistance as a function of the position parameter. As for the method of analysis of thickness-profiled metallic gratings, we apply the multilayered step method, in which the grating region is partitioned into a set of stratified layers having rectangular-profiled gratings, and the flux densities expansion approach. 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 is applied to the resistive boundary condition. Both methods are formulated for the three-dimensional scattering problem by linearly and circularly polarized incidence. From a comparison of the numerical results of thickness-profiled and plane metallic gratings for oblique incidence, the limit of the thickness is investigated.
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 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.
 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].
 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.
2. Setting of the Problem
Figure 1 shows a thickness-profiled metallic grating placed in a conical mounting. We consider the three-dimensional scattering problem by the grating with periodicity Λ and width W that is uniform in the y direction. The thickness profile is defined by d(z) = df(z) as a function of a position parameter z along the boundary. The incident wave is given by a polar angle θi, an azimuthal angle ϕi, and a polarization angle γ. Regions I and III, which have relative permittivity ɛ1 and ɛ3, are lossless materials. The grating layer in region II is described by the relative complex permittivity ɛ2 = ɛ′2 − iɛ″2. Assuming that the thickness profile d(z) is smaller than skin depth and the conductivity σ varies in such a way that the product σd stays finite, the materials of a thin metallic grating can be approximated by a plane grating with surface resistance Rs(z) as
where Rs0 is the resistance value in the maximum thickness d on the thickness profile. The real part of ɛ2 is set as the relative permittivity of the surrounding material and can be stated by ɛ′2 = ɛ = 1 (air) throughout this paper, as in the case of metallic conductors with ωɛ0 ≪ σ, the imaginary part of the permittivity is considerably larger than the real part of it. The complex permittivity ɛ2 is given by
where Z0 = . In the following formulations, the space variables are normalized by the wave number in vacuum k0 = ω such that k0x → x, k0y → y, k0z → z. Using the normalized space variables, Maxwell's equations can be written in dimensionless form as
where Z0 = 1/Y0 and is the rotation of the normalized space variables.
3. Method of Analysis for Thickness-Profiled Gratings
3.1. Formulations in Grating Regions
 In Figure 1, the incident region is the first layer and the region of the semi-infinite substrate is the Lth layer. The grating region is approximated by partitioning it into stratified (L − 2) multilayers having a rectangular profile, and the cross-sectional profile is set to the x-z plane, as shown in Figure 2. As the relative permittivity and permeability in each layer can be expressed by ɛ(z) and μ(z) as a periodic function, respectively, the relative permittivity ɛ(z), its inverse 1/ɛ(z), and the relative permeability μ(z) and its inverse 1/μ(z) in the grating layer are expanded as a Fourier series of Nf terms with Fourier coefficients ɛm, (1/ɛm), μm and (1/μm), respectively:
where s = λ/Λ. The electromagnetic fields Eℓ, Hℓ (ℓ = x, y) and the flux densities Dz, Bz are continuous in the grating region along the z axis. As the structure is periodic, Eℓ, Hℓ, Dz and Bz are expressed in terms of spatial harmonics with the expansion coefficients eim, him, dzm and bzm of mth order using truncated number M:
Substituting equations (5) and (6) into Maxwell's equations (3) and (4), we make 4(2M + 1)-dimensional first-order differential equations for y and z field components directly as
where  is the zero matrix, and [s], [q], [ɛ], [1/ɛ], [μ] and [1/μ] are (2M + 1) × (2M + 1) submatrices. Using the matrix elements in column m and row n, these matrices are expressed by
where δmn is the Kronecker delta. By using 4(2 M + 1)-dimensional column vector a(x) and transforming F = [T] a(x), equation (9) becomes da(x)/dx = i[κ]a(x) where [κ] is a diagonal matrix expressed in terms of the eigenvalue κm of the matrix [C]. The superscripts E and M refer to TE and TM waves, respectively. [T] is a diagonalization matrix for [C] and consists of eigenvectors corresponding to κm, which is numerically calculated by a general subroutine. 4(2M + 1)-dimensional vector a(x) is decomposed into 2(2M + 1)-dimensional vectors a±(x), which represent complex amplitudes in the ± direction and the solution of equation (9) is given by
where x0 is a standard phase position. 2(2M + 1)-dimensional vectors a±(x) can be expressed by (2M + 1)-dimensional vectors Ea±(x) and Ma±(x) for TE and TM waves, respectively, as a±(x) = [Ea±(x) Ma±(x)]t.
3.2. Formulations in Uniform Regions
 Uniform regions I and III are described by the relative permittivity ɛ(z) = ɛ and the relative permeability μ(z) = μ. The electromagnetic fields can be expanded in terms of spatial harmonics. The submatrices can be given by [ɛ] = ɛ and [μ] = μ. Therefore, the coefficient matrix [C] consists of diagonal submatrices [Cm] corresponding to the mth mode. From Maxwell's equations (3) and (4), the differential equations can be rewritten by using the matrix elements of mth mode as follows:
The eigenvalue κm of 4 × 4 matrix [Cm] and the eigenvectors tm are given in a closed form for TE and TM waves in the ±x directions as follows:
The diagonalization matrix [tm] is given by
The eigenvectors tm correspond to TE and TM waves for propagation in the ±x directions and are normalized to Re (eymh*zm − ezmh*ym) = ξm. In order to treat circular polarizations, we have to set the directions of eigenvectors definitely in equation (21). By determining the signs of two elements in each eigenvector for TE or TM waves, as shown in Figure 3, we choose the signs of another two elements according to the ±x directions of propagation, as in Table 1.
Table 1. Determination of the Signs of [tm]
3.3. Boundary Conditions
 By using the transformation matrix [P] defined from ez = [1/ɛ]dz and hz = [1/μ] bz, the continuity of the fields ey, ez, hy and hz at each boundary x = xk (k = 1, …, L − 1) requires that
where x0 and xL corresponding to k = 1 and k = L − 1 equal to x1 = d and xL−1 = 0, respectively. Constant vectors of linearly and circularly polarized incident amplitudes are given by
where the signs “+” and “−” refer to right circularly (RC) and left circularly (LC) polarized waves, respectively. aL+ is the zero vector for the radiation condition. Unknowns are a1+(d) and a3−(0). The diffraction efficiencies of the reflected and transmitted waves E,M,R,Lηmr and E,M,R,Lηmt are given for TE, TM, RC and LC waves as follows:
4. Method of Analysis for Plane Gratings
 A resistive boundary condition can be characterized by using the tangential electric field Eℓtan(y, z) at the interface x = 0, the surface resistance Rs(z) expressed as a function of position parameter z parallel to the boundary, and the surface current density Jℓ(y, z) (ℓ = y, z) as follows:
The tangential electric field is expressed by the sum of the primary field and the scattered field, which can be approximated to be spatial harmonics expansion and written as
The primary and scattered fields can be obtained numerically using the matrix eigenvalue calculation described in section 3. As the primary fields are independent of currents, putting M = 0, the conditions of tangential fields continuity at x = 0 yields linear equations:
where [P] is defined by ez = dz/ɛ and hz = bz/μ. By assuming linearly and circularly polarized incident amplitudes as
the unknowns a1,0+ are determined. Then, the primary fields can be written as
The scattered fields depend on surface currents Jℓ(y, z), which can be expanded in terms of the spatial harmonics:
The unknown coefficients jℓm are related to the jump condition of magnetic fields and the linear equations are obtained as
which have the solutions a1,my+, aL,my− or a1,mz+, aL,mz− corresponding the assumption that jm = [1, 0]t, [0, 1]t. The scattered fields induced from the surface currents jym and jzm, can be expressed as
where e(h)gm is the numerically obtained Green functions in the spectral domain. The current densities jℓm and Jℓ(z) can be approximated in terms of the basis function ϕℓmp and Φℓp of the expansion number K in the spectral and spatial domains, respectively, as
The tangential electric field of equation (31) is obtained from the primary fields of equation (34) and the scattered fields of equation (38) with the approximated currents of equation (40). Applying the Galerkin procedure in a combination of the spectral and spatial domains to the condition (30), yields a system of linear equation as follows:
From the unknown coefficients of Ivp, the surface currents jℓm are obtained. Therefore, the diffraction efficiencies for TE, TM, RC and LC waves are expressed by using the complex amplitudes am±(x) as follows:
5. Numerical Examples
 The aim of this paper is 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. In this paper, we consider sinusoidal and symmetric triangular thickness profiles expressed by continuous functions. Moreover, we consider an asymmetric triangular thickness profile having discontinuity as follows:
Calculations are performed under ɛ1 = 1, ɛ3 = 2.5. In using the Galerkin procedure, step functions are used as the basis functions, a current expansion number of K = 100 is used, and the number of spatial harmonics is truncated to M ≥ 3K from the convergence of solutions for surface resistance distributions.
 In order to investigate the convergence of solutions, we consider sinusoidal thickness profiled gratings having W/Λ = 0.5 and Λ/λ = 0.5. Figure 4 shows the power reflection coefficients against the number of spatial harmonic expansion terms 2M + 1 for oblique incidence θi = ϕi = 45°. It is well known that a method using Fourier series expansion shows poor convergence for a large permittivity profile. Therefore, we investigate convergence by our method using cases with a small resistance of Rs0/Z0 = 0.1 and 0.01 for the large permittivity profile. When the terms are large, the solutions of metallic gratings are close to those of plane gratings. In the following calculations of thickness-profiled gratings, by the convergence of solutions and the computational time, the spatial harmonic expansion terms are truncated to 2M + 1 = 151 and the number of partitioned layers is chosen as L = 32. In the multilayered step method, the partitioning step size of the grating smaller in lower layers and the step distribution function were chosen from the results of planar mounting analysis [Wakabayashi et al., 2003].
Figure 5 shows the Joule loss against surface resistance Rs0/Z0 of sinusoidal thickness profiled gratings for d/λ = 0.01, 0.05 and 0.1. The Joule loss is defined as (1 − Rrp − Lrp − Rtp − Ltp) by using the power reflection coefficients R(L)rp = ∑mR(L)ηmr and the power transmission coefficients R(L)tp = ∑mR(L)ηmt. As surface resistance becomes smaller or larger, the Joule loss becomes smaller. The reason for this is that in the cases of a small or large resistance, the material comes to a lossless conductor or dielectric. The results of thin metallic gratings at a thickness of d/λ = 0.01 are close to those of plane gratings.
Figures 6a and 6b show the power reflection coefficients against thickness d/λ of sinusoidal thickness profiled gratings for θi = 45 and 75°. The results of thin metallic gratings at a thickness below d/λ = 0.01 are close to those for plane gratings, and these tendencies in Figures 6a and 6b are approximately the same. Therefore, the θi dependence is small, and we find that thin metallic gratings with a sinusoidal thickness profile can be treated as plane gratings with surface resistance as a function of the position parameter for incident angles.
 For the comparison of thickness profiles, the power reflection coefficients are calculated for triangular and asymmetric triangular thickness profiled gratings, as shown in Figure 7. In the case of the triangular profile, the thickness when a thickness profiled grating can be treated as a plane grating is less than around 0.01, like a sinusoidal case. In the case of the asymmetric triangular profile, the thickness is small compared with the sinusoidal and triangular cases. From these results and the results of a rectangular thickness profiled grating [Wakabayashi et al., 2004], the thickness of gratings having a discontinuous profile (asymmetric triangular having discontinuity and rectangular) is smaller than that of gratings having a continuous profile (sinusoidal and symmetric triangular) in applicable conditions of approximate modeling by plane grating. To check the cases of diffracted high mode and influence by grating width, the amplitude reflection coefficients of −1st and −2nd mode against periodicity are shown in Figures 8a and 8b for asymmetric triangular profiled gratings and the width of W/Λ = 0.25. As seen in Figures 8a and 8b, Wood's anomalies appear at Λ/λ = 1.0 and 1.5 in the −1st mode and the results of thickness-profiled gratings at d/λ = 0.001 and plane gratings are in close agreement against periodicity with the inclusion of Wood's anomalies. In addition, we can find that the dependence of the grating width is small in the availability of the approximate modeling by plane gratings.
 Finally, in order to investigate the azimuthal characteristics for each polarized component, the power reflection coefficients for TM (γ = 90°) incident on the asymmetric triangular profiled grating are plotted in Figure 9. The thickness-profiled grating at d/λ = 0.001 is in good agreement with the plane grating where the azimuthal angle changes. Furthermore, although the asymmetricity of TM and TE reflected waves becomes small when the thickness is small, asymmetricity caused by the structure is observed in the azimuthal characteristics of extremely thin and plane gratings. Therefore, it can be seen that plane gratings with asymmetric surface resistance show asymmetric characteristics just like thick gratings with asymmetric surface profile at optical wavelength.
 We have shown that a thin metallic grating with a thickness profile placed in a conical mounting can be approximated by a plane grating with surface resistance as a function of position parameter parallel to the boundary. Methods of analyses for thickness-profiled and plane metallic gratings are formulated for the same considered model. By comparing the numerical results, a thin metallic grating around d/λ = 0.001 can be analyzed by the numerical approach for a plane grating. In planar mounting analyses [Wakabayashi et al., 1999] based on the spatial harmonics expansions of electromagnetic fields, although a satisfactory convergence of the solutions of thickness-profiled gratings could not be obtained for small resistance in TM incidence, a resistive boundary condition was considered to be available when the thickness is less than around d/λ = 0.01. In this paper, we have analyzed various thickness-profiled metallic gratings by improving the convergence for conical mounting analyses and small resistance, the resistive boundary condition turned out to be available when the thickness is less than around d/λ = 0.001. From the results of plane gratings with a thickness being disregarded during analysis, it is shown that asymmetric distribution of surface resistance gives asymmetricity to the scattering characteristics. We can obtain the different characteristics for incident directions.