Improved convergence in the analysis of thin metallic gratings with thickness profiles

Authors


Abstract

[1] A combination of the Fourier series expansion method and the multilayered step method is applied to analysis of the scattering problem by a thin metallic grating with a thickness profile. The extremely large permittivity profiles of metallic gratings may affect accuracy of calculations adversely. The convergence of the Fourier expansion method is improved using the spatial harmonics of flux densities instead of electromagnetic fields normal to the surface of a metallic grating. In using the multilayered step method, the scheme used to partition the grating region and the distribution function of dielectric constant within each layer are examined.

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 Dz and Bz instead of the electromagnetic fields Ez and Hz 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.

2. Theory

2.1. Setting of the Problem

[6] Figure 1 shows a one-dimensional grating with periodicity Λ and width W that is uniform in the y-direction. Regions I and III, which have relative permittivities ε1 and ε3, are lossless materials. The grating layer in region II is described by relative permittivity ε2 = ε′2iε″2 and thickness d(z) = df(z) where d is the depth of grooves in the grating. As the thickness is very small and conductivity varies in such a way that the product σd stays finite, the structure of the thin metallic grating can be expressed by surface resistance R(z) as a function of position z:

equation image

where R0 is resistance at the maximum thickness of the surface profile. Complex relative permittivity ε2 of region II can be written in terms of surface resistance as follows:

equation image

where the real part of ε2 is assumed to be ε′2 = ε1 = 1 throughout the paper, since the grating in the region x ≥ 0 is surrounded by air. Let us consider scattering from the grating by a plane wave exp [itequation imagek0(−x cos θi + z sin θi)}] at an incident angle θi.

Figure 1.

Schematic of the cross-sectional profile of a thin metallic grating.

[7] In the following theory, the space variables are normalized by the wave number in vacuum k0 = ω equation image = 2π/λ such that k0xx, k0yy and k0zz. Using the normalized space variables, Maxwell's equations can be rewritten in dimensionless form (which is analytically simple in computational electromagnetics) as

equation image
equation image

where Z0 = 1/Y0 = equation image and equation image is the rotation of the normalized space variables.

2.2. Electromagnetic Fields in the Grating

[8] In the grating region, the permittivity profile ε(x, z) and the permeability profile μ(x, z) cannot be generally separated with respect to the x and z coordinates. Therefore, the grating layer is approximated by partitioning into stratified (L-2) multilayers having rectangular profiles normal to z-axis (see Figure 2). Since the structure of each layer is periodic, the relative permittivity ε(z) and its inverse 1/ε(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 equation imagem, (1/equation image)m, equation imagem and (1/equation image)m, respectively:

equation image
equation image
equation image
equation image

The x and y components of the electromagnetic fields equation imageE and equation imageH (ℓ = x, y) are continuous. Although the z components of the electromagnetic field equation imageEz and the magnetic field equation imageHz are discontinuous, the z components of electric flux density equation imageDz = ε(z)equation imageEz and magnetic flux density equation imageBz = μ(z)equation imageHz are continuous against along the z axis. Accordingly, (equation imageE, equation imageH) and (equation imageDz, equation imageBz) are expanded instead of the electric field (equation imageEz, equation imageHz) normal to the surface of thin metallic grating. equation imageE, equation imageH, equation imageDz and equation imageBz are expressed in terms of spatial harmonics with expansion coefficients em(x), hm(x), dzm(x) and hzm(x):

equation image
equation image
equation image
equation image

where sm and s0 are expressed in terms of the periodicity Λ, wavelength λ, relative permittivity ε1 and incident angle θi by

equation image

The quantities e(x), h(x), dz(x) and bz(x) are defined by the (2M + 1)-dimensional column vectors of the expansion coefficients:

equation image
equation image
equation image
equation image

Substituting equations (9)(12) into Maxwell's equations (3)(4), the differential equations for the y and z field components can be derived as follows:

equation image

TE-waves:

equation image

TM-waves:

equation image

where [C] consists of m × n submatrices:

equation image

[0] is the zero matrix, [ε]−1, [1/ε]−1, [μ] and [1/μ]−1 are the inverse matrices of [ε], [1/ε], [μ] and [1/μ], respectively and δmn is the Kronecker delta. The solutions of the differential equation (18) can be obtained by matrix eigenvalue calculations of the constant matrix [C] in each layer.

Figure 2.

Partitioning a grating into rectangular multilayers.

[9] By using a 2(2M + 1)-dimensional column vector a(x) and transforming

equation image

Equation (18) can be transformed into

equation image

where the matrix [κ] is a diagonal matrix expressed in terms of the eigenvalue κm of the matrix [C] and [T] is a diagonalization matrix for the coefficient matrix [C] and consists of eigenvectors corresponding to κm. The eigenvalues κm can be assigned to (2M + 1) terms of κm± for propagation in the ±x directions. a(x) can also be partitioned into complex amplitudes a±(x) in the ±x direction corresponding to κm. The solution of equation (18) is given by

equation image
equation image

where [U±, xx0)] = [δmn exp{iκm±(xx0)}] and x0 is a standard phase position.

2.3. Electromagnetic Fields in Uniform Regions

[10] Since the electromagnetic fields in uniform region are continuous, the fields can be expanded in terms of spatial harmonics. From Maxwell's equations (3) and (4), the differential equations can be expressed in terms of the coefficient matrix [C] as follows:TE-waves:

equation image
equation image

TM-waves:

equation image
equation image

Uniform regions I and III with no periodicity are described by the relative permittivity ε(z) = ε and the relative permeability μ(z) = μ. The submatrices can be given by [ε] = ε[1] and [μ] = μ[1]. Therefore, the coefficient matrix [C] in equations (27)(29) consist of diagonal submatrices. The eigenvalue κm of 2 × 2 matrix [Cm] corresponding to the m-th mode and the diagonalization matrix [T] can be obtained analytically as

equation image

TE-waves:

equation image

TM-waves:

equation image

When ξm is complex, the sign is chosen such that the imaginary part is negative. The eigenvectors are normalized to eymh*zm = ±ξm (TE-wave) and −ezmh*ym = ±ξm (TM-wave).

2.4. Boundary Conditions

[11] From the transformation equations ez = [1/ε] dz and hz = [1/μ] bz in the grating region, the transformation matrix [M] is defined byTE-wave:

equation image

TM-wave:

equation image

By using the transformation matrix [M], the continuity of the fields e and h (ℓ = y, z) at each boundary requires thatFor x = x1 = d:

equation image

For x = xk (k = 2 ⋯ L − 1):

equation image

For x = xL−1 = 0:

equation image

where a1(x1) and aL+(xL−1) are the incident amplitude in region I and the radiation condition in region III, respectively, and are given by

equation image

Unknowns in equations (35)(37) are a1+(x1) and aL(xL−1). Diffraction efficiency ηmr of the reflected wave and ηmt of the transmitted wave are given by

equation image
equation image

3. Numerical Results

[12] To validate the derivation presented, we give numerical calculations for a one-dimensional, extremely thin metallic grating with a rectangular thickness profile. Permeability was assumed to be μ0 throughout and Dz were expanded in terms of spatial harmonics. Parameter values chosen for numerical verification were ε1 = 1, ε3 = 2.5, Λ/λ = 0.5, W/Λ = 0.5 and θi = 45°. Figures 3a and 3b compare Joule loss of the conventional and present formulations in the cases of R0/Z0 = 0.1 and 0.01. Joule loss is defined as (1 − rptp) using the power reflected coefficient rp (=equation imageηmr) and the power transmitted coefficient tp (=equation imageηmt). From these figures, the derivation presented provides a significant improvement in convergence. In the following calculations, the spatial harmonics expansion of the electric flux density is used to analyze TM waves incident on thin metallic gratings. For d/λ = 0.001 and 2M + 1 > 101, the solutions of the present formulation are in close agreement with those of the Galerkin procedure for plane metallic gratings that are assumed to be thickness of zero. In using Galerkin procedure, a step function was used as the basis function, the number of spatial harmonics was truncated to 2M + 1 = 601, and a current expansion number of N = 100 was used. These results show that differences in scattering properties can be attributed to the thickness of the metallic grating. Therefore, the need to use a parameter representing thickness in the analysis of metallic gratings depends on wavelength.

Figure 3.

Comparison of results obtained for a rectangular profile using the conventional and presented methods with parameters ε1 = 1, ε3 = 2.5, Λ/λ = 0.5, W/Λ = 0.5, TM-incidence and θi = 45°. (a) R0/Z0 = 0.1. (b) R0/Z0 =0.01.

[13] The multilayered step method for treating thin metallic grating of thickness d(z) expressed as a function of z consists of two main steps: (1) the grating region is partitioned into an assembly of stratified layers of some step size and; (2) taking each layer as a modulated dielectric grating, the permittivity profile ε(z) is expanded as a Fourier series. The metallic grating is very thin compared to an incident wavelength, but the permittivity profile varies rapidly. So, to investigate differences in the schemes of partitioning the grating region into multilayers and of approximating discontinuities in each layer by a modulated index profile, sinusoidal and symmetric triangular profiles are considered as defined by a function f(z) as follows:

equation image

[14] For step (1), a number of different step sizes could be chosen. The following three partitioning schemes are considered:

[15] 1. An equal step size is used, so that the thickness dk (k = 1, 2, ⋯, L-2) of each layer is

equation image

This scheme is simple and most commonly used in the analysis of gratings with surface reliefs [e.g., Yamasaki et al., 1991; Wakabayashi et al., 1999].

[16] 2. A step size proportion to the gradient of the grating shape is used, and the thickness dk in each layer is then

equation image

where vertical positions zk are equally spaced.

[17] 3. Since currents and electric fields are well-known to be discontinuous at the edge of a metallic grating at radio wavelengths, step sizes are set smaller in lower layers. The thickness dk in each layer is then given by

equation image

Figures 4a and 4b show Joule loss for TE and TM waves incident on a grating with a sinusoidal profile for R0/Z0 = 0.1 and 0.01. Parameters values chosen were ε1 = 1, ε3 = 2.5, Λ/λ = 0.5, W/Λ = 0.5, d/λ = 0.001 and θi = 45°. The number of spatial harmonic expansion terms 2M + 1 = 151 is determined by the convergence of solutions and the computational time. The accuracy of the convergence was investigated against the number of layers L. In the TE case, when R0/Z0 = 0.1, the convergence tendencies are approximately the same. When R0/Z0 = 0.01, the convergence of scheme 3 is fast. In the TM case, the convergence of scheme 3 is also found to be fast. Figures 5a and 5b show results for a symmetric triangular profile with the same parameters as for the sinusoidal profile. From these figures, the convergence of scheme 3 is clearly the fastest. Scheme 3 is thus used in the following calculations.

Figure 4.

Comparison of partitioning schemes for a grating with a sinusoidal profile and parameters ε1 = 1, ε3 = 2.5, Λ/λ = 0.5, W/Λ = 0.5, d/λ = 0.001, θi = 45° and 2M + 1 = 151. (a) TE-incidence. (b) TM-incidence.

Figure 5.

Comparison of partitioning schemes for a grating with a symmetric triangular profile and parameters ε1 = 1, ε3 = 2.5, Λ/λ = 0.5, W/Λ = 0.5, θi = 45° and 2M + 1 = 151. (a) TE-incidence. (b) TM-incidence.

[18] Having chosen a partitioning scheme, a distribution function g(z) of the dielectric constant at the discontinuities in the ranges z1zz2 and z3 < z < z4 is needed, as shown in Figure 6. Although many functional forms are possible for g(z), the following three functions are considered:A. Step function:

equation image

where the width w of each layer is determined by setting the area under the approximated rectangular profile to be the same as under the actual surface profile in each layer. This function is simple and most commonly used in the multilayered step method [e.g., Wakabayashi et al., 1999].B. Linear function:

equation image

where a and b are constants which can be obtained from g(z1) = ε1, g(z2) = ε2 and g(z3) = ε2, g(z4) = ε1.C. Third-order polynomial function:

equation image

where a, b, c and d are constants which can be obtained from g(z1) = ε1, g(z2) = ε2, g′(z1) = g′(z2) = 0 and g(z3) = ε2, g(z4) = ε1, g′(z3) = g′(z4) = 0.

Figure 6.

Correspondence between the actual structure and the k-th profile in each layer.

[19] In the calculations of the Joule loss of the sinusoidal profile for TE and TM waves, differences in the convergence of the distribution functions cannot be clearly identified. So, to investigate the differences more closely, the 0th-, −1st- and −2nd-order amplitude reflection coefficients equation image were calculated as a function of the number of layers L for the sinusoidal profile, as shown in Figures 7a and 7b. Parameter values used were ε1 = 1, ε3 = 2.5, Λ/λ = 2, W/Λ = 0.5, R/Z0 = 0.01, d/λ = 0.001 and θi = 45°. Although in the case of TE incidence, differences in the speed of convergence cannot be clearly identified, in the case of TM incidence, convergence of the results of the step function A is the fastest. The calculations were then carried out for a symmetric triangular profile with the same parameters as the above sinusoidal profile. The results are plotted in Figures 8a and 8b. Convergence of the step function A is again the fastest. The effects of changing the function can be seen in the 0th-mode results in particular. For the B and C functions, convergence is slower for the triangular profile than for the sinusoidal profile.

Figure 7.

Comparison of the distribution functions of dielectric constant for a grating with a sinusoidal profile and parameters ε1 = 1, ε3 = 2.5, Λ/λ = 2, W/Λ = 0.5, d/λ = 0.001, R/Z0 = 0.01, θi = 45° and 2M + 1 = 151. (a) TE-incidence. (b) TM-incidence.

Figure 8.

Comparison of the distribution functions of dielectric constant for a grating with a symmetric triangular profile and parameters ε1 = 1, ε3 = 2.5, Λ/λ = 2, W/Λ = 0.5, d/λ = 0.001, θi = 45° and 2M + 1 = 151. (a) TE-incidence. (b) TM-incidence.

[20] Finally, Figures 9a and 9b show a comparison of the convergence of Joule loss using the conventional method and present method for sinusoidal and symmetric triangular profiles, respectively. Parameter values were ε1 = 1, ε3 = 2.5, Λ/λ = 0.5, W/Λ = 0.5, R/Z0 = 0.01, d/λ = 0.001 and θi = 45°. The number of layers L = 20 was chosen using the results shown in Figures 5, 6, 8, and 9, and the step function A was used for the distribution function. As can be seen from the figures, improvements achieved using spatial harmonics of electric flux density are noticeable at thicknesses of d/λ = 0.05 and 0.001.

Figure 9.

Comparison of the conventional and present methods for parameters ε1 = 1, ε3 = 2.5, Λ/λ = 0.5, W/Λ = 0.5, R0/Z0 = 0.01, L = 20, TM-incidence and θi = 45°. (a) Sinusoidal profile. (b) Symmetric triangular profile.

[21] The results of metallic gratings having sinusoidal and symmetric triangular profiles are compared with those of plane gratings in Figures 4, 5, 7, 8, and 9. The results of thin metallic grating are in close agreement with those of plane grating. We can confirm the validity of the present approach.

4. Conclusions

[22] A combination of the Fourier expansion series method and the multilayered step method have been applied to analysis of thin metallic gratings with thickness profiles and the convergence of the solutions has been improved. To avoid discontinuities in the electromagnetic fields, spatial harmonics of flux densities were used in the derivation, which was shown to be successful. In addition, the scheme for partitioning the grating region and the distribution function of dielectric constant within each layer were investigated. The numerical results lead to the conclusions that, for the analysis of thin metallic gratings with thickness profiles, partition step size should be smaller in lower layers, and that a step distribution function is the most effective profile for each layer.

[23] Accurate numerical solutions have been obtained for TE and TM cases using the methods presented. In the future, this approach will be applied to thin metallic gratings in conical mounting, and investigate the limits of thickness in the applicability of the resistive boundary condition for plane gratings in the three-dimensional scattering problem.

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