Enhanced Galerkin's analysis of periodic arrays of strips in multilayered substrates

Periodic strip gratings embedded in multilayered media have an application as frequency selective surfaces, polarization discriminators, twist reflectors, hard and soft surfaces, etc. (K. Uchida et al., IEEE Trans. Antennas Propagat., 36, pp. 415-422, 1988; C.-W. Lee et al., IEEE Trans. Antennas Propagat., 47, pp. 1386-1392, 1999). An efficient software tool is presented for the analysis of the scattering of an oblique incident plane wave with arbitrary polarization by a periodic strip grating embedded in a multilayered substrate. This software tool is based on the application of the Galerkin?s version of the method of moments to the solution of mixed potential integral equations for the current density on the strips (K. A. Michalski et al., IEEE Trans. Antennas Propagat., 45, pp. 508-519, 1997). The slowly convergent series leading to the periodic Green?s functions are efficiently computed by applying Kummer?s transformation in the spectral domain, by approximating the asymptotic term by means of the discrete complex image method, and finally, by applying Ewald?s sum method to the asymptotic series (M. J. Park et al., IEEE Trans. Antennas Propagat., 46, pp. 1582-1583, 1998). The entries of Galerkin?s matrix are double integrals involving the Green?s functions and two basis functions. After some manipulations, it can be shown that one of the two integrals can be expressed as a correlation between the two basis functions. This correlation has been obtained in closed form for three different sets of basis functions: entire domain sinusoidal basis functions, entire domain Chebyshev polynomials weighted by edge condition functions, and subsectional rectangular/triangular basis functions. The remaining integral has to be numerically obtained and contains the singularity of the Green?s function. Fortunately, this singularity has a fixed position in the domain of integration, and can be extracted and integrated in closed form for the three sets of basis functions. Three different types of numerical quadratures have been used in the numerical integration process: standard Gauss-Legendre quadrature, Gauss-Kronrod quadrature and the double exponential quadrature rule (Polimeridis et al., IEEE Trans. Antennas Propagat., 58, pp. 1980-1988, 2010). The convenience of each quadrature formula turns out to be dependent on the type of basis function. For a given degree of accuracy, the novel software tool has proven to be faster than the standard spectral domain approach, which requires the evaluation of slowly convergent infinite series