Efficient analysis of periodic structures involving finite dielectric material based on the array scanning method

Abstract

Efficient numerical analysis of finite and infinite periodic structures involving finite dielectric material by the method of moments (MoM) is important for applications of phased arrays and metamaterials [1]. When periodic structures are excited by nonperiodic sources, one may apply the Array Scanning Method (ASM), which replaces the nonperiodic problem with an integral superposition of periodic problems [2]. The integration is performed over the phase shifts between adjacent elements of the structure from 0 to 2π and requires the numerical solution of periodic subproblems [3]-[4]. However, as the inter-element phase shifts range from 0 to 2π, the convergence of the periodic Green's function -and of its gradient- gets very difficult near Rayleigh-Wood's anomaly [5]. This anomaly occurs when dipoles form a beam in the direction of the observation point close to the array plane in the line-by-line formulation of infinite-array Green's function [1]. Therefore, the periodic Green's function needs to be evaluated near Rayleigh-Wood's anomaly accurately to apply the ASM properly. Another application of the ASM is the efficient analysis of the finite periodic structures by the Macro Basis Function (MBF) approach [6]-[8]. The MBF approach reduces the number of unknowns describing each element of a finite periodic structure by at least one order of magnitude. The ASM provides a physically based approach for the proper choice of MBFs, which lead to a sufficiently complete basis for the problem of interest [9]. This paper is organized as follows. In Section 2.1, we revisit Levin extrapolation to accelerate the computation of the doubly periodic Green's function near the Rayleigh-Wood's anomaly to apply the ASM properly. In Section 2.2 we apply the ASM to electromagnetic scattering from a doubly periodic structure composed of dielectric objects excited by a single point source. We observe that the scattered electric field distribution for uniform phase shifts in the spectral domain has singular behavior at the boundary of the visible space due to the singularity of the periodic Green's function. This behavior is responsible for a highly oscillatory response in the spatial domain. To improve the quality of estimated field near the limits of the visible region, we implement a second order polynomial extrapolation on the (MoM) system matrix. Finally, in Section 2.3, a validation example is shown for finite-array solutions of dielectric objects with the help of the combined infinite array solution and MBF approaches. We observe that 4×4 infinite array solutions in spectral domain remain sufficient to form a sufficiently complete set of MBFs for the analysis of a dielectric sphere array.