COMPUTATIONAL ASPECTS OF 2D-QUASI-PERIODIC-GREEN-FUNCTION COMPUTATIONS FOR SCATTERING BY DIELECTRIC OBJECTS VIA SURFACE INTEGRAL EQUATIONS

Abstract

We describe a surface integral-equation (SIE) method suitable for computation of electromagnetic fields scattered by 2D-periodic high-permittivity and plasmonic scatterers. The method makes use of fast evaluation of the 2D-quasi-periodic Green function (2D-QPGF) and its gradient using a tabulation technique in combination with tri-linear interpolation. In particular we present a very efficient technique to create the look-up tables for the 2D-QPGF and its gradient where we use to our advantage that it is very effective to simultaneously compute the QPGF and its gradient, and to simultaneously compute these values for the case in which the role of source and observation point are interchanged. We use the Ewald representation of the 2D-QPGF and its gradient to construct the tables with pre-computed values. Usually the expressions for the Ewald representation of the 2D-QPGF and its gradient are presented in terms of the complex complementary error function but here we give the expressions in terms of the Faddeeva function enabling efficient use of the dedicated algorithms to compute the Faddeeva function. Expressions are given for both lossy and lossless medium parameters and it is shown that the expression for the lossless case can be evaluated twice as fast as the expression for the lossy case. Two case studies are presented to validate the proposed method and to show that the time required for computing the method of moments (MoM) integrals that require evaluation of the 2D-QPGF becomes comparable to the time required for computing the MoM integrals that require evaluation of the aperiodic Green function. Reliable prediction of the interaction of electromagnetic fields with periodic structures plays a key role in numerous applications whether it is in the radio-frequency (RF) regime or in the optical regime. In the RF regime, periodic structures are for instance used to model the scattering properties of large arrays and frequency-selective surfaces. In the optical regime, an application of great importance is the scattering of light by diffraction gratings. To enable accurate and efficient computation of fields scattered by periodic structures, considerable effort has been invested over the years in the development of a range of numerical methods. For many of these methods the performance is excellent as long as the periodically repeated scattering objects have low contrast, but for high permittivity and plasmonic scatterers the problem of obtaining sufficiently accurate results in acceptable computation times arises often. In the RF regime, metals can mostly be modeled by PEC structures but in the optical regime they exhibit plasmonic behavior, see (1) and (2), hence in the optical regime the problem of dealing with plasmonic scatterers is especially important. While the very successful Finite Element Method (FEM) and Volume Integral-Equation (VIE) method apply a volumetric discretization, the Surface Integral Equation (SIE) formulation only uses