Matrix friendly formulation for layered doubly periodic structure and its computation

Abstract

Layered medium doubly periodic structures (LMDPS) have various applications like frequency selective surfaces (FSS), photonic crystal slabs, metamaterials, and reflectarray antennas. They have drawn huge interests among researchers in computational electromagnetics (CEM), but are still posing great challenges due to the complexity of the structure. The method of moments (MoM) as an integral equation solver for CEM is known for its high accuracy and capability to address large and multi-scale problems. For solving LMDPS with MoM, one can employ the equivalence principle algorithm and connection scheme, which avoids the use of layered medium Green's function, but more unknowns are introduced; alternatively, one can also formulate the integral equation using the layered medium doubly periodic Green's function (LMDPGF), which provides more physical insight and involves fewer unknowns. Although Michalski-Zheng's mixed potential integral equation (MPIE) for layered medium is very popular, the recent matrix friendly formulation (MFF) by Chew is an elegant alternative. And it is the subject of this work to extend the matrix-friendly formulation to LMDPS. In detail, the LMDPGF will be derived and its singularity in both spatial and spectral domains will be examined. It will be shown how the original MFF can be modified to address the singularity of the periodic Green's function when the transverse wavenumber equal to zero. Since the LMDPGF is in the form of a double infinite series, which can be very slowly convergent, especially when the source and observation points are close and in the same horizontal plane, we also present an accurate and efficient approach to evaluate it.