Rapidly convergent representations for 2D and 3D Green's functions for a linear periodic array of dipole sources

TL;DR

This paper introduces hybrid spectral-spatial methods for efficiently computing 2D and 3D Green's functions of the Helmholtz equation in periodic arrays, achieving rapid convergence and high numerical efficiency for various array sizes.

Contribution

The paper presents a novel hybrid integral representation approach that combines spectral integrals, Floquet modes, and steepest descent paths for fast Green's function evaluation in periodic structures.

Findings

Achieves high convergence with few quadrature points.

Effective for arrays with small and large periodicities.

Demonstrates efficiency through extensive numerical examples.

Abstract

Hybrid spectral-spatial representations are introduced to rapidly calculate periodic scalar and dyadic Green's functions of the Helmholtz equation for 2D and 3D configurations with a 1D (linear) periodicity. The presented schemes work seamlessly for any observation location near the array and for any practical array periodicities, including electrically small and large periodicities. The representations are based on the expansion of the periodic Green's functions in terms of the continuous spectral integrals over the transverse (to the array) spectral parameters. To achieve high convergence and numerical efficiency, the introduced integral representations are cast in a hybrid form in terms of (i) a small number of contributions due to sources located around the unit cell of interest, (ii) a small number of symmetric combinations of the Floquet modes, and (iii) an integral evaluated along the steepest descent path (SDP). The SDP integral is regularized by extracting the singular behavior near the saddle point of the integrand and integrating the extracted components in closed form. Efficient quadrature rules are established to evaluate this integral using a small number of quadrature nodes with arbitrary small error for a wide range of structure parameters. Strengths of the introduced approach are demonstrated via extensive numerical examples.