Rapidly Convergent Representations for 2D and 3D Green's Functions for a Linear Periodic Array of Dipole Sources

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.

Fast Periodic Interpolation Method for Periodic Unit Cell Problems

“…The at the grids can be evaluated using any available methods, including various acceleration techniques [1]- [8]. However, even simple spectral Floquet mode expansions can be very efficient (see Floquet expansions for arrays in free space in Appendix).…”

Acceleration of Mixed Potentials From Vertical Currents in Layered Media for 2-D Structures With 1-D Periodicity

“…The two series can then be integrated separately: (16) where and are modified spectral and spatial series, defined as -integrals of and , respectively. Both of the standard Ewald series can appropriately be integrated, as shown in next subsections.…”

“…Many numerical algorithms have been proposed for the efficient computation of homogeneous-medium periodic Green's functions [12]- [16]. Again with reference to two-dimensional (2-D), i.e., cylindrical problems, the Ewald method [17]- [19] is one of the most efficient acceleration methods for the computation of the potential due to a 1-D phased array of line sources, with spatial period and phase shift [see Fig.…”

“…In radiation 0018-926X/$31.00 © 2012 IEEE can be accelerated through the Ewald method according to (1), as explained in [19]. (b) 1-D periodic phased array of half-plane sources, which is the quasi-static behavior of the vertical components and ; the relevant Green's function can be accelerated through the Ewald method according to (16), as explained in this paper. and scattering problems, is the phase shift between adjacent sources; in guided-wave problems, is the propagation constant of the Bloch mode under analysis, possibly complex if modes such as leaky waves are considered [4].…”

Scattering and Radiation from/by 1-D Periodic Metallizations Residing in Layered Media

“…During the past decades, a lot of attention has been paid to the construction and the efficient computation of these periodic Green's functions, both for 2-D and 3-D configurations [9]- [12]. Some more recent contributions comprise a mathematical study of quasi-periodic Green's functions for the free-space Helmholtz and Laplace equation with the 3-D Green's function with 1-D periodicity as a particular case [13], an efficient computation of 1-D periodic 3-D Green's function in free space, making use of an Ewald splitting technique to improve the convergence [14], a comparison between several techniques to accelerate the convergence for 2-D and 3-D Green's functions with 1-D and 2-D periodicity [15], an algorithm based on the spectral Kummer-Poisson method for the calculation of 2-D Green's functions with 1-D and 2-D periodicity [16], an interpolation technique for the determination of layered media Green's functions [17], and a calculation method for 2-D and 3-D scalar and dyadic free-space Green's functions with 1-D periodicity [18]. As this paper focuses on a frequency-domain technique, the above cited work refers to frequency-domain periodic Green's functions.…”