## 1. Introduction

[2] In applying numerical full wave methods like the Method of Moments (MoM) or Boundary Integral Equations (BIE) to periodic structures involving conducting or dielectric electromagnetic scatterers, fast and accurate means for evaluating the free-space periodic Green's function (FSPGF) are often needed. This type of Green's function arises in a wide variety of applications, ranging from microwaves to optics, to the study of metamaterials and nanostructures. The Ewald method is one of the fastest methods for calculating the FSPGF. In the Ewald method, the FSPGF is expressed as the sum of a “modified spectral” and a “modified spatial” series. The terms of both series possess Gaussian decay, leading to an overall series representation that exhibits a very rapid convergence rate. The convergence rate is optimum when the “optimum” value of the Ewald splitting parameter is used [*Jordan et al.*, 1986], denoted here as *E*_{opt}. (For some applications, such as when using a periodic Green's function in a MoM solution with full-domain basis functions, one may not wish to have a balanced convergence between the two series, as explained in *Mathis and Peterson* [1996] and *Mathis and Peterson* [1998]. However, when using subdomain basis functions and performing the integrations over the basis and testing functions in the spatial domain, the objective is to minimize the computation time of the periodic Green's function, and this is accomplished by using *E*_{opt}. In this case the singular integrals involved in evaluating the matrix elements can be handled by specially designed numerical quadrature rules [*Khayat and Wilton*, 2005].)

[3] However, the numerical accuracy of the Ewald method degrades very quickly [*Kustepeli and Martin*, 2000] with increasing frequency (i.e., when the periodicity becomes large relative to a wavelength). This is due to a catastrophic loss of significant figures in combining the contributions of the two series, wherein the leading terms (and to a lesser extent, other nearby terms) in each series become very large but nearly equal and opposite in sign.

[4] The method proposed and studied here limits the size of the largest terms in the series relative to that of the total Green's function by modifying the value of the splitting parameter *E* to avoid undue loss of accuracy. Increasing the *E* parameter limits the size of the largest terms in both series at the expense of decreasing the convergence rate. Hence, there is a tradeoff between the size of the largest term allowed, which determines the number of significant figures lost, and the series convergence rate. A value *E*_{L} of the Ewald splitting parameter is then obtained based on the number of significant figures *L* that may be lost. This “best” value, *E*_{L}, then yields the fastest convergence of the Ewald series while limiting the loss of significant figures to the user-defined value *L*.

[5] A preliminary and intuitive analysis for the “best” choice of the Ewald splitting parameter was performed in *Capolino et al.* [2005] for the case of 1-D array of line sources, and in *Capolino et al.* [2007] for the case of a 1-D array of point sources. The case of a 2D-array of point sources has been treated in detail in *Oroskar et al.* [2006]. Here we extend the analysis of the last paper to the two other cases, and provide a unified formalism for the choice of the Ewald splitting parameter and the summation limits that is valid for all three cases.