Efficient computation of the 3D Green’s function for the Helmholtz operator for a linear array of point sources using the Ewald method

Abstract

The Ewald method is applied to accelerate the evaluation of the Green's function (GF) of an infinite equispaced linear array of point sources with linear phasing. Only a few terms are needed to evaluate Ewald sums, which are cast in terms of error functions and exponential integrals, to high accuracy. It is shown analytically that the choice of the standard ''optimal'' Ewald splitting parameter E 0 causes overflow errors at high frequencies (period large compared to the wavelength), and convergence rates are analyzed. A recipe for selecting the Ewald splitting parameter is provided.

Scattering on plasmonic nanostructures arrays modeled with a surface integral formulation

“…The first part of this section generalizes this formulation to periodic systems, and shows that the EFIE and MFIE are restricted to the unit cell with periodic boundary conditions. The EFIE and MFIE involve the pseudo-periodic dyadic Green's function, whose evaluation can be accelerated with Ewald's method [19,[22][23][24]. The MoM is used to discretize and solve the integral equations for the equivalent surface currents.…”

Exceptional points of degeneracy andPTsymmetry in photonic coupled chains of scatterers

“…Furthermore, such additional radiation losses (at high frequencies) could be manipulated and engineered to produce effects related to EPDs, such as loss-induced transmission [39]. Here, the analysis of lossy chains of infinite length is omitted because it would require the use of an infinitely periodic GF [59][60][61][62], whereas our main scope is to provide a simple but intuitive analysis based on the nearest-neighbor approximation, that resembles the TB approach used in [48,50,64,65]. We consider an example of a chain in vacuum (i.e., These values approximately satisfy (19) because of the lowfrequency choice for this EPD to occur.…”

Computation of the one-dimensional free-space periodic Green's function for leaky waves using the Ewald method

“…Each of these two series has Gaussian convergence, and hence is very rapidly converging for all observation points. The Ewald method for the one-dimensional array of point sources in a homogenous medium has been discussed in [1] for the case of a real wavenumber k z0 . The result is given as …”