Windowed Green function method for the Helmholtz equation in the presence of multiply layered media

Abstract

This paper presents a new methodology for the solution of problems of two-and threedimensional acoustic scattering (and, in particular, two-dimensional electromagnetic scattering) by obstacles and defects in presence an arbitrary number of penetrable layers. Relying on use of certain slow-rise windowing functions, the proposed Windowed Green Function approach (WGF) efficiently evaluates oscillatory integrals over unbounded domains, with high accuracy, without recourse to the highly expensive Sommerfeld integrals that have typically been used to account for the effect of underlying planar multi-layer structures. The proposed methodology, whose theoretical basis was presented in the recent contribution (SIAM J. Appl. Math. 76(5), p. 1871, 2016), is fast, accurate, flexible, and easy to implement. Our numerical experiments demonstrate that the numerical errors resulting from the proposed approach decrease faster than any negative power of the window size. In a number of examples considered in this paper the proposed method is up to thousands of times faster, for a given accuracy, than corresponding methods based on use of Sommerfeld integrals.

Inverse design of large-area metasurfaces

“…In Fig. 4, the locally periodic approximate solver (left) is compared to a brute-force surfaceintegral equation (SIE) Maxwell solver [10,9,11] for this optimized solution, showing good quantitative agreement. More precisely, at right we compare the computed intensities |E z (x, y)| 2 for several separations y from the surface.…”

Convolution quadrature methods for time-domain scattering from unbounded penetrable interfaces

“…In this section we present the WGF method for the solution of the two-layer transmission problem (6). As is shown in [10,13,30] and in the numerical examples presented in Section 5, the proposed WGF method approach can be easily extended to tackle more general configurations involving unbounded material interfaces, such as multiply layered media and waveguide branches.…”

“…Although this CQ-BIE approach has proven to be competitive to volume discretization methods in the context of obstacle scattering problems [3,4,34], its extension to problems involving unbounded material interfaces is severely hindered by the fact that standard BIE formulations require the knowledge of problem-specific Green functions to deal with the unboundedness of the material interfaces. These Green functions, however, are often unavailable (in terms of tractable mathematical expressions) or are given in terms of computationally expensive Sommerfeld integrals 1 [27,29,30].Recent advances on BIE methods for time-harmonic problems of scattering from unbounded material interfaces have led to the development of highly efficient solvers that completely bypass the use of problem-specific Green functions [10,12,13,24,30,37]. The windowed Green function (WGF) method, in particular, has successfully been used in layered media [12,13,30], dielectric waveguides [10] and all-dielectric metasurfaces [31] simulations in the frequency domain.…”

“…These Green functions, however, are often unavailable (in terms of tractable mathematical expressions) or are given in terms of computationally expensive Sommerfeld integrals 1 [27,29,30].Recent advances on BIE methods for time-harmonic problems of scattering from unbounded material interfaces have led to the development of highly efficient solvers that completely bypass the use of problem-specific Green functions [10,12,13,24,30,37]. The windowed Green function (WGF) method, in particular, has successfully been used in layered media [12,13,30], dielectric waveguides [10] and all-dielectric metasurfaces [31] simulations in the frequency domain. The method relies on a certain "second-kind" BIE-given in terms of free-space Helmholtz Green functions-posed on all the (bounded and unbounded) interfaces.…”

Sideways adiabaticity: beyond ray optics for slowly varying metasurfaces

“…This appendix is devoted to the derivation of the integral representation formula (10). In order to achieve that, we show first the symmetry of the exact Green's function (9). Consider then the functions w(r) = G(r|r 1 ) and v(r) = G(r|r 2 ) for the r 1 , r 2 ∈ Ω + .…”