A 3D spatial spectral integral equation method for electromagnetic scattering from finite objects in a layered medium

Abstract

The generalization of a two-dimensional spatial spectral volume integral equation to a three-dimensional spatial spectral integral equation formulation for electromagnetic scattering from dielectric objects in a stratified dielectric medium is explained. In the spectral domain, the Green function, contrast current density, and scattered electric field are represented on a complex integration manifold that evades the poles and branch cuts that are present in the Green function. In the spatial domain, the field-material interactions are reformulated by a normal-vector field approach, which obeys the Li factorization rules. Numerical evidence is shown that the computation time of this method scales as on the number of unknowns. The accuracy of the method for three numerical examples is compared to a finite element method reference.

Computational aspects of a spatial-spectral domain integral equation for scattering by objects of large longitudinal extent

“…We use the formulation for electromagnetic scattering as explained in [3,4]. In these articles, the branch cuts and poles in the Green function are evaded by representing the contrast current density, the electric field and the Green function in the spectral domain on a path…”

“…(3) makes it unsuitable for a generalization to two dimensions. Therefore, the Taylor-series approach was replaced by a piecewise-linear (PWL) discretization for the 3D algorithm in [3]. With the PWL discretization, a function…”

“…Previously a 2D and 3D spatial spectral integral equation methods for electromagnetic scattering from dielectric objects in multilayered media were proposed [1][2][3]. This approach is based on handling the field-material interaction in the spatial domain and the Green function in the spectral domain.…”

A Normal-vector-field-based Preconditioner for a Spatial Spectral Domain-integral Equation Method for Multi-Layered Electromagnetic Scattering Problems

“…In [28], the matrix C −1 and its approximations have been proposed as preconditioners and promising improvements were obtained after deploying these preconditioners. For the nonperiodic case, the spatial spectral method based on Gabor frames and an auxiliary field in combination with a normal-vector field formulation [14,15] bears a close resemblance to the case of fully spectral methods for periodic structures. Therefore, it is a natural idea to extend the application of the C −1 preconditioner in [28] to the Gabor-frame based spatial spectral solver, which is the main objective of this paper.…”

“…In [13][14][15], a spatial spectral method is proposed to solve two-dimensional (2D) transverse electric (TE), 2D transverse magnetic (TM) and three-dimensional (3D) scattering problems in a layered medium, respectively. The main differences between this method and other volume integral equation solvers are: (1) a Gabor frame is used as a discretization in the transverse plane, which brings a fast and accurate Fourier transformation; and (2) a spectral integration path is chosen to avoid the singularities of the Green function in the spectral domain.…”