An FFT-accelerated direct solver for electromagnetic scattering from penetrable axisymmetric objects

Abstract

Fast, high-accuracy algorithms for electromagnetic scattering from axisymmetric objects are of great importance when modeling physical phenomena in optics, materials science (e.g. meta-materials), and many other fields of applied science. In this paper, we develop an FFT-accelerated separation of variables solver that can be used to efficiently invert integral equation formulations of Maxwell's equations for scattering from axisymmetric penetrable (dielectric) bodies. Using a standard variant of Müller's integral representation of the fields, our numerical solver rapidly and directly inverts the resulting second-kind integral equation. In particular, the algorithm of this work (1) rapidly evaluates the modal Green's functions, and their derivatives, via kernel splitting and the use of novel recursion formulas, (2) discretizes the underlying integral equation using generalized Gaussian quadratures on adaptive meshes, and (3) is applicable to geometries containing edges and points. Several numerical examples are provided to demonstrate the efficiency and accuracy of the aforementioned algorithm in various geometries.

On analyticity of scattered fields in layered structures with interfacial graphene

“…Consequently, these approaches are an extremely important alternative and are becoming more widely used by practitioners. Paramount among these interfacial methods are those based upon Integral Equations (IEs), [33][34][35][36][37][38][39][40][41][42] however, these face difficulties. Most have been addressed in recent years through (i) the use of sophisticated quadrature rules to deliver HOS accuracy, (ii) the design of preconditioned iterative solvers with suitable acceleration, 43 and (iii) new strategies to avoid periodizing the Green function.…”

Comparison of integral equations for the Maxwell transmission problem with general permittivities

“…Focus has been on avoiding densemesh and topological low-frequency breakdown, on avoiding false resonances, and on providing unique solutions for wider ranges of material parameters. Among later contributions we mention [10,12,13,19,22,25,26,33].…”

“…This is more than other popular IERs use. Typical numbers are four [10,25,27,28,34] or six [12,26,33]. The major advantage with our new IERs, however, is that they offer unique solutions, that is they are free from false eigenwavenumbers, in particular for plasmonic scattering.…”

An Extended Charge-Current Formulation of the Electromagnetic Transmission Problem

“…This is in contrast to indirect formulations [5,6,16,26], where the surface densities have no immediate physical interpretation. Our paper, and many other papers [9,17,20,21,23,26], use integral representations of the electric and magnetic fields, but it is also possible to start with representations of scalar and vector potentials and antipotentials [5,6,18].…”

“…The original four-scalar-density Müller system [21, p. 319] uses the surface current densities M s and J s and contains compact differences of hypersingular operators. These operator differences are quite hard to implement numerically in three dimensions, even though it definitely is possible on axisymmetric surfaces[17]. Our variant of the Müller system is derived from the original Müller system via integration by parts and relating the surface divergence of M s and J s to M and E , see[9, Eqs.…”