An Extended Charge-Current Formulation of the Electromagnetic Transmission Problem

Abstract

A boundary integral equation formulation is presented for the electromagnetic transmission problem where an incident electromagnetic wave is scattered from a bounded dielectric object. The formulation provides unique solutions for all combinations of wavenumbers for which Maxwell's equations have a unique solution. This includes the challenging combination of a real positive wavenumber in the outer region and an imaginary wavenumber inside the object. The formulation, or variants thereof, is particularly suitable for numerical field evaluations as confirmed by examples involving both smooth and nonsmooth objects.

Modified Representations for the Close Evaluation Problem

“…One can use this technique to modify any other wave problems, including sound-hard, penetrable obstacles. Future work includes applying those techniques to plasmonic scattering problems [46,47], deriving an asymptotic analysis to quantify the limit behavior of the error as the evaluation point approaches the boundary, as well as extensions to other partial differential equations such as Stokes problems and others.…”

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

“…For "Dirac" it is proven that there exist unique solutions when Ω + is a bounded Lipschitz domain with connected exterior Ω − and (k − , k + ) satisfies the conditions of [22,Proposition 8.3]. Existence of unique solutions for "HK 8-dens" is proven for (k − , k + ) as described in [19,Section 8.3]. It is assumed in [19] that Γ is smooth and Ω + simply connected but the proof holds also for objects that are multiply connected.…”

“…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].…”

“…"Dirac" is derived in [22] by embedding the time-harmonic Maxwell's equations into a Dirac equation and by tuning the six free parameters of this equation as to optimize numerical performance. "HK 8-dens" is derived in [19] by extending a classic IER of the Helmholtz transmission problem [23] via the use of four uniqueness parameters. Both our IERs use eight unknown scalar surface densities for modeling.…”

“…The original papers [19,22] use mutually different notation and contain numerical examples for reduced and two-dimensional versions of the IERs. The purpose of this work is to present "Dirac" and "HK 8-dens" in a unified and programming-friendly notation and to conduct a series of numerical experiments for their full versions in three dimensions as to see which IER is the most efficient.…”