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

Abstract

Two recently derived integral equations for the Maxwell transmission problem are compared through numerical tests on simply connected axially symmetric domains for non-magnetic materials. The winning integral equation turns out to be entirely free from false eigenwavenumbers for any passive materials, also for purely negative permittivity ratios and in the static limit, as well as free from false essential spectrum on non-smooth surfaces. It also appears to be numerically competitive to all other available integral equation reformulations of the Maxwell transmission problem, despite using eight scalar surface densities.

The quasi-static plasmonic problem for polyhedra

“…Our description of σ ess (K , H 1/2 (∂ )) ∩ R + is particularly simple for convex polyhedra, since one only needs to consider the double layer potential of − S 2 + 1/4 to compute this interval. Note that spectral parameters λ ∈ (0, 1/2) correspond to permittivities satisfying < −1; in [18,Sect. 8], it is suggested that < −1 is likely to be a necessary condition for the existence of surface plasmon waves on ∂ .…”

“…If is Lipschitz and U is assumed to be of finite energy, then any plasmonic eigenvalue must satisfy that < 0, since Green's formula implies that R 3 \ |∇U | 2 dx = − |∇U | 2 dx. Plasmonic problems, where Re < 0, appear as quasi-static approximations of electrodynamical problems where the scatterer is much smaller than the wavelength of the scattered electromagnetic wave, see [1] and [18,Sect. 8].…”