Dirac Integral Equations for Dielectric and Plasmonic Scattering

Abstract

A new integral equation formulation is presented for the Maxwell transmission problem in Lipschitz domains. It builds on the Cauchy integral for the Dirac equation, is free from false eigenwavenumbers for a wider range of permittivities than other known formulations, can be used for magnetic materials, is applicable in both two and three dimensions, and does not suffer from any low-frequency breakdown. Numerical results for the two-dimensional version of the formulation, including examples featuring surface plasmon waves, demonstrate competitiveness relative to state-of-the-art integral formulations that are constrained to two dimensions. However, our Dirac integral equation performs equally well in three dimensions, as demonstrated in a companion paper.

An efficient full-wave solver for eddy currents

“…holds by the jump relations and the divergence theorem. For a discussion of the 𝐿 2 topology considered for the fields and the corresponding trace space, we refer to [18,Sec. 5].…”

“…The field k−1 𝐻 + , appearing in (7), is the magnetic field rescaled by the wave impedance √ 𝜇 + ∕𝜖 + in Ω + . This field is natural when formulating Maxwell's equations as a Dirac equation, and was denoted 𝐵 + in [18].…”

“…The Dirac derivative 𝐃𝐹 is the left-hand side in (10), and replacing ∇ by the vector 𝑥 in this matrix yields the Clifford product 𝑥𝐹 appearing in the Dirac radiation condition. (For a short explanation of the underlying multivector formalism we refer to [18,Sec. 3], and for the long explanation we refer to [26].)…”

“…We point out that although arg(𝑘 − ) = 0 and arg(𝑘 + ) = 𝜋∕4 in all our numerical examples, there is no approximation of the full MTP ( 7) involved, and a known permittivity can be included in 𝑘 + . The Dirac BIE from [18,17] is the starting point for the present work and from now on referred to as Dirac (A). This BIE is based on the embedding of Maxwell's equations into an elliptic Dirac equation and a Cauchy integral representation for the fields (Eq.…”

“…It has 12 free parameters, as recalled in (24) below, and these can be chosen to avoid false eigenwavenumbers for all passive materials. As for low-frequency breakdown, the only regimes found in [18,17] where the Dirac (A) exhibits false eigenwavenumbers is when 𝑘 ± → 0 at the same time as k = 𝑘 + ∕𝑘 − → ∞ or k → 0. (We refer to [17,Sec.…”