Spectral theory for Maxwell’s equations at the interface of a metamaterial. Part I: Generalized Fourier transform

Abstract

We explore the spectral properties of the time-dependent Maxwell's equations for a plane interface between a metamaterial represented by the Drude model and the vacuum, which fill respectively complementary half-spaces. We construct explicitly a generalized Fourier transform which diagonalizes the Hamiltonian that describes the propagation of transverse electric waves. This transform appears as an operator of decomposition on a family of generalized eigenfunctions of the problem. It will be used in a forthcoming paper to prove both limiting absorption and limiting amplitude principles.

Simulating transient wave phenomena in acoustic metamaterials using auxiliary fields

“…At these frequencies, the limiting absorption principle fails or holds only on a subspace of source terms that needs to be characterized. The question of the existence of a limiting absorption principle has been also studied in the context of an interface problem involving a dispersive electromagnetic media in [5,7].…”

“…To tackle broadband acoustic experiments it is advantageous to consider time-domain formulations. To this end, a set of auxiliary fields can be introduced as it has been done in a number of studies on Maxwell's equations, see [45,18,19,6,7]. Based on this formalism, various methods have been proposed in the transient regime for computational electromagnetism, based on the finite-element method [27], the discontinuous Galerkin method [28] and the staggered-grid finite-difference method [46].…”

Negative Index Materials: Some Mathematical Perspectives

“…Another interesting question would be to investigate the limiting amplitude principle, which concerns the behavior of the fields in the time domain generated by a harmonic forcing term for large time. In some particular settings, the limiting amplitude principle was already considered in [13,16], but the question for a general setting remains open.…”

Mathematical models for dispersive electromagnetic waves: An overview

“…A second requirement is that if D(x, t) (or B(x, t)) is real then E(x, t) (or H(x, t)) is real too. According to (12) and (11)…”

“…where (33(b)) is completed with properly chosen initial conditions compatible with (12). The above justifies the term local, since differential operators are local in time (they 'see' only the behaviour of a function around a given time).…”