Spectral theory for Maxwell's equations at the interface of a metamaterial. Part II: Limiting absorption, limiting amplitude principles and interface resonance

TL;DR

This paper analyzes Maxwell's equations at a negative material interface, proving principles related to wave behavior and revealing a unique resonance phenomenon causing unbounded responses at specific frequencies.

Contribution

It introduces the use of a generalized Fourier transform to establish limiting principles and identifies a novel interface resonance linked to embedded eigenvalues.

Findings

Proved limiting absorption and amplitude principles for Maxwell's equations at the interface.

Discovered a unique interface resonance causing linear blow-up in response.

Linked the resonance to a non-zero embedded eigenvalue of the operator.

Abstract

This paper is concerned with the time-dependent Maxwell's equations for a plane interface between a negative material described by the Drude model and the vacuum, which fill, respectively, two complementary half-spaces. In a first paper, we have constructed a generalized Fourier transform which diagonalizes the Hamiltonian that represents the propagation of transverse electric waves. In this second paper, we use this transform to prove the limiting absorption and limiting amplitude principles, which concern, respectively, the behavior of the resolvent near the continuous spectrum and the long time response of the medium to a time-harmonic source of prescribed frequency. This paper also underlines the existence of an interface resonance which occurs when there exists a particular frequency characterized by a ratio of permittivities and permeabilities equal to $-1$ across the interface. At this frequency, the response of the system to a harmonic forcing term blows up linearly in time. Such a resonance is unusual for wave problem in unbounded domains and corresponds to a non-zero embedded eigenvalue of infinite multiplicity of the underlying operator. This is the time counterpart of the ill-posdness of the corresponding harmonic problem.