Spectral analysis of polygonal cavities containing a negative-index material

Hazard, Christophe; Paolantoni, Sandrine

Annales Henri Lebesgue, Tome 3 (2020), pp. 1161-1193.

L’objet de cet article est d’étudier les effets spectraux d’une interface entre le vide et un matériau à indice négatif (NIM), c’est-à-dire un matériau dispersif dont la permittivité électrique et la perméabilité magnétique deviennent négatives dans une certaine gamme de fréquences. Nous considérons ici une situation élémentaire, à savoir, 1) un modèle très simple de NIM : le modèle de Drude non dissipatif, pour lequel la négativité se produit à basse fréquence ; 2) une équation de propagation scalaire bidimensionnelle déduite des équations de Maxwell ; 3) le cas d’une cavité bornée occupant un domaine polygonal partiellement rempli d’une portion de matériau Drude. En raison de la dispersion fréquentielle (la permittivité et la perméabilité dépendent de la fréquence), l’analyse spectrale d’une telle cavité conduit à un problème aux valeurs propres non linéaire. Grâce à l’utilisation d’une inconnue supplémentaire, nous linéarisons le problème et nous présentons une description complète du spectre. Nous montrons en particulier que l’interface entre le NIM et le vide est à l’origine de divers phénomènes de résonance liés aux différentes composantes d’un spectre essentiel.

The purpose of this paper is to investigate the spectral effects of an interface between vacuum and a negative-index material (NIM), that is, a dispersive material whose electric permittivity and magnetic permeability become negative in some frequency range. We consider here an elementary situation, namely, 1) the simplest existing model of NIM: the non dissipative Drude model, for which negativity occurs at low frequencies; 2) a two-dimensional scalar model derived from the complete Maxwell’s equations; 3) the case of a simple bounded cavity: a polygonal domain partially filled with a portion of Drude material. Because of the frequency dispersion (the permittivity and permeability depend on the frequency), the spectral analysis of such a cavity is unusual since it yields a nonlinear eigenvalue problem. Thanks to the use of an additional unknown, we linearize the problem and we present a complete description of the spectrum. We show in particular that the interface between the NIM and vacuum is responsible for various resonance phenomena related to various components of an essential spectrum.

Classification : 35P05, 35P30, 35Q60, 47A10, 78A25
Mots clés : dispersion, Drude model, essential spectrum, resonance

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