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.
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.
Revised:
Accepted:
Published online:
Keywords: dispersion, Drude model, essential spectrum, resonance
@article{AHL_2020__3__1161_0, author = {Hazard, Christophe and Paolantoni, Sandrine}, title = {Spectral analysis of polygonal cavities containing a negative-index material}, journal = {Annales Henri Lebesgue}, pages = {1161--1193}, publisher = {\'ENS Rennes}, volume = {3}, year = {2020}, doi = {10.5802/ahl.58}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/ahl.58/} }
TY - JOUR AU - Hazard, Christophe AU - Paolantoni, Sandrine TI - Spectral analysis of polygonal cavities containing a negative-index material JO - Annales Henri Lebesgue PY - 2020 SP - 1161 EP - 1193 VL - 3 PB - ÉNS Rennes UR - http://archive.numdam.org/articles/10.5802/ahl.58/ DO - 10.5802/ahl.58 LA - en ID - AHL_2020__3__1161_0 ER -
%0 Journal Article %A Hazard, Christophe %A Paolantoni, Sandrine %T Spectral analysis of polygonal cavities containing a negative-index material %J Annales Henri Lebesgue %D 2020 %P 1161-1193 %V 3 %I ÉNS Rennes %U http://archive.numdam.org/articles/10.5802/ahl.58/ %R 10.5802/ahl.58 %G en %F AHL_2020__3__1161_0
Hazard, Christophe; Paolantoni, Sandrine. Spectral analysis of polygonal cavities containing a negative-index material. Annales Henri Lebesgue, Volume 3 (2020), pp. 1161-1193. doi : 10.5802/ahl.58. http://archive.numdam.org/articles/10.5802/ahl.58/
[ACK + 13] Spectral theory of a Neumann–Poincaré-type operator and analysis of cloaking due to anomalous localized resonance, Arch. Rational Mech. Anal., Volume 208 (2013) no. 2, pp. 667-692 | DOI | Zbl
[AK16] Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann–Poincaré operator, J. Math. Anal. Appl., Volume 435 (2016) no. 1, pp. 162-178 | DOI | Zbl
[AL95] Spectral properties of a class of rational operator valued functions, J. Oper. Theory, Volume 33 (1995) no. 2, pp. 259-277 | MR | Zbl
[AMRZ17] Mathematical analysis of plasmonic nanoparticles: the scalar case, Arch. Ration. Mech. Anal., Volume 224 (2017) no. 2, pp. 597-658 | DOI | MR | Zbl
[BCC12] T-coercivity for scalar interface problems between dielectrics and metamaterials, ESAIM, Math. Model. Numer. Anal., Volume 46 (2012) no. 6, pp. 1363-1387 | Numdam | MR | Zbl
[BCC14a] T-coercivity for the Maxwell problem with sign-changing coefficients, Comm. Part. Diff. Eq., Volume 39 (2014) no. 6, pp. 1007-1031 | MR | Zbl
[BCC14b] Two-dimensional Maxwell’s equations with sign-changing coefficients, Appl. Numer. Math., Volume 79 (2014), pp. 29-41 | DOI | MR
[BCCC16] On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients, J. Comput. Phys., Volume 322 (2016), pp. 224-247 | DOI | MR | Zbl
[BDR99] Analyse spectrale et singularités d’un problème de transmission non coercif, C. R. Acad. Sci. Paris, Série I, Math., Volume 328 (1999) no. 8, pp. 717-720 | DOI
[BGD16] Calculation and analysis of the complex band structure of dispersive and dissipative two-dimensional photonic crystals, J. Opt. Soc. Am. B, Volume 33 (2016) no. 4, pp. 691-702 | DOI
[BGK79] Minimal Factorization of Matrix and Operator Functions, Operator Theory: Advances and Applications, 1, Birkhäuser, 1979 | MR
[BZ19] Characterization of the essential spectrum of the Neumann–Poincaré operator in 2D domains with corner via Weyl sequences, Rev. Mat. Iberoam., Volume 35 (2019) no. 3, pp. 925-948 | DOI | Zbl
[CHJ17] Spectral theory for Maxwell’s equations at the interface of a metamaterial. Part I: Generalized Fourier transform, Commun. Part. Differ. Equations, Volume 42 (2017) no. 11, pp. 1707-1748 | DOI | MR | Zbl
[CJK17] Mathematical models for dispersive electromagnetic waves: an overview, Comput. Math. Appl., Volume 74 (2017) no. 11, pp. 2792-2830 | DOI | MR | Zbl
[CMM12] On the existence and uniqueness of a solution for some frequency-dependent partial differential equations coming from the modeling of metamaterials, SIAM J. Math. Anal., Volume 44 (2012) no. 6, pp. 3806-3833 | DOI | MR | Zbl
[CPP19] Self-adjoint indefinite Laplacians, J. Anal. Math., Volume 139 (2019) no. 1, pp. 155-177 | DOI | MR | Zbl
[CS85] A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl., Volume 106 (1985), pp. 367-413 | DOI | MR | Zbl
[EE87] Spectral Theory and Differential Operators, Oxford Mathematical Monographs, Oxford University Press, 1987
[ELT17] Rational eigenvalue problems and applications to photonic crystals, J. Math. Anal. Appl., Volume 445 (2017) no. 1, pp. 240-279 | DOI | MR | Zbl
[FS07] Hamiltonian treatment of time dispersive and dissipative media within the linear response theory, J. Comput. Appl. Math., Volume 204 (2007) no. 2, pp. 199-208 | DOI | MR | Zbl
[GM12] Negative index materials and time-harmonic electromagnetic field, C.R. Physique, Volume 13 (2012) no. 8, pp. 786-799 | DOI
[Gri14] The plasmonic eigenvalue problem, Rev. Math. Phys., Volume 26 (2014), 1450005, p. 26 | MR | Zbl
[GT10] Macroscopic Maxwell’s equations and negative index materials, J. Math. Phys., Volume 51 (2010) no. 5, 052902, p. 28 | MR | Zbl
[GT17] The nonlinear eigenvalue problem, Acta Numer., Volume 26 (2017), pp. 1-94 | DOI | Zbl
[Kat13] Perturbation Theory for Linear Operators, Grundlehren der mathematischen wissenschaften in einzeldarstellungen, 132, Springer, 2013 | Zbl
[Nag89] Towards a “matrix theory” for unbounded operator matrices, Math. Z., Volume 201 (1989) no. 1, pp. 57-68 | DOI | MR | Zbl
[Ngu16] Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients, J. Math. Pures Appl., Volume 106 (2016) no. 9, pp. 342-374 | DOI | MR | Zbl
[Ola95] Remarks on a transmission problem, J. Math. Anal. Appl., Volume 196 (1995) no. 2, pp. 639-658 | MR | Zbl
[Pan19] On self-adjoint realizations of sign-indefinite Laplacians, Rev. Roum. Math. Pures Appl., Volume 64 (2019) no. 2-3, pp. 345-372 | MR | Zbl
[Pen00] Negative refraction makes a perfect lens, Phys. Rev. Lett., Volume 85 (2000), pp. 3966-3969 | DOI
[PP17] The essential spectrum of the Neumann–Poincaré operator on a domain with corners, Arch. Ration. Mech. Anal., Volume 223 (2017) no. 2, pp. 1019-1033 | DOI | Zbl
[SB11] Solving rational eigenvalue problems via linearization, SIAM J. Matrix Anal. Appl., Volume 32 (2011) no. 1, pp. 201-216 | MR | Zbl
[Tip98] Linear absorptive dielectrics, Phys. Rev. A, Volume 57 (1998), pp. 4818-4841
[Tip04] Linear dispersive dielectrics as limits of Drude–Lorentz systems, Phys. Rev. E, Volume 69 (2004), 016610, p. 5
[Tre08] Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, 2008 | Zbl
[Ves68] The electrodynamics of substance with simultaneously negative values of and , Sov. Phys. Usp., Volume 10 (1968) no. 4, pp. 509-514 | DOI
Cited by Sources: