Comptes Rendus
Negative index materials: at the frontier of macroscopic electromagnetism
Comptes Rendus. Physique, Volume 21 (2020) no. 4-5, pp. 343-366.

The notions of negative refraction and negative index, introduced by V. Veselago more than 50 years ago, have appeared beyond the frontiers of macroscopic electromagnetism and purely formal during 30 years, until the work of J. Pendry in the late 1990s. Since then, the negative index materials and the metamaterials displayed extraordinary properties and spectacular effects which have tested the domain of validity of macroscopic electromagnetism. In this article, several of these properties and phenomena are reviewed. First, mechanisms underlying the negative index and negative refraction are briefly presented. Then, it is shown that the frame of the time-harmonic Maxwell’s equations cannot describe the behavior of electromagnetic waves in the situations of the perfect flat lens and corner reflector due to the presence of essential spectrum at the perfect -1 index frequency. More generally, it is shown that simple corner structures filled with frequency dispersive permittivity have a whole interval of essential spectrum associated with an analog of “black hole” phenomenon. Finally, arguments are provided to support that, in passive media, the imaginary part of the magnetic permeability can take positive and negative values. These arguments are notably based on the exact expression, for all frequency and wave vector, of the spatially-dispersive effective permittivity tensor of a multilayered structure.

Les notions de réfraction négative et d’indice négatif, imaginées par V. Veselago il y a plus de 50 ans, ont semblé au-delà des frontières de l’électromagnétisme macroscopique et sont restées purement formelles pendant 30 ans, jusqu’aux travaux de J. Pendry à la fin des années 1990. Depuis lors, les matériaux à indice négatif et les métamatériaux ont montré des propriétés extraordinaires et des effets spectaculaires qui ont mis à l’épreuve le domaine de validité de l’électromagnétisme macroscopique. Dans cet article, plusieurs de ces propriétés et phénomènes sont passés en revue. Tout d’abord, les mécanismes sous-jacents aux indices négatifs et à la réfraction négative sont brièvement présentés. Ensuite, il est montré que le cadre des équations de Maxwell harmoniques en temps ne peut pas décrire le comportement des ondes électromagnétiques dans les situations de la lentille plate et du réflecteur en coin parfaits en raison de la présence de spectre essentiel à la fréquence où l’indice prend la valeur -1. Plus généralement, il est montré que de simples structures en coin remplies d’une permittivité dispersive en fréquence ont un intervalle entier de spectre essentiel associé à un analogue du phénomène de «  trou noir  ». Enfin, des arguments sont fournis pour soutenir que, dans les milieux passifs, la partie imaginaire de la perméabilité magnétique peut prendre des valeurs positives et négatives. Ces arguments reposent notamment sur l’expression exacte, pour toutes les fréquences et tous les vecteurs d’onde, du tenseur de permittivité effective avec dispersion spatiale d’une structure multicouche.

Published online:
DOI: 10.5802/crphys.29
Keywords: Negative index, Metamaterials, Frequency dispersion, Corner mode, Spatial dispersion, Passivity, Permeability
Mot clés : Indice négatif, Métamatériaux, Dispersion en fréquence, Mode de coin, Dispersion spatiale, Passivité, Perméabilité

Boris Gralak 1

1 CNRS, Aix Marseille Univ, Centrale Marseille, Institut Fresnel, Marseille, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Boris Gralak. Negative index materials: at the frontier of macroscopic electromagnetism. Comptes Rendus. Physique, Volume 21 (2020) no. 4-5, pp. 343-366. doi : 10.5802/crphys.29. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.29/

[1] V. G. Veselago The electrodynamics of substances with simultaneously negative values of ε and μ, Sov. Phys. Usp., Volume 10 (1968), p. 509 | DOI

[2] J. B. Pendry; A. J. Holden; W. J. Robbins; D. J. Stewart Magnetism from conductors and enhanced nonlinear phenomena, IEEE Trans. Microw. Theory Tech., Volume 47 (1999), p. 2075 | DOI

[3] L. D. Landau; E. M. Lifshitz; L. P. Pitaevskiĭ Electrodynamics of Continuous Media, Courses of Theoretical Physics, vol. 8, Robert Maxwell, M. C., 1984

[4] Z. Hashin; S. Shtrikman A variational approach to the theory of the effective magnetic permeability of multiphase materials, J. Appl. Phys., Volume 33 (1962), p. 3125 | DOI | Zbl

[5] D. J. Bergman The dielectric constant of a composite material—a problem in classical physics, Phys. Rep., Volume 43 (1978), p. 377 | DOI | MR

[6] G. W. Milton Bounds on the complex dielectric constant of a composite material, Appl. Phys. Lett., Volume 37 (1980), p. 300 | DOI

[7] G. W. Milton Bounds on the complex permittivity of a two-component composite material, J. Appl. Phys., Volume 52 (1981), p. 5286 | DOI

[8] G. W. Milton Bounds on the electromagnetic, elastic, and other properties of two-component composites, Phys. Rev. Lett., Volume 46 (1981), p. 542 | DOI

[9] G. W. Milton The Theory of Composites, Cambridge University Press, Cambridge, 2002 | Zbl

[10] A. Bensoussan; J.-L. Lions; G. Papanicolaou Asymptotic Analysis for Periodic Structures, American Mathematical Society Chelsea Publishing, Providence, RI, 2011 | DOI

[11] J. B. Pendry; A. J. Holden; W. J. Stewart; I. Youngs Extremely low frequency plasmons in metallic mesostructures, Phys. Rev. Lett., Volume 76 (1996), p. 4773 | DOI

[12] J. B. Pendry; A. J. Holden; D. J. Robins; W. J. Stewart Low frequency plasmons in thin-wire structures, J. Phys.: Condens. Matter., Volume 10 (1998), p. 4785

[13] D. R. Smith; W. J. Padilla; D. C. Vier; N. C. Nemat-Nasser; S. Schultz Composite medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett., Volume 84 (2000), p. 4184 | DOI

[14] R. A. Shelby; D. R. Smith; S. Schultz Experimental verification of a negative index of refraction, Science, Volume 292 (2001), p. 77 | DOI

[15] R. M. Walser Electromagnetic metamaterials, Proc. SPIE, Complex Mediums II: Beyond Linear Isotropic Dielectr (A. Lakhtakia; W. S. Weiglhofer; I. J. Hodgkinson, eds.), SPIE, San Diego, CA, USA, 2001, 4467 pages

[16] E. Yablonovitch Inhibited spontaneous emission in solid-state physics and electronics, Phys. Rev. Lett., Volume 58 (1987), 002059 | DOI

[17] J. Sajeev Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett., Volume 58 (1987), 002486

[18] B. Gralak; S. Enoch; G. Tayeb Anomalous refractive properties of photonic crystals, J. Opt. Soc. Am. A, Volume 17 (2000), p. 1012 | DOI

[19] M. Notomi Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap, Phys. Rev. B, Volume 62 (2000), 10696 | DOI

[20] P. Yeh Electromagnetic propagation in birefringent layered media, J. Opt. Soc. Am., Volume 69 (1979), 000742 | MR

[21] B. Gralak; S. Enoch; G. Tayeb Superprism effects and EBG antenna applications, Metamaterials: Physics and Engineering Explorations, Chapter 10 (N. Engheta; R. W. Ziolkowski, eds.), John Wiley and Sons, Hoboken, NJ, USA

[22] B. Gralak; M. Cassier; G. Demésy; S. Guenneau Electromagnetic waves in photonic crystals: laws of dispersion, causality and analytical properties, Compendium of Electromagnetic Analysis — From Electrostatics to Photonics, Volume 4: Optics and Photonics I, Chapter 4, World Scientific, Hackensack, NJ, USA, 2020 (Editor-in-chief I. Tsukerman) | DOI

[23] E. Cubukcu; K. Aydin; E. Ozbay; S. Foteinopoulou; C. M. Soukoulis Negative refraction by photonic crystals, Nature, Volume 423 (2003), p. 604 | DOI

[24] C. R. Simovski; P. A. Belov; S. He Backward wave region and negative material parameters of a structure formed by lattices of wires and split-ring resonators, IEEE Trans. Antennas Propag., Volume 51 (2003), 002582

[25] C. R. Simovski Material parameters of metamaterials (a review), Opt. Spectrosc., Volume 107 (2009), 000726

[26] V. V. Zhikov On an extension of the method of two-scale convergence and its applications, Sb. Math., Volume 191 (2000), p. 973 | DOI | Zbl

[27] G. Bouchitté; D. Felbacq Homogenization near resonances and artificial magnetism from dielectrics, C. R. Math., Volume 339 (2004), p. 377 | DOI | MR | Zbl

[28] D. R. Smith; S. Schultz; P. Markos; C. M. Soukoulis Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients, Phys. Rev. B, Volume 65 (2002), 195104

[29] C. R. Simovski; S. A. Tretyakov Local constitutive parameters of metamaterials from an effective-medium perspective, Phys. Rev. B, Volume 75 (2007), 195111

[30] M. G. Silveirinha Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters, Phys. Rev. B, Volume 75 (2007), 115104

[31] M. G. Silveirinha Generalized lorentz-lorenz formulas for microstructured materials, Phys. Rev. B, Volume 76 (2007), 245117

[32] A. Alù First-principles homogenization theory for periodic metamaterials, Phys. Rev. B, Volume 84 (2011), 075153

[33] J. D. Jackson Classical Electrodynamics, Whiley, New York, 1998 | Zbl

[34] Y. Liu; S. Guenneau; B. Gralak Artificial dispersion via high-order homogenization: magnetoelectric coupling and magnetism from dielectric layers, Proc. R. Soc. A, Volume 469 (2013), 20130240 | MR | Zbl

[35] V. M. Agranovich; Y. N. Gartstein Spatial dispersion and negative refraction of light, Usp. Phys. Nauk. (UFN), Volume 176 (2006), 001051

[36] R. V. Craster; J. Kaplunov; A. V. Puchugin High-frequency homogenization for periodic media, Proc. R. Soc. A, Volume 466 (2010), 002341 | MR | Zbl

[37] R. V. Craster; J. Kaplunov; E. Nolde; S. Guenneau High-frequency homogenization for checkerboard structures: defect modes, ultrarefraction, and all-angle negative refraction, J. Opt. Soc. Am. A, Volume 28 (2011), 001032

[38] P. A. Belov; C. R. Simovski Boundary conditions for interfaces of electromagnetic crystals and the generalized ewald-oseen extinction principle, Phys. Rev. B, Volume 73 (2006), 045102

[39] M. Silveirinha Additional boundary condition for the wire medium, IEEE Trans. Antennas Propag., Volume 54 (2006), 001766 | DOI | MR | Zbl

[40] W. Smigaj; B. Gralak Validity of the effective-medium approximation of photonic crystals, Phys. Rev. B, Volume 77 (2008), 235445 | DOI

[41] R. Pierre; G. Gralak Appropriate truncation for photonic crystals, J. Mod. Opt., Volume 55 (2008), 001759 | DOI | Zbl

[42] C. R. Simovski On electromagnetic characterization and homogenization of nanostructured metamaterials, J. Opt., Volume 13 (2011), 103001

[43] V. A. Markel; J. C. Schotland Homogenization of Maxwell’s equations in periodic composites: Boundary effects and dispersion relations, Phys. Rev. E, Volume 85 (2012), 066603

[44] V. A. Markel; I. Tsukerman Current-driven homogenization and effective medium parameters for finite samples, Phys. Rev. B, Volume 88 (2013), 125131

[45] I. Tsukerman Classical and non-classical effective medium theories: New perspectives, Phys. Lett. A, Volume 381 (2017), 001635 | DOI | MR

[46] A. Demetriadou; J. Pendry Taming spatial dispersion in wire metamaterial, J. Phys.: Condens. Matter., Volume 20 (2008), 295222

[47] A. I. Cabuz; D. Felbacq; D. Cassagne Spatial dispersion in negative-index composite metamaterials, Phys. Rev. A, Volume 77 (2008), 013807

[48] V. A. Markel Can the imaginary part of permeability be negative?, Phys. Rev. E, Volume 78 (2008), 026608

[49] M. G. Silveirinha Poynting vector, heating rate, and stored energy in structured materials: A first-principles derivation, Phys. Rev. B, Volume 80 (2009), 235120

[50] A. Alù Restoring the physical meaning of metamaterial constitutive parameters, Phys. Rev. B, Volume 83 (2011), 081102(R)

[51] M. G. Silveirinha Examining the validity of Kramers–Kronig relations for the magnetic permeability, Phys. Rev. B, Volume 83 (2011), 165119

[52] A. Alù; A. D. Yaghjian; R. A. Shore; M. G. Silveirinha Causality relations in the homogenization of metamaterials, Phys. Rev. B, Volume 84 (2011), 054305

[53] Y. Liu; S. Guenneau; B. Gralak Causality and passivity properties of effective parameters of electromagnetic multilayered structures, Phys. Rev. B, Volume 88 (2013), 165104

[54] J. B. Pendry Negative refraction makes a perfect lens, Phys. Rev. Lett., Volume 85 (2000), pp. 3966-3969 | DOI

[55] J. B. Pendry; D. Shurig; D. R. Smith Controlling electromagnetic fields, Science, Volume 312 (2006), p. 1780 | DOI | MR | Zbl

[56] D. Shuring; J. J. Mock; B. J. Justice; S. A. Cummer; J. B. Pendry; A. F. Starr; D. R. Smith Metamaterial electromagnetic cloak at microwave frequencies, Science, Volume 314 (2006), p. 977

[57] U. Leonhard Optical conformal mapping, Science, Volume 312 (2006), p. 1777 | DOI | MR

[58] M. Cassier; G. W. Milton Bounds on Herglotz functions and fundamental limits of broadband passive quasistatic cloaking, J. Math. Phys., Volume 58 (2017), 071504 | DOI | MR | Zbl

[59] B. Gralak Analytic properties of the electromagnetic Green’s function, J. Math. Phys., Volume 58 (2017), 071501 | DOI | MR | Zbl

[60] G. W. ‘t Hooft Comment on “Negative refraction makes a perfect lens”, Phys. Rev. Lett., Volume 87 (2001), 249701

[61] J. M. Williams Some problems with negative refraction, Phys. Rev. Lett., Volume 87 (2001), 249703

[62] N. Garcia; M. Nieto-Vesperinas Left-handed materials do not make a perfect lens, Phys. Rev. Lett., Volume 88 (2002), 207403 | DOI

[63] M. Nieto-Vesperinas Problem of image superresolution with a negative-refractive-index slab, J. Opt. Soc. Am. A, Volume 21 (2004), 000491 | DOI

[64] D. Maystre; S. Enoch Perfect lenses made with left-handed materials: Alice’s mirror?, J. Opt. Soc. Am. A, Volume 21 (2004), p. 122 | DOI

[65] I. Stockman Criterion for negative refraction with low optical losses from a fundamental principle of causality, Phys. Rev. Lett., Volume 98 (2007), 177404 | DOI

[66] P. M. Valanju; R. M. Walser; A. P. Valanju Wave refraction in negative-index media: Always positive and very inhomogeneous, Phys. Rev. Lett., Volume 88 (2002), 187401

[67] B. Gralak; A. Tip Macroscopic Maxwell’s equations and negative index materials, J. Math. Phys., Volume 51 (2010), 052902 | DOI | MR | Zbl

[68] A. V. Dorofeenko; A. A. Zyablovsky; A. A. Pukhov; A. A. Lisyansky; A. P. Vinogradov Light propagation in composite materials with gain layers, Phys.-Usp., Volume 55 (2012), 001080

[69] H. O. Hagenvik; J. Skaar Fourier–Laplace analysis and instabilities of a gainy slab, J. Opt. Soc. Am. B, Volume 32 (2015), 001947

[70] H. O. Hagenvik; M. E. Malema; J. Skaar Fourier theory of linear gain media, Phys. Rev. A, Volume 91 (2018), 043826 | MR

[71] A. Tip Linear absorptive dielectric, Phys. Rev. A, Volume 57 (1998), 004818

[72] J.-M. Combes; B. Gralak; A. Tip Spectral properties of absorptive photonic crystals, Contemporary Mathematics (Waves in Periodic and Random Media), Volume 339, American Mathematical Society, Providence, RI, 2003, 1 pages | DOI | MR | Zbl

[73] M. Cassier; P. Joly; M. Kachanovska Mathematical models for dispersive electromagnetic waves: an overview, Comput. Math. Appl., Volume 74 (2017), 002792 | DOI | MR | Zbl

[74] A. Tip; L. Knöll; S. Scheel; D.-G. Welsch Equivalence of the Langevin and auxiliary-field quantization methods for absorbing dielectrics, Phys. Rev. A, Volume 63 (2001), 043806

[75] C.-A. Guerrin; B. Gralak; A. Tip Singularity of the dyadic Green’s function for heterogeneous dielectrics, Phys. Rev. E, Volume 75 (2007), 056601

[76] A. Figotin; J. H. Schenker Spectral theory of time dispersive and dissipative systems, J. Stat. Phys., Volume 118 (2005), 000199 | DOI | MR | Zbl

[77] R. E. Collin Frequency dispersion limits resolution in veselago lens, Progr. Electromagn. Res. B, Volume 19 (2010), p. 233 | DOI

[78] B. Gralak; D. Maystre Negative index materials and time-harmonic electromagnetic field, C. R. Phys., Volume 13 (2012), 000786 | DOI

[79] M. Cassier; C. Hazard; P. Joly Spectral theory for Maxwell’s equations at the interface of a metamaterial. Part I: Generalized Fourier transform, Commun. Partial Differ. Equ., Volume 42 (2017), 001707 | DOI | MR | Zbl

[80] S. Guenneau; B. Gralak; J. B. Pendry Perfect corner reflector, Opt. Lett., Volume 30 (2005), 001204 | DOI

[81] S. Guenneau; A. C. Vutha; S. A. Ramakrishna Negative refraction in 2d checkerboards related by mirror anti-symmetry and 3d corner lenses, New J. Phys., Volume 7 (2005), 000164 | DOI

[82] S. Guenneau; S. A. Ramakrishna; S. Enoch; S. Chakrabarti; G. Tayeb; B. Gralak Cloaking and imaging effects in plasmonic checkerboards of negative ε and μ and dielectric photonic crystal checkerboards, Photon. Nanostruct.-Fundam. Appl., Volume 10 (2007), 000083

[83] B. Gralak; S. Guenneau Transfer matrix method for point sources radiating in classes of negative refractive index materials with 2n-fold antisymmetry, Waves Random Complex Media, Volume 17 (2007), 000581 | DOI | MR | Zbl

[84] A.-S. Bonnet-Ben Dhia; L. Chesnel; P. Ciarlet Jr. T-coercivity for scalar interface problems between dielectrics and metamaterials, J. Math. Mod. Num. Anal., Volume 46 (2012), 001363 | MR | Zbl

[85] A.-S. Bonnet-Ben Dhia; L. Chesnel; X. Claeys Radiation condition for a non-smooth interface between a dielectric and a metamaterial, Math. Models Methods Appl. Sci., Volume 23 (2013), 001629 | MR | Zbl

[86] A.-S. Bonnet-Ben Dhia; L. Chesnel; P. Ciarlet Jr. Two-dimensional Maxwell’s equations with sign-changing coefficients, Appl. Numer. Math., Volume 79 (2014), 000029 | MR | Zbl

[87] A.-S. Bonnet-Ben Dhia; L. Chesnel; P. Ciarlet Jr. T-coercivity for the Maxwell problem with sign-changing coefficients, Commun. Part Differ. Equ., Volume 37 (2014), 001007 | MR | Zbl

[88] C. Hazard; S. Paolantoni Spectral analysis of polygonal cavities containing a negative-index material, Annales Henri Lebesgue, Volume 3 (2020), pp. 1161-1193 | DOI

[89] P. Yeh Optical Waves in Layered Media, John Wiley and Sons, New York, 1988

[90] J. G. Van Bladel Electromagnetic Fields, IEEE Press Series on Electromagnetic Wave Theory, Whiley-Interscience, Hoboken, NJ, USA, 2007

[91] Y. Brûlé; B. Gralak; G. Demésy Calculation and analysis of the complex band structure of dispersive and dissipative two-dimensional photonic crystals, J. Opt. Soc. Am. B, Volume 33 (2016), 000691 | DOI

[92] P. Lalanne; W. Yan; A. Gras; C. Sauvan; J.-P. Hugonin; M. Besbes; G. Demésy; M. D. Truong; B. Gralak; F. Zolla; A. Nicolet; F. Binkowski; L. Zschiedrich; S. Burger; J. Zimmerling; R. Remis; P. Urbach; H. T. Liu; T. Weiss Quasinormal mode solvers for resonators with dispersive materials, J. Opt. Soc. Am. A, Volume 36 (2019), 000686 | DOI

[93] G. Demésy; N. Nicolet; B. Gralak; C. Geuzaine; C. Campos; J. E. Roman Non-linear eigenvalue problems with GetDP and SLEPc: Eigenmode computations of frequency-dispersive photonic open structures, Comput. Phys. Commun., Volume 257 (2020), 107509 | DOI | MR

[94] A.-S. Bonnet-Ben Dhia; C. Carvalho; L. Chesnel; P. Ciarlet Jr. On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients, J. Comput. Phys., Volume 322 (2016), 000224 | MR | Zbl

[95] V. Veselago; L. Braginsky; V. Shklover; C. Hafner Negative refractive index materials, J. Comput. Theor. Nanosci., Volume 3 (2006), p. 1 | DOI

[96] T. Koschny; P. Markos; D. R. Smith; C. M. Soukoulis Resonant and antiresonant frequency dependence of the effective parameters of metamaterials, Phys. Rev. E, Volume 68 (2003), 065602(R) | DOI

[97] R. A. Depine; A. Lakhtakia Comment i on “resonant and antiresonant frequency dependence of the effective parameters of metamaterials”, Phys. Rev. E, Volume 70 (2004), 048601

[98] A. L. Efros Comment ii on “resonant and antiresonant frequency dependence of the effective parameters of metamaterials”, Phys. Rev. E, Volume 70 (2004), 048602

[99] A. D. Boardman Electromagnetic Surface Modes, Whiley, New York, 1982

[100] A. I. Fernández-Domínguez; A. Wiener; F. J. García-Vidal; S. A. Maier; J. B. Pendry Transformation-optics description of nonlocal effects in plasmonic nanostructures, Phys. Rev. lett., Volume 108 (2012), 106802

[101] A. A. Rukhadze; V. P. Silin Electrodynamics of media with spatial dispersion, Usp. Fiz. Nauk, Volume 74 (1961), 000223

[102] D. Forcella; C. Prada; R. Carminati Causality, nonlocality, and negative refraction, Phys. Rev. Lett., Volume 1180 (2017), 134301

[103] A. Tip; A. Moroz; J.-M. Combes Band structure of absorptive photonic crystals, J. Phys. A: Math. Gen., Volume 33 (2000), 006223 | MR | Zbl

[104] N. Yu; F. Capasso Flat optics with designer metasurfaces, Nat. Mater., Volume 13 (2014), 000139

[105] D. Lin; P. Fan; E. Hasman; M. L. Brongersma Dielectric gradient metasurface optical elements, Science, Volume 345 (2014), 000298

[106] M. C. Rechtsman; J. M. Zeuner; Y. Plotnik; Y. Lumer; D. Podolsky; F. Dreisow; A. Szameit Photonic floquet topological insulators, Nature, Volume 496 (2013), 000196

[107] C. L. Fefferman; J. P. Lee-Thorp; M. I. Weinstein Honeycomb Schrödinger operators in the strong binding regime, Commun. Pure Appl. Math., Volume 71 (2018), 001178 | Zbl

[108] R. C. Craster; S. Guenneau Metamaterials and Plasmonics, Volume 2: Elastic, Acoustic, and Seismic Metamaterials, World Scientific Series in Nanoscience and Nanotechnology, World Scientific Publishing, London, UK, 2017

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